supplementary information - nature · supplementary information ... were integrated using a...
TRANSCRIPT
W W W N A T U R E C O M N A T U R E | 1
SUPPLEMENTARY INFORMATIONdoi101038nature12128
SUPPLEMENTARY INFORMATION
The Magnetohydrodynamic Simulation
We generated the MHD turbulence data that was analyzed in this letter by a numerical
simulation of the incompressible MHD equations in a [0 2π]3 periodic spatial
domain The simulation used a formulation of the MHD equations in terms of the
Elsasser variables zplusmn = u plusmn b
parttzplusmn = plusmn(b0middotnabla)z
plusmn minus nablap + (12)[ (z+ timesω
minus + z
minustimesω+) plusmn nablatimes(z
minustimesz+)]
1113089111308911130911113091 + (12) (ν + λ)nabla2z
plusmn + (12) (ν minus λ)nabla2
z∓ + F (S1)
with nablamiddotzplusmn = 0 and ω
plusmn=nablatimesz
plusmn The magnetic field b in the simulation has been assigned
Alfveacuten velocity units so that it is given by b=Bradic4πρ in terms of the magnetic field in
cgs units The vector b0 is an externally imposed uniform magnetic field p is the
kinematic pressure field determined by incompressibility ν is kinematic viscosity and
λ = c24πσ is magnetic diffusivity The magnetic Prandtl number is unity ν = λ =
11 times 10minus4
The body force which stirs the fluid was taken to be a Taylor-Green flow
F = f0 [sin(kf x)cos(kf y)cos(kf z)ex minus cos(kf x) sin(kf y) cos(kf z)ey]
applied at modes kf = 2 with an amplitude f0 = 025 Since the same forcing is applied
to both Alfveacuten wave modes the resulting turbulence is balanced (negligible cross-
helicity) and there is no forcing in the corresponding magnetic induction equation
Equations (S1) were integrated using a pseudospectral parallel code on a 10243
periodic grid with nonlinear terms evaluated in physical space and with the pressure
field and linear dissipation terms evaluated in Fourier space The simulation was
dealiased using phase-shift and 2radic23 spherical truncation so that the effective
maximum wavenumber is about kmax = 1024radic23 asymp 482 Time-integration was carried
out by a slaved second-order Adams-Bashforth method In this scheme the linear
terms in (S1) are solved exactly in time using an integration factor to reduce
numerical stiffness The computational time-step was ∆t = 25 times10minus4
In the simulations archived in the database there is no external field b0 = 0
The magnetic fluctuations were instead seeded with initial small-scale noise and
allowed to grow by dynamo action All data from the simulation was taken after it had
reached a statistically stationary state The Ohmic contribution to the total electric
field E=-utimesBc+Jσ is plotted for a frame of this data in Fig2a in the main text with
J=cnablatimesB4π from Ampegraverersquos law and with the Ohmic field normalized by the rms
value of the motional electric field The three-dimensional structure is seen more
clearly in the Supplementary Movie 1 which provides a rotating view Both this
movie and Fig2a in the main text present a volume-rendering of the electric field
with color associated to the vector magnitude and with partial transparency to reveal
some of the internal structure The imprint of the Taylor-Green forcing can be clearly
seen in the large-scale spatial organization of the small-scale current sheets The
small magnitudes of the normalized Ohmic field reflect the high-conductivity of
the turbulent MHD flow
From the statistical steady state of the simulation 1024 frames of data were
generated in physical space and ingested into the database including the 3
components of the velocity vector u 3 components of the magnetic field vector b and
SUPPLEMENTARY INFORMATION
2 | W W W N A T U R E C O M N A T U R E
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the pressure p Also calculated and archived were the 3 components of the magnetic
vector potential a = curlminus1
b in the Coulomb gauge nablamiddota = 0 The data were stored at
every 10 DNS time-steps ie the samples are stored at time-step δt =00025
Extensive tests showed that this provides sufficient temporal resolution to allow
accurate particle tracking The total duration of the stored data is 1024 times 00025 =
256 ie about one large-eddy turnover time The energy spectra of the velocity and
magnetic fields time-averaged over this interval are plotted in Fig2b in the main text
Other statistical parameters of the archived flow are listed here
Velocity (w=u) Magnetic (w=b)
Total energy
euro
Ew = Ewint (k)dk =1
2|w |
2 Eu=77 times 10-2
Eb=85 times 10-2
Dissipation
euro
εw
= 2ν k2int Ew(k)dk εu= 11times 10
-2 εb= 22times 10
-2
Rms field component
euro
prime w = |w |23
1 2
uprime= 023 b = 024
Taylor microscale
euro
λw
= (15ν εw)1 2 prime w λu= 89times10
-2 λb= 66times 10
-2
Taylor-scale Reynolds Reλw=wprimeλwν Re u= 186 Re b= 144
Kolmogorov time scale τw=(νεw)12
u =01 b = 007
Kolmogorov length scale ηw=(ν3εw)
14 u= 33times 10
-3 b= 28times 10
-3
Integral scale
euro
Lw =π
2 prime w 2
kminus1int Ew(k)dk Lu= 056 Lb= 035
Large eddy turnover time Tw=Lwwprime Tu=243 Tb=146
Cross-helicity HC=〈umiddotb〉 HC= 13times10-3
Magnetic-helicity HM=〈amiddotb〉 HM= -07times10-3
Description of the JHU Turbulence Database Cluster
The JHU Turbulence database cluster stores the output of high-resolution direct
numerical simulations (DNS) of turbulent flows in a cluster of relational databases
At present it archives two distinct datasets The first dataset is the entire 10244 space-
time history of a DNS of an isotropic turbulent flow in an incompressible fluid in
3D The second is the 10244 space-time history of a DNS of the incompressible
magneto-hydrodynamic (MHD) equations In total the database clusters stores
over 80 Terabytes of data across 8 database nodes Users of the database may write
and execute analysis programs on their host computers while the programs make
subroutine-like calls (getFunctions) requesting desired variables from the archived
datasets over the network Built-in 1st- and 2nd-order space differentiation as well as
spatial and temporal interpolations are implemented on the database
We describe the architecture of the database cluster its user interface the data
analysis functionality that it supports and provide some implementation details of the
query execution framework The database cluster makes extensive use of database
technology to partition index and query multi-Terabyte simulation data Archiving
these data in a database cluster serves several purposes ndash it preserves the
computational effort it provides for easy verification and repeatability of the results
of experiments and provides public access to high-resolution simulations
Additionally it allows for new types of experimentation that are either not possible or
difficult to execute in the traditional high-performance computing (HPC) setting
W W W N A T U R E C O M N A T U R E | 3
SUPPLEMENTARY INFORMATION RESEARCH
employed to perform DNS of turbulent flows One such example presented in this
letter is tracking particles or the evolution of fields backward in time This process is
naturally supported by the database cluster and can be performed just as easily as
iterating forward in time This is due to the fact that experiments that move from
time-step to time-step are executed by only accessing stored data in a localized region
in space and do not require performing calculations associated with the dynamic
advancement of the DNS (which may require operating over large portions of the data
volume)
The user interface provides public access to the complete 10244 space-time histories
of the turbulence simulations It is based on a Web-services model which allows
users to make subroutine-like requests (getFunctions) that are automatically
transferred over the network and executed on the database cluster The Web-services
methods are implemented using the standard SOAP protocol (Simple Object Access
Protocol) Invoking the methods with modern programming languages such as Java or
C can be done easily For FORTRAN and C codes we provide client wrapper
interfaces to the gSOAP library and sample code that includes calls to the Web-
services methods Additionally we provide a MATLAB interface that uses the
MATLAB-Fast-SOAP package with several routines with calls to the Web-services
methods The list of getFunctions that are currently implemented is available at
httpturbulencephajhueduserviceturbulenceasmx This includes evaluating each
SupplementaryFig1ArchitectureoftheJHUTurbulencedatabasecluster
SUPPLEMENTARY INFORMATION
4 | W W W N A T U R E C O M N A T U R E
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of the simulated fields (velocity u pressure p magnetic field b and vector potential a)
at arbitrary locations in space and time computing their first and second derivatives
evaluating box filters of arbitrary width computing the sub-grid stress tensor and
tracking particles forward and backward in time
The set of processing functions are implemented as stored procedures on the database
servers in C using Microsoft SQL Serverrsquos Common Language Runtime Spatial
interpolation is performed using Lagrange polynomial interpolation of 4th
6th
and 8th
order Temporal interpolation is performed using Cubic Hermite Interpolation First
and second derivatives are evaluated using centered finite differencing of 4th
6th
and
8th
orders Spatial filtering is implemented with a box filter of user specified filter
width Finally fluid particle tracking is performed by means of a second order
accurate Runge Kutta integration scheme The processing functions set was designed
with careful consideration of the trader-offs between generality and efficiency The
set includes the basic common tasks described above as these tasks are the essential
building blocks of most data analysis techniques and are also the most data-intensive
in nature Hence they can be efficiently executed on the database cluster and are
general enough to support a variety of data analysis techniques specific to each client
The architecture of the database cluster is presented in Supplementary Figure 1 The
cluster consists of several database nodes each running Microsoft SQL Server and a
Web servermediator The Web servermediator hosts the frontend Website and the
Web services methods It also acts as a mediator ndash it processes user queries breaks
them down according to the spatial partitioning of the data submits each part for
execution to the appropriate database node and finally assembles the results and
