mhd turbulance: relaxation processes and variational principles

Physica D 37 (1989) 215-226 North-Holland, Amsterdam

MHD TURBULENCE: RELAXATION PROCESSES AND VARIATIONAL PRINCIPLES

David MONTGOMERY* Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Lee PHILLIPS Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA

Turbulent relaxation processes seem to play a more prominent role in magnetohydrodynamics (MHD) than in hydrodynamics and exhibit a wider variety of behavior. In decaying turbulence, "relaxed" states can result from highly unequal decay rates of extensive, cascadable ideal invariants such as energy, magnetic helicity, cross helicity, etc. In externally-driven MHD systems, the relaxed states may result from other processes, less well understood. We are exploring a formulation based on a principle of the minimum rate of energy dissipation. This is a nineteenth century principle from which seems to descend the more modern (and less well accepted) principle of " the minimum rate of entropy production". Consequences of the conjectured principle are described for two cases: (1) for the reversed-field pinch (RFP), an externally-applied electric field supplies magnetic helicity at a constant rate; (2) for the case of a constant rate of dissipation of cross-helicity, a relaxed state with aligned vorticity and current densities is predicted. The former problem models the steady-state operation of a familiar fusion confinement device. The latter models certain features of solar wind turbulence, regarded as a driven, steady-state system.

1. Introduction

For at least two reasons, the understanding and manipulation of magnetohydrodynamic (MHD) turbulence is more in need of theory than is the case for Navier-Stokes fluids. (1) In the two applications thought to be of the most importance, controlled thermonuclear confinement and inter- pretation of spacecraft measurements, experimental diagnostics are and are likely to remain far less complete than they are for, say, hydrodynamics and aerodynamics. We have much less information, of any kind, about what is actually going on. (2) Because of the long collision mean free paths, the MHD description itself has a less convincing derivation, for parameter ranges of interest, than does the Navier-Stokes description for fluids. What the equations say should happen and what really does happen are rather more distinct issues for magnetofluids than they are for water or air.

*Permanent address: Dartmouth Collef, e, Hanover, NH 03755, USA.

We are still at an early stage in formulating what the central theoretical MHD turbulence questions ought to be. The subiect is less formal- ized and well staked out then is the case for non-conducting fluids, in addition to resting on less solid experimental knowledge. The more abstract and mathematicized issues characteristic of homogeneous turbulence theory or "dynamical systems" theory are mostly present in unsolved simpler problems that originate with neutral fluids, and there is not much incentive to focus on them in MHD at this moment.

Magnetohydrodynamic turbulence also often fails to satisfy the assumptions of symmetry which characterize standard homogeneous turbulence theory. It is highly anisetropie and inhomoge- n e o u s , a n d "~'- ~ . . . . . . . 1__ -~-.~: . . . . ~ k, , t i le lu~ ~c a re

boundary conditions. If an externally maintained dc magnetic field is present, the mean values of the various fields (zero in homogeneous turbulence theory) are often larger than the rms fluctuations about them. The only good reason for studying MHD may be that we need to know the answers to a lot of practical questions, rather than that it is

0167-2789/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

216 D. Mom gomery and L. Phillips / MHD relaxation process

elegant or simple or a natural spin-off of closely related hydrodynamic problems.

In this article, we describe a class of turbulent MHD processes that are believed to be of interest for space physics and controlled fusion. Evidence for them, not always welcome, has been accumu- lating for over two decades. It appears that, under a variety of circumstances, a magnetofluid can pass through a highly unpredictable and variable evolution before settling in to a recognizable "relaxed" state. This relaxed state may itself be quite turbulent and characterized only by regularities that are far less than a complete determination of all the fields; but it is quite distinct from the trivial state of spatial uniformity to which all turbulent fluids tend if they are not driven by an outside agency. Variational principles seem to be successful in predicting some of the properties of the relaxed states, to an extent that is not yet entirely clear. Many of these relaxation processes have in common that they result from the dissipation of some global, cascadable quantity at small spatial scales, simultaneously with the approxi- mate conservation, or constant rate of supply, of another. The relaxed state is sometimes characterized by the near absence of nonfinear modal transfer.

