Sanjoy Ghosh

Anisotropic three‐dimensional MHD turbulence

Direct spectral method simulation of the three-dimensional magnetohydrodynamics (MHD) equations is used to explore anisotropy that develops from initially isotropic fluctuations as a consequence of a uniform applied magnetic field. Spectral and variance anisotropies are investigated in both compressible and incompressible MHD. The nature of the spectral anisotropy is consistent with the model of Shebalin et al. [1983] in which the spectrum broadens in the perpendicular wavenumber direction, the anisotropy being greater for smaller wavenumbers. Here this effect is seen for both incompressible and polytropic compressible MHD. In contrast, the longitudinal (compressive) velocity fluctuations remain isotropic. Variance anisotropy is observed for low plasma beta compressible MHD but not for incompressible MHD. Solar wind observations are qualitatively consistent with both variance and spectral anisotropies of the type discussed here.

Simulation of high-frequency solar wind power spectra using Hall magnetohydrodynamics

Solar wind frequency spectra show a distinct steepening of the ƒ−5/3 power law inertial range spectrum at frequencies above the Doppler-shifted ion cyclotron frequency. This is commonly attributed to dissipation due to wave-particle interactions. We consider the extent to which this steepening can be described, using a magnetohydrodynamic formulation that includes the Hall term. An important characteristic of Hall MHD is that although the ion cyclotron resonance is included, there is no wave-particle dissipation of energy. In this study we use a compressible Hall MHD code with a constant magnetic field and a polytropic equation of state. Artificial dissipation in the form of a bi-Laplacian operator is used to suppress numerical instabilities, allowing for a clear separation of the dissipative scales from the ion cyclotron scales. A distinct steepening appears in the simulation power spectra near the cyclotron resonance for certain types of initial conditions. This steepening is associated with the appearance of right circularly polarized fluctuations at frequencies above the ion cyclotron resonance. Similar steepenings and polarization enhancements are observed in solar wind magnetic field data.

A numerical study of the nonlinear cascade of energy in magnetohydrodynamic turbulence

Power spectra of solar wind magnetic field and velocity fluctuations more closely resemble those of turbulent fluids (spectral index of −5/3) than they do predictions for magnetofluid turbulence (a −3/2 index). Furthermore, the amount the solar wind is heated by turbulence is uncertain. To aid in the study of both of these issues, we report numerically derived energy cascade rates in magnetohydrodynamic (MHD) turbulence and compare them with predictions of MHD turbulence phenomenologies. Either of the commonly predicted spectral indices of 5/3 and 3/2 are consistent with the simulations. Explicit calculation of inertial range energy cascade rates in the simulations show that for unequal levels of fluctuations propagating parallel and antiparallel to the magnetic field, the majority species always cascades faster than does the minority species, and the cascade rates are in better agreement with a Kolmogoroff-like MHD turbulence phenomenology than with a generalized Kraichnan phenomenology even in situations where the fluctuations are much smaller than the mean magnetic field. The “Kolmogoroff constant” for MHD turbulence for small normalized cross helicity is roughly 6.7 in two dimensions and 3.6 for one calculation in three dimensions. For large normalized cross helicity, however, none of the existing models can account for the numerical results, although the Kolmogoroff-like case still works somewhat better than the Kraichnan-like. In particular, the applied magnetic field has much less influence than expected, and Alfvenicity is more important than predicted. These results imply the need for better phenomenological models to make clear predictions about the solar wind.

Waves, structures, and the appearance of two-component turbulence in the solar wind

Spacecraft observations of magnetic field fluctuations in the solar wind reveals a “Maltese Cross” pattern in the two-dimensional correlation function measurements of solar wind fluctuations [Matthaeus et al., 1990]. This pattern suggests the presence of two components: fluctuations with their (Fourier) wave vector approximately parallel to the ambient magnetic field (e.g., slab turbulence) and fluctuations with their (Fourier) wave vector approximately perpendicular to the ambient magnetic field (e.g., quasi two-dimensional turbulence). To date, the appearance of such a pattern has never been reproduced from numerical simulation studies. Here we present results of several MHD simulations that address this issue using both two-and-one-half dimensional and three-dimensional compressible models and a wide variety of initial states and plasma parameters. Slab turbulence and quasi two-dimensional turbulence appear in various runs; however, their simultaneous appearance is difficult to achieve and seems to rely upon their separate existence in the initial data. In contrast, the presence of transverse pressure-balanced magnetic structures causes slab turbulence to evolve in such a manner that a two-component correlation function emerges through time averaging. We suggest that the Maltese Cross and similar observations may be a consequence of either the initial data or of averaging over different parcels of solar wind.