returns them to the user The data are distributed across the nodes of the cluster based
on the Morton z-order space-filling curve This curve also governs the partitioning
and indexing of the data within a database node We employ the Morton z-curve for
the organization of the data because it has good spatial locality (nearby regions in the
3D space are mapped to nearby locations in the 1D index space) it is easy to compute
and the ordering that it produces is both stable and admissible (at each step at least
one horizontal and one vertical neighbor have already been encountered) The entire
data volume of each dataset is subdivided into data voxel or ldquoatomsrdquo of size 83 for a
total of 221
atoms These data atoms are stored as binary large objects (blobs) in a
database table and are indexed using a standard B+-tree database index The key for
the B+-tree clustered index is a combination of the time step and Morton z-index of
the lower left corner of each atom This data structure ensures fast access to nearby
regions in the physical space because such regions will reside in nearby locations on
the Morton z-curve and will therefore be placed close together on disk
We have developed efficient query processing techniques for the rapid execution of
the batch requests submitted by clients of the service For the evaluation of batch
queries (queries consisting of multiple target locations) that perform decomposable
kernel computations we have developed an IO streaming method of execution which
evaluates the queries incrementally and in parts by means of partial-sums
Decomposable kernel computations are computations that evaluate a linear
combination of the data values in a particular region (kernel) and a set of coefficients
In one dimension such computations can be modeled as follows
W W W N A T U R E C O M N A T U R E | 5
SUPPLEMENTARY INFORMATION RESEARCH
g(x ) = li (x ) f (xnN
2+i
i=1
N
)
where x is the target location xn
is the location on the grid closest to the target
location N is the kernel width f (xi ) are the data values stored at grid nodes and
l(x ) are the coefficients of the computation Such computations are decomposable
because they can be executed in parts where each part is evaluated separately and the
results are summed together Interpolation differentiation and filtering all fall in this
category We make use of this incremental evaluation to efficiently execute a batch of
individual queries at the same time We preprocess all of the target locations and their
data requirements in a dictionary (map) data structure This data structure stores key-
value pairs where each key is the index of a database atom and each value is a list of
target locations that need data from this atom When all of the target locations are
processed we create a temporary table of all of the indexes of the database atoms that
have to be retrieved We retrieve the atoms by executing a join between this
temporary table and the table storing the data As each atom is retrieved from disk it is
routed to each target location that needs data from it Subsequently for each target
location a partial sum of the computation is evaluated over the intersection of the
locationrsquos kernel and the atomrsquos data This partial sum is added to the running total
and produces the result when all data atoms have been processed This mode of
execution allows us to stream over the data atoms in a single pass that performs IOs
to increasing offsets which is at least as efficient as a sequential pass Data atoms are
retrieved only once even if need by multiple queries and since they are small enough
to fit in cache the data is effectively reused from cache for all of the associated partial
sum computations Evaluating by partial sums also supports distributed computations
where parts of the computation can be performed on different database servers and
added together at the mediator
Stochastic Flux-Freezing
The mathematical theory of stochastic line-motion underlying the database calculation
is contained in Ref19 and is briefly reviewed here To put this work into physical
context it is important to emphasize that magnetic field lines do not really move
As has long been understood313233
magnetic line motion is just a convenient fiction
useful for intuitive understanding of the MHD solutions but without any physical
reality For smooth solutions of ideal MHD there are generally infinitely many
consistent line-motions and any of these may be used to interpret the solutions This is
analogous to the freedom to choose a gauge in electrodynamics calculations While
a particular line-motion can never be distinguished from any consistent alternative
a law of motion of field-lines has observable consequences (eg conservation of
magnetic flux through co-moving loops) that allow it to be empirically falsified
When one adds non-ideal terms to the Ohmrsquos law on the other hand there is in
general no deterministic line-motion whatsoever that is consistent with 3D MHD
(and generally no smooth line-motion in 2D) For a proof of this assertion by explicit
example for the case of resistive MHD see Ref 34 However once one realizes that
ldquoline-motionrdquo is a purely theoretical construct there is no need to restrict attention
only to deterministic ldquomotionsrdquo of lines Ref19 showed that a stochastic motion of
ldquovirtualrdquo magnetic fields is always consistent with resistive MHD and gives an
intuitive way to understand its solutions
SUPPLEMENTARY INFORMATION
6 | W W W N A T U R E C O M N A T U R E
RESEARCH
The stochastic line-motion law for resistive MHD in any dimension can be stated
precisely as follows19
the solution of the resistive induction equation
parttB=nablatimes(utimesB) +λnabla2B
with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula
euro
B(xt) = 〈B(at 0) sdot nablaa
˜ x (at)det[nablaa
˜ x (at)] |˜ a (xt)〉
where
euro
˜ x (at) is the solution of the initial-value problem for the stochastic differential
equation
euro
d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)
euro
˜ x (at 0) = a
where
euro
˜ a (xt) is the inverse function of the flow map
euro
˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions
euro
˜ W (t) This formula can be written also as
euro
B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by
euro
˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)
and where
euro
˜ J (att 0) =nablaa˜ x (at)det[nabla
a˜ x (at)] As in quantum theory these virtual
fields
euro
˜ B (xt)have meaning only as intermediate states that must be summed over
(averaged) to give physical results The matrix
euro
˜ J (att 0) satisfies the differential
equation
euro
d
dt˜ J (att 0) = ˜ J (att 0)nabla
xu( ˜ x (at)t) - ˜ J (att 0)(nabla
xsdot u)( ˜ x (at)t)
euro
˜ J (at 0t 0) = I
forward in time from t0 to t along the stochastic trajectories which arrive at x at time t
Using this matrix equation together with the stochastic differential equation for the
trajectories one may in principle calculate the ensemble of virtual fields
euro
˜ B (xt)at time
t for any specified initial data
euro
B(at 0) at time t0 For more details see Refs1922
The formal similarity of the above theorems with the textbook results for ideal MHD
(eg the standard Lundquist formula) suggests that the usual formulas should be
recovered in the limit λrarr0 For example in the stochastic equation for
euro
˜ x (t) if
one simply drops the term involving λ then it reduces to the deterministic equation
dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines
ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic
flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for
smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous
| u(xt) ndash u(xt) | le K |x-x|
(corresponding to Houmllder exponent h=1) then it is not hard to show35
that
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
SUPPLEMENTARY INFORMATION
2 | W W W N A T U R E C O M N A T U R E
RESEARCH
the pressure p Also calculated and archived were the 3 components of the magnetic
vector potential a = curlminus1
b in the Coulomb gauge nablamiddota = 0 The data were stored at
every 10 DNS time-steps ie the samples are stored at time-step δt =00025
Extensive tests showed that this provides sufficient temporal resolution to allow
accurate particle tracking The total duration of the stored data is 1024 times 00025 =
256 ie about one large-eddy turnover time The energy spectra of the velocity and
magnetic fields time-averaged over this interval are plotted in Fig2b in the main text
Other statistical parameters of the archived flow are listed here
Velocity (w=u) Magnetic (w=b)
Total energy
euro
Ew = Ewint (k)dk =1
2|w |
2 Eu=77 times 10-2
Eb=85 times 10-2
Dissipation
euro
εw
= 2ν k2int Ew(k)dk εu= 11times 10
-2 εb= 22times 10
-2
Rms field component
euro
prime w = |w |23
1 2
uprime= 023 b = 024
Taylor microscale
euro
λw
= (15ν εw)1 2 prime w λu= 89times10
-2 λb= 66times 10
-2
Taylor-scale Reynolds Reλw=wprimeλwν Re u= 186 Re b= 144
Kolmogorov time scale τw=(νεw)12
u =01 b = 007
Kolmogorov length scale ηw=(ν3εw)
14 u= 33times 10
-3 b= 28times 10
-3
Integral scale
euro
Lw =π
2 prime w 2
kminus1int Ew(k)dk Lu= 056 Lb= 035
Large eddy turnover time Tw=Lwwprime Tu=243 Tb=146
Cross-helicity HC=〈umiddotb〉 HC= 13times10-3
Magnetic-helicity HM=〈amiddotb〉 HM= -07times10-3
Description of the JHU Turbulence Database Cluster
The JHU Turbulence database cluster stores the output of high-resolution direct
numerical simulations (DNS) of turbulent flows in a cluster of relational databases
At present it archives two distinct datasets The first dataset is the entire 10244 space-
time history of a DNS of an isotropic turbulent flow in an incompressible fluid in
3D The second is the 10244 space-time history of a DNS of the incompressible
magneto-hydrodynamic (MHD) equations In total the database clusters stores
over 80 Terabytes of data across 8 database nodes Users of the database may write
and execute analysis programs on their host computers while the programs make
subroutine-like calls (getFunctions) requesting desired variables from the archived
datasets over the network Built-in 1st- and 2nd-order space differentiation as well as
spatial and temporal interpolations are implemented on the database
We describe the architecture of the database cluster its user interface the data
analysis functionality that it supports and provide some implementation details of the
query execution framework The database cluster makes extensive use of database
