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We investigate by numerical simulation the rotational flows in a toroid confining a conducting magnetofluid in which a current is driven by the application of externally supported electric and magnetic fields. The computation involves no microscopic instabilities and is purely magnetohydrodynamic (MHD). We show how the properties and intensity of the rotations are regulated by dimensionless numbers (Lundquist and viscous Lundquist) that contain the resistivity and viscosity of the magnetofluid. At the magnetohydrodynamic level (uniform mass density and incompressible magnetofluids), rotational flows appear in toroidal, driven MHD. The evolution of these flows with the transport coefficients, geometry, and safety factor are described.

Ideal magnetohydrodynamics (MHD) still provides the mathematical framework and the textbook vocabulary in which the possible states of a toroidal plasma are discussed, generally regarded as static equilibria. This is so, despite the increasing realization that virtually all toroidal magnetofluids have nontrivial fluid flows (finite velocity fields) in them. A very different perspective results from nonideal MHD, including both resistivity and viscosity and invoking nonideal boundary conditions. There, it has been shown that if Ohm’s law and Faraday’s law are given equal importance with force balance, flows are an inevitable consequence of the assumptions of time independence and axisymmetry. Previous treatments of the toroidal steady states for such systems have been based on perturbation theory in which the flow velocity was assumed small, as a consequence of high viscosity (or in dimensionless terms, low Hartmann number H). Here, recently newly available numerical programs are used to lift this limitation and to solve nonlinearly for the allowed steady states of an axisymmetric, current-carrying, toroidal magnetofluid without such an expansion in the Hartmann number. Flow patterns for values of H from ≪1 to ≫1 have been calculated. As H is raised, the flow pattern goes from the predominantly poloidal pair of counter-rotating “convection cells” revealed by the perturbation theory to a pattern in which the toroidal kinetic energy of flow considerably exceeds the poloidal kinetic energy. In no case is the flow discovered a simple rotation.

Numerical and analytical studies of decaying, two-dimensional (2D) Navier-Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Both formulations define an entropy through a somewhat ad hoc discretization of vorticity to the "particles" of which statistical mechanical methods are employed to define an entropy, before passing to a mean-field limit. In one case, the particles are delta-function parallel "line" vortices ("points" in two dimensions), and in the other, they are finite-area, mutually-exclusive convected "patches" of vorticity which in the limit of zero area become "points." We use time-dependent, spectral-method direct numerical simulation of the Navier-Stokes equations to see if initial conditions which should relax to different late-time states under the two formulations actually do so., Comment: 21 pages, 24 figures: submitted to "Physics of Fluids"

The influence of the curvature of the imposed magnetic field on reversed field pinch dynamics is investigated by comparing the flow of a magnetofluid in a torus with aspect ratio 1.83, with the flow in a periodic cylinder. It is found that an axisymmetric toroidal mode is always present in the toroidal, but absent in the cylindrical configuration. In particular, in contrast to the cylinder, the toroidal case presents a double poloidal recirculation cell with a shear localized at the plasma edge. Quasi-single-helicity states are found to be more persistent in toroidal than in periodic cylinder geometry.

It is a well known fact that in the presence of a dc applied magnetic field, magnetohydrodynamic (MHD) turbulence develops spectral anisotropy from isotropic initial conditions. Typically, the reduced spectrum is steeper in the direction of the magnetic field than it is in any transverse direction. One might expect that a dc field is not essential, and it is the local mean field that is responsible. To address this issue, three-dimensional MHD pseudo-spectral incompressible relaxation simulations are performed, and structure functions computed according to whether the separation is parallel to, or transverse to, the local mean magnetic field. Correlation lengths are longer in the locally averaged magnetic field direction than in any perpendicular direction, even when the global mean magnetic field is zero. Local anisotropy is observed to be stronger in regions of strong magnetic field. A general definition of anisotropy angles and a methodology to study local anisotropy are proposed.

Certain unresolved ambiguities surround pressure determinations for incompressible flows, both Navier-Stokes and magnetohydrodynamic. For uniform-density fluids with standard Newtonian viscous terms, taking the divergence of the equation of motion leaves a Poisson equation for the pressure to be solved. But Poisson equations require boundary conditions. For the case of rectangular periodic boundary conditions, pressures determined in this way are unambiguous. But in the presence of "no-slip" rigid walls, the equation of motion can be used to infer both Dirichlet and Neumann boundary conditions on the pressure P, and thus amounts to an over-determination. This has occasionally been recognized as a problem, and numerical treatments of wall-bounded shear flows usually have built in some relatively ad hoc dynamical recipe for dealing with it, often one which appears to "work" satisfactorily. Here we consider a class of solenoidal velocity fields which vanish at no-slip walls, have all spatial derivatives, but are simple enough that explicit analytical solutions for P can be given. Satisfying the two boundary conditions separately gives two pressures, a "Neumann pressure" and a "Dirichlet pressure" which differ non-trivially at the initial instant, even before any dynamics are implemented. We compare the two pressures, and find that in particular, they lead to different volume forces near the walls. This suggests a reconsideration of no-slip boundary conditions, in which the vanishing of the tangential velocity at a no-slip wall is replaced by a local wall-friction term in the equation of motion., Comment: 8 pages, 4 figures, to appear in Journal of Plasma Physics, send e-mail to bkress@darthmouth.edu