The emphasis here will be on separating, for the first time, the relaxation processes into decay processes, which lead to long-rived transient states for isolated initial-value problems, and steady-state processes which occur for driven, dissipative systems. The first class of processes have been explored for a decade [1-8] under such headings as "selective decay" [2], "dynamic afignment" [4], the loose and somewhat anthropomorphic "self- organization" [9, 10], and (a term that is quite new) "nonlinearity depression" [!!], T h e ~wecmA

class of processes is more recently identified [12], less well understood, and apparently physically distinct from the first, though the results for the relaxed states can sometimes look similar. Previ- ously-discussed "self-organization" processes for isolated systems have been based on the relatively rapid decay rate of certain ideal invariants (rela-

tive to others) at high Reynolds numbers. For steady-state, driven systems, all quantities decay at the same rate at which they are supplied, by definition. Inequalities among free decay rates provide no guidance.

Section 2 reviews the by-now familiar processes of selective decay and dynamic alignment. These are related to inverse cascade processes, which originated in the theory of two-dimensional Navier-Stokes (2D NS) turbulence. Section 3 con- siders the less well explored area of driven MHD turbulence and, in an effort to account for some new results, exhumes and generalizes a proposed variational principle for this case: a principle of minimum rate of energy dissipation, subject to constraints. The principle acquires its content from the constraints and boundary conditions which keep the dissipation rates away from zero and seems to be easy to implement only for the case in wlfich the time-averaged mean fields are large compared with the rms fluctuations about them. Two applications are described. In the first (al- ready presented in detail elsewhere [12]), an external electric field supplies magnetic hericity, at a constant average rate, to a reversed-field pinch (RFP). In the second situation, motivated by the solar wind, cross-helicity is supplied at a constant average rate by a pressure drop along a mean dc magnetic field.

2. Decay relaxation processes

2.1. Basic MHD vocabulary

We try to provide a simple review [1-7] of selective decay and dynamic alignment in the con-

sion will be limited to incompressible MHD with a uniform mass density p, which is not to say that compressible MHD is uninteresting or unimpor- tant, just more co~npricated and les~ developed.

The basic fields are a fluid velocity v with characteristic speed v 0 and a magnetic field B with characteristic magnitude B 0. We work in a

D. Montgome~ and L Phillips / MHD re'axation process 217

familiar set of dimensionless units in which an O(1) macroscopic characteristic length L 0 is the length unit; L o is a distance over which v and B vary significantly. The time unit is the time needed for either a typical fluid flow or a typical Alfv6n

wave to traverse L 0 (i.e. either Lo/v o or Lo/C A, where C 2 = Bg/4~ro in cgs units and 0 is the mass density). Velocities are in units of o o or C A, depending upon circumstances. The dimensionless electric current density is j = V x B, and the dimensionless vorticity to is to = V x v; to and ] may be thought of as analogous "sources" of v and B. It is often convenient to represent B in terms of a vector potential A: B = V x A.

In these dimensionless units, we may write the MHD equation of motion as

O.-7-+v,Vv= -Vpm+jXB+vV2v. (2.1)

Eq. (2.1) is an obvious generalization of the Navier-Stokes equation: only the Lorentz force j x B is different. The kinematic viscosity v is a recipro:al Reynolds number, with a defining velocity that is either v 0 or C A, and for simplicity v will be ~reated as a constant, uniform, scalar, asually << i Pm is the mechanical pressure.

The magnetic field is advanced in parallel to v

by [13]

~B Oi,, = V X(v×B)+~V2B, (2.2)

where 71 is a magnetic diffusivity, also assumed uniform, constant, and scalar, and in the dimensionless units, also << 1 in the cases of most

interest. !t is sometimes convenient to r~ . . . . . . cur!

from (2.2) and advance A instead of B:

~A 8 - - 7 - = e x g + V ¢ - ~ V x ( v XA), (2.3)

where ~ is a scalar potential. It is often conve-

nient to choose the Coulomb gauge, V ,A = 0. Then taking the divergence of (2.3) provides a

Poisson equ~tio, for ¢ that determines it once a boundary condition for ~ is chosen. The determi- nations of O and Pm are similar.

In realistic situations, there are distinct boundary conditions to be imposed upon r, B, ./, A, ~, and Pro" Most of the problems being discussed become much more involved if such realistic boundary conditions are enforced. This has been begun elsewhere [8, 12], but here we are interested in making our point as simply as possible, and we will in general not consider material boundaries with realistic boundary ¢~nditions, but rather, spatially periodic ones.