technology to partition index and query multi-Terabyte simulation data Archiving
these data in a database cluster serves several purposes ndash it preserves the
computational effort it provides for easy verification and repeatability of the results
of experiments and provides public access to high-resolution simulations
Additionally it allows for new types of experimentation that are either not possible or
difficult to execute in the traditional high-performance computing (HPC) setting
W W W N A T U R E C O M N A T U R E | 3
SUPPLEMENTARY INFORMATION RESEARCH
employed to perform DNS of turbulent flows One such example presented in this
letter is tracking particles or the evolution of fields backward in time This process is
naturally supported by the database cluster and can be performed just as easily as
iterating forward in time This is due to the fact that experiments that move from
time-step to time-step are executed by only accessing stored data in a localized region
in space and do not require performing calculations associated with the dynamic
advancement of the DNS (which may require operating over large portions of the data
volume)
The user interface provides public access to the complete 10244 space-time histories
of the turbulence simulations It is based on a Web-services model which allows
users to make subroutine-like requests (getFunctions) that are automatically
transferred over the network and executed on the database cluster The Web-services
methods are implemented using the standard SOAP protocol (Simple Object Access
Protocol) Invoking the methods with modern programming languages such as Java or
C can be done easily For FORTRAN and C codes we provide client wrapper
interfaces to the gSOAP library and sample code that includes calls to the Web-
services methods Additionally we provide a MATLAB interface that uses the
MATLAB-Fast-SOAP package with several routines with calls to the Web-services
methods The list of getFunctions that are currently implemented is available at
httpturbulencephajhueduserviceturbulenceasmx This includes evaluating each
SupplementaryFig1ArchitectureoftheJHUTurbulencedatabasecluster
SUPPLEMENTARY INFORMATION
4 | W W W N A T U R E C O M N A T U R E
RESEARCH
of the simulated fields (velocity u pressure p magnetic field b and vector potential a)
at arbitrary locations in space and time computing their first and second derivatives
evaluating box filters of arbitrary width computing the sub-grid stress tensor and
tracking particles forward and backward in time
The set of processing functions are implemented as stored procedures on the database
servers in C using Microsoft SQL Serverrsquos Common Language Runtime Spatial
interpolation is performed using Lagrange polynomial interpolation of 4th
6th
and 8th
order Temporal interpolation is performed using Cubic Hermite Interpolation First
and second derivatives are evaluated using centered finite differencing of 4th
6th
and
8th
orders Spatial filtering is implemented with a box filter of user specified filter
width Finally fluid particle tracking is performed by means of a second order
accurate Runge Kutta integration scheme The processing functions set was designed
with careful consideration of the trader-offs between generality and efficiency The
set includes the basic common tasks described above as these tasks are the essential
building blocks of most data analysis techniques and are also the most data-intensive
in nature Hence they can be efficiently executed on the database cluster and are
general enough to support a variety of data analysis techniques specific to each client
The architecture of the database cluster is presented in Supplementary Figure 1 The
cluster consists of several database nodes each running Microsoft SQL Server and a
Web servermediator The Web servermediator hosts the frontend Website and the
Web services methods It also acts as a mediator ndash it processes user queries breaks
them down according to the spatial partitioning of the data submits each part for
execution to the appropriate database node and finally assembles the results and
returns them to the user The data are distributed across the nodes of the cluster based
on the Morton z-order space-filling curve This curve also governs the partitioning
and indexing of the data within a database node We employ the Morton z-curve for
the organization of the data because it has good spatial locality (nearby regions in the
3D space are mapped to nearby locations in the 1D index space) it is easy to compute
and the ordering that it produces is both stable and admissible (at each step at least
one horizontal and one vertical neighbor have already been encountered) The entire
data volume of each dataset is subdivided into data voxel or ldquoatomsrdquo of size 83 for a
total of 221
atoms These data atoms are stored as binary large objects (blobs) in a
database table and are indexed using a standard B+-tree database index The key for
the B+-tree clustered index is a combination of the time step and Morton z-index of
the lower left corner of each atom This data structure ensures fast access to nearby
regions in the physical space because such regions will reside in nearby locations on
the Morton z-curve and will therefore be placed close together on disk
We have developed efficient query processing techniques for the rapid execution of
the batch requests submitted by clients of the service For the evaluation of batch
queries (queries consisting of multiple target locations) that perform decomposable
kernel computations we have developed an IO streaming method of execution which
evaluates the queries incrementally and in parts by means of partial-sums
Decomposable kernel computations are computations that evaluate a linear
combination of the data values in a particular region (kernel) and a set of coefficients
In one dimension such computations can be modeled as follows
W W W N A T U R E C O M N A T U R E | 5
SUPPLEMENTARY INFORMATION RESEARCH
g(x ) = li (x ) f (xnN
2+i
i=1
N
)
where x is the target location xn
is the location on the grid closest to the target
location N is the kernel width f (xi ) are the data values stored at grid nodes and
l(x ) are the coefficients of the computation Such computations are decomposable
because they can be executed in parts where each part is evaluated separately and the
results are summed together Interpolation differentiation and filtering all fall in this
category We make use of this incremental evaluation to efficiently execute a batch of
individual queries at the same time We preprocess all of the target locations and their
data requirements in a dictionary (map) data structure This data structure stores key-
value pairs where each key is the index of a database atom and each value is a list of
target locations that need data from this atom When all of the target locations are
processed we create a temporary table of all of the indexes of the database atoms that
have to be retrieved We retrieve the atoms by executing a join between this
temporary table and the table storing the data As each atom is retrieved from disk it is
routed to each target location that needs data from it Subsequently for each target
location a partial sum of the computation is evaluated over the intersection of the
locationrsquos kernel and the atomrsquos data This partial sum is added to the running total
and produces the result when all data atoms have been processed This mode of
execution allows us to stream over the data atoms in a single pass that performs IOs
to increasing offsets which is at least as efficient as a sequential pass Data atoms are
retrieved only once even if need by multiple queries and since they are small enough
to fit in cache the data is effectively reused from cache for all of the associated partial
sum computations Evaluating by partial sums also supports distributed computations
where parts of the computation can be performed on different database servers and
added together at the mediator
Stochastic Flux-Freezing
The mathematical theory of stochastic line-motion underlying the database calculation
is contained in Ref19 and is briefly reviewed here To put this work into physical
context it is important to emphasize that magnetic field lines do not really move
As has long been understood313233
magnetic line motion is just a convenient fiction
useful for intuitive understanding of the MHD solutions but without any physical
reality For smooth solutions of ideal MHD there are generally infinitely many
consistent line-motions and any of these may be used to interpret the solutions This is
analogous to the freedom to choose a gauge in electrodynamics calculations While
a particular line-motion can never be distinguished from any consistent alternative
a law of motion of field-lines has observable consequences (eg conservation of
magnetic flux through co-moving loops) that allow it to be empirically falsified
When one adds non-ideal terms to the Ohmrsquos law on the other hand there is in
general no deterministic line-motion whatsoever that is consistent with 3D MHD
(and generally no smooth line-motion in 2D) For a proof of this assertion by explicit
example for the case of resistive MHD see Ref 34 However once one realizes that
ldquoline-motionrdquo is a purely theoretical construct there is no need to restrict attention
only to deterministic ldquomotionsrdquo of lines Ref19 showed that a stochastic motion of
ldquovirtualrdquo magnetic fields is always consistent with resistive MHD and gives an
intuitive way to understand its solutions
SUPPLEMENTARY INFORMATION
6 | W W W N A T U R E C O M N A T U R E
RESEARCH
The stochastic line-motion law for resistive MHD in any dimension can be stated
precisely as follows19
the solution of the resistive induction equation
parttB=nablatimes(utimesB) +λnabla2B
with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula
euro
B(xt) = 〈B(at 0) sdot nablaa
˜ x (at)det[nablaa
˜ x (at)] |˜ a (xt)〉
where
euro
˜ x (at) is the solution of the initial-value problem for the stochastic differential
equation
euro
d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)
euro
˜ x (at 0) = a
where
euro
˜ a (xt) is the inverse function of the flow map
euro
˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions
euro
˜ W (t) This formula can be written also as
euro
B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by
euro
˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)
and where
euro
˜ J (att 0) =nablaa˜ x (at)det[nabla
a˜ x (at)] As in quantum theory these virtual
fields
euro
˜ B (xt)have meaning only as intermediate states that must be summed over
(averaged) to give physical results The matrix
euro
˜ J (att 0) satisfies the differential
equation
euro
d
dt˜ J (att 0) = ˜ J (att 0)nabla
xu( ˜ x (at)t) - ˜ J (att 0)(nabla
xsdot u)( ˜ x (at)t)
euro
˜ J (at 0t 0) = I
forward in time from t0 to t along the stochastic trajectories which arrive at x at time t
Using this matrix equation together with the stochastic differential equation for the
trajectories one may in principle calculate the ensemble of virtual fields
euro
˜ B (xt)at time
t for any specified initial data
euro
B(at 0) at time t0 For more details see Refs1922
The formal similarity of the above theorems with the textbook results for ideal MHD
(eg the standard Lundquist formula) suggests that the usual formulas should be
recovered in the limit λrarr0 For example in the stochastic equation for
euro
˜ x (t) if
one simply drops the term involving λ then it reduces to the deterministic equation
dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines
ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic
flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for
smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous
| u(xt) ndash u(xt) | le K |x-x|
(corresponding to Houmllder exponent h=1) then it is not hard to show35
that
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
W W W N A T U R E C O M N A T U R E | 3
SUPPLEMENTARY INFORMATION RESEARCH
employed to perform DNS of turbulent flows One such example presented in this
letter is tracking particles or the evolution of fields backward in time This process is
naturally supported by the database cluster and can be performed just as easily as
iterating forward in time This is due to the fact that experiments that move from
time-step to time-step are executed by only accessing stored data in a localized region
in space and do not require performing calculations associated with the dynamic
advancement of the DNS (which may require operating over large portions of the data
volume)
The user interface provides public access to the complete 10244 space-time histories
of the turbulence simulations It is based on a Web-services model which allows
users to make subroutine-like requests (getFunctions) that are automatically
transferred over the network and executed on the database cluster The Web-services
methods are implemented using the standard SOAP protocol (Simple Object Access
Protocol) Invoking the methods with modern programming languages such as Java or
C can be done easily For FORTRAN and C codes we provide client wrapper
interfaces to the gSOAP library and sample code that includes calls to the Web-
services methods Additionally we provide a MATLAB interface that uses the
MATLAB-Fast-SOAP package with several routines with calls to the Web-services
methods The list of getFunctions that are currently implemented is available at
httpturbulencephajhueduserviceturbulenceasmx This includes evaluating each
SupplementaryFig1ArchitectureoftheJHUTurbulencedatabasecluster
SUPPLEMENTARY INFORMATION
4 | W W W N A T U R E C O M N A T U R E
RESEARCH
of the simulated fields (velocity u pressure p magnetic field b and vector potential a)
at arbitrary locations in space and time computing their first and second derivatives
evaluating box filters of arbitrary width computing the sub-grid stress tensor and
tracking particles forward and backward in time
The set of processing functions are implemented as stored procedures on the database
servers in C using Microsoft SQL Serverrsquos Common Language Runtime Spatial
interpolation is performed using Lagrange polynomial interpolation of 4th
6th
and 8th
order Temporal interpolation is performed using Cubic Hermite Interpolation First
and second derivatives are evaluated using centered finite differencing of 4th
6th
and
8th
orders Spatial filtering is implemented with a box filter of user specified filter
width Finally fluid particle tracking is performed by means of a second order
accurate Runge Kutta integration scheme The processing functions set was designed
with careful consideration of the trader-offs between generality and efficiency The
set includes the basic common tasks described above as these tasks are the essential
building blocks of most data analysis techniques and are also the most data-intensive
in nature Hence they can be efficiently executed on the database cluster and are
general enough to support a variety of data analysis techniques specific to each client
The architecture of the database cluster is presented in Supplementary Figure 1 The
cluster consists of several database nodes each running Microsoft SQL Server and a
Web servermediator The Web servermediator hosts the frontend Website and the
Web services methods It also acts as a mediator ndash it processes user queries breaks
them down according to the spatial partitioning of the data submits each part for
execution to the appropriate database node and finally assembles the results and
returns them to the user The data are distributed across the nodes of the cluster based
on the Morton z-order space-filling curve This curve also governs the partitioning
and indexing of the data within a database node We employ the Morton z-curve for
the organization of the data because it has good spatial locality (nearby regions in the
3D space are mapped to nearby locations in the 1D index space) it is easy to compute
and the ordering that it produces is both stable and admissible (at each step at least
one horizontal and one vertical neighbor have already been encountered) The entire
data volume of each dataset is subdivided into data voxel or ldquoatomsrdquo of size 83 for a
total of 221
atoms These data atoms are stored as binary large objects (blobs) in a
database table and are indexed using a standard B+-tree database index The key for
the B+-tree clustered index is a combination of the time step and Morton z-index of
the lower left corner of each atom This data structure ensures fast access to nearby
regions in the physical space because such regions will reside in nearby locations on
the Morton z-curve and will therefore be placed close together on disk
We have developed efficient query processing techniques for the rapid execution of
the batch requests submitted by clients of the service For the evaluation of batch
queries (queries consisting of multiple target locations) that perform decomposable
kernel computations we have developed an IO streaming method of execution which
evaluates the queries incrementally and in parts by means of partial-sums
Decomposable kernel computations are computations that evaluate a linear
combination of the data values in a particular region (kernel) and a set of coefficients
In one dimension such computations can be modeled as follows
W W W N A T U R E C O M N A T U R E | 5
SUPPLEMENTARY INFORMATION RESEARCH
g(x ) = li (x ) f (xnN
2+i
i=1
N
)
where x is the target location xn
is the location on the grid closest to the target
location N is the kernel width f (xi ) are the data values stored at grid nodes and
l(x ) are the coefficients of the computation Such computations are decomposable
because they can be executed in parts where each part is evaluated separately and the
results are summed together Interpolation differentiation and filtering all fall in this
category We make use of this incremental evaluation to efficiently execute a batch of
individual queries at the same time We preprocess all of the target locations and their
data requirements in a dictionary (map) data structure This data structure stores key-
value pairs where each key is the index of a database atom and each value is a list of
target locations that need data from this atom When all of the target locations are
processed we create a temporary table of all of the indexes of the database atoms that
have to be retrieved We retrieve the atoms by executing a join between this
temporary table and the table storing the data As each atom is retrieved from disk it is
routed to each target location that needs data from it Subsequently for each target
location a partial sum of the computation is evaluated over the intersection of the
locationrsquos kernel and the atomrsquos data This partial sum is added to the running total
and produces the result when all data atoms have been processed This mode of
execution allows us to stream over the data atoms in a single pass that performs IOs
to increasing offsets which is at least as efficient as a sequential pass Data atoms are
retrieved only once even if need by multiple queries and since they are small enough
to fit in cache the data is effectively reused from cache for all of the associated partial
sum computations Evaluating by partial sums also supports distributed computations
where parts of the computation can be performed on different database servers and
added together at the mediator
Stochastic Flux-Freezing
The mathematical theory of stochastic line-motion underlying the database calculation
is contained in Ref19 and is briefly reviewed here To put this work into physical
context it is important to emphasize that magnetic field lines do not really move
As has long been understood313233
magnetic line motion is just a convenient fiction
useful for intuitive understanding of the MHD solutions but without any physical
reality For smooth solutions of ideal MHD there are generally infinitely many
consistent line-motions and any of these may be used to interpret the solutions This is
analogous to the freedom to choose a gauge in electrodynamics calculations While
a particular line-motion can never be distinguished from any consistent alternative
a law of motion of field-lines has observable consequences (eg conservation of
magnetic flux through co-moving loops) that allow it to be empirically falsified
When one adds non-ideal terms to the Ohmrsquos law on the other hand there is in
general no deterministic line-motion whatsoever that is consistent with 3D MHD
(and generally no smooth line-motion in 2D) For a proof of this assertion by explicit
example for the case of resistive MHD see Ref 34 However once one realizes that
ldquoline-motionrdquo is a purely theoretical construct there is no need to restrict attention
only to deterministic ldquomotionsrdquo of lines Ref19 showed that a stochastic motion of