It has become common to formulate theories and computations of magnetohydrodynamic turbulent effects in rectangular periodic boundary conditions, proceeding by analogy with what is seen as a useful framework for Navier–Stokes fluid turbulence. It is shown here that because of certain features of Maxwell’s equations for electrodynamics, it is inconsistent to invoke three-dimensional, rectangular, periodic boundary conditions and symmetry at the same time that the displacement current is neglected. The difficulty does not arise in the two-dimensional case. In three dimensions, the difficulty can be remedied by a reformulation in cylindrical geometry, imposing symmetry in the azimuthal and axial directions, but not in the radial one; a geometry that is closer to laboratory possibilities than the wholly three-dimensional periodic assumption. The reformulation seems particularly necessary in cases with a net flux of magnetic field and/or electric currents through the system. These cases no longer seem disconti...

For shock waves propagating in materials with nonideal equations of state, a variety of nonstandard phenomena can occur. Here, we present numerical studies of two such exotic shock effects: (i) "anomalous" behavior, in the terminology of Zel’dovich and Raizer; and (ii) a search for "acoustic emission instabilities." The motivation is in part the possibility of such phenomena in the implosion of inertial confinement fusion (ICF) pellet materials, whose equations of state are currently far from well known. In shock wave theory, anomalous materials are those whose isentropes have regions of negative curvature (in the plane of pressure versus specific volume) through which the shock adiabatic passes. The existence of such regions is significant because they can interfere with the steepening of compressive pulses into shocks, lead to the formation of rarefactive shock waves, and even cause shocks to "split." A van der Waals fluid with a large heat capacity is one example of a material possessing such anomalous properties. Acoustic emission instability—the second exotic shock mechanism considered—may occur when the slope of the shock adiabatic lies below a critical value. In this phenomenon, perturbations of a two-dimensional planar shock front can render it unstable, and lead to the downstream emission of acoustic waves. In addition to the van der Waals fluid, an equilibrium dissociation model for strong shocks in diatomic hydrogen is shown to fulfill the theoretical criteria for this instability, but its numerical verification has been hard to achieve, suggesting that further study is needed. Both classes of phenomena may be expected to play a role in ICF compression scenarios.

It has long been common to use ideal magnetohydrodynamic (MHD) steady states as a zeroth approximation for confined plasmas, even when the behaviour of `resistive' instabilities was under discussion. Implicitly, the zeroth order resistive terms discarded are formally larger than the first order ones kept. Most of our normal mode vocabulary derives from these (ideal, zero-flow) MHD steady states and their perturbed behaviour. If the plasma is regarded as resistive, with both Ohm's law and Faraday's law being taken seriously along with the equation of motion and boundary conditions are included, then the situation becomes much more complex. It is far from clear which, if any, of the ideal zero-flow steady states are close to a realizable resistive one. We have been reconsidering this problem in the spirit of hydrodynamic shear flows, and have reached somewhat different conclusions than have been reached in previous decades. In a toroid, vortical flows seem to be a universal feature of the resistive steady states that are found. These flows involve both toroidal vorticity and (higher order) toroidal velocity and have characteristic patterns that are more or less unique for a given set of boundary conditions. Arbitrary `source' terms for the supply of mass and the maintenance of pressure gradients are unnecessary. No expansions in the inverse aspect ratio are involved. The numerical value of the kinematic viscosity and the correct form of the viscous stress tensor are important and reliable experimental information on these seems to be in short supply. A reconsideration of the fundamentals of confinement theory seems to be in order.

Numerical simulation is used to study the turbulence generated by the passage of strong shocks (typical Mach number 7.3) through an inhomogeneous fluid at moderate Reynolds numbers. Before passage of the shock, the material consists of mass-density inhomogeneities embedded in a background fluid. The entire system is initially at uniform temperature, pressure, and number density, with the nonuniform mass density resulting from differing mass species in different regions. In the present application, the substances are treated as ideal gases, though in the motivating physical problems they are more complex materials. The shock retains its identity and a sharp front, but leaves behind it a turbulent state whose locally averaged properties only slowly become spatially uniform. The shock acquires a turbulent “thickness” (the linear dimension of the nonuniform region behind the shock front) that seems ultimately damped by viscous and thermally conducting properties that are dependent on transport coefficients and (highly uncertain) Reynolds numbers. Typically, the turbulence is highly compressible, with comparable mean divergences and curls in the velocity field, and fractional rms density fluctuations of the order of 0.25 in the parameter ranges studied. The rms vorticity generated can be estimated reasonably well from dimensional considerations. The effect of the high density inhomogeneities is primarily to create a wide region of compressible turbulence behind the shock. The inhomogeneities create both a succession of reflected shocks and considerable vorticity.