Of course, eqs. (2.1) to (2.3) need to be supple- mented by V • v = 0 = x7 • B. The condition V • v = 0, upon taking the divergence of eq. (2.1), leads in the usual way to a Poisson equation that determines Pro, the mechanical pressure. V " B = 0, if enforced initially, is preserved by eq. (2.2).

Eqs. (2.1) to (2.3) are perhaps the simplest ver- sion of dissipative MHD that contains its essential features. They would not command universal agreement as an adequate description of any of the phenomena to which they might be applied: they leave out compressibility, radiation, charge separation, anisotropic (tensor) dissipation coef- ficients, kinetic effects, etc. However, the),' are al- ready considerably more difficult than the Navier- Stokes equation, which they contain as a special case (B-- , 0), and which has not been fully di- gested, even after 150 years. So, maybe they are complicated enough for the moment.

A great deal of work has been done with eqs. (2.1) to (2.3) and closely related descriptions as applied to nonlinear problems and turbulence. Refs. [14]-N0] provide representative samples, but are not by any means a comprehensive bibliogra- phy of the field. No attempt is made here to provide a self-contained discussion of such topics as cascades, inverse cascades, spectral power laws, magnetic reconnection, or dynamo theory. In- stead, we only remark upon what happens if different decay rates exist for the expressions which in the absence of dissipation (v = 0 = 7) are ideal extensive constants of ~he motion for (2.1) to 12.3),

218 D. Montgomery and L. Phillips/MHD relaxation process

and fully developed turbulence is present initially. Later, we address the driven, steady-state case.

The most important ideal invariants for eqs. (2.1) to (2.3), under periodic (and some other) boundary conditions, are the global, ideal, quadratic invariants

v. - f d3x(o + 8"),

14 - ½ f

"energy", (2.4)

"cross-helicity", (2.5)

and

fd3xA ° B , "magnetic helicity". (2.6)

The proofs are easy, as far as the iavari'ance under periodic boundary conditions is concerned [18, 191.

Eqs. (2.4) to (2.6) define' ideal invariants for three-dimensional (3D) flows. For two-dimensional (2D) flows, (2.4) and (2.5) are still good ideal invariants, but (2.6) disappears and is re- placed by [91

A= f ' d'xA~., "mean square vector potential",

(2.7)

where for 2D flows, A = (0, 0, At(x, y, t)) only. A is not an invariant for 3D flows and H m vanishes in the usual 2D geometry, but there is a magnetic ideal invariant in both cases. For 2D Navier- Stokes flows there is the well-known ideal invariant "enstrophy":

~2 -= ~ f d 2 x J . (2.8)

The enstrophy is not an ideal invariant for 3D Navier-Stokes, or for MHD.

Ti~ere are other ideal invariants such as (for 2D Navier-Stokes)

(,o n) = f d2x ", n> 2, (2.9)

or (for 2D MHD)

< A : > - fd xA:, , > 2, (2.10)

or the Alfv6n flux invariant in 3D MHD, or Kelvin's circulation integral for 3D Navier-Stokes flow, that seem to be less significant than the quadratic ("rugged") invariants like (2.4) to (2.8) in determining the properties of turbulence. The reasons why are less than completely clear [26]. The general feature in which quadratic global invariants differ from others like (~") or (A~") is that once they and the equations of motion are expanded in a complete orthogonal set of basis functions and tke expansion is truncated, quadratic invariants are still invariant and others are not. (One exceptional case is known [41] of a cubic invariant which is preserved under truncation.) This has been discussed at length elsewhere and no fully satisfactory resolution has been 'given.

It is by now well known that systems such as 2D Navier-St~kes [42-44], 3D MHD [19, 20, 28], and 2D MHD [9, 21-23] are susceptible to inverse cascade behavior (the jury is still out on the case of 3D Navier-Stokes, apparently). The presence of band-limited sources of excitation (band-limited in Fourier wavenumber space) that supply the fight "inversely cascadable" quantity will lead to the systematic transfer of that quantity to longer and longer wavelengths (smaller and smaller k). If the box size is finite, the excitations apparently pile up [32] at the longest wavelength allowed by the boundary conditions until they are limited by their own dissipation. The three clear-cut cases in which an inversely-cascadable quantity has been identified are: 2D Navier-Stokes (E), 3D MHD (H,) , and 2D MHD (A). These have been explored in a rather abstract format with periodic boundary conditions and a random-number-gen- erator-driven forcing function. In what follows, we will presuppose some familiarity with these re- suits. Literally hundreds of relevant graphs have been published but we will not reproduce them here.

