implications of navier-stokes turbulence theory for plasma turbulence

Proc. Indian Ac.ad. Sci., Vol. 86 A, No. 2, August 1977, pp. 87-110, �9 Printed in India.

Implications of Navier-Stokes turbulence theory for plasma turbulence

DAVID MONTGOMERY Department of Physics and Astronomy, The University of lowa, Iowa City, Iowa 52242, U.S.A.

1. Introduction

In writing about plasma turbulence, it is almost as important to say what is not to be discussed as it is to say what is to be discussed. Therefore, in a few proliminary paragraphs, wo shall eliminate somo items that we might have chosen to consider.

When one talks about the turbulent motions of any continuum (liquid, plasma, magneto-fluid, etc.) the theoretical framework can often be represcnted in the sche- matic form

(~t + iH) f = N(f), (1)

which stands symbolically for all of the equations of motion. By f , we mean a column rector which contains all the unknowns in the problem, f c o u l d be a column rector of fluid variables and magnetic field components in magnetohydrodynamics; or f o r a Vlasov plasma ir could be a column vector of particlo distribution functions. iH represents a linear operator, possibly containing derivatives or integrals, and N represents a nonlinear operator, frequently a bilinear one. N(f) is what we throw away when we linearize the system, so that

(8 H) f(o) + i = 0 (2)

stands for the linearized equations of motion. A standard and straightforward method of solving eq. (2) is to seek eigenfunctions and corresponding eigenvalues oJj such that

tIf~=w~f:, j----l, 2, 3, ....

The general solution to eq. (2) is then f (o) = ~'jajf~ exp ( - ~ o ~ t),

(3)

(4)

88 David Montgomery

with a set of amplitudes aj which are determined by initial and/or boundary conditions. When the real part of oJ~ is non-zero we have waves, and when the imaginary part is non-zero, we have damping or unstable growth. In short, eq. (4) represents most of what is comfortably familiar in linear plasma theory, and provides much of our vocabulary for the subject.

The thing that creates the difficulties in the subject of turbulence is that the nonlinear terms in eq. (1) must be taken into account. The crucial question is: How much? Whether explicitly stated or not, the overwhelming majority o f ' p l a s m a turbulence' calculations have been concerned with tases for which the nonlinear terms are comparable with, or smaller than, the linear ones,

II Hfll ~ II N(f)!l ,

so that ir is not unreasonable to believe that some of the qualitative properties of eq. (4) are shared by the turbulent solutions. It is only in this limit that ir makes sense to talk about ' waves ', ' frequencies ', ' instabilities ', ' normal modes ', or the other terminology that derives from linear mathematics and that most plasma papers have been phrased in terms of. (This is the only meaning that can be given to the expression ' weak turbulence '.)

It should be said at the outset that this weak turbulence limit is not what is under discussion here. Instead, what we are interested in is the fully nonlinear limit in which

[I N(f) l l ~" II Hfl l ,

and if one wants to discard something, throwing away the linear terms in eq. (1) would be a more reasonable procedure than trying to iterate around eq. (4), or modify it in some way.

Considering this limit is a more drastic step than has been realized by plasma theorists. Ir means the abandonment of the comfortable and familiar vocabulary of waves, frequencies, resonance, growth rates, and so forth. There is no longer any reason to expect anything to oscillate sinusoidally in time, even approximately. Efforts to stretch our linear concepts to cover this new situation can only result in a lack of clarity and precision in what we say, and can only obscure the very new and exciting physics that is there, by keeping us from thinking about it in adequate ways.

It is fortunate for us that others have passed this way first. Investigators of Navier- Stokes (hydrodynamic) turbulence have had to consider this highly nonlinear limit, due to the almost accidental fact that in the limit of high Reynolds number (the case of most practical interest) their linear operator iH is in effect zero. They have not had the delightful parlor game of linearize-the-equations-of-motion to play. They have been forced to meet the fully nonlinear problem head-on, and have in so doing advanced Navier-Stokes turbulence theory about fifteen years beyond where we are now with plasma turbulence. The literature is difficult to read and its most proli¡ authors are difficult to approach. But there is no escaping the fact that they know a lot ofthings we need to know right away, so we had just as well begin. A good place to start reading is in the 1973 Les Houches lectures of Orszag (1974). A recent book by Leslie is of some use (Leslie 1973). Batchelor's classic monograph (Batehelor 1959) is a somewhat older and more elementary survey, but it is very accurate and readable.

Navier-Stokes and plasma turbulence 89

The next section is a brief and undoubtedly over-simplified discussion of Navier- Stokes turbulence theory. After that, we will pass on to some discussiort of the two-dimensional case, which has marked differences from the three-dimensional one. In section 4 we will introduce the subject of M H D turbulence, with an eye to applying as muchas possible the techniques that have beert developed in Navier-Stokes theory. Finally in section 5 we shall introduce the subject of turbulence in Vlasov plasmas from the point of view of the so-called ' direct interaction approximation ' developed for Navier-Stokes fluids by Kraichnan (1958, 1959).

2. Navier-Stokes turbulence

Eq. (1), specialized to a Navier-Stokes fluid such as water, becomes

(-~t q - v " V ) v : = - V p -[-v~72v, (5)

where v : v ( x , t) is the fluid velocity, p:---p(x, t) is the pressure, and vis the kinematic viscosity. Eq. (5) is supplemented by the incompressibility cortdition,

V "v =0 . (6)

Taking the divergence of eq. (5) and using eq. (6) implies that

V2p = - V ' ( v �9 Vv) , (7)

which amounts to a Poisson eq. for the pressure in terms of v. The solution is a functional p=p(v) which makes use of any imposed boundary condition, and its substitution back into eq. (5) yields an expression for Ov/Ot in terms of v alone.

The simplest boundary conditions from a theoretical viewpoint are spatially periodic ones, and it is convenient to expand v over a large but finite box in a Fourier series, so that

v = Zv(k, t) exp ( ik. x). k

The equation for v (R, t) (in component notation, with repeated indices summed over) takes the form

dv~(k)dt : - ~ A/,B~(k) vp(p) v,(r) -- vk2v,(k). (8)

p , - - r = k

M~p,(k) is a coupling coefficient defined by

i M, p, (k) ~--- ~ P, p, (k),

P,p,(k) = kpP,,(k) + k,P,p(k),

p,B(k) : 8, B _ k,k~___. (9) k ~

90 David Montgomery

The sum is over all wave numbers p and r (allowed by the boundary conditions) which can be added vectorially to give k. The column vector of eq. (1) is now the column voctor of all the different Fourier coetticients va(k, t), a = l , 2, and 3. The time argument has been omitted for economy. If an external force is stirring the fluid, its Fourier transform should be added to the right-hand sido of eq. (8). k ,v , (k)=0 means that there aro only two independent components of v(k) for each k.

