A Few Issues in MHD Turbulence

Alex Alexakis*, Bill Matthaeus%, Pablo Mininni^ , Duane Rosenberg and Annick Pouquet

NCAR / CISL / TNT

* Observatoire de Nice % Bartol, U. Delaware

^ Universidad de Buenos Aires

Warwick, September 17, 2007 pouquet@ucar.edu

A Few Issues in MHD Turbulence:The strong and the weak

* Introduction• Some examples of MHD turbulence in astro, geophysics & the lab.

• Equations, invariants, exact laws, phenomenologieies

* Direct Numerical Simulations, mostly for the decay case

• Is there any measurable difference with fluid turbulence?• What are the features of an MHD flow, both spatially and spectrally?

•Discussion • Accessing high Reynolds numbers for better scaling laws

• Can modeling of MHD flows help understand their properties?• An example : The generation of magnetic fields at low PM

• Can adaptive mesh refinement help unravel characteristic features of MHD?

* Conclusion

Observations of galactic magnetic fields (after Brandenburg & Subramanian, 2005)

• Prediction of the next solar cycle, because of long-term memory in the system (Dikpati, 2007)

Cyclical reversal of the solar magnetic field over the last 130 years

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HINODE / Sunrise November 2006

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Surface (1 bar) radial magnetic fields for

Jupiter, Saturne & Earth, Uranus & Neptune

(16-degree truncation, Sabine Stanley, 2006)

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Axially dipolar

Quadrupole ~ dipole

Brunhes Jamarillo Matuyama Olduvai

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Reversal of the Earth’s magnetic field over the last 2Myrs (Valet, Nature, 2005)

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Temporal assymmetry of reversal process

Experimental dynamo within a constrained flow: Riga

Gailitis et al., PRL 84 (2000)

Karlsruhe, with a Roberts flow (see e.g. Magnetohydrodynamics 38,

2002, special issue)

Taylor-Green turbulent flow at Cadarache

Numerical dynamo at a magnetic Prandtl number PM=/=1 (Nore et al., PoP, 4, 1997) and PM ~ 0.01 (Ponty et al., PRL, 2005).In liquid sodium, PM ~ 10-6 : does it matter?

R

H=2R

Bourgoin et al PoF 14 (‘02), 16 (‘04)…

Experimental dynamo in 2007

ITER (Cadarache)

  

 

Small-scale

The MHD equations [1]

• P is the pressure, j = ∇ × B is the current, F is an external force, ν is the viscosity, η the resistivity, v the velocity and B the induction (in Alfvén velocity units); incompressibility is assumed, and .B = 0.

______ Lorentz force

The MHD invariants ( = =0) * Energy: ET=1/2< v2 + B2 > (direct cascade to small

scales, including in 2D)

* Cross helicity: HC= < v.B > (direct cascade)

And: * 3D: Magnetic helicity: HM=< A.B > with B= x A (Woltjer, mid ‘50s)

* 2D: EEAA= < A2 > (+) [A: magnetic potential]

Both HM and EA undergo an inverse cascade (evidence: statistical mechanics, closure models and numerical simulations)

The ElsässerElsässer variables zz± ± = v ± b= v ± b

t z+ + z- .z+ = - P + + ∆ z+ + - ∆ z- + F±

with with ±± = [ = [ ± ± ]/2 and F]/2 and F± = FV ± FM

Invariants:

E± = < z ±± 22 >/2 = < v >/2 = < v22 + B + B22 ± 2 v.B >/2 ± 2 v.B >/2

Parameters in MHD

• RV = Urms L0 / ν >> 1

• Magnetic Reynolds number RM = Urms L0 / η

* Magnetic Prandtl number: PM = RM / RV = ν / η

PM is high in the interstellar medium.

PM is low in the solar convection zone, in the liquid core of the Earth, in liquid metals and in laboratory experiments.