Numerical simulation of Alfvénic turbulence in the solar wind

Low-frequency fluctuations in the solar wind magnetic field and plasma velocity are often highly correlated, so much so that the fluctuations can be thought of as nearly perfect Alfven waves. Evidence from the Helios and Ulysses spacecraft suggest strongly that these fluctuations emanate from the solar corona with high correlation and flat power spectra (∼f−1). These fluctuations constitute a source of free energy for a turbulent cascade of magnetic and kinetic energy to high wave numbers, a cascade that evolves most rapidly in the vicinity of velocity shears and the heliospheric current sheet. Numerical solutions of both the compressible and incompressible equations of magnetohydrodynamics (MHD) in Cartesian geometry showed that sharp gradients in velocity would decrease substantially the Alfvenicity of initially pure Alfvenic fluctuations; however, the effects of solar wind expansion on this turbulent evolution is, as yet, undetermined. We demonstrate that as was the case in Cartesian geometry, in an expanding volume, velocity shears and pressure-balanced flux tubes still reduce the Alfvenicity of parallel propagating wave packets. These three-dimensional spherically expanding simulations include velocity shears separating fast and slow flows, pressure-balanced flux tubes, and a central current sheet which is the site of magnetic reconnection. Two-dimensional spectra constructed in the r – θ plane resemble closely those resulting from similar initial conditions in Cartesian geometry.

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.

The evolution of slab fluctuations in the presence of pressure-balanced magnetic structures and velocity shears

The traditional view that solar wind fluctuations are well-described as a spectrum of parallel-propagating Alfven waves has been challenged many times but is still a frequently encountered perspective. Here we examine whether it remains consistent to view most of the fluctuation energy as resident in parallel-propagating Alfven waves in situations in which there are also present either transverse pressure-balanced (PB) magnetic structures or transverse velocity shears. We address these questions through direct simulation of compressible magnetohydrodynamics, with expansion effects neglected. We show that parallel-propagating Alfven waves are redirected to large oblique angles after refractive interactions with PB structures or advective interactions with velocity shears, reflecting the nonequilibrium nature of the initial spectral distribution. The timescale for these processes ranges from 2–8 eddy-turnover times or characteristic nonlinear times. Relatively small amounts of PB structure and/or shear energy can redirect initially parallel-propagating Alfven waves to highly oblique angles. Velocity microstreams appear to be particularly efficient at creating highly oblique waves. Even though the excited wave vectors are eventually primarily oblique, the magnetic variance ratios show a minimum variance in the mean magnetic field direction.

Nonlinear decay of magnetic helicity in magnetohydrodynamic turbulence with a mean magnetic field

We show that the magnetic helicity associated with fluctuations in homogeneous incompressible magnetohydrodynamic (MHD) turbulence with a mean magnetic field decays in time because of nonlinear processes. Evidence is obtained numerically, by use of both dissipative and nondissipative spectral method simulations. The described effect stands in contrast to expectations based on studies of MHD turbulence without an applied mean field, in which magnetic helicity is transferred nonlinearly to long wavelengths and is preserved in time because of selective decay when dissipation is present. The process of nonlinear decay is described in terms of a generalized ideal helicity invariant and is characterized by the transient production of a mean induced electric field aligned with the applied magnetic field, an effect reminiscent of the alpha effect in dynamo theory. A simple phenomenological model for the decay process is proposed.

Parametric instabilities of a large-amplitude circularly polarized Alfvén wave: Linear growth in two-dimensional geometries

The growth of parametric instabilities, which may lead to the development of a turbulent cascade, is studied using a magnetohydrodynamic (MHD) code that permits nonlinear couplings in the parallel direction to the ambient magnetic field and one perpendicular direction. Compressibility is included in the form of a polytropic equation of state. Parametric instabilities associated with a parallel-propagating decay instability are found to dominate the low-beta case. An obliquely propagating filamentationlike instability dominates the high-beta case. The nonlinear growth of the nth harmonic of a daughter wave growing as a factor of n times the fundamentals growth rate is found in both cases. Nonlinear saturation is caused by the parallel decay instability in the low-beta case and by the oblique filamentationlike instability in the high-beta case.

A kinematic analysis of the role of velocity shear in expanding plasmas

We present a kinematic analysis of the effects on waves of the shear due to variations in radial flow and field-aligned propagation speeds in an expanding medium. We show that although the expansion limits the extent to which wave vectors transverse to the shear flow are produced, shear can be very effective at turning wave vectors and producing off-radial fluctuations. Any initial wave vector spectrum will become spread in wave vector space with no change in spectral index. For typical spectra of the sheared radial speeds, the small-scale variability in the speed is more important than the large-scale in the sense of effectively turning the wave vectors. Radial spectra will be unchanged by stream or microstream shear, and although field-aligned Alfven speed changes will produce some observable effects, nonlinear processes are still needed to fully explain observations. This work suggests a model for near-Sun solar wind turbulence in which modes are “driven” nonlocally at all scales rather than the Kolmogoroff model, in which only large-scale driving is important; this is consistent with well-developed turbulence spectra in highly structured regions farther out in the wind.