Poloidal velocity fields seem to be a fundamental feature of resistive toroidal magnetohydrodynamic (MHD) steady states. They are a consequence of force balance in toroidal geometry, do not require any kind of instability, and disappear in the "straight cylinder" (infinite aspect ratio) limit. If a current density j results from an axisymmetric toroidal electric field that is irrotational inside a torus, it leads to a magnetic field B such that ¿×(j×B) is nonvanishing, so that the Lorentz force cannot be balanced by the gradient of any scalar pressure in the equation of motion. In a steady state, finite poloidal velocity fields and toroidal vorticity must exist. Their calculation is difficult, but explicit solutions can be found in the limit of low Reynolds number. Here, existing calculations are generalized to the more realistic case of no-slip boundary conditions on the velocity field and a circular toroidal cross section. The results of this paper strongly suggest that discussions of confined steady states in toroidal MHD must include flows from the outset.

We consider the resistive steady states of a uniformly conducting magnetofluid inside a toroidal boundary. The problem becomes tractable in the limit of slow flow: i.e., low Reynolds number, which may be in turn justified when the viscous Lundquist number is small. Previous calculations are extended to apprehend the toroidal component of the necessary flow. The emerging pattern is one of helical vortices which seem likely to be ubiquitous in toroidal geometry, and which disappear in the ``straight--cylinder approximation.''

We report numerical computations of decaying two-dimensional Navier--Stokes turbulence inside a circular rigid boundary. We summarize previously reported calculations involving no-slip boundary conditions and present results with higher spatial resolution than achieved before (with, however, no qualitative changes in the observed behavior). We then report new results with stress-free boundary conditions (for a viscous fluid, but bounded by a perfectly slippery wall). The method used is spectral, involving expansions of the fields in orthonormal sets of functions which obey two boundary conditions (circular analogues of the Chandrasekhar–Reid functions). The computation takes place entirely in the spectral space. Large-scale Reynolds numbers are typically less than a thousand. Interest focuses on the role played by angular momentum, in determining the decay of the turbulence with no-slip boundary conditions, and the role of possible other ideal invariants in the stress-free case.

It was recently demonstrated that static, resistive, magnetohydrodynamic equilibria, in the presence of spatially-uniform electrical conductivity, do not exist in a torus under a standard set of assumed symmetries and boundary conditions. The difficulty, which goes away in the ``periodic straight cylinder approximation,'' is associated with the necessarily non-vanishing character of the curl of the Lorentz force, j x B. Here, we ask if there exists a spatial profile of electrical conductivity that permits the existence of zero-flow, axisymmetric r esistive equilibria in a torus, and answer the question in the affirmative. However, the physical properties of the conductivity profile are unusual (the conductivity cannot be constant on a magnetic surface, for example) and whether such equilibria are to be considered physically possible remains an open question., 17 pages, 4 figures

Most theoretical and computational studies of turbulence in Navier—Stokes fluids and/or guiding-centre plasmas have been carried out in the presence of spatially periodic boundary conditions. In view of the frequently reproduced result that two-dimensional and/or MHD decaying turbulence leads to structures comparable in length scale to a box dimension, it is natural to ask if periodic boundary conditions are an adequate representation of any physical situation. Here, we study, computationally, the decay of two-dimensional turbulence in a Navier—Stokes fluid or guiding-centre plasma in the presence of circular no-slip rigid walls. The method is wholly spectral, and relies on a Galerkin approximation by a set of functions that obey two boundary conditions at the wall radius (analogues of the Chandrasekhar— Reid functions). It is possible to explore Reynolds numbers up to the order of 1250, based on an RMS velocity and a box radius. It is found that decaying turbulence is altered significantly by the no-slip boundaries. First, strong boundary layers serve as sources of vorticity and enstrophy and enhance the early-time energy decay rate, for a given Reynolds number, well above the periodic boundary condition values. More importantly, angular momentum turns out to be an even more slowly decaying ideal invariant than energy, and to a considerable extent governs the dynamics of the decay. Angular momentum must be taken into account, for example, in order to achieve quantitative agreeement with the predicition of maximum entropy, or ‘most probable’, states. These are predicitions of conditions that are established after several eddy turnover times but before the energy has decayed away. Angular momentum will cascade to lower azimuthal mode numbers, even if absent there initially, and the angular momentum modal spectrum is eventually dominated by the lowest mode available. When no initial angular momentum is present, no behaviour that suggests the likelihood of inverse cascades is observed.

At a college where I taught, there was a ‘Last Lecture’ series, in which the faculty took turns giving a public lecture that would be the lecture they would give if it were to be their very last one. The invitation to contribute a few nontechnical pages to this issue of Journal of Plasma Physics has a somewhat similar valedictory ring, and inspires mixed feelings.