• Energy ratio EM/EV

• Uniform magnetic field B0

• Amount of magnetic helicity HM

< < z z-/+-/+L L i i z zii

±± 2 2 > = - [ 4/d ] > = - [ 4/d ] ±± rr

Two scaling laws in MHD [1] à la Yaglom (1949), and Antonia et al. (1997)

F(r) = F(x+r) - F(x) : structure function for field F ;

longitudinal component FL(r) = F . r/ |r|

in dimension d, with zz± ± = v ± b= v ± b,, ± = - dtE± = T ± C,

omitting dissipation and forcing and assuming stationarity [Politano and AP, 1998, Geophys. Res. Lett. 25 (see also Phys. Rev. E 57)]

Note: < z+ z+ z- > ~ < ( v + B)2 ( v - B) > ~ + +

AA lack of equipartition between kinetic and magnetic energy is observed in many instances (e.g., in the solar wind)

Two scaling laws in MHD [2]

In terms of velocity and magnetic field, these two scaling laws become:

< vLvi2 >+ <vLbi

2 > -2 < bLvibi > = - (4/d) Tr

- <bLbi2 > - < bLvi

2 >+2 < vLvibi > = - (4/d) c r with T = - dtET and c = - dtHc • Strong V regime, or strong B regime or Alfvénic (v~B) regime (Ting et al 1986, …);

in the latter case, v-B correlations play a dynamical role (Politano+AP,GRL 25, 1998; see also Boldyrev, 2006)

• Where such laws apply, the energy input TT can be measured, e.g. in the solar wind

Phenomenologies for MHD turbulence

• If MHD is like fluids Kolmogorov spectrum EK41(k) ~ k-5/3

Or

• Slowing-down of energy transfer to small scales because of Alfvén waves

propagation along a (quasi)-uniform field B0: EIK(k) ~ (T B0)1/2 k-3/2

(Iroshnikov - Kraichnan (IK), mid ‘60s)

transfer ~ NL * [NL/A] , or 3-wave interactions but still with isotropy.

Eddy turn-over time NL~ l/ul and wave (Alfvén) time A ~ l/B0

And

• Weak turbulence theory for MHD (Galtier et al PoP 2000): anisotropy develops

and the exact spectrum is: EWT(k) = Cw k-2 f(k//)

Note: WT is IK -compatible when isotropy (k// ~ k ) is assumed: NL~ l/ul and A~l// /B0

Or k-5/3 (Goldreich Sridhar, APJ 1995) ? Or k

-3/2 (Nakayama, ApJ 1999; Boldyrev, PRL 2006) ?

Spectra of three-dimensional MHD turbulence

• EK41(k) ~ k-5/3 as observed in the Solar Wind (SW) and in DNS (2D & 3D) Jokipii, mid 70s, Matthaeus et al, mid 80s, …

• EIK(k) ~ k-3/2 as observed in SW, in DNS (2D & 3D), and in closure models Müller & Grappin 2005; Podesta et al. 2007; Mason et al. 2007; Yoshida 2007

• EWT(k) ~ k-2 as may have been observed in the Jovian magnetosphere, and

recently in a DNS, Mininni & AP arXiv:0707.3620v1, astro-ph (see later)

• Is there a lack of universality in MHD turbulence (Boldyrev; Schekochihin)? If so, what are the parameters that govern the (plausible) classes of universality? The presence of a strong guiding uniform magnetic field?

* Can one have different spectra at different scales in a flow? * Is there it a lack of resolving power (instruments, computers)? * Is an energy spectrum the wrong way to analyze / understand MHD?

Recent results using direct numerical simulations and models of MHD

Numerical set-up

• Periodic boundary conditions, pseudo-spectral code, de-aliased with the 2/3 rule

• From 643 to 15363 grid points

• No imposed uniform magnetic field• Either decay runs or forcing

• ABC flow (Beltrami) + random noise at small scale, with V and B in equipartition (EV=EM), or Orszag-Tang (X-point configuration), or Taylor-Green flow

• Energy transfer and non-local interactions in Fourier space• Temporal evolution of maximum of current and vorticity• Energy dissipation and scaling laws• Piling, folding & rolling-up of current and vorticity sheets• Alignment of v and B fields in small-scale structures• Energy spectra and anisotropy• Intermittency

• Beyond Moore’s law

Energy TransferLet uK(x) be the velocity field with wave numbers in the range K < |k| < K+1