I f all variables aro written in dimensionless units, with a mean velocity U measuring the velocities a n d a characteristic length scale L measuring the lengths, the equations again take the form of eq. (8), but with v replaced by R -a, where R is the Reynolds number: R=UL/v. Typically R is ~ 1 for both atmospheric and oceanographic applications, and is usually �87 1 for industrial processes. Thus the linear terms in eq. (8) aro often orders of magnitude smaller than the nonlinear ones, except at the smallest spatial scales (largest values of k~), where the viscous term begins to increase relativo to the nonlinear term and for large enough k ~, eventually dominates.

The linear temas in eq. (8) aro dissipative and the non-linear ones aro conservative, so that if we drop the viscous term altogether, we can readily prove conservation of energy:

_a ~ I v.(k)[ ~:0 (v=0). di k

(10)

If we keep the viscous terms, we can show that

dtd ~ [ v'(k) [ a =--2v ~ k~ [ v'(k)[ 2" k Ir

(11)

The principal function of the linear terms is to drain off energy at short wavelengths, while the (much larger) nonlinear terms have the effect of moving energy from one place to another in k space ('non-linear scrambling', in Orszag's phrase). In the applications of interest, many different k vectors get excited simultaneously and feed each other. Eq. (8) can stand for literally millions of equations.

For this reason, a full solution of eq. (8), starting from sharply-defined initial condi- tious, is neither possible nor desirable. One must fall back on statistical mechanies, as elsewhere in the physics of systems with very large numbers of degrees of freedom. We may treat the v,(k) in eq. (8) as random variables, with the randomness introduced by initial conditions which aro probabilistically distributed (an ensemble). Averages over the ensemble, or expectation values, will be represented by angular braekets ( ) . Each member of the ensemble, or ' realization ', evolves deterministically according to eq. (8). Ensemble averages are regarded as the best possible predictions for a single experiment. Fluctuations about the ensemble averages ate regarded us moasuros of the predictive power of the ensemble averages.

There are real differences between eq. (8) and the single-particle equations of motion from which orto often starts classical equilibfium and non-equilibrium statistieal mechanics. For one thing, there are an infinito number of them, so any manageabl• description involves a truncation, or restriction to a largo but finito subset, usually by introducing a maximum wave number kma ~. That has the implication, that must be constantly kept in mind, that no spatial features with a lengtla scalo less than ~ k -x

lna•

lVŸ and plasma turbulence 91

can be resolved. If there were, for example, some kind of diseontinuity (such as shoeks in compressible flow) which developed, we could not see it clearly in sucia a truncated representation.

Lee (1952) proposed the introduction of a phase space whose coordinates are the real and imaginary parts of the v,(k). If we drop the dissipative term in eq. (8), we can see at a glance that

O [dv.(k)] =0, OA,(k) l_ dt J

(12)

which is the condition for incompressible flow in this phase space. A Liouville theorem immediately follows, which states that

/ 3 P[vl(kl), vs(kl), va(k1), 5(ks) .... ] =0.

Dt (13)

In eq. (13), P is the probability distribution of all the different Fourier coet¡ in the phase space, and D/Dt means a convective time derivative following the motion of a typical phase point for any realization.

The possibility exists of trying to do an equilibrium statistical mechanics of the Fourier coefficients in analogy with the equilibrium statistical mechanics of point partMes which we learn in school. Unfortunately this turns out not to give realistic or useful answers, but we learn something important by trying ir anyway.

Constructing thermal equilibrium ensembles by the combinatorial methods of Boltzmann and Gibbs involves the constants of the motion in ah important way. One seeks the most probable state, subject to constraints provided that the expectation values of the constants match prescribed values. The resulting canonical ensemble depends upon the constants of the motion (represented by their expressions in terms of phase space variables) in an exponential way. Usually there is only one constant of the motion of the proper type, namely the energy, and the resulting canonical distribution

P = const • exp (--e/T), (14)

with the Lagrange multiplier 1/T identified as the reciprocal of the temperature, is familiar to everyone.

Ir is instructive to see what this program leads to in the present case. The' energy ' e is now just

e = El v.(k) l ~q (15) k

(This quantity is still constant if kmax is finite.) The canonical distribution (14) then factors into a product of single-k Gaussian distributions, one for each k, and ir is elementary to show that equipartition results:

( I v.(k) ] s) = 2T, all k. (16)

92 David Montgomery

In fact, nothing like eq. (16) is observed. What is observed is fr•quently a propor- tionality something liko

( I v.(k) I ~) ~ const • k -11/3 (17)

corresponding to an omni-directional energy spectrum (i.e., 4~rk 2 ( [ v,(k) [2 )) which vades as k-5/3 (Kolmogoroff's law). This remains true, for aU but the highest k's, no matter how small we make the dissipative terms, so long as they are not iderttically z e r o .

There is no reason to doubt the accuracy of the prediction (16) for the truncated version of (8) with strictly zero dissipation (v=0). What is being illustrated is that the properties of (8) with v # 0 are totally different from those with v--0, no matter how small v is. The real system never exhibits thermal equilibrium behavior, or anything close to it, although its equations of motion over most o f k space may differ insignificantly from the strictly non-dissipative equations which do lead to thermal equilibrium behavior. This is the most important difference between turbulence theory and most of statistical mechanics: turbulence is always far from anything that can accurately be called thermal equilibrium.

Because of the fact that the Navier-Stokes eq. (8)dissipates energy (see eq. (11)), the only steady-state behavior possible is one in which energy is supplied at the same ra tea t which it is dissipated. We have also observed that the dissipation occurs only at the upper end of the wave number range for high enough Reynolds number. Ir also frequently turns out that energy is supplied at characteristic spatial scales that are toward the longer-wavelength (smaU k) end of the range, by some macroscopic stirring mechanism. If the places in k-space where energy is injected and dissipated are far apart, the spectrum in between becomes isotropic over direction, and assumes, a universal shape which has been closely approximated by the Kolmogoroff (1941) universal k-S/a omnidirectional energy spectrum.