Two-dimensional steady states of plane Poiseuille flow are found by following the evolution of a two-dimensional Navier-Stokes fluid numerically for long times. Two runs are presented, one for Reynolds number R = 4000, the other for R = 15000. The Reynolds number is based on the pressure gradient and channel half-width. The final states are characterized by an array of alternating-sign vortices. The states are accurately time independent in a frame moving with the vortices. In this frame of reference the stream function and the vorticity display an interesting correlation, showing that the flow is divided into three spatially distinct regions. Theoretical understanding of the state is sought in terms of the statistical mechanics of large numbers of discrete line vortices in the mean field limit, but without great success. Attention is devoted to characterizing the relaxed state.

Magnetohydrodynamic activity near the threshold of instability is studied numerically for resistive equilibria in periodic cylinders. Attention focuses on the role of the viscosity (and its reflection in the Hartmann number) in determining the existence and nonlinear evolution of instabilities. Three identical axisymmetric resistive equilibria without flow are investigated, with three viscosities. The highest viscosity leads to stability of the axisymmetric state. The intermediate value leads to a helically deformed equilibrium with flow, with an (m,n)=(2,1) deformation. The lowest viscosity leads to a 'mixed' helically deformed, final, approximately steady state with flow, with (m,n)=(2,1) and (3,2) deformations and their harmonics, plus other, adjacent, modes near m=2n. It is concluded that it is likely that the numerical value of the viscosity, and possibly the form of the viscous stress tensor, must never be ignored in discussions of near-threshold incompressible MHD activity in driven, dissipative MHD plasmas.

Two-dimensional Navier-Stokes turbulent decay has been followed numerically for very long times. The code is spectral, satisfies periodic boundary conditions, and does not make use of hyperviscosity or any small-scale smoothing. The resolution is (512) 2 , the initial Reynolds number based on the box dimension is about 14000, and the run continues for over 190 initial eddy turnover times. Isolated vortices form, and the turbulence has become highly intermittent in the manner seen by McWilliams and Brachet et al., for times greater than about 30. However, it is not the case that merger of like-signed vortices stops; it only slows down. By t = 210, the final merger is complete, and the vorticity distribution is dominated by one large vortex of either sign. The negative (positive) vortices each occupy about two percent of the box area, and together account for over 98 percent of the total enstrophy. Their alignment suggests the formation of an Ewald lattice with a basic cell containing two point vortices. The ratio of enstrophy to energy continues to decrease monotonically, and the picture is consistent with a “selective decay” process, as described some time ago. A not-entirely-understood phenomenon is the concentration of the vorticity into two cores, suggestive of a negative-temperature state of the discrete line vortex model.

A driven dissipative magnetohydrodynamic state is calculated that is consistent with a non-ideal boundary condition relevant to magnetically and mechanically active boundaries of laboratory or astrophysical plasmas. The solution is based on the results from previous analytical and computational investigations of electrically driven states, and is characterized by finite helical distortions and a small flow velocity. It is shown that such a configuration may persist even in the absence of previously assumed rigid perfectly conducting boundaries. It is proposed as a model for an astrophysical ‘flux rope’, and suggests an additional in situ diagnostic for such flux ropes.

We present a computational method to solve the magnetohydrodynamic equations in spherical geometry. The technique is fully nonlinear and wholly spectral, and uses an expansion basis that is adapted to the geometry: Chandrasekhar-Kendall vector eigenfunctions of the curl. The resulting lower spatial resolution is somewhat offset by being able to build all the boundary conditions into each of the orthogonal expansion functions and by the disappearance of any difficulties caused by singularities at the center of the sphere. The results reported here are for mechanically and magnetically isolated spheres, although different boundary conditions could be studied by adapting the same method. The intent is to be able to study the nonlinear dynamical evolution of those aspects that are peculiar to the spherical geometry at only moderate Reynolds numbers. The code is parallelized, and will preserve to high accuracy the ideal magnetohydrodynamic (MHD) invariants of the system (global energy, magnetic helicity, cross helicity). Examples of results for selective decay and mechanically-driven dynamo simulations are discussed. In the dynamo cases, spontaneous flips of the dipole orientation are observed., Comment: 15 pages, 19 figures. Improved figures, in press in Physics of Fluids

We explore some consequences of the ``alpha model,'' also called the ``Lagrangian-averaged'' model, for two-dimensional incompressible magnetohydrodynamic (MHD) turbulence. This model is an extension of the smoothing procedure in fluid dynamics which filters velocity fields locally while leaving their associated vorticities unsmoothed, and has proved useful for high Reynolds number turbulence computations. We consider several known effects (selective decay, dynamic alignment, inverse cascades, and the probability distribution functions of fluctuating turbulent quantities) in magnetofluid turbulence and compare the results of numerical solutions of the primitive MHD equations with their alpha-model counterparts' performance for the same flows, in regimes where available resolution is adequate to explore both. The hope is to justify the use of the alpha model in regimes that lie outside currently available resolution, as will be the case in particular in three-dimensional geometry or for magnetic Prandtl numbers differing significantly from unity. We focus our investigation, using direct numerical simulations with a standard and fully parallelized pseudo-spectral method and periodic boundary conditions in two space dimensions, on the role that such a modeling of the small scales using the Lagrangian-averaged framework plays in the large-scale dynamics of MHD turbulence. Several flows are examined, and for all of them one can conclude that the statistical properties of the large-scale spectra are recovered, whereas small-scale detailed phase information (such as e.g. the location of structures) is lost., 22 pages, 20 figures