D. Montgomery and L. Phillips/MHD relaxation process 219

2.2. Selective decay and dynamic alignment

The preceding remarks are intended as a back- ground for this section. Consider now the case of a turbulent MHD system with broadly excited spectra in 3D, which is being allowed to evolve in the absence of external forcing. Suppose that it is in the nature of the turbulent evolution that E and/arc both decay rapidly compared with H m, so much so that before long they reach the minimum values that they can have compatible with the value of H m that they have at that time. There is then a variational problem, defined by minimizing (2.4) and the absolute value of (2.5) subject to a given value of (2.6). This may clearly be achieved for v = 0, since (2.6) does not contain ~. Using a Lagrange undetermined multiplier h -~, the re- quired minimum should be contained among the solutions to the variational problem

~tf (B 2 - X-XA • B) d3x = 0. (2.11)

We will address the consequences of (2.11) in rectangular periodic boundary conditions only, for simplicity. The problem gets much more involved if material boundaries, vAth boundary conditions on B, j, A, #,, etc., are considered. L e t t i n g B ~ B + S B , A ~ A + ~ A , 8 B = V xSA, and integrating by parts leads to the Euler- Lagrange equation

B = X-XA. (2.12)

Taking the curl of (2.12) leads to

~, x B = 2 = X-~B, (2.13)

which in the case of periodic boundary conditions is equivalent to (2A2). ~dT4r~ ~t, t~ ¢,~ ,~,h;,.h j X B = 0, as in (2.13), are called "force free" [15].

There are many possible solutions of (2A2) or (2.13) in rectangular periodic geometry. For each Fourier mode k, there are two orthogonal pOar- izations of A (k) for wbfich k • A (k) = 0. What (2.12) says is that they must be combined addi- tively so that their sum satisfies i k x A ( k ) =

X-1A(k), and this can b:. done if X - a= +k. For each such solution, E = X-1H m. The minimurr is attained for X -~= +_ the minimum wavenumber allowed by the boundary conditions, depending upon the sign of H m (E is > 0). All the excitations are packed into the longest allowed wavelength.

This is perhaps the simplest example of a "relaxed" state which might result from the unequal decay rates of ideal invariants. It is in fact a caricature of a much more physically motivated calculation due to Taylor [45-48], which was in effect the first MHD "selective decay" theory. It dealt with possible relaxed states of a magnetofluid inside a toroidal conducting shell, idealized as a straight cylinder with periodically identified ends. It is only geometrical complica- tions and boundary conditions that make one selective decay calculation much more complicated than another. Taylor's success at predicting several features of reversed-field pinch (RFP) behavior using (2.13) in more realistic geometry was what led to our own interest in this class of processes, and indeed may be said to have beel~ the beginning of this subject.

We have not yet discussed the reasons for the possibility of highly unequal decay rates, but the foregoing problem illustrates an important feature. In the case of a driven inverse cascade, it is known [19, 28, 32] that H m is carried to long wavelengths where dissipation is relatively inef- fective and that E is cascaded to short wavelengths where it disappears into heat. There is a fundamental tendency for the nonlinear terms of (2.1) to (2.3) to transfer E spectraUy in the dissipative direction and H m in the direction where it will accumulate. It would be surprising if this

decaying initial-value prob]em. "['he oldest known example of such a selective

decay (and the only one easily proved) is for decaying 2D Navier-Stokes turbulence [42-44, 6],

d f ~ dt E < 0, (2.~4)

220 D. Montgomery and L. Phillips / MHD relaxation process

which applies for penodic boundary conditions. The equality sign holds if, and only if, all the excitations are in a single wavenumber k. It has been demonstrated numerically that the single-k states are all unstable against perturbations except k = k~n, corresponding to the longest wavelength allowed by the boundary conditions. What has been said amounts to a demonstration that for 2D Navier-Stokes decaying turbulence,

(~2/E) ---, k~in as t ~ oo. (2.15)

In one sense, this is not such an interesting result, since both E and $2 approach zero as t -~ ee. But for low ~, or high Reynolds numbers, the near- dominance of the k = k mi n contributions can set in a long time before the decay becomes so far advanced as to become uninteresting [2].