Very roughly, Kolmogoroff 's picture was of a steady, pred0minately local, flux of energy through k space. Far from the injection or 'forcing' wave number (kF, say), but still well below the values o fk for which the viscous term in (8) becomes important, a universal forro for the wavelength distribution of this ' cascading ' energy is sought. More speci¡ it is assumed that the omnidirectional spectrum of kinetic energy per unit mass, e(k)-~4zrk 2 ( I v~(k) 19 )q can depend only upon two quantities: e, the kinetic energy per unir mass supplied to and dissipated by the fluid, per unit time; and k itself. The omnidirectional spectrum, which has dimensions LaT ~, is set equal to

( ~ ) == c~'k B, (18)

where a and fl are exponents to be determined, and C is a universal dimensionless constant, e has dimensions L2T ~3. Equating powers of L and T on opposite sides of (! 8) gives a=2/3 , f l~--5/3. This prediction has been verified repeatedly experi- mentally [by Grant et al (1962) in perhaps the most elegant measurement] and is discussed from a more sophisticated perspectivc by Kraichnan (1973).

There are also a wide variety of time dependent problems that are interesting, and on which great progress has been made. Most of this progress stems from work by Kraichnan (1958, 1959, 1964, 1967, 1975) and bis collaborators, in what may prove


to be the most significant set of contributions to any kind of continuum mechanics in the twentieth century. These papers are long, formidably ditticult to read, and in some cases lack a clear analytical justification for their results. They might not have attracted much attention if they had existed only as theoretical exercises. But the agreement they have achieved with experimental data (Kraichnan 1964) and with the results of numerical solutions of Navier-Stokes equat… (Herring and Kraichnan 1972) is unprecedented in the subject of turbulence and unknown in the case of plasma physics. The most intuitive elementary treatment of the first of these time-dependent theories, the 'direct-interaction approximation' (DIA), is by Betchov (1966). We have given our own derivation, which is at least no worse than some others, in Seyler et al (1975).

The questions the time-dependent theories try to answer are more detailed than steady-state spectral laws. A central kind of calculation to make has been a decay catculation, i.e., the evolution of ( I v,(k, t) / 2) as a fullction of time for ah initial value problem. The subject of time-dependent theories would require more space than is available here to expose. Again, Orszag's (1974) notes may be the best in. troductory exposition now available. We shall return to the subject of time-dependent turbulence theory for the case of Vlasov turbulence later (see section 5).

3. Two-dimensional turbulenee: inverse eascades

In dealing with large-scale atmospheric motions, we notice right away that the characteristic wavelengths are much greater than the thickness of the earth's atmosphere. That fact, combined with certain peculiar features of the' Coriolis force (Pedlosky 1971), means that for purposes of discussing atmospheric turbulence, the case of Navier-Stokes flow which is independent of one coordinate, say z, and with velocity vectors only in the x and y directions, becomes of interest. For this case, the vorticity vector,

to : V • v : oJ (x, y, t)~~ (19)

is constant in direction and normal to the plane of the velocity vector, oJ may be thought of a sa source for v.

Fjq~rtoft (1953) noticed some time ago that there were some strikingly different properties for two-dimensional turbulence. For the result of taking the curl of eq. (5) is just

( �91 v ' V ) to = vV2to (20)

so that to the extent that viscosity is negligible, to is constant fo ra given fluid element. In particular, the enstrophy, or mean square vorticity, is constant:

~2 = f ( v • v) ~ dxdy = fto2 dxdy = const. (21)

The k-space expressions for energy and enstrophy,

e = Z l v , ( k ) l 2, k

: ~ k ~ [ v,(k) I z, (22) k

94 David Montgomery

ate both constant in time according to the zero-viscosity version of eq. (20), and even ate constant in the face of a truncated Fourier representatiort with a finite kma,- The analog of eq. (8) is

dtoOk)dt -- ~ Mt(r' p) to(p) to(r) - vk* to(k) (23)

p+r=k

where M x (r, p)=�89 • r) (r ~z -- p-~), and when we truncate, we omit all terms with k, p, or r greater than kmax from both sides of eq. (23).

Neglecting viscosity for the time being, what we notice about eq. (22) is the following. Suppose, as in three-dimensional turbulence, the spectrum evolves so that the excitations in the I v,(k) [z pass, in time, to higher and higher k-values. Then what the simultaneous constancy of both e and ~ tells us is that any such transfer of [ v.0k) l s to higher k must be accompanied by a simultaneous transfer to lower k.

The implications of this aro that there is in the medium some capacity to organize itself on larger and larger spatial scales. This idea had antecedents in a discrete- vortex model of Onsager (1949) (see also Joyce and Montgomery 1973, Montgo- mery and Joyce 1974, and Seyler 1976), and flies in the face of many of our preconcep- tions of turbulence as a mechanism which degrades long-range order. The process is presently believed to play an important tole in the generation of weather patterns at the largest spatial scales.

When one tries to do an equilibrium statistical theory for two-dimensional statistical theory for two-dimensional turbulence by analogy with eqs. (14) to (17), the question arises as to which global constants of the motion should be used in constructing the canonical dist¡ Any integral of the forro ( n : a n y integer)

In ~ f to" dxdy

is eonstant, according to the inviscid version of eq. (20). Kraichnan (1967) made the important observation that of the various possibilities for In, only n = 2 gives ah integral which is still constant when the truncated zero-viseosity versiort of eq. (23) is used for the equation of motion. Constaats of the motion which survive the k- space truncation are, for reasons not altogether clear, more central in the statistical behavior of the fluid than the others; we will call them ' rugged ' constants of the motion to distinguish them. If, as is now believed, the canonical distribution for two-dimensional turbulence involves only two rugged constants of the motion, e and ~ the analogue of the Gibbs distfibution eq. (14) is

P = const • exp ( - a e -- fil/) (24)

with two temperature-Iike Lagrange multipliers a -x, B -t, This leads to the modal energy distribution (Kraichnan 1967)

( I v.Ot)] s) = (a + flk~) -x. (25)

a and t8 ate deternª as the solution of the pair of simultaneous algebraic equations for given (e) and ( ~ ) :