Previous computations concerning the allowed magnetohydrodynamic steady states of a visco-resistive magnetofluid in a toroid are extended. The current is supported by an externally imposed toroidal electric field, and a scalar resistivity and viscosity are assumed. Emphasis is on the character of the necessary velocity fields (mass flows) that toroidal geometry demands. Non-ideal boundary conditions are imposed at the toroidal boundary. One of the more interesting results to emerge is the sensitive dependence of the flow pattern on the shape of the toroidal cross-section boundary: the dipolar poloidal flow that had appeared for cross sections that were symmetric about the midplane is seen to deform continuously into a monopolar pattern for a ‘D-shaped’ cross section as the viscous Lundquist number $M$ is increased. A net toroidal mass flow also develops. A boundary layer whose properties scale with fractional powers of $M$ is also studied. The interior of the magnetofluid is approximately force-free, with current densities and magnetic fields that are nearly parallel for JET-like parameters. Steep velocity derivatives and a steep pressure drop in this boundary layer become steeper with increasing $M$ (decreasing viscosity). The magnetic quantities do not reflect the rapid velocity and vorticity variations in the boundary layer. The maximum velocities, in the region where the viscosity is large enough for the numerics to work, are of the order of a few hundreds of centimetres per second. Measurements of velocity fields confined to the boundary layer would misrepresent the interior plasma conditions. Uncertainties in the magnetized plasma viscosity remain as an obstacle to unambiguous tests of the results in the case of real plasmas.

Numerical studies of the effect of a dc magnetic field on dynamo action (development of magnetic fields with large spatial scales), due to helically-driven magnetohydrodynamic turbulence, are reported. The apparent effect of the dc magnetic field is to suppress the dynamo action, above a relatively low threshold. However, the possibility that the suppression results from an improper combination of rectangular triply spatially-periodic boundary conditions and a uniform dc magnetic field is addressed: heretofore a common and convenient computational convention in turbulence investigations. Physical reasons for the observed suppression are suggested. Other geometries and boundary conditions are offered for which the dynamo action is expected not to be suppressed by the presence of a dc magnetic field component., Comment: To appear in Physics of Plasmas

We consider a model of viscoresistive incompressible magnetohydrodynamics in a periodic cylinder, with boundary conditions meant to idealize in a tractable way those of a laboratory plasma. The resistivity is described by a tensor presenting a field-dependent anisotropic part suggested by kinetic theory, controlled by a certain anisotropy parameter. An explicit analytical description of the corresponding axisymmetric zero-flow equilibria is given, and it is shown how tokamak-like or paramagnetic-pinch-like field profiles are obtained as the anisotropy parameter is changed. The study of the stability properties of such equilibria is deferred to a later paper.

Resistive steady states in toroidal magnetohydrodynamics (MHD), where Ohm's law must be taken into account, differ considerably from ideal ones. Only for special (and probably unphysical) resistivity profiles can the Lorentz force, in the static force-balance equation, be expressed as the gradient of a scalar and thus cancel the gradient of a scalar pressure. In general, the Lorentz force has a curl directed so as to generate toroidal vorticity. Here, we calculate, for a collisional, highly viscous magnetofluid, the flows that are required for an axisymmetric toroidal steady state, assuming uniform scalar resistivity and viscosity. The flows originate from paired toroidal vortices (in what might be called a ``double smoke ring'' configuration), and are thought likely to be ubiquitous in the interior of toroidally driven magnetofluids of this type. The existence of such vortices is conjectured to characterize magnetofluids beyond the high-viscosity limit in which they are readily calculable., 17 pages, 4 figures

A brief overview is provided as an introduction to hydrodynamic-like turbulence that characterizes the dynamics of plasmas in several parameter regimes. This includes magnetohydrodynamics (MHD), the electron fluid plasma, which is closely related to two-dimensional hydrodynamics, and the solar wind, which is usually viewed as a laboratory for three-dimensional MHD, with more involved plasma physics at the dissipative scales. An emphasis is placed on energy decay, spectra, relaxation processes, coherent structures, and higher statistics with selected applications in solar wind and laboratory plasmas.

By reinterpreting a recent successful closure procedure in terms of local spatial averaging and the neglect of fluctuations about that average, it is shown how the results of that closure scheme (the “alpha model”) may be more simply derived.

During the last four decades, a considerable number of investigations has been carried out into the evolution of turbulence in two dimensional Navier-Stokes flows. Much of the information has come from numerical solution of the (otherwise insoluble) dynamical equations and thus has necessarily required some kind of boundary conditions: spatially periodic, no-slip, stress-free, or free-slip. The theoretical framework that has proved to be of the most predictive value has been one employing an entropy functional (sometimes called the Boltzmann entropy) whose maximization has been correlated well in several cases with the late-time configurations into which the computed turbulence has relaxed. More recently, flow in the unbounded domain has been addressed by Gallay and Wayne who have shown a late-time relaxation to the classical Oseen vortex (also sometimes called the Lamb-Oseen vortex) for situations involving a finite net circulation or non-zero total integrated vorticity. Their proof involves powerful but...

