Spectrum of magnetohydrodynamic turbulence

We propose a phenomenological theory of strong incompressible magnetohydrodynamic turbulence in the presence of a strong large-scale external magnetic field. We argue that in the inertial range of scales, magnetic-field and velocity-field fluctuations tend to align the directions of their polarizations. However, the perfect alignment cannot be reached; it is precluded by the presence of a constant energy flux over scales. As a consequence, the directions of shear-Alfvén fluid and magnetic-field fluctuations at each scale lambda become effectively aligned within the angle phi(lambda) proportional to lambda (1/4), which leads to scale-dependent depletion of the nonlinear interaction and to the field-perpendicular energy spectrum E(k(perpendicular)) proportional to k(perpendicular)(-3/2). Our results may be universal, i.e., independent of the external magnetic field, since small-scale fluctuations locally experience a strong field produced by large-scale eddies.     

Ideal and resistive magnetohydrodynamic modes

This paper gives a further discussion of the analytical properties of both discrete and continuous Alfven wave spectra in an incompressible as well as compressible plasma. Although the continuous MHD modes produced by a well-behaved initial perturbation decay according to a power law, some singular solutions exist and are found to behave differently. In particular, it is possible to exhibit the existence of a new continuous mode which decays exponentially, and not as an inverse power of time, and this exponential damping is not the consequence of a continuous variation of the magnetic field. Even the set of discrete magnetohydrodynamic modes is shown to be empty unless certain conditions are satisfied. Next, we consider resistive modes and give explicit solutions for them which are valid in the neighborhood of the Alfven resonance layer and discuss their implications for plasma heating schemes. Finally, we study discrete and continuous Alfven wave spectra in a compressible plasma and discuss how they behave differently from those in an incompressible plasma. In particular, we show that though compressibility of the plasma is responsible for the slow mode continuum, strong compressibility will eliminate it. The discrete modes in a compressible plasma undergo an exponential damping even in an ideal plasma if the compressibility is strong.     

Shell models of magnetohydrodynamic turbulence

Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though shell models cannot account for the spatial complexity of MHD turbulence, their dynamics are not over simplified and do reflect those of real MHD turbulence including intermittency or chaotic reversals of large-scale modes. Furthermore, these models use realistic values for dimensionless parameters (high kinetic and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing extended inertial range and accurate dissipation rate. Using modern computers it is difficult to attain an inertial range of three decades with direct numerical simulations, whereas eight are possible using shell models.     

Anisotropic scaling of magnetohydrodynamic turbulence

We present a quantitative estimate of the anisotropic power and scaling of magnetic field fluctuations in inertial range magnetohydrodynamic turbulence, using a novel wavelet technique applied to spacecraft measurements in the solar wind. We show for the first time that, when the local magnetic field direction is parallel to the flow, the spacecraft-frame spectrum has a spectral index near 2. This can be interpreted as the signature of a population of fluctuations in field-parallel wave numbers with a k(-2)_(||) spectrum but is also consistent with the presence of a "critical balance" style turbulent cascade. We also find, in common with previous studies, that most of the power is contained in wave vectors at large angles to the local magnetic field and that this component of the turbulence has a spectral index of 5/3.     

Oscillatory disintegration of nonevolutionary magnetohydrodynamic discontinuities

Trans-Alfvénic shock waves are considered in the approximation of small amplitude and almost parallel propagation of the magnetic field. Such shocks are nonevolutionary, since the problem of time evolution of their small perturbation does not have a unique solution. Therefore, they cannot exist as stationary configurations and must disintegrate or transform to some more general, nonsteady flow. The disintegration configuration necessarily includes an Alfvén discontinuity that is also nonevolutionary. It is shown that the contradiction inherent in the nonevolutionary configuration is removed if its time evolution has the form of oscillatory disintegration, i.e., reversible transformation of one type of discontinuity to the other. In this process fast and slow shock or rarefaction waves as well as contact discontinuities are emitted. Zh. éksp. Teor. Fiz. 113, 615–628 (February 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Impulse formulations of Hall magnetohydrodynamic equations

Impulse formulations of Hall magnetohydrodynamic (MHD) equations are developed. The Lagrange invariance of a generalized ion magnetic helicity is established for Hall MHD. The physical implications of this Lagrange invariant are discussed. The discussion is then extended to compressible Hall MHD and a generalized ion magnetic potential helicity Lagrange invariant is established. The physical implications of the generalized ion magnetic potential helicity Lagrange invariant are shown to be the same, as to be expected, as those of the generalized ion magnetic helicity Lagrange invariant.     

Moving inhomogeneity in a magnetohydrodynamic medium

The integral equations of ideal magnetohydrodynamics are derived by the Green’s function method, taking into account the velocity of the medium inside the inhomogeneity before and after its disturbance by the incident field. The extinction principle of of magnetohydrodynamics is demonstrated for a moving half space. A comparative analysis is made with the results of an analogous problem in which only the velocity of the interface between the two media is taken into account. Zh. Tekh. Fiz. 67, 6–11 (May 1997)     

Stability of magnetohydrodynamic stratified shear flows

Summary The linear stability of a stratified shear flow of a perfectly conducting bounded fluid in the presence of a magnetic field aligned with the flow and buoyancy forces has been studied under Boussinesq approximation. A new upper bound has been obtained for the range of real and imaginary parts of the complex wave velocity for growing perturbations. The upper bound depends on minimum Richardson number, wave number, Alfvén velocity and basic flow velocity. H?iland's necessary criterion for instability of hydrodynamic stratified homogeneous shear flow is modified and its analog for nonhomogeneous magnetohydrodynamic cases is derived. Finally the upper bound for the growth rate ofKC _i and its variants, whereK is the wave number andC _i the imaginary part of complex wave velocity, is derived as the necessary condition of instability. All estimates remain valid even when the minimum richardson numberJ ₁, for some practical problems, exceeds 1/4 for growing perturbations. The authors of this paper have agreed to not receive the proofs for correction.     