(~> = ,~(~ + ~k~) -t, 1,

( f~ ) = ,W,'(~+#kW1, (26) k


in terms of the expectations ofe and ~ . Even though omission of viscosity from (20) is expected to lead to unphysical behavior, it is still of interest to test the predictions of (25) for the various k's. This has been done to a considerable extent by Seyler et al (1975) and Basdevant and Sadourny (1975). Either a or q can be < 0, so long as ~ q- q ~ is > 0 for k in the interval km~ >~ k >~ kmin, where kmi a is the minimum wave number allowed by the boundary conditions. For fixed e, ~ , letting km~ x get hrger and larger always takes us into the regime a<0, q For this regime, ( I v,(k)12) can be quite sharply peaked at kmi n. This situation shows up macrosco- pically as a pair of large counterrotating vortices which fill up the basie periodic box. (The appearance of such patterns in numerical solutions of the two-dimensional Navier-Stokes equations by Deem and Zabusky (1971) and Tappert and Hardin (1971) at Bell Laboratories was what initially aroused our interest in this subject. Montgomery (1972) was the first to connect these results with the phenomena of negative temperatures.)

Kraichnan (1967) also constructed Kolmogoroff-like spectral predictions for possible quasi-steady states involving smaU but finite viscosity and external forces which inject energy near a forcing wave number k F. Kraichnan conjectured a dual cascade,

with energy moving away from k F toward smaller k, and enstrophy moving toward

large k in the opposite direction. Kolmogoroff dimensional-analysis arguments like those which lead to eq. (18) give omnidirectional energy spectra for the two ranges of k-S/3 and k -z, respectively. These predictions have been tested by numerical simulation by Lilly (1969), and by Fyfe et al (1976). The results are not inconsistent with the Kraichnan predictions, but the last word has not been said. It is uncertain what happens to the energy when ir gets to the lowest allowed wave numbers. Does ir accumulate there or bounce back? No simulation has yet run accurately for long enough to find out. And the scatter in the data is large.

Time-dependent theories, sucia as the direct interaction approximation and its descendants, have been applied with some success to two-dimensional Navier-Stokes decaying turbulence (Her¡ et al 1974). Most of the emphasis has been on situations in which the back transfer was not expected to be a dramatie effect.

Our next task is to try and see what the implications of this body of theory are for magnetohydrodynamic turbulence and Vlasov turbulence.

4. MHD turbulence: the possibility of inverse magnetic eascades

In the case of incompressible MHD, the column vectorf of eq. (1) can be thought ofas

where v, B are the velocity field and magnetic field. Written out in an appropriate set of dimensionless variables, with velocities expressed in units of mean Alfv› speexl, eq. (l) becomes

~3v + v . Vv = (V • 13) • B -- VP + vV%, (27) 0t

0B -- V • (v • B) q- q (28) Ot

96 David Montgoraery

v--l, ]/.-1, are mechanical and magnetic Reynolds numbers, respectively. Supplo- mentary eonditions aro

V " v--0 and V " B = 0, (29)

white the electric current density J--:V x B a n d the vorticity vector ~- - -V x v can be regarded as derived quantities, p is regarded asa function of v and B which is given by the boundary conditions and the Poisson equation which results from taking the divergence of (27) and using V " v--0.

Ir is worth remarking that unless one includes ah external d.c. magnetic field, the incompressible MHD equations linearized about the quiescent equilibrium (v=:-:0, B : 0 ) have no normal mode solutions. They share this property with the Navier- Stokes equation, which can also be regarded a s a special case of eqs (27) and (28) with B----0. Thus if we try to approach the qualitative behavior of the incompressible MHD fluid from the point of view of its normal mode solutions (4), we are gua- ranteed to miss virtually everything iuteresting about ir!

Putting the compressibility of ah external d.c. magnetic field obscures these intrinsically nonlinear excitations by giving us some normal ruedes to conjure with, but they are still there. Putting the compressiblity in, if done with proper care, will lead us to some interesting unsolved problems which are complicated by two difficulties which are troublesome. One is mathematical: the presence of fractional powers of the density on the right-hand side of eq. (27). The other is physical: the virtually certain appearance of shocks in the solutions which lead to singular behavior and which will be difficult to treat by any kind of decompositiort into Fourier series or other orthogonal functions.

Much of the work on MHD turbulence has been motivated by the 'dynamo problem' in which astrophysical objects may be conjectured to generate their own macroscopic magnetic fields. The volume of Roberts and Stix (1971) contains reports on many of the less recent investigations in this area. There is also no short- age of laboratory examples where the conversion of kinetic energy to magnetic energy may be a dominant process (generation of megagauss magnetie fields in laser plasmas, magnetic island formation in tokamaks, for instance). The stimulus has existed for some time to discuss the evolution of turbulent MHD fields, and the astrophysicists have been busy at this for some time. (see, e.g. Spitzer 1957.)

Somewhere about 1973, Frisch et al (1975) had the idea that in MHD fluids, inverse turbulent cascades could perhaps exist that would bear some similarity to the inverse Navier-Stokes cascades described in Section 3. They concentrated on three- dimensional geometries, and followed the Kraichnan (1967) program more or less as ir has been outlined here for two-dimensional geometries. As a first step, they identified three ' rugged' constants of the motion for eqs (27) and (28). They are the energy

e T ~ ~ f dxdydz (B z + v2), (30)

the magnetic heltcity

H M - - �89 f dxdydz A" B (31)


where A is the vector potential (V • A --- B), and the cross helicity

H C ~ �89 f dxdydz v" B. (32)

Absolute equilibrium Gibbs distributions depending exponentially upon the truncated Fourier representations of e T, HM, and H C were constructed, each invariant

being multiplied by the reciprocal of its associated (possibly negative)temperature. For some initial values of eT, H M, H C, the magnetic spectra were found to be sharply

peaked at the longest wavelengths. It was conjectured that, in the presence of dissipation and external forcing terms, eqs (27) and (28) might support (with v ~ 0, /.�91 -r 0) dual cascades, in which a cascade of magnetic helicity to low k might accom- pany a cascade of energy and cross helicity to high k A conjecture concerning equipartition between magnetic and kinetic energies at high k, anda suggested procedure for generalizing the Kolmogoroff-Kraichnan arguments concerning cascades, led to predictions of omnidirectionai magnetic energy spectra in the inverse cascade region of k -1 and omnidirectional magnetic and kinetic energy spectra of k-31 " in the direct cascade region (Pouquet et al 1975). These predictions have not as yet been tested by either numerical solution of the MHD equations or by experiment. Some verification has been achieved within the framework of the so-called ' eddy- damped quasi-normal ' model equations (Pouquet et al 1975) which is less than wholly satisfactory because of the sensitivity of the spectral predictions of these model equations to the choice of certain expressions for relaxation times; these are at present physically undetermined and have a great deal of arbitrariness in them, which can reflect itself in different possibilities for the spectral exponents.