The rate of enstrophy decay in two-dimensional Navier–Stokes turbulence is explored numerically with high resolution pseudospectral simulations. Initial conditions and all other physical parameters are held fixed while viscosity is decreased through about three orders of magnitude, while increasing the numerical resolution as required to resolve the smallest scales that develop. The attempt is to discern whether or not a regime is being approached in which the behavior of the vorticity decay could be approximated as ideal and nondissipative. Although the enstrophy decay shows a (logarithmic) tendency to go to zero as the viscosity is decreased, at fixed times, the extrapolated values of viscosity to approach a zero enstrophy decay regime lie many orders of magnitude out of our currently available numerical resolution.

The development of anisotropy in an initially isotropie spectrum is studied numerically for two-dimensional magnetohydrodynamic turbulence. The anisotropy develops through the combined effects of an externally imposed d.c. magnetic field and viscous and resistive dissipation at high wavenumbers. The effect is most pronounced at high mechanical and magnetic Reynolds numbers. The anisotropy is greater at the higher wavenumbers.

The equations of motion for turbulent two-dimensional magnetohydrodynamic flows are solved in the presence of finite viscosity and resistivity, for the case in which external forces (mechanical and/or magnetic) act on the fluid. The goal is to verify the existence of a magnetohydrodynamic dynamo effect which is represented mathematically by a substantial back-transfer of mean square vector potential to the longest allowed Fourier wavelengths. External forces consisting of a random part plus a fraction of the value at the previous time step are employed, after the manner of Lilly for the Navier–Stokes case. The regime explored is that for which the mechanical and magnetic Reynolds numbers are in the region of 100 to 1000. The conclusions are that mechanical forcing terms alone cannot lead to dynamo action, but that dynamo action can result from either magnetic forcing terms or from both mechanical and magnetic forcing terms simultaneously. Most real physical cases seem most accurately modelled by the third situation. The spatial resolution of the 32 × 32 calculation is not adequate to test accurately the predictions of the spectral power laws previously arrived at on the basis of the assumption of simultaneous cascades of energy and vector potential. Some speculations are offered concerning possible relations between turbulent cascades and the ‘disruptive instability’.

An incompressible, dissipative numerical code of the spectral type is used to follow the nonlinear evolution of a magnetohydrodynamic sheet pinch in two spatial dimensions. The evolution involves considerable turbulent activity in the electric current field, with the excited spatial scales ranging from the size of the containing volume down to the dissipation lengths of the magnetic and velocity fields. Strong current filamentation near magnetic X-points is observed, as is lsquo;jetting’, or expulsion of magnetofluid from the vicinity of the X-point parallel to the current sheet.

A spectral-method numerical code is used to compute mass-density fluctuation spectra in turbulent magnetofluids. The computations are used to test and extend a recent analytical theory of density variations in slightly compressible magnetofluids given by Montgomery, Brown and Matthaeus, and used by them to infer inertial-range density-fluctuation spectra for the nearby interstellar medium and solar wind. A local equation of state is assumed, relating density to pressure. Constant, scalar resistivities and viscosities are used. In the limit of low Mach numbers and high mechanical-to-magnetic pressure ratios, the fit of the computations to the analytical theory is seen to be close.

Current-carrying plasmas in a strong d.c. magnetic field are subject to violent disruptions above certain thresholds. At present difficult to verify, explanations are typically sought in terms of "tearing modes". An alternative explanation is in terms of inverse magnetic helicity cascades, generated from a variety of possible sources of small-scale MHD turbulence. Strongly anisotropic MHD plasmas may be described by the Strauss equations. Indications of turbulent inverse cascade behavior for the Strauss equations are sought, in parallel with earlier examples from MHD and fluid mechanics.

Incompressible turbulent flows are investigated in the framework of ideal magnetohydrodynamics. All the field quantities vary with only two spatial dimensions. Equilibrium canonical distributions are determined in a phase space whose co-ordinates are the real and imaginary parts of the Fourier coefficients for the field variables. In the geometry considered, the magnetic field and fluid velocity have variable x and y components, and all field quantities are independent of z. Three constants of the motion are found (one of them new) which survive the truncation in Fourier space and permit the construction of canonical distributions with three independent temperatures. Spectral densities are calculated. One of the more novel physical effects is the appearance of macroscopic structures involving long wavelength, self-generated, magnetic fields (‘magnetic islands’) for a wide range of initial parameters. Current filaments show a tendency toward consolidation in much the same way that vorticity filaments do in the guiding-centre plasma case. In the presence of finite dissipation, energy cascades to higher wavenumbers can be accompanied by vector potential cascades to lower wavenumbers, in much the same way as, in the fluid dynamic (Navier-Stokes) case, energy cascades to lower wavenumbers accompany enstrophy cascades to higher wavenumbers. It is suggested that the techniques may be relevant to theories of the magnetic dynamo problem and to the generation of megagauss magnetic fields when pellets are irradiated by lasers.