K+1K+1Q+1Q+1

QQ KK

^ Isotropy ^ Sharp filters

Fourier space

- Rate of energy transfer in MHD10243 runs, either T-G or ABC forcing

(Alexakis, Mininni & AP; Phys. Rev. E 72, 0463-01 and 0463-02, 2005)

R~ 800

Advection terms

All scales contribute to energy transfer through the Lorentz force

This plateau seems to be absent in decay runs (This plateau seems to be absent in decay runs (Debliquy et al., PoP Debliquy et al., PoP 12,12, 2005 2005))

Rate of energy transfer Tub(Q,K) from u to b for different K shells

The magnetic field at a given scale receives energy in equal The magnetic field at a given scale receives energy in equal amounts from the velocity field from all larger scales amounts from the velocity field from all larger scales (but more (but more from the forcing scale).from the forcing scale).

The nonlocal transfer represents ~25% of the total energy transfer at this Reynolds number (the scaling of this ratio with Reynolds number is not yet determined).

K= 10

K= 20

K= 30

• Growth of the current maximum: linear phase, followed by an algebraic tn regime with n ~ 3

Jmax for the Orszag-Tang flow Mininni et al., PRL 97, 244503 (2006) (res.: 1283 to 10243)

• Growth of the current maximum: linear phase followed by an algebraic tn regime with n ~ 3

Jmax for a random flow, resolutions up to 15363 grid points (RV from 690 to 10100)

OT flow

AMR in incompressible 2D - MHD turbulence at R~1000

• Adaptive Mesh Refinement using spectral elements of different orders

• Accuracy matters when looking at Max norms, here the current, although there are no noticeable differences on the L2 norms (for both the energy and enstrophy)

Rosenberg et al., New J. Phys. 2007

Orszag-Tang simulations at different Reynolds numbers (X factor of 10)• Is the energy dissipation rate T constant in MHD at large Reynolds number

(Mininni + AP, arXiv:0707.3620v1, astro-ph), as presumably it is in 2D-MHD in the reconnection phase?

There is evidence of constant in the hydro case (Kaneda et al., 2003)

Energy dissipation rate in MHD for several RV

OT- vortexLow Rv High Rv

Scaling with Reynolds number of times for max. current & dissipation

Time Tmax(1) at which

global maximum of dissipation is reached in (ABC+ random) flow

and

Time Tmax(2) at which the current

reaches its first maximum

Both scale as Rv0.08

Scaling with Reynolds number of dissipation

MHD decay simulation @ NCAR on 15363 grid pointsVisualization freeware: VAPOR http://www.cisl.ucar.edu/hss/dasg/software/vapor

Zoom on individual current structures: folding and rolling-up Mininni et al., PRL 97, 244503 (2006)

Magnetic field lines in brown

At small scale, long correlation length along the local mean magnetic field (k// ~ 0)

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Hasegawa et al., Nature (2004); Phan et al., Nature (2006), …

Recent observations (and computations as well) of Kelvin-Helmoltz roll-up of current sheets

Current and vorticity are strongly correlated in the rolled-up sheet

Current J2 Vorticity 2

15363 run, early time

V and B are aligned in the rolled-up sheet, but not equal (B2 ~2V2): Alfvén vortices?

(Petviashvili & Pokhotolov, 1992. Solar Wind: Alexandrova et al., JGR 2006)

Current J2 cos(V, B)15363 run, early time

Hc : Velocity - magnetic field correlation

PdFs of cos(v,B):

• Flow with weak normalized total cross helicity Hc

• Flow with strong Hc

Matthaeus et al. arXiv.org/abs/ 0708.0801

Velocity - magnetic field correlation [3]

• Local map in 2D of v & B alignment: |cos (v,B)| > 0.7 (black/white) (otherwise, grey regions).

even though the global normalized correlation coefficient is ~ 10-4

Quenching of nonlinear terms in MHD

(Meneguzzi et al., J. Comp. Phys. 123, 32, 1996)

similar to the Beltramisation (v // ) of fluids

Vorticity =xv & Relative helicity intensity h=cos(v,)• Local v- alignment (Beltramization). Tsinober & Levich, Phys. Lett. (1983);

Moffatt, J. Fluid Mech. (1985); Farge, Pellegrino, & Schneider, PRL (2001), Holm & Kerr PRL (2002).