Our efforts have centred on the two-dimensional MHD case, which is as different from the three-dimensional case as the two-dimensional Navier-Stokes case is from the three-dimensional one. It is also possible (Montgomery 1976) to argue that under a strong enough external d.c. magnetic field, MHD turbulence ought to become approximately two-dimensional, with the plane of variation normal to the external ¡ This conjecture had been previously put forward by Kit and Tsinober (1971) on the basis of the analysis of some experiments, and Schumann (1976) saw something similar numerically for somewhat different physical parameter ranges. Finally, it is possible to visualize the addition of the cascaded (non-helical) quantities in two dJmensions within the framework of an isotropic homogeneous turbulence theory, sometking which it has not yet proved possible to do in three dimensions. Helicity is ah intrinsically anisotropic concept.

The theoretical framework was provided by Fyfe and Montgomery (1976) and concerns a situation in which v=(vx, vy, 0) only and B=(Bx, By, Be) only, where v~, vy, B~, and By depend upon x, y, and t, but not upon z. Be is a constant, uniform, externally-applied magnetic field which, once the assumption of two dimensionality has been made, no longer enters the equations of motion in any way.

Fyfe identified three ' rugged ' constants of the motion in this geometry: the energy

e _~ �89 f dxdy (B 2 ~- v2), (33)

the cross-helicity

V -=- �89 f dxdy v- B, (34)

98 David Montgomery

and the mean square vector potential

A ----- �89 f dxdy a 2 (35)

where B = V x a. The equatiorts of motion for this geometry collapse to the pair of scalar relations

/9oJ___.__~ = --v. V~o, + B" VJz, (36) St

-- - -v. Vaz, (37) 8t

where co = ~7 x v = oJz› only, j = V x B =A› only, and a~ is the only non- vanishing component of the vector potential a = a~›

For ( P ) = 0 situations (the main case treated so far) the modal energy expectations predicted on the basis of the canonical ensemble constructed from (33) and (35) are

and

([B(k) I~) = (e + 7k--Z) -1 (38)

( iv(k) l ~ ) = ~-1,

with energy and vector potential temperatures ct -1, y-1. These have been verified in considerable detail by Fyfe et al (1976), by a numerical solution of the two-dimensional MHD equations, using the Orszag spectral method.

Introduction of finite q v, and external forcing terms in eqs (36) and (37) lead to the conjecture of a dual cascade situation in which the mean square vector potential A cascades away from kF in the direction of longer wavelengths while e cascades in the direction ofever shorter wavelengths. A different generalization of the Koimogoroff- Kraichnan program of dimensional analysis from that of Pouquet et al (1975) leads to conjectured wave number spectra (omnidirectional) of k-l/3 for the magnetic energy spectrum in the inverse cascade region and k-S~ 3 for the magnetic and kinetic energy spectra in the direct cascade region. Direct simulation tests of these power laws were made by Fyfe et al (1976), without a large enough number of independent k-veetors to draw any firm conclusions about exponents, but with indisputable evidenc• of a substantial back-transfer of vector potential in k-space, with the attendant generation of long-wavelength magnetic fields.

It is a fair statement that no numerical simulation has as yet been performed for either MHD or Navier-Stokes fluids with sufficient wave number range (i.e., suffieient k~z[k~n) to draw accurate conclusions about exponents in power laws for dual cascade situations. Experiments, also, have a long way to go before adequate tests of these exponents are reasonable to expect. It is more meaningful at I this stage to concen- trate on the qualitative features of dual cascades and avoid being drawn into the ' battle of the exponents '.

There is a time-dependent theory of MHD turbulence based on the direct interaction equations (Kraiehnan and Nagarajan 1967; Fyfe 1976) but it has notas yet been subjeeted to any numerical treatmvnt.


5. The DIA for Vlasov plasmas

In this section, some results are presented which have not been previously published. They concern the so-calted ' direct interaction approximation' (DIA) for a Vlasov plasma. The principal antecedent of this work is a paper by Orszag and Kraiclman (1967). Related work less directly connected with the present development is duo to Dupreo (1966, 1967) and Weinstock (1969, 1970, 1972).

The goal is to generalize the DIA from Navier-Stokes turbulence theory (Kraichnan 1958, 1959) in order to acquire a computable set of closed differentio-integral equations for which a detailed numerical integration, paralleling that of Kraichnan (1964) or Herring and Kraichnan (1972), can be carried out. It should be kept in mind that DIA equations are intrinsically of such a character that virtuaUy no analytical consequences of them have been deduced. In the light of their algebraic consequences alone, they would have little demonstrated worth. Ir has been only by the closest kind of interplay between analytical theory and numerical solution of the resulting approximate equations that the DIA has been mado to yield anything worth- while. This interplay may represent the most important event thus lar in the history of turbulence theory. It is timo tosee if a parallel program is possibl• for a Vlasov plasma. Ir is not obvious that it is possible; the higher dimensionality of the phase space description required for a Vlasov plasma puts the problem on the periphery of what is at present computationally feasible, even for the one-dimensional caso. Nevertheless it should bo kept in mind that between 1959 and 1964, ir was also doubted that the DIA equations could be handled numerically; and likewise, that a recurring feature of computational physics has been the ease with which 'unsolvable ' numerical problems have become tractablo as advances in computor technology and programming techniques became available.

Wo start with the one-dimensional Vlasov-Poisson systom for ah electron plasma, with electric field E=E(x, t) and distribution function f= f (x , r, t). In approp¡ dimensionless units, these are

~f (39)

S E 1 - f fdv. (40) Ox

Wr consideran ensemble of sinª Vlasov plasmas and donote avorag•s ov•r the ensemble by ( ) . By 8fi we mean the fluctuation about ( f ) ; thus

f = < f ) + S f (41)

and

E = (E) + SE. (42)

We shall restrict attention to homogr turbulence in the pmsence of zoro average eloctric fiold, so that (E ) = 0 and

E = �91 (43)

Wo also assume that O<f )[Ox=O.