In this paper, some multiple scale perturbation calculations of turbulent MHD transport coefficients begun in earlier papers are first completed. These generalize 'alpha effect' calculations by treating the velocity field and magnetic field on the same footing. Then the problem of rendering such calculations self-consistent is addressed, generalizing an eddy-viscosity hypothesis similar to that of Heisenberg for the Navier-Stokes case. The method also borrows from Kraichnan's direct interaction approximation. The output is a set of integral equations relating the spectra and the turbulent transport coefficients. Previous 'alpha effect' and 'beta effect' coefficients emerge as limiting cases. A treatment of the inertial range can also be given, consistent with a -5/3 energy spectrum power law. In the Navier-Stokes limit, a value of 1.72 is extracted for the Kolmogorov constant. Further applications to MHD are possible.

A pseudo-spectral three-dimensional Strauss-equations code is used to describe internal disruptions in a strongly magnetized, electrically-conducting fluid, with and without an externally applied, axial electric field. Rigid conducting walls form a square (x, y) boundary, and periodic boundary conditions are assumed in the axial (z) direction. Typical resolution is 64 × 64 × 32 and the maximum Lundquist number is approximately 400. The dynamics are dominated by a helical current filament which wraps itself around the axis of the cylinder; parts of this filament can sometimes become strongly negative. The ratio of turbulent kinetic energy to total poloidal magnetic energy rises from very small values to values of the order of a few hundredths, and executes ‘bounces’ as a function of time in the absence of the external electric field. In the presence of the external electric field, the first bounce is by far the largest, then the plasma settles into a non-uniform quasi-steady state characterized by a poloidal fluid velocity flow. At the large scales, this flow has the shape of a pair of counter-rotating bean-shaped vortices. The subsequent development of this fluid flow depends strongly upon whether or not a viscous term is added to the equation of motion. Inclusion of viscosity tends to damp the flow and leads to pronounced subsequent bounces suggestive of sawtooth oscillations, though the first bounce remains substantially the largest. By means of a three-mode (Lorenz-like) truncation of the Strauss equations, the evolution of the largest spatial scales alone may be examined. Some time-dependent solutions of the low-order truncation system suggest qualitative agreement with fully resolved solutions of the Strauss equations, while other solutions exhibit interesting dynamical-systems behaviour which is thus far unparalleled in the fully-resolved simulation results.

A theory is presented for the rate of thermal relaxation of a two-dimensional plasma in a strong uniform d.c. magnetic field. The Vahala—Montgomery kinetic description is completed by providing a cut-off time for the time of interaction of two particles which contribute to the collision term. The kinetic equation preclicts that thermal relaxation occurs as a function of the dimensionless time (ωpt) (ωp/Ω) (n0λ2D)−½, where ωp, is the plasma frequency, Ω is the gyrofrequency, and n0 λ2D is the number of particles per Debye square. By contrast, in the absence of an external magnetic field, a two-dimensional plasma relaxes as a function of (ωpt) (n0λ2D)−1.

The pseudospectral three-dimensional MHD code of Dahlburg et al. (1986 and 1987) is used to compute the dynamical behavior of a channel of magnetofluid carrying an axial current and magnetic flux. This situation contains the essential MHD behavior of the reversed-field pinch (RFP). An externally imposed electric field is applied to an initially current-free magnetofluid and drives currents that rise and eventually fluctuate about values corresponding to pinch ratios Theta of about 1.3, 2.2, and 4.5. A period of violent turbulence leads to an approximately force-free core, surrounded by an active MHD boundary layer that is not force-free. A steady state is reached that can apparently be sustained indefinitely (for several hundred Alfven transit times or longer). The turbulence level and time variability in the steady state increase with increasing Theta. The average toroidal magnetic field at the wall reverses for Theta = 2.2 and 4.5, but not for Theta = 1.3. Negative toroidal current filaments are observed. The Lundquist numbers are of the order of a few hundred.

The thermal relaxation process for a spatially uniform two-dimensional plasma in a uniform d.c. magnetic field is simulated numerically. Thermal relaxation times are defined in terms of the time necessary for the numerically computed Boltzmann H function to decrease through a given part of the distance to its minimum value. Dependence of relaxation time on two parameters is studied: number of particles per Debye square n0 λ2D and ratio of gyrofrequency to plasma frequency Ω/ωp. When Ω2/ω2p becomes ≫[ln (L/2πλD)]−½, where L is the linear dimension of the system, it is found that the relaxation time varies to a good approximation as (n0 λ2D)½ and Ω/ωp.