ldquovirtualrdquo magnetic fields is always consistent with resistive MHD and gives an
intuitive way to understand its solutions
SUPPLEMENTARY INFORMATION
6 | W W W N A T U R E C O M N A T U R E
RESEARCH
The stochastic line-motion law for resistive MHD in any dimension can be stated
precisely as follows19
the solution of the resistive induction equation
parttB=nablatimes(utimesB) +λnabla2B
with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula
euro
B(xt) = 〈B(at 0) sdot nablaa
˜ x (at)det[nablaa
˜ x (at)] |˜ a (xt)〉
where
euro
˜ x (at) is the solution of the initial-value problem for the stochastic differential
equation
euro
d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)
euro
˜ x (at 0) = a
where
euro
˜ a (xt) is the inverse function of the flow map
euro
˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions
euro
˜ W (t) This formula can be written also as
euro
B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by
euro
˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)
and where
euro
˜ J (att 0) =nablaa˜ x (at)det[nabla
a˜ x (at)] As in quantum theory these virtual
fields
euro
˜ B (xt)have meaning only as intermediate states that must be summed over
(averaged) to give physical results The matrix
euro
˜ J (att 0) satisfies the differential
equation
euro
d
dt˜ J (att 0) = ˜ J (att 0)nabla
xu( ˜ x (at)t) - ˜ J (att 0)(nabla
xsdot u)( ˜ x (at)t)
euro
˜ J (at 0t 0) = I
forward in time from t0 to t along the stochastic trajectories which arrive at x at time t
Using this matrix equation together with the stochastic differential equation for the
trajectories one may in principle calculate the ensemble of virtual fields
euro
˜ B (xt)at time
t for any specified initial data
euro
B(at 0) at time t0 For more details see Refs1922
The formal similarity of the above theorems with the textbook results for ideal MHD
(eg the standard Lundquist formula) suggests that the usual formulas should be
recovered in the limit λrarr0 For example in the stochastic equation for
euro
˜ x (t) if
one simply drops the term involving λ then it reduces to the deterministic equation
dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines
ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic
flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for
smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous
| u(xt) ndash u(xt) | le K |x-x|
(corresponding to Houmllder exponent h=1) then it is not hard to show35
that
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
SUPPLEMENTARY INFORMATION
4 | W W W N A T U R E C O M N A T U R E
RESEARCH
of the simulated fields (velocity u pressure p magnetic field b and vector potential a)
at arbitrary locations in space and time computing their first and second derivatives
evaluating box filters of arbitrary width computing the sub-grid stress tensor and
tracking particles forward and backward in time
The set of processing functions are implemented as stored procedures on the database
servers in C using Microsoft SQL Serverrsquos Common Language Runtime Spatial
interpolation is performed using Lagrange polynomial interpolation of 4th
6th
and 8th
order Temporal interpolation is performed using Cubic Hermite Interpolation First
and second derivatives are evaluated using centered finite differencing of 4th
6th
and
8th
orders Spatial filtering is implemented with a box filter of user specified filter
width Finally fluid particle tracking is performed by means of a second order
accurate Runge Kutta integration scheme The processing functions set was designed
with careful consideration of the trader-offs between generality and efficiency The
set includes the basic common tasks described above as these tasks are the essential
building blocks of most data analysis techniques and are also the most data-intensive
in nature Hence they can be efficiently executed on the database cluster and are
general enough to support a variety of data analysis techniques specific to each client
The architecture of the database cluster is presented in Supplementary Figure 1 The
cluster consists of several database nodes each running Microsoft SQL Server and a
Web servermediator The Web servermediator hosts the frontend Website and the
Web services methods It also acts as a mediator ndash it processes user queries breaks
them down according to the spatial partitioning of the data submits each part for
execution to the appropriate database node and finally assembles the results and
returns them to the user The data are distributed across the nodes of the cluster based
on the Morton z-order space-filling curve This curve also governs the partitioning
and indexing of the data within a database node We employ the Morton z-curve for
the organization of the data because it has good spatial locality (nearby regions in the
3D space are mapped to nearby locations in the 1D index space) it is easy to compute
and the ordering that it produces is both stable and admissible (at each step at least
one horizontal and one vertical neighbor have already been encountered) The entire
data volume of each dataset is subdivided into data voxel or ldquoatomsrdquo of size 83 for a
total of 221
atoms These data atoms are stored as binary large objects (blobs) in a
database table and are indexed using a standard B+-tree database index The key for
the B+-tree clustered index is a combination of the time step and Morton z-index of
the lower left corner of each atom This data structure ensures fast access to nearby
regions in the physical space because such regions will reside in nearby locations on
the Morton z-curve and will therefore be placed close together on disk
We have developed efficient query processing techniques for the rapid execution of
the batch requests submitted by clients of the service For the evaluation of batch
queries (queries consisting of multiple target locations) that perform decomposable
kernel computations we have developed an IO streaming method of execution which
evaluates the queries incrementally and in parts by means of partial-sums
Decomposable kernel computations are computations that evaluate a linear
combination of the data values in a particular region (kernel) and a set of coefficients
In one dimension such computations can be modeled as follows
W W W N A T U R E C O M N A T U R E | 5
SUPPLEMENTARY INFORMATION RESEARCH
g(x ) = li (x ) f (xnN
2+i
i=1
N
)
where x is the target location xn
is the location on the grid closest to the target
location N is the kernel width f (xi ) are the data values stored at grid nodes and
l(x ) are the coefficients of the computation Such computations are decomposable
because they can be executed in parts where each part is evaluated separately and the
results are summed together Interpolation differentiation and filtering all fall in this
category We make use of this incremental evaluation to efficiently execute a batch of
individual queries at the same time We preprocess all of the target locations and their
data requirements in a dictionary (map) data structure This data structure stores key-
value pairs where each key is the index of a database atom and each value is a list of
target locations that need data from this atom When all of the target locations are
processed we create a temporary table of all of the indexes of the database atoms that
have to be retrieved We retrieve the atoms by executing a join between this
temporary table and the table storing the data As each atom is retrieved from disk it is
routed to each target location that needs data from it Subsequently for each target
location a partial sum of the computation is evaluated over the intersection of the
locationrsquos kernel and the atomrsquos data This partial sum is added to the running total
and produces the result when all data atoms have been processed This mode of
execution allows us to stream over the data atoms in a single pass that performs IOs
to increasing offsets which is at least as efficient as a sequential pass Data atoms are
retrieved only once even if need by multiple queries and since they are small enough
to fit in cache the data is effectively reused from cache for all of the associated partial
sum computations Evaluating by partial sums also supports distributed computations
where parts of the computation can be performed on different database servers and
added together at the mediator
Stochastic Flux-Freezing
The mathematical theory of stochastic line-motion underlying the database calculation
is contained in Ref19 and is briefly reviewed here To put this work into physical
context it is important to emphasize that magnetic field lines do not really move
As has long been understood313233
magnetic line motion is just a convenient fiction
useful for intuitive understanding of the MHD solutions but without any physical
reality For smooth solutions of ideal MHD there are generally infinitely many
consistent line-motions and any of these may be used to interpret the solutions This is
analogous to the freedom to choose a gauge in electrodynamics calculations While
a particular line-motion can never be distinguished from any consistent alternative
a law of motion of field-lines has observable consequences (eg conservation of
magnetic flux through co-moving loops) that allow it to be empirically falsified
When one adds non-ideal terms to the Ohmrsquos law on the other hand there is in
general no deterministic line-motion whatsoever that is consistent with 3D MHD
(and generally no smooth line-motion in 2D) For a proof of this assertion by explicit
example for the case of resistive MHD see Ref 34 However once one realizes that
ldquoline-motionrdquo is a purely theoretical construct there is no need to restrict attention
only to deterministic ldquomotionsrdquo of lines Ref19 showed that a stochastic motion of
ldquovirtualrdquo magnetic fields is always consistent with resistive MHD and gives an
intuitive way to understand its solutions
SUPPLEMENTARY INFORMATION
6 | W W W N A T U R E C O M N A T U R E
RESEARCH
The stochastic line-motion law for resistive MHD in any dimension can be stated
precisely as follows19
the solution of the resistive induction equation
parttB=nablatimes(utimesB) +λnabla2B
with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula
euro
B(xt) = 〈B(at 0) sdot nablaa
˜ x (at)det[nablaa
˜ x (at)] |˜ a (xt)〉
where
euro
˜ x (at) is the solution of the initial-value problem for the stochastic differential
equation