It has been implicit in the discussions so far that /4 c did not play much of a role in the selective decay. Suppose now that Hm -~ 0 and I-/¢ had an initially large valae. Suppose [30, 31] that for some reason E rnigt:t decay rapidly relative to H¢ in either 2D or 3D MHD turbulence, so rapidly that it would attain its minimum value, relative to H c. Then, a straightforward variational calculation shows that this is achieved by

v = +_ B, (2.16)

a state we have called "dynamically aligned," and a totally different state than the v = 0 state (2.12). Arguments not apparently related to im'erse cascade processes have been given [4-6, ,~0] as to why E should dissipate more rapidly than H c and seem to be well verified numerically [4-6, 36]. Notice that (2.16), like (2.12), is not a unique determination of the relaxed state, only a property of it.

Unbidden, turbulent MHD processes leading to (2.12) or (2.16) seem to have appeared in rather different contexts. Taylor made the pivotal obser- vation that the state (2.13) (appropriately adapted to more realistic material boundary conditions) could explain several diverse operating character-

istics of the ZETA toroidal pinch device [45-48]. Our work with inverse cascades led us [1] to propose the selective dec,,y hypothesis as the dynamical basis for Taylor's minimization. Numeri- cal [2] investigations of the conceptual analogue for 2D MHD (minimum E/A) confirmed the essence of the basic transfer process. Other related numerical investigations accumulated additional evidence [3, 4, 7]. The hydrodynamic calculation of Bretherton and Haidvogel [49] was also very influential.

Clearly, the states (2.12), (2.13) (with v = 0) and (2.16) cannot be achieved at the same time. Yet even earlier, well-documented examples exist of the achievement of (2.16), particularly the obser- vation of so-called "Alfvtnic" periods in the solar wind plasma. As early as 1971, Belcher and Davis reported [50] periods in excess of 24 hours in which magnetometer and particle energy analyzer data led to the conclusion that v and B were tracking each other according to (2.16) to a high accuracy, despite the highly chaotic nature of both fields. This second relaxation process was actually more directly diagnosed experimentally than had been the selective decay in toroidal Z pinches.

The phase space of initial conditions for decaying MHD turbulence contains regions which lead either to the state (2.12), (2.13) or to the state (2.16), or equivalently in two dimensions to the states of minimum E/A and (2.16). Ting, Matthaeus, and Montgomery [6] explored and at- tempted to identify phase-space boundaries that would separate states of eventual "dynamic alignment" and "selective decay." Roughly speaking, in decaying MHD turbulence, if kinetic energy dominates initially, it will continue to do so there- after. States in which the magnetic energy dominates will continue to be primarily mag_netic~ There is reason to believe that in the absence of dissipation, something like equipartition will occur, but not in decays, unless the kinetic and magnetic energies are comparable to start with. States w.hich are mostly magnetic to start with, as in fusion confinement devices, are candidates for selective decay, and states which have an initial H e compa-

D. Montgomery and L. Phillips/MHD relaxation process 221

rable with E are candidates for dynamic alignment. States which are initially primarily kinetic usually behave like Navier-Stokes fluids and whatever magnetic excitations there are usually seem to evolve passively. States with comparable magnetic and kinetic excitations initially but nc, t much H e can exhibit erratic "transition" behavior before deciding which regime to fall into. The phase-space boundaries are at present only vaguely delineated. Little effort has gone into studying the different regimes as a function of boundary conditions.

3. Driven, dissipative, MILD; Possible relaxation to states of minimum dissipation

The principles of the last section were all based on likely inequalities among decay rates for more than one global, cascadable, ideal invariant. Such a formulation is inadequate for externally driven systems in which (by definition) all quantities are supplied at their average dissipation rates and nothir~,~ decays. The question arises as to what conceptual framework, variational or not, might be used to describe such systems.

At first sight, an attractive paradigm seems to be cascades/inverse cascades; but for at least two reasons, it soon becomes clear that the framework cannot be a very useful one.