100 David Montgomery

We will find it convenient to introduce the notations

f(1, ti) =--.f(xl, v l, t O,

3]'(1, t o ~ 8f(x l, v l, ti),

f(2, t2) =---f(x z, v 2, t2),

f(3, t l) -~f (x a, va, ti),

etc. Note that ( f ( l , ti) ) is a function of v t but not of x i. Note also that in general, 3lis not understood as ' small ' compared to ( f ) .

Both E and 3]" will be assumed to obey spatially periodic boundary conditions over a la rge but finito interval. We write

3]'(1, t o -~ l k 3fk (1, ti) exp (ikxl),

E(1, t O = lkE~(ta) exp (ikxl) (44)

where 3fk(1, t o depends upon v i, t l, and k, but not x l, and Ek(ti) depends upon t a and k but not xi or vi. We also have that

Eo( tl) = O,

(Ek(ta)> = O,

(8]'(1, tO) = 0, (45)

and that

Ek(ti)----- k f dvi �91 ti) = Lk(1) �91 tO. (46)

Eq. (46) defines the linear operator Lk -~ - - L ~ . The cortfiguration-space version of the same solution to Poisson's equation that eq. (46) represents wiU be written as

E(I, ti) = L(I) 8f(l, ti).

Ensemble-averaging Vlasov's equation gives

_0 ( f (1 , ti) ) - ~ (E(I , ti) �91 ti) ) ~tl 0vi

• L(2) (�91 ti) �91 ti) ), 0vi

or in Fourior-space,

Ot---- x ( f ( I , ti) ) = Lk(2) (Sf_k(l, ta)3fk(2, tx)), k

(47)

(48)


where there is no k = 0 term in the sum. For shorthand we introduce the notation Qk(1, ti, 2, t2) = (�91 ti) 8f_k(2, tz))=Q_k(2, t~, 1, ti) = Qk*(2, t~, 1, ti). Then eq. (48) can be written as

0t-1 <f(1, ti) > -- �91 1 k(2) Qk(1, tr, 2, ti). (49) k

Eq. (49) is the simplest of the three DIA eqs to be derived. In order to get an equation for advancing Q~, we must first introduce the unit

infinitesimal response function, or more colloquially, the 'Green's function '. To introduce this properly, we need to w¡ the Vlasov equation in a way that makes it look as much like the Fourier-transformed Navier-Stokes equation as possible. We first write Vlasov's equation as

� 9 1 + � 9 1 + v X_( ( f> + �91 -- E ~ (<f>) + �91 l q

~x Ov (50)

and subtract from it O<.f)[c~t = O(fE )/Ov, to get

[ L +vi ~ - - 3 ( f ( 1 , ti)>L(1)] 3f(l, ti) �91 ~xx ~ j

= ~ [E(1, ti) �91 ti) -- (E(1, ti) �91 ti))]. (51)

Fourier-transformed, eq. (51) becomes

[~tx + ikvi -- �91 f ~ t i ) ) Lk(1)] ~f~(l' tz)

_ OviO ~ L,(3) [�91 ti) �91 ti)-- ((�91 ti) �91 ti))]. p+r=k

(52)

Because of the assumption of homogeneity in space, the last term of eq. (52) is zero for k # 0, so that (52) can at last be written as

I~ ] ~ + Hk(1, ti) �91 ti) -- bvl Lp(3)�91 tl)�91 ti), p+r=k

(53)

where by Hk, we mean the linear operator defined by

Hg(1, t i )g( l , t i) ~ ikvig(l, t l) - - ~ <f(1 , ti))Lk(1 ) g(1, t o

for any function g(t, ti). If ( f ( l , ti) ) were time-independent, Hk(1, ti) would be the linearized Vlasov operator, but in fact, there is nothing about eq. (49) to indicate that (f(1, ti) ) is even approximately independent of t 1.


Wo aro interestod momentarily in the solutions of eq. (53) linearized about any single realization in the ensemblo. If we call �91 m (1, ta) an infinitesimal perturbation on 8fk(l, ta), then eq. (53) gives, discarding products of perturbation quantities

[~tt q Hk(1, ta)] 3f~x)(1, ta)

�91 ~ L,(3) [3f,(3, ti) ~f, _ a m (1, ti) Ovl

p+r=k

+ �91 t o �91 ti) ]. (54)

In particular, wo aro interested in the solution of eq. (54) which approaches 8(vl-- v o) 3(k -- k 0) as ti --> to from above, and which vanishes for q < t o. This is the ' unit infinitesimal response function '. It is generally a function of k, k0, v, vo, q, t 0. We call the part with k = k o the ' diagonal part ', and the part with k ~ k 0 the ' off-diagonal par t ' (Seyler et al 1975). Sometimes when thero is no possibility of confusion, we shall abbreviate Hk(1, ta) as simply Hk(ta).

The diagonal part obeys the equation

[ 0 ] G ~ (vz, tz , yo ' to ) + ~~o (ta) ~o

- - ~ ~ L,(3) [8f,(3, ta) G ~ (v x, t 1, v o, t o) 8v~

p+r=ko

+~f,(1, tO ~ (v3, t~, yo, to):]

= 8 ( v ~ - yo) �91 ( t ~ - to) (55)

with G~o=--0 for t l < t o. The superscript 0 moans that k = k o is the wavenumber which

has beon impulsively porturbed. The last term on the left-hand sido of eq. (55) involves only off-diagonal parts of G ~

Wo write the off-diagonal equation for G ~ as (k # ko):

[~t~ d- Hk(ta)] G~ (vx, tl, Vo, to)

-- ~ V ' L,(3) [�91 ta) G ~ (v 1, t 1, v o, t o) ~PI ~ �9

p+r--k

+ �91 t o G~ t x, v o, to)]

= -~ILL% (3) 8f~_k o (3, t o G~o(v 1, t x, v e, t o)

+ ~ z~ o (3) ~f~o (1, ta) a ~ (v~, t~, yo, to) 8v z kO

(56)


The primo on the summation sign in eq. (56) means to omit the term with r -- k 0 from the first sum and the term p = k o from tho second. These terms have been singled out and written on the right; they aro the only ones involving diagonal parts of G ~ Again, G ~ -~ 0 for tl<t o.

