Reduced Magnetohydrodynamics Research Articles

Saturation of a tearing mode in zero-β full magnetohydrodynamics

Within the cylindrical approximation, the nonlinear saturation of a tearing mode island in the full magnetohydrodynamics (MHD) model with no pressure and a uniform resistivity profile is described using a recent perturbation method based on the technique of matched asymptotic expansion, where the perturbation parameter is the island width ws. The matching condition provides the saturation equation for ws that is given to the adequate order and does not depend on the normalization length. This enables to prove that the errors due to reduced MHD (RMHD) are negligible in the calculation of tearing mode saturation in the tokamak. It also provides a way to benchmark MHD numerical simulations of magnetic self-reversal for the reversed-field pinch where RMHD does not apply.

Coronal Heating, Weak MHD Turbulence, and Scaling Laws

Long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry are carried out, within the framework of reduced magnetohydrodynamics (RMHD), to understand coronal heating driven by motion of field lines anchored in the photosphere. We unambiguously identify MHD anisotropic turbulence as the physical mechanism responsible for the transport of energy from the large scales, where energy is injected by photospheric motions, to the small scales, where it is dissipated. As the loop parameters vary different regimes of turbulence develop: strong turbulence is found for weak axial magnetic fields and long loops, leading to Kolmogorov-like spectra in the perpendicular direction, while weaker and weaker regimes (steeper spectral slopes of total energy) are found for strong axial magnetic fields and short loops. As a consequence we predict that the scaling of the heating rate with axial magnetic field intensity $B_0$, which depends on the spectral index of total energy for given loop parameters, must vary from $B_0^{3/2}$ for weak fields to $B_0^{2}$ for strong fields at a given aspect ratio. The predicted heating rate is within the lower range of observed active region and quiet Sun coronal energy losses.

Direct comparisons of compressible magnetohydrodynamics and reduced magnetohydrodynamics turbulence

Direct numerical simulations of low Mach number compressible three-dimensional magnetohydrodynamic (CMHD3D) turbulence in the presence of a strong mean magnetic field are compared with simulations of reduced magnetohydrodynamics (RMHD). Periodic boundary conditions in the three spatial coordinates are considered. Different sets of initial conditions are chosen to explore the applicability of RMHD and to study how close the solution remains to the full compressible MHD solution as both freely evolve in time. In a first set, the initial state is prepared to satisfy the conditions assumed in the derivation of RMHD, namely, a strong mean magnetic field and plane-polarized fluctuations, varying weakly along the mean magnetic field. In those circumstances, simulations show that RMHD and CMHD3D evolve almost indistinguishably from one another. When some of the conditions are relaxed the agreement worsens but RMHD remains fairly close to CMHD3D, especially when the mean magnetic field is large enough. Moreover, the well-known spectral anisotropy effect promotes the dynamical attainment of the conditions for RMHD applicability. Global quantities (mean energies, mean-square current, and vorticity) and energy spectra from the two solutions are compared and point-to-point separation estimations are computed. The specific results shown here give support to the use of RMHD as a valid approximation of compressible MHD with a mean magnetic field under certain but quite practical conditions.

Nonlinear simulation of tearing mode based on 4-field RMHD model

Simulations of dynamics of tearing modes based on a 4-field reduced magneto hydro dynamics model are performed, laying an emphasis on interaction with microscopic and transport processes. The simulation results show the importance of turbulent fluctuations for the onset of tearing mode. The helical current perturbation associated with a magnetic island is studied. The perturbed bootstrap current has a structure in the magnetic island (including a phase difference), indicating the need to improve assumptions in the theory of neoclassical tearing mode.

Reduced magnetohydrodynamics and parallel spectral transfer

The self-consistency of the reduced magnetohydrodynamics (RMHD) model is explored by examining whether (parallel) spectral transfer might invalidate the assumptions employed in deriving it. Using direct numerical simulations we find that transfer of energy to structures with high parallel wavenumber is in fact limited by ongoing perpendicular transfer. Thus, the dynamics associated with RMHD models remains consistent with the underlying assumptions of RMHD. In particular, in well-resolved simulations it is neither necessary nor correct to introduce additional dissipation terms that (artificially) damp spectral transfer parallel to the mean magnetic field B0.

A simplified numerical model of coronal energy dissipation based on reduced MHD

A 3D model intermediate between cellular automata (CA) models and the reduced magnetohydrodynamic (RMHD) equations is presented to simulate solar impulsive events generated along a coronal magnetic loop. The model consists of a set of planes distributed along a magnetic loop between which the information propagates through Alfven waves. Statistical properties in terms of power-laws for energies and durations of dissipative events are obtained, and their agreement with X-ray and UV flares observations is discussed. The existence of observational biases is also discussed.

What can we infer about the underlying physics from burst distributions observed in an RMHD simulation?