First, most MHD forcing mechanisms tend to operate at large spatial scales rather than at small or intermediate ones. Common mechanical forcing mechanisms include pressure drops across which the fluids or magnetofluids are forced to flow, or moving bound?xies. Common electromagnetic forcing is an applied voltage drop, or emf, which results in the flow of an electric current. If one attempts to associate an "injection length scale" with any of these, the resulting length usually turns out to be something ~ e the dimensions of the system. Cascade/inverse cascade beha~4or re- mains undefined for situations in which more than one global, ideal invariant is supplied at the large scales.

Second, implicit in the cascade/inverse cascade picture is a high degree of randomness, manifested as a lack of correlation among different Fourier modes: a necessary consequence of spatial homo- geneity. In numerical computations of cascade/ inverse cascade behavior, this is often helped along by using a random forcing function to inject the cascadable quantities. In fact, most imaginable mechanical and electromagnetic driving mechanisms are rather coherent in character. Such randomness as the fields exhibit seems to appear primarily at the small scales and to be due to the dynamical behavior of the magnetofluid itself.

Thus, the typical MHD turbulence situation is one of rather coherent, large-scale forcing, any- thing but spatially homogeneous or isotropic, with shapes and orientations ordained by the contain- ers or boundary conditions. It may or may not homogenize and isotropize itself at the small scales; often not, in the presence of a finite dc magnetic field [33]. The large- and intermediate- scale behavior is often more noticeable and more practically important than the small-scale behavior. The difficulties in applying the cascade/ inverse cascade picture to any occurring situation are formidable.

It appears that some recent driven, steady-state, MHD turbulence computations have revealed some features that are similar to those appearing in decay calculations (the state of the experimen- talist's art does not at present permit the test of either case in the laboratory). For example, for the reversed-field pinch [8] ("RFP") configuration, a state w,hich is close to force-free (i.e. L/× B I << ~']" IBU) is maintained in the steady state over a large interior volume of the magnetofluid. In this respect, the prediction (2.13) might be thought to apply, except for the computational result that the ratio X now depends strongly upon space instead of beir.,g simply a constant Lagrange multiplier. A rather detailed list of computational simila1~ities. and differences from the selective-decay predic- tions for the RFP can be assembled [8, 12], clearly emphasizing the need for a reformulation of the variational problem.

222 D. Montgomery and L. Phillips~ MHD relaxation process

A principle we have been exploring is the classic hydrodynamic I:rinciple of the minimum rate of energy dissipation [51, 52]. Its application [12] to the electrically driven, steady-state, reversed- field-pinch configuration seems to have yielded a number of agreements with the results of the computational studies [8] of the same phenomena. We shall state the principle below and then apply it to the case of a mechanically driven magnetofluid in a de magnetic field, an application which we believe to be new.

The intuitive content of the minimum-dissipation-rate principle is simple. The driven turbulent magnetofluid smooths itself toward a uniform thermal equilibrium state by dissipation, inher- ently most effective at the small scales. However, it is prohibited from reaching the uniform equilibrium state by boundary conditions, external forces, and constraints. The dissipation rates are always positive-definite integrals of squares of field gradi- ents and thus measure directly the departure of the fields from spatial uniformity. The hypothesis is that these become as small as they can become, for the given boundary conditions, external forces and constraints, at least for weak enough driving.

The principle itself originates in the nineteenth century in the work of Helmholtz, Korteweg, and Lord Rayleigh (see e.g. [51, 52]). It should be regarded as a primitive ancestor of the more general (and more controversial) "principle of minimum entropy production" [52-54], though they are quite distinct. The principle has been proved for fluids [51] and magnetofluids [12] under much more restrictive assumptions that might be desired and its limits of applicability are not known.

It has been regarded first as applying to laminar time-independent states, but it has been shown to lead to plausible average profiles for time-depen- dent steady states when the turbulent fluctuation levels remai~ fractionally small relative to the mean fields [12]. In this sense, it cannot apply to the usual ease of homogeneous turbulence at all (where the mean fields are all zero). It may also not be relevant for fluid situations, such as three- dimensional wall-bounded shear flows, where ex-

periments reveal that fluctuations refuse to remain small relative to the mean fields. However, for magnetically-supported RFP plasmas there is ev- ery reason to believe in large regions of parameter space in which the rms MHD fluctuations (SB/B, say) remain below 10% and in the ease of the tokamak, well below that. The principle has been applied in detail to the RFP [12]. A ease can be made for the minimization of Ohmic dissipation subject to the constraints of constant rate of supply and dissipation of magnetic helicity (i.e. constant /~ j . Bd3x, time-averaged over longer than the time scales of the fluctuations), and constant toroidal magnetic flux. The Euler-Lagrange equation for this variational problem is [12]

V × j + a j + V s = 0 , (3.1)

where a is a Lagrange multiplier and V 2s = 0. Solutions to (3.1) subject to the boundary conditions j x d = 0 , B . ~ = 0 , where t$ is the unit normal to the perfectly conducting wall, have been given in ref. [12] and show several features in common with the driven [8] computations. To repeat this rather detailed comparison would be to lengthen the present paper prohibitively.