We now formally solve (56) for the off-diagonal part of G ~ in terms of the diagonal o

part. The Green's function gk for solving (56) oboys

[ 8 + ] o ~ t H~(tt) g~ (vi' tx' v~ to)

- - ~ ~ L,(3) [�91 t Og;(v t,t l ,v o,to) 8vx

p+r=k o

+ 8f,(1, ti) gp (va, t x, ro, to) ]

= 8(tl -- to) �91 -- Yo) (57)

with g0 ___ 0, if t t < t 0. In omitting the primo from the summation, we have made our first (and least severe) approximation: wo have put in the two omitted torms of the summation in (56). We can now recognize that the eq. (57) for g0 is formally identical with eq. (55) for the diagonal part of G ~ and that the two functions obey identieal initial conditions. Therefore they must be the same function. Since there is nothing special about k 0, the result should hold for any wavenumber. We may drop the superscript 0 and the use of G for the diagonal part altogether, and write for the 011"-

diagonal part of G: the formal solution

t t

G: (v x, t x, v o, t o) = f to

ds f dv' gk (v~, tx, v', s)

[ 0 (4) 8 (4, s) (v', to) • Lk~, o f ~ o g% s, I~o~

q + 8 Lk (4) �91 (v', S) g% (V o S, V o, to) J. r 0

(58)

Finally, (58) is to be substituted into (55) giving an equation for the diagonal part of G alone:

[o ] + H~ ~ (I, tx) g~o (vx, tx, yo, to)

tl

_ ~_ ~ ,~, ~,, I~:, ~3. ,, f 4~,.,. ,,. ,. ,.. ,, p+r=ko to

[ 8 (4, s) (v', s, v o, t o) • ~.., t,,_~ o (4) 8f,_~ o g~o

P. (A. ) - -2


+ ~v, Lko (4) �91 (V', g) g~o (V4, S, Vo, to) l

t i + 8f, (1, ta) f

to ds f dv' g, (va, t 1, v', s)

• I ~ L% (4) �91 (4, s) g% (v', S, Vo, to)

+ " L ~ 0 (4)�91 (v', s)gk o (v o s, v o, to)J~ tgv'

= 8(v 1 - - yo)8(t 1 - - to). (59)

We now ensemble average (59), and make a more severe approximation that has not as yet been tested, even for the Navier-Stokes case. Namely, we assume that in the ensemble averages which result from averaging (59), & can be satisfactorily replaced by its ensemble average; i.e., we are assuming it is a satisfactory approximation to replace gk by its ensemble average (gk). The result is

I~ 1 + Hk (1, t o (g~ (vi, tx, Yo, to))

tx

p + r = k to

&f&, <~, <v~, tx, v,, s))

• 0-~ [L-r (4) (gk (v', s, v 0, to) ) Q, (3, t 1, 4, s)

+ Lk (4) (& (v 4, s, v o, to) ) Qp (3, t 1, v', s)]

t t

+ f dsf av' (g, (va, ti, v', ,)> to

• 0-~ [L_, (4) (gk (v', s, v 0, to) ) Q, (1, ti, 4, s)

+ L~ (4) (gk (v4, s, v o, to) ) Q, (1, h, v', s)] f

= 8(vl- yo) 8(ti-t0). (60)

Eqs (60) and (49) are two of the three DIA relations. In order to close the system, we now need to derive an equation to advance Qk in time, involving (gg) and ( f ) . This is the mysterious part of the theory ! Nothing up to this point shouid seem parti- cularly shocking, but there are some steps in what follows that may be most accurately described as mystical.


Multiplying (53) by 8f_~(2, t2) and ensomblo averaging #ves an equation for Qk:

[ 0 1 + Hk(1. tl)j Q,(I. ti. 2. t~)

a ~ Lp(3) (�91 tx) 3.f,(1, t o 3f_k(2, tO), (61)

p+r=k

which is exact. To close the system requires an approximato expression for tho bracket on the r.h.s, of (61) in terms of ( f ) and (gk).

In fluid turbulence, an experimental fact is that to a good zeroth approximation, the turbulent variables are Gaussianly distributed. We assume the same to be truo of the 8fk. But ir they were exactly Gaussianly distributod, the r.h.s, of (61) would be identically zero. Ir is the small departure from the Gaussian nature of the distri- burlon of the �91 that permits any evolution of Qk. We writo symbolically that

8/, =ss~ + ~s~ G (62)

whero the superscripts stand for the ' Gaussian ' and ' non-Gaussian ' parts of 8f~. This is purely a symbolic notation for what one hopes can be made systematie in

terms of probability distributions of 8fk. 8f 7 - is assumed to be ~, than �91 in the

sense that � 91 makes the dominant contributions to even moments:

Q.(1, tx, 2, ti) ~-- (8f7(1, tl)Sf~ (2, t~)), (63)

but the ffrst non-zero contribution to the r.h.s, of (61) is

[o ] + Hi0, tO Q.(1, h, 2, t,)

-- ~ ~�91 Lp(3){(sfNG(3, ti) ~f~(l, t,) 8f~ (2, tO) Ov 1 ~ -

p+r=k

+ (SfpG(3, ti) 8fNG(1, ex) 8fG(2, tz)}

+ (�91 ti)~f?(1, t,)8f~G(2, tO)}. (64)

Cle, arly the problem is how to represent 8f NG in terms of 8f7. This is done by returning to

[~ } o2 -~1 H~(tx) �91 tx) = 0vi-- L,(3) 8f,(3, q) �91 (1, ti). (65)

p+r~=k


We hypothesize that for purposes of calculating, e.g.,

(sfNG(I, ti) 8fG(3, ti) 8fG(2, t~)), we can calculate 8fNG(I, ti) in the following way.

Namely, let

H,(t 8f (1, ti) _~ --0v 1 L,(4) 8fG(4,, t~) �91 (1, t o (66)

l + A = r and

- - 0 E L,(4)[�91 ti)+sfNG(4, ti)] ~vt l+ ;t=r

X [Sf G (1, ti) q- 8f~NG(1, ti)], (67)

where the tilde ,~ means that in (66), the triad interaction involving (I, ,~) = (k, --p) is omitted from the sum, and the tildes in (67) mean that this triad has been turned on again, so that all the triads are included in the sum. (This fictitious problem, in whieh we have the option of disabling the triad interactions one at a time, plays a central role in the calculation.)