euro
d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)
euro
˜ x (at 0) = a
where
euro
˜ a (xt) is the inverse function of the flow map
euro
˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions
euro
˜ W (t) This formula can be written also as
euro
B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by
euro
˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)
and where
euro
˜ J (att 0) =nablaa˜ x (at)det[nabla
a˜ x (at)] As in quantum theory these virtual
fields
euro
˜ B (xt)have meaning only as intermediate states that must be summed over
(averaged) to give physical results The matrix
euro
˜ J (att 0) satisfies the differential
equation
euro
d
dt˜ J (att 0) = ˜ J (att 0)nabla
xu( ˜ x (at)t) - ˜ J (att 0)(nabla
xsdot u)( ˜ x (at)t)
euro
˜ J (at 0t 0) = I
forward in time from t0 to t along the stochastic trajectories which arrive at x at time t
Using this matrix equation together with the stochastic differential equation for the
trajectories one may in principle calculate the ensemble of virtual fields
euro
˜ B (xt)at time
t for any specified initial data
euro
B(at 0) at time t0 For more details see Refs1922
The formal similarity of the above theorems with the textbook results for ideal MHD
(eg the standard Lundquist formula) suggests that the usual formulas should be
recovered in the limit λrarr0 For example in the stochastic equation for
euro
˜ x (t) if
one simply drops the term involving λ then it reduces to the deterministic equation
dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines
ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic
flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for
smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous
| u(xt) ndash u(xt) | le K |x-x|
(corresponding to Houmllder exponent h=1) then it is not hard to show35
that
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
W W W N A T U R E C O M N A T U R E | 5
SUPPLEMENTARY INFORMATION RESEARCH
g(x ) = li (x ) f (xnN
2+i
i=1
N
)
where x is the target location xn
is the location on the grid closest to the target
location N is the kernel width f (xi ) are the data values stored at grid nodes and
l(x ) are the coefficients of the computation Such computations are decomposable
because they can be executed in parts where each part is evaluated separately and the
results are summed together Interpolation differentiation and filtering all fall in this
category We make use of this incremental evaluation to efficiently execute a batch of
individual queries at the same time We preprocess all of the target locations and their
data requirements in a dictionary (map) data structure This data structure stores key-
value pairs where each key is the index of a database atom and each value is a list of
target locations that need data from this atom When all of the target locations are
processed we create a temporary table of all of the indexes of the database atoms that
have to be retrieved We retrieve the atoms by executing a join between this
temporary table and the table storing the data As each atom is retrieved from disk it is
routed to each target location that needs data from it Subsequently for each target
location a partial sum of the computation is evaluated over the intersection of the
locationrsquos kernel and the atomrsquos data This partial sum is added to the running total
and produces the result when all data atoms have been processed This mode of
execution allows us to stream over the data atoms in a single pass that performs IOs
to increasing offsets which is at least as efficient as a sequential pass Data atoms are
retrieved only once even if need by multiple queries and since they are small enough
to fit in cache the data is effectively reused from cache for all of the associated partial
sum computations Evaluating by partial sums also supports distributed computations
where parts of the computation can be performed on different database servers and
added together at the mediator
Stochastic Flux-Freezing
The mathematical theory of stochastic line-motion underlying the database calculation
is contained in Ref19 and is briefly reviewed here To put this work into physical
context it is important to emphasize that magnetic field lines do not really move
As has long been understood313233
magnetic line motion is just a convenient fiction
useful for intuitive understanding of the MHD solutions but without any physical
reality For smooth solutions of ideal MHD there are generally infinitely many
consistent line-motions and any of these may be used to interpret the solutions This is
analogous to the freedom to choose a gauge in electrodynamics calculations While
a particular line-motion can never be distinguished from any consistent alternative
a law of motion of field-lines has observable consequences (eg conservation of
magnetic flux through co-moving loops) that allow it to be empirically falsified
When one adds non-ideal terms to the Ohmrsquos law on the other hand there is in
general no deterministic line-motion whatsoever that is consistent with 3D MHD
(and generally no smooth line-motion in 2D) For a proof of this assertion by explicit
example for the case of resistive MHD see Ref 34 However once one realizes that
ldquoline-motionrdquo is a purely theoretical construct there is no need to restrict attention
only to deterministic ldquomotionsrdquo of lines Ref19 showed that a stochastic motion of
ldquovirtualrdquo magnetic fields is always consistent with resistive MHD and gives an
intuitive way to understand its solutions
SUPPLEMENTARY INFORMATION
6 | W W W N A T U R E C O M N A T U R E
RESEARCH
The stochastic line-motion law for resistive MHD in any dimension can be stated
precisely as follows19
the solution of the resistive induction equation
parttB=nablatimes(utimesB) +λnabla2B
with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula
euro
B(xt) = 〈B(at 0) sdot nablaa
˜ x (at)det[nablaa
˜ x (at)] |˜ a (xt)〉
where
euro
˜ x (at) is the solution of the initial-value problem for the stochastic differential
equation
euro
d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)
euro
˜ x (at 0) = a
where
euro
˜ a (xt) is the inverse function of the flow map
euro
˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions
euro
˜ W (t) This formula can be written also as
euro
B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by
euro
˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)
and where
euro
˜ J (att 0) =nablaa˜ x (at)det[nabla
a˜ x (at)] As in quantum theory these virtual
fields
euro
˜ B (xt)have meaning only as intermediate states that must be summed over
(averaged) to give physical results The matrix
euro
˜ J (att 0) satisfies the differential
equation
euro
d
dt˜ J (att 0) = ˜ J (att 0)nabla
xu( ˜ x (at)t) - ˜ J (att 0)(nabla
xsdot u)( ˜ x (at)t)
euro
˜ J (at 0t 0) = I
forward in time from t0 to t along the stochastic trajectories which arrive at x at time t
Using this matrix equation together with the stochastic differential equation for the
trajectories one may in principle calculate the ensemble of virtual fields
euro
˜ B (xt)at time
t for any specified initial data
euro
B(at 0) at time t0 For more details see Refs1922
The formal similarity of the above theorems with the textbook results for ideal MHD
(eg the standard Lundquist formula) suggests that the usual formulas should be
recovered in the limit λrarr0 For example in the stochastic equation for
euro
˜ x (t) if
one simply drops the term involving λ then it reduces to the deterministic equation
dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines
ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic
flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for
smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous
| u(xt) ndash u(xt) | le K |x-x|
(corresponding to Houmllder exponent h=1) then it is not hard to show35
that
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
SUPPLEMENTARY INFORMATION
6 | W W W N A T U R E C O M N A T U R E
RESEARCH
The stochastic line-motion law for resistive MHD in any dimension can be stated
precisely as follows19
the solution of the resistive induction equation
parttB=nablatimes(utimesB) +λnabla2B
with initial condition B(xt0) at time t0 is given by a stochastic Lundquist formula
euro
B(xt) = 〈B(at 0) sdot nablaa
˜ x (at)det[nablaa
˜ x (at)] |˜ a (xt)〉
where
euro
˜ x (at) is the solution of the initial-value problem for the stochastic differential
equation
euro
d˜ x (at) = u( ˜ x (at)t)dt + 2λd ˜ W (t)
euro
˜ x (at 0) = a
where
euro
˜ a (xt) is the inverse function of the flow map
euro
˜ x (at) and where the average 〈〉 is over the ensemble of Brownian motions
euro
˜ W (t) This formula can be written also as
euro
B(xt) = 〈 ˜ B (xt)〉 where the ldquovirtual magnetic fieldrdquo at (xt) is defined by
euro
˜ B (xt) = B(at 0) sdot ˜ J (att 0) |˜ a (xt)
and where
euro
˜ J (att 0) =nablaa˜ x (at)det[nabla
a˜ x (at)] As in quantum theory these virtual
fields
euro
˜ B (xt)have meaning only as intermediate states that must be summed over
(averaged) to give physical results The matrix
euro
˜ J (att 0) satisfies the differential
equation
euro
d
dt˜ J (att 0) = ˜ J (att 0)nabla
xu( ˜ x (at)t) - ˜ J (att 0)(nabla
xsdot u)( ˜ x (at)t)
euro
˜ J (at 0t 0) = I
forward in time from t0 to t along the stochastic trajectories which arrive at x at time t
Using this matrix equation together with the stochastic differential equation for the
trajectories one may in principle calculate the ensemble of virtual fields
euro
˜ B (xt)at time
t for any specified initial data
euro
B(at 0) at time t0 For more details see Refs1922
The formal similarity of the above theorems with the textbook results for ideal MHD
(eg the standard Lundquist formula) suggests that the usual formulas should be
recovered in the limit λrarr0 For example in the stochastic equation for
euro
˜ x (t) if
one simply drops the term involving λ then it reduces to the deterministic equation
dxdt=u(xt) which describes the standard line-motion law of Alfveacuten (ie field-lines
ldquofrozen-inrdquo to the bulk plasma fluid velocity) It is rigorously correct that stochastic
flux-freezing reduces to standard flux-freezing in the limit of infinite conductivity for
smooth laminar MHD solutions Eg if the velocity field is Lipschitz continuous
| u(xt) ndash u(xt) | le K |x-x|
(corresponding to Houmllder exponent h=1) then it is not hard to show35
that
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
W W W N A T U R E C O M N A T U R E | 7