3.1. Mechanical forcing; Constant "cross-helicity" dissipation rate

A second minimum-d~s,:apation-rate calculation which has a relatively simple answer is that of a constant rate of supply and dissipation of "cross- helicity," (v . B). The dissipation rate of cross helicity is

( ( r l+ p ) j ' ~ ) , (3.2)

while the rate of energy dissipation is

( ( r l j 2 + p~o 2)), (3.3)

and the minimization of (3.3) subject to the con- stancy of (3.2) is a well-posed problem once a set of boundary conditions is adopted.

D. Montgomery and L Phillips/MHD relaxation process 223

In periodic boundary conditions the problem is easy, as we will see momentarily. It is not so easy to make up a more realistic problem to treat numerically in which the rate of cross-helicity supply depends only upon external parameters in the same way that the rate of magnetic helicity supply is fixed for the RFP case by fixing the toroidal flux and voltage drop. Dot eq. (2.2) with v and add to it the result of dotting (2.1) with B:

~ ( O ' B ) = - V "(pm B) + V °[(12X B) X 12]

- v .[,,,,, x n+ ix v] - ( ~ + v ) e . j , (3.4)

upon use of some farni!iar vector identities. If we integrate (3.4) over a large volume V and use the divergence theorem,

d d c-- -dT f v v° Bd3x

= f s d a . { - p m B + ( v x B) x v

x B- jxv}

- fvdaX(V + ~)~o'j, (3.5a)

where fs de is a surface integral over the surface S bounding V. The volume integral on the right of (3.5a) is the rate of dissipation of cros~-heficity. The surface integral contributions are representa- tions of processes by which cross-helicity can en- ter or leave the volume V. The two terms involving v vanish either at material walls or in the presence of periodic boundary conditions in the relevant direction, leaving

d d--7 Hc = d .{-pmB- B }

- + rl),o.jd3x. (3.5b)

Eq. (3.5b) strongly suggests that the straightforward way to inject cross-heficity is by arranging

an externally supported mechanical pressure drop down the direction of a mean dc magnetic field. (This is the direct analogue of the injection of magnetic helicity via a voltage drop down a mean de magnetic field.) If it were not for the trouble- some surface integral involving vorticity on the fight-hand side of (3.5b), the rate of ero:ss- helicity injection would be governed solely by these two (externally adjustable)parameters. The fs d a . vo x B term vanishes for periodic boundary conditions in the do direction but not at material walls unless B is purely normal to those walls, or ~ obeys stress-free boundary conditions.

Despite the lack of external control of the second surface integral in (3.5b), it is still instructive to inquire as to what happens if the mean injection rate of cross-helicity is assumed to be constant and we minimize the energy dissipation rate. As in ref. [12], we will assume that the various fields consist of non-zero time averages indicated by overbars, plus fluctuations about those averages. Thus, B = B + 8B, v = ~ + By, etc. We also assume that fluctuations are small enough that time averages of their products are negligible compared with products of the time-averaged quantities: 18~. 8~ i << ~i" I~l, etc.

We are led, then, to the variational problem of minim~ng the dissipation rate,

(3.6)

subject to the constraint:

cross-heficity dissipation rate = f ~ , + ~)]" = d3x

= const. (3.7)

Treating the va6ations in 8j and 8~0 as independent and taking (3.7) into account by a Lagrange multipfier 2a, the Euler-Lagrange equations are simple:

+,7)3 = o, (3.8a)

(3.Sb)

224 D. Montgomery and L Phillips / MHD relaxation process

Eqs. (3.8) have a non-trivial solution only if a = + (l, rl)l/2(~ + v)- x, in which case

3 = + (v/~)t /2~. (3.9)

Eq. (3.9) should be compared with eq. (2.16). If there were no net space- and time-averaged fields { ~), { B), as in the case of purely periodic boundary cenditions, then (3.9) would imply B = ± (v/~)xiz~, or alignment of the velocity and magnetic fields, and would reduce to (2.16) for the case of unit magnetic Prandfl number. Non-zero mean (~) and ( B ) can keep ~ and B from linil, g up, however, even in the presence of the alignment of their curls according to (3.9).