--> no mirror symmetry, together with weak nonlinearities in the small scales

Blu

e h

> 0

.95

Red

h

<-0

.95

Blu

e h

> 0

.95

Red

h

<-0

.95

Strong relative magnetic helicity (~ 1): change of topology across sheet

Current J2 cos(A, B) , with B=xA15363 run, early time

Current at peak of dissipation

Zoom

Global view

MHD scaling at peak of dissipation [1]

Energy spectra compensated by k3/2

Solid: ET

Dash: EM

Dot: EV

Insert: energy flux

Dash-dot: k5/3-compensated

LintM ~ 3.1

M ~ 0.4

MHD scaling at peak of

dissipation [2]

• Anomalous exponents of structure functions for Elsässer variables, with isotropy assumed (similar results for v and B)

Note 4 ~ 0.98 0.01, i.e. far from fluids and with

more intermittency

MHD decay run at peak of dissipation [3]

Isotropy ratio

RS = S2b / S2

b//

Isotropy obtains in the large scales, whereas anisotropy develops at smaller scales

RS is proportional to the so-called Shebalin angles Lint

M ~ 3.1, M ~ 0.4

Structure functions at peak of dissipation [4]

L1/2 -compensation of S2b

and // structure fns.

Flat at large scales,

with equipartition of the and // components, hence

E(k) ~ k-3/2 ,

isotropic IK (1965) spectrum

Insert: l 2/3-compensated

Solid: perpendicularDash: parallel

(Mininni + AP, arXiv:0707.3620v1, astro-ph)

LintM ~ 3.1 , M ~ 0.4

Structure functions at peak of dissipation [5]

Structure function S2 ,with 3 ranges:

L2 (regular) at small scale

L at intermediate scale, as for weak turbulence: Ek~k

-2. Is it the signature of weak MHD

turbulence?

L1/2 at largest scales (Ek~ k-3/2)

Insert: anisotropy ratio

Solid: perpendicularDash: parallel

(Mininni + AP, arXiv:0707.3620v1, astro-ph)

LintM ~ 3.1 , M ~ 0.4

Solid: parallel, Dash: perpendicular, LintM ~ 3.1, M ~ 0.4

Solid: perpendicularDash: parallel

Kolmogorov-compensated Energy Spectra: k5/3 E(k)

Navier-Stokes, ABC forcing

Small Kolmogorov k-5/3 law (flat part of the spectrum)

K41 scaling increases in range, as the Reynolds number increases

• Bottleneck at dissipation scale

Solid: 20483, Rv= 104, R~ 1200

Dash: 10243, Rv=4000

Linear resolution: X 2Cost: X 16

Kolmogorov k-5/3 law

Mininni et al., submitted

Evidence of weak MHD turbulence

with a k-2 spectrum

• in Reduced MHD computations (Dmitruk et al., PoP 10, 2003)

• in the Jovian magnetosphere (Saur et al., Astron. Astrophys. 386, 2002)

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Extreme events at high Rv unraveled by high-resolution runs (grid from 483 to 15363

points)

Discussion• Can we derive a dynamic model for the Kelvin-Helmoltz rolling-up of current and vorticity sheets? • Are Alfvén vortices, as observed for example in the magnetosphere, present in MHD at high Reynolds number?• What is the physical origin of the tn ~ 3 evolution of the current and vorticity maxima in MHD (time-dependent velocity shear?)?• Is another scaling range possible at scales smaller than where the weak turbulence spectrum is observed?

• What is the effect of the non-locality of nonlinear energy transfer observed in MHD on the flow dynamics, e.g. on the dynamo problem (generation of magnetic fields)?• Quantification of anisotropy in MHD, including in the absence of a large-scale magnetic field? How much // vs. perp. transfer is there?• Large-scale coherent forcing versus random forcing, i.e. universality?

Thank you for your attention!