SUPPLEMENTARY INFORMATION RESEARCH
euro
˜ x (at) minus x(at)2le
2dλ
K(e
Ktminus1) (S2)
where the average 〈〉 is over the ensemble of Brownian motions and d is the space
dimension In that case the ensemble of stochastic flows
euro
˜ x (at) converges with
probability 1 to the deterministic flow x(at) as λrarr0 With a somewhat stronger
differentiability assumption on the velocity field the gradients
euro
nablaa˜ x (at) also
converge to
euro
nablaax(at) and the standard Lundquist formula of ideal MHD
euro
B(xt) = B(at 0) sdot nablaax(at)det[nabla
ax(at)] |a(xt)
is rigorously recovered as λrarr0 Notice that this discussion includes smooth chaotic
flows in which the constant K can be taken to be the leading Lyapunov exponent and
the inequality (S2) becomes an equality asymptotically at long times
The above results need not hold however if the Lipschitz constant K (or the norm
nablau of the velocity gradient) diverges to infinity as λ approaches zero Textbook
proofs of Alfveacutenrsquos Theorem always implicitly assume that nablau remains finite as λrarr0
However this is not true for turbulent solutions of the MHD equations in the limit
of infinite conductivity with the magnetic Prandtl number νλ fixed In that case
gradients of both velocity and magnetic field diverge so that the energy dissipation
rate stays finite in the limit (the ldquozeroth law of turbulencerdquo Ref16) Not only do the
textbook proofs of Alfveacutenrsquos Theorem not apply when the velocity andor magnetic
field become singular but it has been known for some time that the theorem itself
can then fail Already in 1978 an exact solution of the ideal induction equation
was constructed by HGrad which exhibits reconnection at an X-point where the
advecting velocity is singular36
The necessary conditions for solutions of ideal MHD
to violate standard flux-freezing were established in Ref 37 The numerical results in
this Letter indicate that MHD turbulence not only does not satisfy the assumptions of
the textbook proofs but that the standard flux-freezing relation actually fails to hold
even as conductivity increases without bound Note however that the stochastic form
of flux-freezing established in Ref19 holds for λ finite but arbitrarily small Very
interestingly the field-line ldquomotionrdquo seems to remain random in the limit of infinite
conductivity because of the physical phenomenon of ldquospontaneous stochasticityrdquo
Stochastic flux-freezing thus appears to be a property of the rough or singular
solutions of ideal MHD that are obtained in the limit λrarr0 We have here considered
vanishing resistivity but the same limiting behavior of turbulent MHD solutions is
expected for any sort of small non-ideal term in the Ohmrsquos law This type of
universality has been rigorously proved to hold in the Kraichnan-Kazantsev model of
kinematic dynamo13
Stochasticity of field-line motion in high Reynolds-number
MHD turbulence is not a consequence of resistive diffusion but is instead an effect of
advection by a ``roughrsquorsquo velocity field See Refs 8131522 for more discussion
The analysis above has important implications for the reconnection problem It has
generally been assumed that magnetic field lines act as if ``frozen-inrsquorsquo for ideal MHD
Thus microscopic plasma physics mechanisms are thought necessary to explain how
field-lines ldquobreakrdquo or ldquosliprdquo relative to the plasma However it turns out that
solutions of ideal MHD need not obey Alfveacutenrsquos flux-freezing relation when the
velocity and magnetic fields become ldquoroughrdquo or singular Of course ideal MHD
never strictly holds because there are always some non-ideal terms in the generalized
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
SUPPLEMENTARY INFORMATION
8 | W W W N A T U R E C O M N A T U R E
RESEARCH
Ohmrsquos law however small Physically speaking the turbulence accelerates the tiny
violations of flux-freezing due to such non-idealities so efficiently in fact that the
violations persist in the limit of vanishing non-ideal terms and are independent of the
exact form of the non-ideality This is possible just because the ldquoroughrdquo mathematical
solutions obtained in the limit do not conserve magnetic flux
Numerical Implementation of Stochastic Flux-Freezing
The mathematical formulation of stochastic flux-freezing presented above is not
convenient for numerical implementation because of the necessity of inverting
euro
˜ x (at) to find
euro
˜ a (xt) As noted in Ref22 it is easier to find directly the stochastic
trajectories that end at x by solving the stochastic equation
euro
d˜ a (τ) = u(˜ a (τ )τ )dτ + 2λd ˜ W (τ )
euro
˜ a (t) = x (S3)
backward in time from τ=t to τ=t0 The matrix
euro
˜ J ( ˜ a tτ) for each trajectory is obtained
by solving simultaneously
euro
d
dτ˜ J ( ˜ a tτ) = -nabla
xu( ˜ a (τ)τ) ˜ J ( ˜ a tτ) +(nabla
xsdot u)( ˜ a (τ)τ) ˜ J ( ˜ a tτ)
euro
˜ J ( ˜ a tt) = I (S4)
from τ=t to τ=t0 and calculating the ldquovirtual fieldrdquo
euro
˜ B (xtτ) = B(˜ a (τ )τ ) sdot ˜ J ( ˜ a tτ) for
all τ between t0 and t in a single backward integration The average
euro
B(xt) = 〈 ˜ B (xtτ )〉
so calculated is independent of time τ and coincides with the solution of equation (4)
The above-described algorithm is the same as that employed previously in Ref22 to
study the kinematic dynamo but using the new MHD turbulence database it fully
incorporates the effects of the Lorentz force
The stochastic flux-freezing calculation is implemented numerically in the database
by solving the stochastic differential equation (S3) with the Euler-Maruyama scheme
for a time-step Δτ=195times10-5
chosen conservatively so that
euro
prime u Δτ + 2λΔτ lt 01ηb
where ηb=28times10-3
is the resistive length The matrix equation (S4) is solved with a
corresponding Euler scheme for the same time-step The simultaneous backward
integration of (S3)(S4) requires calling the velocity u and the velocity-gradient
nablau from the database at each time step Note that the database MHD flow is
incompressible so that nablasdotu=0 and the term involving the velocity divergence can be
dropped from (S4) (In practice the velocity gradient matrices nablau obtained by calls to
the database are only traceless to a fraction of a percent because of errors introduced
by Lagrange interpolation and finite-difference approximation In our calculation we
thus use the velocity-gradient
euro
nablauminus1
3(nabla sdot u)I with the trace removed to make the
Lagrange-interpolated velocity-gradient more consistent with the pseudo-spectral
simulation results) An N-sample ensemble of independent stochastic trajectories
euro
˜ a n(τ) matrices
euro
˜ J n( ˜ a tτ)and virtual fields
euro
˜ B n(xtτ) is generated in this manner for
n=1hellipN and the empirical average calculated as
euro
1
N˜ B n(xtτ)
n=1
N
sum This average
should converge to the archived magnetic field B(xt) when Δτ is taken sufficiently
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
W W W N A T U R E C O M N A T U R E | 9
SUPPLEMENTARY INFORMATION RESEARCH
small and N is taken sufficiently large
The process is illustrated in Supplementary Movie 2 The movie begins showing the
stochastic trajectories
euro
˜ a n(τ) n=1hellipN going backward in time from spacetime point
(xt) The magnetic fields at the locations of the ldquocloudrdquo of particles at time τ are
those which contribute significantly to the magnetic field B(xt) After reaching an
(arbitrarily) chosen time τ=t0 the physical fields
euro
B( ˜ a n(t 0)t 0) n=1hellipN in the
ldquocloudrdquo are then taken as initial conditions to calculate an ensemble of ldquovirtualrdquo fields
euro
˜ B n(τ) n=1hellipN by solving along the stochastic trajectories the ldquofrozen-inrdquo equation
euro
d
dτ˜ B n(τ) = ˜ B n(τ ) sdot nablau(an(τ)τ )
euro
˜ B n(t 0) = B(˜ a n(t 0)t 0) (S5)
from τ=t0 to τ=t These ldquovirtual fieldsrdquo are shown in the movie evolving from τ=t0 to
τ=t along the stochastic trajectories stretched and rotated by the flow to the final
point (xt) In practice we do not solve the equation (S5) but use the mathematically
equivalent formula
euro
˜ B n(τ) = ˜ B n(t 0) sdot ˜ J n( ˜ a τt 0) = ˜ B n(t 0) sdot ˜ J n( ˜ a tt 0) sdot ˜ J n-1
( ˜ a tτ ) since
thematrices
euro
˜ J n( ˜ a tτ) havealreadybeencalculatedandstoredfort0leτletFinally
theldquovirtualfieldsrdquo
euro
˜ B n(t) = ˜ B n(xtt 0) n=1N at the spacetime point (xt) are
averaged in the last frame of the movie to obtain
euro
1
N˜ B n(t)
n=1
N
sum recovering the archived
magnetic field B(xt)
In Figure 3b of the Letter are plotted the relative errors
euro
RelErr(xtτ) =1
| B(xt) |B(xt) minus
1
N˜ B n(xtτ)
n=1
N
sum
for a typical point (xt) in the database as a function of times τ (called t0 in the text
figure) The small errors in this figure for large N illustrate that the stochastic flux-
freezing relation successfully recovers the magnetic field point by point This is a
very stringent test of the accuracy of the archived data and the convergence of our
numerical integration of (S3)(S4) It is important however to demonstrate that
similar convergence holds at all points in the database and not just for a particular
chosen point In Supplementary Figure 2 is plotted
euro
1
PRelErr(xptτ)p=1
P
sum averaged
over P=512 diagnostic points in the database as a function of times τ between t0 and t
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
SUPPLEMENTARY INFORMATION
1 0 | W W W N A T U R E C O M N A T U R E
RESEARCH
The errors decrease for greater N but also increase for earlier τ because larger N
values are required at earlier times to properly sample the more extended ldquocloudsrdquo of
points and also integration errors in Δτ build up going backward in time For times τ over half a turnover time the errors decrease to values lt5 as N increases This
residual error is due to the Lagrange interpolation of velocity gradients as well as to
errors from finite Δτ and N By contrast the relative error in the magnetic field using
standard flux-freezing averaged over the same P points rises rapidly going to earlier τ and is well above 100 by half an eddy turnover time
Observational Tests of Turbulent Reconnection
ThethesisofthisLetterthatturbulenceleadstoabreakdownofflux‐freezing
isintimatelyconnectedtopreviousworkonstochasticorturbulent
reconnection78InparticularRef8discussesindetailtherelationshipof
Richardsondispersionoffield‐linestoturbulentreconnectiontheoryWhile
therehasbeensomenumericalworkaimedattestingthismodel2838ultimately