�9 This is the essence of the DIA procedure. We turn offthe triad couplings one at a time against a fully developed turbulent background, instead of turning them on one at a time against ah empty (vacuum) background, as we would do in normal perturbation theory. Then we superpose the results as if the corttributions were indel~ndent. This procedure has at presenta cook-book status, and has not been justified from first principles (cf., however, Kraichnan 1959, 1975).

Subtracting (66) from (67) and discarding products of perturbatiort quantities gives

[~tx + Hr(ti)] 8fNG(I, ti)

-- E L,(4)[Sf?(4, ti)8fNG(l, t l ) + 8fNG(4, tz)Sf G (1, tx) ]

l+A=r

_ 0 L~(4) 8fG(4, t 0 �91 tx). (68) Ov 1 -p

The way (gk) makes its entrance should now be obvious. For the Green's function

for (68) obeys exactly (57), with k replaced by r. Calculating ff'fNG (1, t i) from (68) and superposing the contributions of all possible triads gives

l+A=r

• /9 Li(4) f G (4; s)fG(v ', s) (~y' l

(69)


where we have again replaced g, by its ensemble average. We derive two similar expressions for 8fNG(3, t 1) and �91 t~).

p -

To compute (Sf NG (1, t O �91 ta)3f_Gk(2 , ta)), which aceording to (69)is equal to �9 p

l+ ~==r

• ~ L , (4) ( f7 (4, s)�91 s)�91 t l) 8f~ (2, ti)). (70) DV' P

we make use of the well-known result that for any four Gaussian rartdom variablos of zero mean (say �91 �91 8f(3), and �91 the following relatiort holds:

~3f(l) �91191 � 91 (SŸ 3.]'(2)) ~8f(3)�91

+ (�91 8./'(3)) (�91 8f(4))

+ (�91 �91 (�91 8f(3)). (71)

This has the implication, aceording to (63), that the expression (70) ean be expressed wholly in terms of Qk's and (gk)'s, thus providing the elosure we need. When the algebra (now straightforward but tedious) is all done, we fiad that eq. (64) takes on the forro:

] Hk(I, ta)] Qk(1, q , 2, t~)

tx 0

--~3v, ~ L,(3)I f ds f dv' (g,(v a, t, v', s)) p--r=k 0

• _0 LLk(4) Qk(4, s, 2, ta) Q, (1, t 1, v', s) t3v'

- - L,(4) Q,(1, t 1, 4, s) Qk(V', s, 2, ta) ] t i

+ fdsfdv' ( g r ( V i , t l , 1,", S)>

0

• ~ [Lk(4) Qk(4, S, 2, te) Q,(3, t 1, v', s) �91

- Lp(4) Q�98 t l, 4, s) Qk(v', s, 2, ta)] t .

+ f dsfdv' (g~(v2, te, v', s)/x 0

• _~O [--L,(4) Q,(1, q, 4, s) Qp( 3, t 1, v', s) Ov'

- Lp(4) Q,(I, t 1, v', s) Qp(3, t x, 4, s,)] ~. )

(72)


Eqs (72), together with (60) and (49) provide a closed, well-defmed set of differentio- integral equations which involve no implicifly defined functions of exponentiated operators. The high dimensionality (there are five running indices) means that they aro not trivial to imagine computing with. But they are, to the author's knowledge, the first sucia closed description that has been obtained, that does not have to be supplemented by any additional hypotheses or physical ' insight', and which leads by a meebanical numo¡ procedure to the evolution of eovariances and spectra. The eomputations to be performed with these three equations will require largo but not prohibitivo amounts of computational skill and machine capacity. It is our hopo to be able to report their solution at some time in the futuro, and make a comparison with a manywave number direct solution of eq. (53).

Two additional features of the DIA for a Vlasov plasma are perhaps worth men- tioning. First, the inclusion of eollisions may be necessary in order to achieve a physically realistic description, oyen at arbitrarily low values of the plasma parameter. The example of the Navier-Stokes equation suggests the possibility that Vlasov turbulenee without collisions may have nothing more to do with real life than Navier- Stokes turbulence without viscosity. This may be true no matter how small the plasma parameter is, so long as it is not zero. The inclusion of a linear (Bhatnagar- Gross-Krook) collision terna on the ¡ sido of (39), and its subsequent inclu- sien in the formalism, would be a minor operation. But the inclusion of a proper (Fokker-Planck) nonlinear collision term would be a significant additional compli. cation. Second, there is in this problem a constraint which has nowhero been imposed on the formalista which has no parallel in Navier-Stokes turbulento. Namely, the quantity ( f ) + �91 is constrained to be non-negativo. For the case of mean-squaro fluctuations 8f which aro small compared to ( f ) , this may not be a serious matter; but for 8f's which aro comparable to ( f ) , the seriousness of the constraint is not at this point easy to assess. (Seo also Orszag and Kraichnan 1967.)

6. Discussion

We have attempted, in the foregoing sections, to create a basis from which plasma physicists unused to the methodology and con~ptualizations of Navier-Stokes fluid turbulence theory, can find a basis for access to that subject. But it would be over- selling the subject to imply that such considorations are likely to lr in the short run, to major contributions to the design of technical devices. Fluid turbulenco theory may havo another lesson to teach, in that the actual contribution of basie homogo- neous turbulencr theory to the design of fluid machinery has seldom been largo. Engineering capability has continually out-run basic theoretical understanding. We can only hopr the sarao will be true for plasma seience. A tragically long wait might be in store for us if we rely on a first-pfincipl•s understanding for input into the design of devices. By the same argument, the experimentalists should not either make unrr dr of this kind on theory, or be too frightened by the threat of thr disapproval.

Navier-S t okes and plasma turbulence 109

Acknowledgements

The author wishes to acknowledge valuable conversat ions with David E Fyfe, Jackson R Herring, and Ann ick Pouquet. This work was supported in par t by the Nat ional

Aeronautics and Space Adminis t ra t ion G r a n t NGL-16-001-043.

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implications of navier-stokes turbulence theory for plasma turbulence

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