Incompressible magnetohydrodynamic turbulence is treated in the presence of cylindrical boundaries which are perfectly conducting and rigidly smooth. The model treated is non-dissipative and two-dimensional, the variation of all quantities in the axial direction beingignored. Equilibrium Gibbs ensemble predictions are explored assuming the constraint of constant axial current (appropriate to tokamak operation). No small-amplitude approximations are made. The expectation value of the turbulent kinetic energy is found to approach zero for the state of maximum mean-square vector potential to energy ratio. These are the only states for which large velocity fluctuations are not expected.

A method is developed for selecting 'most probable' MHD states compatible with certain global constraints (e.g., the energy, the toroidal magnetic flux, the total toroidal electric current, or the magnetic helicity). These states occur as the solutions of nonlinear partial differential equations which are far simpler than the original MHD equations. The stabilized Z pinch is discussed by way of illustration of the theory of probable states.

Three-dimensional, pseudo-spectral computation is used to follow the evolution of a resistive, incompressible magnetofluid. The magnetofluid is confined by rigid, free-slip, perfectly-conducting square boundaries in the x, y directions (‘poloidal’ boundaries), and periodic boundary conditions are assumed in the z direction (‘toroidal’ direction). A constant, uniform d.c. magnetic field B0 is assumed in the z direction and a non-uniform current density j flows along it initially. Starting from a non-equilibrium hollow current profile, the evolution is followed for several tens of Alfvén transit times. Considerable small-scale turbulence develops, which causes energy to decay more rapidly than magnetic helicity. The average toroidal magnetic field at the (x, y) boundary reverses sign spontaneously. The near spatial constancy of the ratio jB/(jB) ≡ cos θ, in the relaxed state at late times, suggests that the state is nearly force-free. However, the ratio j. B/B2 ≡ α is considerably less uniform than is cos θ suggesting more residual disorder than a pure minimum-energy state would display.

The Strauss equations of reduced resistive magnetohydrodynamics are solved pseudo-spectrally, inside a cylinder of square cross-section. Conducting, free-slip, boundary conditions are imposed at the boundaries normal to the direction of the imposed d.c. magnetic field and net current, and periodic boundary conditions are imposed in the third direction. The emphasis is on the development of disruptions. Initial conditions are not analytical equilibria, and are characterized by wide-band noise perturbations in many Fourier modes. Typical spatial resolution is 32 × 32 × 16. At this resolution, the code takes approximately 0·7 seconds per time step on a CRAY-1 computer. Lundquist numbers are limited by the need to resolve the small-scale turbulence which develops. Disruptions are characterized by (i) a burst of kinetic fluid activity which is roughly equipartitioned with the magnetic fluctuations at the small scales, but which involves overall kinetic energies which are much less than the magnetic energies; (ii) a helical ‘m = l, n = 1’ current filament which develops out of the wide-band turbulent noise and wraps itself around the magnetic axis; and (iii) relatively mild disturbances of the magnetic field lines, at least at these low values of Lundquist number (S ≃ 100). The results are compared with those from similar codes which solve the linearized Strauss equations.

Two-dimensional magnetohydrodynamic turbulence is explored by means of numerical simulation. Previous analytical theory, based on non-dissipative constants of the motion in a truncated Fourier representation, is verified by following the evolution of highly non-equilibrium initial conditions numerically. Dynamo action (conversion of a significant fraction of turbulent kinetic energy into long-wavelength magnetic field energy) is observed. It is conjectured that in the presence of dissipation and external forcing; a dual cascade will be observed for zero-helicity situations. Energy will cascade to higher wavenumbers simultaneously with a cascade of mean square vector potential to lower wavenumbers, leading to an omni-directional magnetic energy spectrum which varies ask-⅓at lower wavenumbers, simultaneously with a build-up of magnetic excitation at the lowest wavenumber of the system. Equipartition of kinetic and magnetic energies is expected at the highest wavenumbers in the system.

Linear stability calculations conducted in plasma physics are based on the theory of hydrodynamic stability of neutral fluids. The present investigation is concerned with the validity of procedures based on linear stability analysis in six much-studied situations. All have the property that the fluid goes from laminar to turbulent at critical values of some dimensionless number, such as the Reynolds number or the Rayleigh number. The question is, at what threshold values of the dimensionless number do the unstable motions set in, and can these thresholds be predicted by a linear analysis of the stability of the laminar state as the threshold is approached from the stable side. In three cases linear stability analysis clearly seems to fail. These cases include Plane Poiseuille Flow, Plane Couette Flow, and Cylindrical Pipe Flow (Hagen-Poiseuille Flow). Situations in which predictions provided by linear stability analysis are correct are related to Rotating Couette Flow, Thermally-Driven Convection, and Instability of Laminar Boundary Layers.

The consequences of the quasilinear equations are explored. Particular attention is paid to the differences between the one-dimensional and the two- and three-dimensional cases, and to the differences between the cases of discrete and continuous wave-number spectra. The possibilities of and problems associated with including damped waves are treated. The relation between conservation laws and the ‘resonance approximation’, in which the limit of zero growth rate for the unstable waves is taken at finite times, is clarified. Numerical solutions for the one-dimensional case with finite growth rate are presented.