Scaling properties of three-dimensional magnetohydrodynamic turbulence

The scaling properties of three-dimensional magnetohydrodynamic turbulence with finite magnetic helicity are obtained from direct numerical simulations using 512(3) modes. The results indicate that the turbulence does not follow the Iroshnikov-Kraichnan phenomenology. The scaling exponents of the structure functions can be described by a modified She-Leveque model zeta(p) = p/9+1-(1/3)(p/3), corresponding to basic Kolmogorov scaling and sheetlike dissipative structures. In particular, we find zeta(2) approximately 0.7, consistent with the energy spectrum E(k) approximately k(-5/3) as observed in the solar wind, and zeta(3) approximately 1, confirming a recent analytical result.     

Small-scale structures in three-dimensional magnetohydrodynamic turbulence

We investigate using direct numerical simulations with grids up to 1536(3) points, the rate at which small scales develop in a decaying three-dimensional MHD flow both for deterministic and random initial conditions. Parallel current and vorticity sheets form at the same spatial locations, and further destabilize and fold or roll up after an initial exponential phase. At high Reynolds numbers, a self-similar evolution of the current and vorticity maxima is found, in which they grow as a cubic power of time; the flow then reaches a finite dissipation rate independent of the Reynolds number.     

Calculations of two-fluid magnetohydrodynamic axisymmetric steady-states

M3D-C¹$C^1$ is an implicit, high-order finite element code for the solution of the time-dependent nonlinear two-fluid magnetohydrodynamic equations [S.C. Jardin, J. Breslau, N. Ferraro, A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions, J. Comp. Phys. 226 (2) (2007) 2146–2174]. This code has now been extended to allow computations in toroidal geometry. Improvements to the spatial integration and time-stepping algorithms are discussed. Steady-states of a resistive two-fluid model, self-consistently including flows, anisotropic viscosity (including gyroviscosity) and heat flux, are calculated for diverted plasmas in geometries typical of the National Spherical Torus Experiment (NSTX) [M. Ono et al., Exploration of spherical torus physics in the NSTX device, Nucl. Fusion 40 (3Y) (2000) 557–561]. These states are found by time-integrating the dynamical equations until the steady-state is reached, and are therefore stationary or statistically steady on both magnetohydrodynamic and transport time-scales. Resistively driven cross-surface flows are found to be in close agreement with Pfirsch-Schlüter theory. Poloidally varying toroidal flows are in agreement with comparable calculations [A.Y. Aydemir, Shear flows at the tokamak edge and their interaction with edge-localized modes, Phys. Plasmas 14]. New effects on core toroidal rotation due to gyroviscosity and a local particle source are observed.     

Nonlinear magnetohydrodynamic waves in compensated metals

We show that if a conductor is placed in a quantizing magnetic field H ₀, nonlinear small-amplitude electromagnetic waves can propagate in the conductor. For compensated metals we find the solution of the Maxwell equations when the field H ₀ is perpendicular to the direction in which the waves propagate. Zh. éksp. Teor. Fiz. 112, 1841–1846 (November 1997)     

Spectral energy dynamics in magnetohydrodynamic turbulence

Spectral direct numerical simulations of incompressible MHD turbulence at a resolution of up to 1024(3) collocation points are presented for a statistically isotropic system as well as for a setup with an imposed strong mean magnetic field. The spectra of residual energy, E(R)k=|E(M)k - E(K)k|, and total energy, Ek=E(K)k+E(M)k, are observed to scale self-similarly in the inertial range as E(R)k approximately k(-7/3), E(k)approximately k(-5/3) (isotropic case) and E(R)(k(perpendicular) approximately k(-2)(perpendicular), E(k(perpendicular))approximately k(-3/2)(perpendicular) (anisotropic case, perpendicular to the mean field direction). A model of dynamic equilibrium between kinetic and magnetic energy, based on the corresponding evolution equations of the eddy-damped quasinormal Markovian closure approximation, explains the findings. The assumed interplay of turbulent dynamo and Alfvén effect yields E(R)k approximately kE2(k), which is confirmed by the simulations.     

Global magnetohydrodynamic simulations of the magnetosphere

The authors demonstrate the use of a global MHD (magnetohydrodynamic) simulations to study the magnetospheric configuration by reviewing some of the results obtained from the Ogino model. The authors start by considering the steady-state configuration of the magnetosphere in the absence of an IMF (interplanetary magnetic field), and then demonstrate how that configuration is changed when a northward or southward IMF is introduced. It is noted that the magnetosphere is very dynamic and since global MHD simulations are intrinsically time-dependent, they offer the possibility of modeling the time-sequence of events in the magnetosphere. Finally results of a calculation in which a magnetospheric substorm is modeled are presented     

Dynamic alignment in driven magnetohydrodynamic turbulence

Motivated by recent analytic predictions, we report numerical evidence showing that in driven incompressible magnetohydrodynamic turbulence the magnetic- and velocity-field fluctuations locally tend to align the directions of their polarizations. This dynamic alignment is stronger at smaller scales with the angular mismatch between the polarizations decreasing with the scale lambda approximately as theta(lambda) is proportional to lambda(1/4). This can naturally lead to a weakening of the nonlinear interactions and provide an explanation for the energy spectrum E(k) is proportional to k(-3/2) that is observed in numerical experiments of strongly magnetized turbulence.