We determine that the sizes of bursts in mean-square current density in a reduced magnetohydrodynamic (RMHD) simulation follow a power-law probability density function (PDF). The PDFs for burst durations and waiting time between bursts are clearly not exponential and could also be power-law. This suffices to distinguish their behaviour from the original Bak et al. sandpile model which had exponential waiting time PDFs. However, it is not sufficient to distinguish between turbulence, some other SOC-like models, and other red noise sources.

A Reduced Magnetohydrodynamic Model of Coronal Heating in Open Magnetic Regions Driven by Reflected Low‐Frequency Alfven Waves

A reduced magnetohydrodynamic (RMHD) description is employed to examine a suggestion made by W. H. Matthaeus and colleagues in 1999 that coronal heating might be sustained by a cascade of lowfrequency MHD turbulence. Here RMHD simulations show that the low-frequency cascade to high transverse wavenumbers can be driven by an externally maintained —ux of low-frequency propagating waves, in combination with re—ection caused by an inhomogeneous background medium. The Alfvec n simulations show that the suggestions made previously on the basis of a phenomenology are indeed realizable. In addition, the eUect is seen to sensitively depend on the presence of re—ection, as the background turbulence level needed to maintain the cascade can be sustained only when re—ection is imposed. The steady level of turbulence is insensitive to the initial seed turbulence level (provided it is nonzero). Consequences of this model for realistic models of coronal heating in open —eld-line regions are discussed.

Nonlinear coupling of rotating magnetic island triplets

The role and mechanism of nonlinear coupling of finite size rotating magnetic islands in a tokamak are analyzed in a finite dimensional model obtained from the continuum Reduced Magnetohydrodynamic (RMHD) description of resistive tearing modes. General properties of the electrodynamic interaction of modes of different helicities are found and a phase space analysis of the effects of inertia and eddy currents in the tokamak wall is presented. The results obtained are relevant for the definition of strategies of feedback control of resistive instabilities and a possible help for the interpretation of tokamak performances linked to the onset of a spectrum of magnetic perturbations.

The rate of flux pile-up magnetic reconnection in reduced MHD

The rate of two-dimensional flux pile-up magnetic reconnection is known to be severely limited by gas pressure in a low-beta plasma of the solar corona. As earlier perturbational calculations indicated, however, the pressure limitation should be less restrictive for three-dimensional flux pile-up. In this paper the maximum rate of reconnection is calculated in the approximation of reduced magnetohydrodynamics (RMHD), which is valid in the solar coronal loops. The rate is calculated for finite-magnitude reconnecting fields in the case of a strong axial field in the loop. Gas pressure effects are ignored in RMHD but a similar limitation on the rate of magnetic merging exists. Nevertheless, the magnetic energy dissipation rate and the reconnection electric field can increase by several orders of magnitude as compared with strictly two-dimensional pile-up. Though this is still not enough to explain the most powerful solar flares, slow coronal transients with energy release rates of order 1025– 1026 erg s−1and heating of quiet coronal loops are within the compass of the model.

Scaling of spectral anisotropy with magnetic field strength in decaying magnetohydrodynamic turbulence

Space plasma measurements, laboratory experiments, and simulations have shown that magnetohydrodynamic (MHD) turbulence exhibits a dynamical tendency towards spectral anisotropy given a sufficiently strong background magnetic field. Here the undriven decaying initial-value problem for homogeneous MHD turbulence is examined with the purpose of characterizing the variation of spectral anisotropy of the turbulent fluctuations with magnetic field strength. Numerical results for both incompressible and compressible MHD are presented. A simple model for the scaling of this spectral anisotropy as a function of the fluctuating magnetic field over total magnetic field is offered. The arguments are based on ideas from reduced MHD (RMHD) dynamics and resonant driving of certain non-RMHD modes. The results suggest physical bases for explaining variations of the anisotropy with compressibility, Reynolds numbers, and spectral width of the (isotropic) initial conditions.

Weakly Compressible Magnetohydrodynamic Turbulence in the Solar Wind and the Interstellar Medium