This alignment of vorticity and current, as an effect distinct from alignment of velocity field and magnetic fields, is somewhat subtle and is not distinguishable in the presence of purely periodic boundary conditions. There may be some implications for the solar wind plasma in this calculation. It seems to be becoming increasingly clear that cross-helicity plays a central role in the evolution of solar wind and stellar wind turbulence, to the extent that the MHD description apphes at all. The injection of cross-helicity by the outward- streaming plasma from the sun, undoubtedly due ~o processes represented by the first surface integral ~n (3.5b), is intN~siea!ly corr_espo, di~g magnetic helicity for the RFP [12] and is harder cally. It seems, however, to be problem worth undertaking.

messier than the injection problem to model numeri- a central physical

4. Discussion

No propc..y of the MHD turbulence has been used, in the calculations of the relaxed states, except the fractional smallness of the fluctuations about the time averages (SB/B << 1, 8j/j << 1, etc.). This fractional smallness is not to be ex- pected for homogeneous turbulence, or for most Navier-Stokes flows, or for MHD in the absence

of a mean dc magnetic field. The turbulence has as its principal dynamical role the production of the relaxed state, but properties of the turbulence such as its spectra are not regarded as of paramount importance.

The study of possible steady states for driven magnetofluids has historically proceeded from considerations of ideal MHD equilibria. The emphasis has been on the instabilities of such equilibria and their likely evolution, with almost no emphasis on how such equilibria might be pro- duced. Needless to say, the possibilities for such ideal equilibrium profiles are virtually limitless, and analytic speculation on how they might evolve l~.onlinearly is particularly open-ended.

From a hydrodynamicist's point of v'~ew, such a program can only seem bizarre. Attempting it for, say, plane Poiseuille flow or pipe flow could only be a non-convergent process, whatever the parameter ranges. Yet in those flows the addition of a pressure head, correct non-ideal boundary conditions, and a viscosity, however small, renders the time-independent profile not only simple but uniquely parabolic.

It may be that what we have done, particularly wit~ the RFP calculation [12], is simply to take the first step at obtaining correct non-ideal steady states with a full contingent of proper boundary conditions and toroidal voltage drop. If so, all we are doing is discovering, painfully late in the day, the voltage-driven MHD analogue of laminar pipe flow.

However, there are reasons to believe that more is involved. A true steady state in MHD requires that OB/Ot = - V x E = 0, so that E = - v x B + rlV × B = at most the gradient of a scalar. In the presence of negligible fluid velocities and uniform resistivities this requirement restricts j = V × B greatly. It requires, in fact, that j = VqJ, where x7 2 4, = 0. There are no non-trivial axisymmetric solutions for which j x ~ = O at a cylindrical wall, and indeed, all the solutions of (3.1) except the uniform j = (constant)~z solution have a prominent V × j and therefore an O(r/) "dynamo" term v × B. It appears that the mean-field dissipation

D. Montgomery and L Phillips~ MHD relaxation process 225

rate may be minimized by a state which is not time-independent !

MHD has a great need for a series of experi- ments in whit:h the elementary driven states (indeed, the electrically driven analogues of Poiseuille and pipe flow!) are accurately measured and calcu- lated. Attempting to build thermonuclear reactors without having done this may eventually come to seem not so much amusing as unforgivable. In the meantime, the best information we have is likely to come, as it has in the recent past, from a combination of numerical computation and what might be described as analytical speculation. The task of deciding what role minimum dissipation states really play in highly evolved MHD turbulence is a logical candidate as a next task for MHD computation.

Acknowledgements

This work was supported in part at Dartmouth under NASA Grant NAG-W-710 and US Depart- ment of Energy Grant FG02-85ER53194, and in part at Los Alamos National Laboratory under the auspices of the US Department of Energy. Gratitude is expressed to collaborators in the cited research articles for their indispensable contributions, in particular ,LP. Dahlburg and W.H. Matthaeus.

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mhd turbulance: relaxation processes and variational principles

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