thismodelneedstobecomparedwithastrophysicalobservationsThisisnot
perfectlystraightforwardbecausethemodelwasoriginallyconceivedinthe
idealizedcontextofexternallydrivenspatiallyhomogeneousturbulenceacting
onacurrentsheetwithanassociatedlarge‐scalemagneticfieldwhoseAlfveacuten
velocityisatleastasgreatastheturbulentrmsvelocityTheconvectionzonesof
starsandionizedpartsofaccretiondisks(subjecttothemagnetorotational
instability)arelikelyexamplesbutdirectdetailedobservationsareunavailable
forbothcasesWecanalsoapplythismodeltoopticallythincollisionless
plasmasassumingasaminimalconditionthatthelarge‐eddyscaleis
sufficientlylargerthananyrelevantmicroscale8Thisopensupthepossibility
oftestingthemodelagainstreconnectioneventsinthesolarcoronaSince
turbulenceinthesolarcoronaappearshighlyintermittentinspaceandtimeany
SupplementaryFig2Relativeerrorinnumericalimplementationofstochasticfluxshy
freezingaveragedoverPpointsinthedatabasePlottedaretheerrorsatfourvaluesofN
Forcomparisonisplottedtheaveragerelativeerrorusingstandardflux‐freezing(black)
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
W W W N A T U R E C O M N A T U R E | 1 1
SUPPLEMENTARY INFORMATION RESEARCH
testhastoinvolvelocalmeasurementsofthermsturbulentvelocitytheAlfveacuten
speedthelengthofthecurrentsheetandeitherthereconnectionspeedorthe
thicknessofthereconnectinglayerHerewediscusstherelationshipbetween
modelpredictionsandobservationsandgiveabriefoverviewofprospectsfora
definitiveanswer
Theoriginalworkonthistopic7assumedanisotropicforcingofturbulent
motionswithvelocityfluctuationsuLdrivenonarathersharplydefinedscale
ItwasfurtherassumedthattheAlfveacutenicMachnumberMA=uLvAisatleast
somewhatsmallerthanunityOnthelargestscalestheresultingvelocityfield
isthusonlyweaklyturbulentandcanbeapproximatedasanensembleof
interactingwavesThewaveenergycascadesanisotropicallywithfield‐parallel
wavelengthfixedbutwithperpendicularwavelengthdecreasingstepwiseuntil
itisoforderMA2Onallscalessmallerthanthistheeddiesarestrongly
nonlinearinthesensethattheircoherencetimeor``eddy‐turnovertimersquorsquois
comparabletothecorrespondingAlfveacutenwaveperiod(so‐calledcritical
balance26)Thesestronglyturbulenteddiesarethesourceofcross‐field
diffusionandtheconsequentenhancementofthereconnectionrateGivena
large‐scalefieldreversalwithatransversesizeLgtthepredictionsforthe
currentlayerwidthandreconnectioninflowspeedare78
euro
Δ ~ MA2(L)
12and
euro
vrec~ M
A
2vAL( )
12
(S6)
Here~denotesequalityuptoaprefactoroforderaboutunitythatisprobably
somewhatdependentonlarge‐scalegeometryofreconnectingfluxstructures
Theexactnatureofthesmallscalephysicsontheotherhandwillbeirrelevant
aslongasthecurrentsheetthicknessΔexceedsthemicrophysicalscalesbya
comfortablemargin
InthesituationdescribedabovethermsvelocityuisequaltouLonorderof
magnitudeThismightsuggestnaiumlvelythatinastrophysicalapplicationsthe
formulas(S6)shouldbeappliedwithuLtakentobetheturbulentrmsvelocityu
inferredfromobservationsegnon‐thermalDopplerbroadeningofline
spectra3040Howeveritshouldbeborneinmindthattheseexpressionsare
dependentontheassumedidealizedformofenergyinjectionAmorerobust
expressionfollowsfromnotingthattheturbulentvelocityatthestrong‐eddy
scaleisvT=MA2vA(denotedvtransinRef8)Ifweintroducethecorresponding
turbulentAlfveacutenicMachnumberMT=vTvA~MA2andifwespecifyasthelargest
lengthscale(integrallength)oftheturbulenceparalleltothemagneticfieldthen
wehave
euro
Δ ~ MT(L)12and
euro
vrec~ v
TL( )
12
(S7)
Forcomparisonwithobservationsequation(S7)shouldusuallygivea
reasonableestimateofthepropertiesofthereconnectionlayerifvTistakento
beofordertheestimatedturbulentvelocitydispersionuprimeIndeedlocally
generatedturbulenceshouldbedominatedatitslargestscalesbystrongly
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
SUPPLEMENTARY INFORMATION
1 2 | W W W N A T U R E C O M N A T U R E
RESEARCH
nonlinearturbulenceandtherewillbenoweaklyturbulentcascadeWhenthe
turbulenceisgeneratedexternallyandreachesthereconnectinglayeras
interactingwavesequation(S7)maysomewhatoverestimatethecurrentsheet
thicknessandthereconnectionspeedMorenumericalworkalongthelinesof
Refs2838willbehelpfultocharacterizewithbetteraccuracytheeffectsof
differenttypesofturbulencegenerationmechanismonthereconnectionrates
Neverthelesstheorder‐of‐magnituderesults(S7)arethekeypredictionsofthe
theoryespeciallythescalingonthevariousphysicalparametersandthe
independenceofthemicroscales
Thebestastrophysicalobservationsofmagneticreconnectioneventsboth
currentlyandinthenearfutureareinvariousregionsoftheheliospherein
particularintheEarthrsquosmagnetosphereandinthesolarcoronaAswenow
explain(seealsoRef8)thereconnectiontheorybasedonMHDturbulencewill
generallynotapplyinthemagnetospherebutverylikelydoesholdincertain
situationsinthesolarcorona
InthemagnetosphereHallreconnectionorfullykineticreconnectiondominates
becauseinthatenvironmentusually∆ltρiwhere∆isgivenby(S7)andρiisthe
iongyroradiusThisisbothbecausethelength‐scaleofreconnectingstructures
Lisoftenclosetoρiandalsobecauseturbulenceinthemagnetosphereis
transientorinsufficientlystrong39InthesolarwindimpingingontheEarthrsquos
magnetosphereegtheiongyroradiusisaboutoneandonehalfordersof
magnitudesmallerthanthesizeofreconnectionstructures39Fora
turbulentAlfveacutenicMachnumber~01theexpectedcurrentsheetthicknessis
closetotheiongyroradiusandtheMHDturbulencemodelisinapplicable
LargerscalereconnectingfluxstructuresoccurinthemagnetotailwithL~103ρi
butthelowvaluesofMTandLstillleadto∆in(S7)oforderorsmallerthanρi
Itisimportanttoemphasizethatvariousformsofturbulencecanoccurinthe
magnetosphereandmayenhancereconnectionratesForexampleRef41
documentsaneventinthemagnetotailwhereturbulentemfappearstosupply
thereconnectionelectricfieldThisiskineticplasmaturbulencehowevernot
MHDturbulenceIntheeventofRef41thereconnectionlayerthicknessδwas
observedtobearoundahundredkmthickoftheorderofρiThusthe
wavenumbersgt1δofmodesinsidethelayerwerelocatedinthepartofthe
observedenergyspectrumwithexponentminus286abovetheldquokneerdquoattheion
gyroradiusandnotintheMHDrangewithminus173spectrumextendingabouta
decadebelowthekneeSuchareconnectionlayercanobviouslynotbeanalyzed
intheframeworkofMHDTheremayneverthelessbesomesignificant
commonalitiesofsuchkineticturbulentreconnectionwithMHDturbulent
reconnection78whichdeservetobeexploredForexampleeventslikethatin
Ref41mightbeexplainedatleastqualitativelybygeneralizingthetheoryin
Refs78towhistlerturbulencedescribedbyelectronmagnetohydrodynamics
(EMHD)42
OntheotherhandtheconditionsforanMHDturbulencetheoryoftenholdinthe
solarcoronaForexampleveryextendedpost‐CMEcurrentsheetsinthesolar
corona304043havealmostsevenordersofmagnitudeseparationinscalebetween
Landρiandcurrentsheetwidthsandreconnectionspeedsshouldfollow
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
W W W N A T U R E C O M N A T U R E | 1 3
SUPPLEMENTARY INFORMATION RESEARCH
equation(S7)Measurementsofthepropertiesofapost‐CMEcurrentsheet30
revealacurrentsheetthicknesstolengthratiointherange016to008while
theturbulentAlfveacutenicMachnumberestimatedfromobservationsofDoppler
line‐broadeningisapproximately01ingoodagreementwith(S7)for~L(Note
thattheauthorsofRef30suggestedthatthepredictedcurrentsheetthickness
wasactuallysomewhatlessthanobservedbasedonanaiumlveuseofequation
(S6))Thisapplicationofthetheoryisconsistentsincethereisstillaverylarge
scaleratioΔρi~106Theparallelscaleoftheturbulencewasnotmeasured
butislikelytobesmallerthanLWeconcludefromthisthatthepredicted
currentsheetthicknessisatmostcomparabletothelowerboundfrom
observationsSubsequentobservationsintheX‐ray43suggestthattheactual
thicknessofthecurrentsheetmaybetypicallyoverestimatedbyanorderof
magnitudeusingUVmeasurementsThebottomlineisthatobservationsof
post‐CMEcurrentsheetsinthesolarcoronashowthatthecurrentsheetismuch
broaderthanwouldbeexpectedfrommicroscopicprocessesandconsistentwith
thebroadeningexpectedfromturbulenceTheuncertaintiesinthescaleofthe
strongeddiesandneglectedfactorsoforderunityinthetheorymakeit
impossibletoassertmorethanaroughconsistencyThelattermaybeovercome
byanalysisofafullcatalogueofCMEs44witharangeofpropertiessothatthe
predictedscalingscanbetestedindependentofnumericalcoefficientsThelack
ofknowledgeofismorechallengingAreasonableestimatemaybepossible
usingtheMHDturbulencerelationfortheparallelscale~vAvT2εwithεthe
turbulentenergydissipationrateaquantitymeasurableinprincipleusingonly
coarse‐grainedobservations8Fortunatelythedependenceofthereconnection
ratesoniscomparativelyweak
SUPPLEMENTARY REFERENCES
31 W A Newcomb ldquoMotion of magnetic lines of forcerdquo Ann Phys 3 347-385
(1958)
32 V M Vasyliunas ldquoNonuniqueness of magnetic field line motionrdquo J Geophys
Res 77 6271-6274 (1972)
33 H Alfveacuten ldquoOn frozen-in field lines and field-line reconnectionrdquo J Geophys Res
81 4019-2021 (1976)
34 AL Wilmot-Smith ER Priest and G Hornig ldquoMagnetic diffusion and the
motion of field linesrdquo Geophys amp Astrophys Fluid Dyn 99 177-197 (2005)
35 M M Freidlin and A D Wentzell Random Perturbations of Dynamical Systems
(Springer New York 1998)
36 H Grad ldquoReconnection of magnetic lines in an ideal fluidrdquo Courant Institute of
Mathematical Sciences preprint COO-3077-152 MF-92 New York University New
York 1978 httparchiveorgdetailsreconnectionofma00grad
37 G L Eyink amp H Aluie ldquoThe breakdown of Alfveacutenrsquos theorem in ideal plasma
SUPPLEMENTARY INFORMATION
1 4 | W W W N A T U R E C O M N A T U R E
RESEARCH
flows Necessary conditions and physical conjecturesrdquo Physica D 223 82-92 (2006)
38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list
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38GKowalALazarianETVishniacampKOtmianowska‐MazurldquoReconnection
studiesunderdifferenttypesofturbulencedrivingrdquoNonlinProcGeophys19
297‐314(2012)
39GZimbardoAGrecoLSorriso‐ValvoSPerriZVoumlroumlsGAburjania
KChargaziaampOAlexandrovaldquoMagneticturbulenceinthegeospace
environmentrdquoSpaceSciRev15689‐134(2010)
40ABemporadldquoSpectroscopicdetectionofturbulenceinpost‐CMEcurrent
sheetsrdquoAstrophysJ689572‐584(2008)
41SYHuangetalldquoObservationsofturbulencewithinreconnectionjetinthe
presenceofguidefieldrdquoGeophysResLett39L11104(2012)
42JChoandALazarianldquoTheanisotropyofelectronmagnetohydrodynamic
turbulencerdquoAstrophysJ615L41‐L44(2004)
43SLSavageDEMcKenzieKKReevesTGForbesampDWLongcope
ldquoReconnectionoutflowsandcurrentsheetobservedwithHINODEXRTinthe
2008April9ldquoCartwheelCMErdquoflarerdquoAstrophysJ722329‐342(2010)
44httpcdawgsfcnasagovCME_list