A new four-field system of equations is derived from the compressible magnetohydrodynamic (MHD) equations for low Mach number turbulence in the solar wind and the interstellar medium, permeated by a spatially varying magnetic field. The plasma beta is assumed to be of order unity or less. It is shown that the full MHD equations can be reduced rigorously to a closed system for four fluctuating field variables: magnetic flux, vorticity, pressure, and parallel flow. Although the velocity perpendicular to the magnetic field is shown to obey a two-dimensional incompressibility condition (analogous to the Proudman-Taylor theorem in hydrodynamics), the three-dimensional dynamics exhibit the effects of compressibility. In the presence of spatial inhomogeneities, the four dynamical equations are coupled to each other, and pressure fluctuations enter the weakly compressible dynamics at leading order. If there are no spatial inhomogeneities and or if the plasma beta is low, the four-field equations reduce to the well-known equations of reduced magnetohydrodynamics (RMHD). For pressure-balanced structures, the four-field equations undergo a remarkable simplification which provides insight on the special nature of the fluctuations driven by these structures. The important role of spatial inhomogeneities is elucidated by 2.5-dimensional numerical simulations. In the presence of inhomogeneities, the saturated pressure and density fluctuations scale with the Mach number of the turbulence, and the system attains equipartition with respect to the kinetic, magnetic, and thermal energy of the fluctuations. The present work suggests that if heliospheric and interstellar turbulence exists in a plasma with large-scale, nonturbulent spatial gradients, one expects the pressure and density fluctuations to be of significantly larger magnitude than suggested in nearly incompressible models such as pseudosound.

Electron inertia and small-scale magnetic structures in a nonuniform collisionless plasma

In collisionless plasmas, electron inertia has a strong influence on the formation of magnetic islands, through magnetic field line reconnection, and on the dynamics of coherent nonlinear structures such as magnetic vortices. We present a physical model for the nonlinear dynamics of such magnetic structures in configurations with a strong magnetic field. This model includes diamagnetic velocities and ion gyro-radius and electron inertia effects and yields the so-called Reduced MagnetoHydroDynamic (RMHD) equations in the appropriate limit.

The equations of reduced magnetohydrodynamics

The equations of high- and low-beta reduced magnetohydrodynamics (RMHD) are considered anew in order to elucidate the relationship between compressible MHD and RMHD and also to distinguish RMHD from recently developed models of nearly incompressible MHD. Our results, summarized in two theorems, provide the conditions under which RMHD represents a valid reduction of compressible MHD. The equations for low-beta RMHD and high-beta RMHD are shown to be identical. Furthermore, as a direct consequence of our analysis, the conditions under which both two-dimensional incompressible MHD (in terms of the spatial co-ordinates as well as the fluid variables) and 2½ dimensional incompressible MHD (i.e. only two-dimensional in the spatial co-ordinates) represent a valid reduction of three-dimensional compressible MHD are also formulated. It is found that the elimination of all high-frequency and long-wavelength modes from the magneto-fluid reduces the fully compressible MHD equations to either two-dimensional incompressible MHD in the plasma beta (β) limit β ≪ 1, or 2½-dimensional incompressible MHD for β ≈ 1. Our approach clarifies several inconsistencies to be found in previous investigations in that the reduction is exact. Our results and analysis are expected to be of interest for plasma fusion and space and solar physics.

Renormalization group analysis of reduced magnetohydrodynamics with application to subgrid modeling

The technique for obtaining a subgrid model for Navier–Stokes turbulence introduced by Yakhot and Orszag [J. Sci. Comput. 1, 3 (1986); Phys. Rev. Lett. 57, 1772 (1986)], based on renormalization group analysis (RNG), is extended to the reduced magnetohydrodynamic (RMHD) equations. A RNG treatment of the Alfvén turbulence (perpendicular scale k−1⊥≪k−1∥ parallel scale) supported by the RMHD equations leads to effective values of the viscosity and resistivity at large scales, k→0, dependent on the amplitude of turbulence. When the RNG analysis is augmented by the Kolmogorov argument for energy cascade the effective viscosity and resistivity become independent of the molecular quantities. This leads to a ‘‘universal’’ subgrid model, which all models approach at the largest scales. A self-contained system of equations is derived for the range of scales, 0<k<K, where K=π/Δ, is the maximum wave number for a grid size Δ. In this system the resistive and viscous dissipation is represented by differential operators, whose coefficients depend upon the amplitudes of the large-scale quantities being computed.

Hamiltonian formulation of reduced magnetohydrodynamics

Reduced magnetohydrodynamics (RMHD) is a principal tool for understanding nonlinear processes, including disruptions, in tokamak plasmas. Although analytical studies of RMHD turbulence are useful, the model’s impressive ability to simulate tokamak fluid behavior has been revealed primarily by numerical solution. A new analytical approach, not restricted to turbulent regimes, based on Hamiltonian field theory is described. It is shown that the nonlinear (ideal) RMHD system, in both its high-beta and low-beta versions, can be expressed in Hamiltonian form. Thus a Poisson bracket, { , }, is constructed such that each RMHD field quantity ξi evolves according to ξ̇i ={ξi,H}, where H is the total field energy. The new formulation makes RMHD accessible to the methodology of Hamiltonian mechanics; it has lead, in particular, to the recognition of new RMHD invariants and even exact, nonlinear RMHD solutions. A canonical version of the Poisson bracket, which requires the introduction of additional fields, leads to a nonlinear variational principle for time-dependent RMHD.