Evolution Of Cross Helicity Research Articles

Frequency-dependent Alfvén-wave Propagation in the Solar Wind: Onset and Suppression of Parametric Decay Instability

Abstract Using numerical simulations we investigate the onset and suppression of parametric decay instability (PDI) in the solar wind, focusing on the suppression effect by the wind acceleration and expansion. Wave propagation and dissipation from the coronal base to 1 au is solved numerically in a self-consistent manner; we take into account the feedback of wave energy and pressure in the background. Monochromatic waves with various injection frequencies, f 0, are injected to discuss the suppression of PDI, while broadband waves are applied to compare the numerical results with observation. We find that high-frequency ( ) Alfvén waves are subject to PDI. Meanwhile, the maximum growth rate of the PDI of low-frequency ( ) Alfvén waves becomes negative due to acceleration and expansion effects. Medium-frequency ( ) Alfvén waves have a positive growth rate but do not show the signature of PDI up to 1 au because the growth rate is too small. The medium-frequency waves experience neither PDI nor reflection so they propagate through the solar wind most efficiently. The solar wind is shown to possess a frequency-filtering mechanism with respect to Alfvén waves. The simulations with broadband waves indicate that the observed trend of the density fluctuation is well explained by the evolution of PDI while the observed cross-helicity evolution is in agreement with low-frequency wave propagation.

Comparison of forcing functions in magnetohydrodynamics

Results are presented of direct numerical simulations of incompressible, homogeneous magnetohydrodynamic turbulence without a mean magnetic field, subject to different mechanical forcing functions commonly used in the literature. Specifically, the forces are negative damping (which uses the large-scale velocity field as a forcing function), a nonhelical random force, and a nonhelical static sinusoidal force (analogous to helical ABC forcing). The time evolution of the three ideal invariants (energy, magnetic helicity and cross helicity), the time-averaged energy spectra, the energy ratios and the dissipation ratios are examined. All three forcing functions produce qualitatively similar steady states with regards to the time evolution of the energy and magnetic helicity. However, differences in the cross helicity evolution are observed, particularly in the case of the static sinusoidal method of energy injection. Indeed, an ensemble of sinusoidally-forced simulations with identical parameters shows significant variations in the cross helicity over long time periods, casting some doubt on the validity of the principle of ergodicity in systems in which the injection of helicity cannot be controlled. Cross helicity can unexpectedly enter the system through the forcing function and must be carefully monitored.

Two-fluid and finite Larmor radius effects on helicity evolution in a plasma pinch

The evolution of magnetic energy, helicity, and hybrid helicity during nonlinear relaxation of a driven-damped plasma pinch is compared in visco-resistive magnetohydrodynamics and two-fluid models with and without the ion gyroviscous stress tensor. Magnetic energy and helicity are supplied via a boundary electric field which initially balances the resistive dissipation, and the plasma undergoes multiple relaxation events during the nonlinear evolution. The magnetic helicity is well conserved relative to the magnetic energy over each event, which is short compared with the global resistive diffusion time. The magnetic energy decreases by roughly 1.5% of its initial value over a relaxation event, while the magnetic helicity changes by at most 0.2% of the initial value. The hybrid helicity is dominated by magnetic helicity in low-β pinch conditions and is also well conserved. Differences of less than 1% between magnetic helicity and hybrid helicity are observed with two-fluid modeling and result from cross helicity evolution. The cross helicity is found to change appreciably due to the first-order finite Larmor radius effects which have not been included in contemporary relaxation theories. The plasma current evolves towards the flat parallel current state predicted by Taylor relaxation theory but does not achieve it. Plasma flow develops significant structure for two-fluid models, and the flow perpendicular to the magnetic field is much more substantial than the flow along it.

TURBULENT CROSS-HELICITY IN THE MEAN-FIELD SOLAR DYNAMO PROBLEM

We study the dynamical and statistical properties of turbulent cross-helicity (correlation of the aligned fluctuating velocity and magnetic field components). We derive an equation governing generation and evolution of the turbulent cross-helicity and discuss its meaning for the dynamo. Using symmetry properties of the problem we suggest a general expression for the turbulent cross-helicity pseudo-scalar and compute the turbulent coefficients in this expression. Effects of the density stratification, large-scale magnetic fields, differential rotation and turbulent convection are taken into account. We investigate the relative contribution of these effects to the cross-helicity evolution for two kinds of dynamo models of the solar cycle including a distributed mean-field model and a flux-transport dynamo model. We show that the contribution from the density stratification follows the evolution of the radial magnetic field, while large-scale electric currents produce a more complicated pattern of the cross-helicity of the comparable magnitude. We suggest that the results of observational analysis of the cross-helicity will depend on the averaging scales. Our results show that the pattern of the cross-helicity evolution strongly depends on details of the dynamo mechanism. Thus, we anticipate that direct observations of the cross-helicity on the Sun may serve for the diagnostic purpose of the solar dynamo process.

Modeling the turbulent cross-helicity evolution: production, dissipation, and transport rates

It has been recognized that the turbulent cross helicity (correlation between the velocity and magnetic-field fluctuations) can play an important role in several magnetohydrodynamic plasma phenomena such as the global magnetic-field generation, turbulence suppression, and so on. Despite its relevance to the cross-helicity evolution, little attention has been paid to the dissipation rate of the turbulent cross helicity, ϵ W . In this paper, we consider the model expression for ϵ W . In addition to the algebraic model, an evolution equation of ϵ W is proposed on the basis of the statistical analytical theory of inhomogeneous turbulence. A turbulence model with the modeling of ϵ W is applied to the solar-wind turbulence. Numerical results on the large-scale evolution of cross helicity are compared with the satellite observations. It is shown that, as far as the solar-wind application is concerned, the simplest possible algebraic model for ϵ W is sufficient for elucidating the large-scale spatial evolution of the solar-wind turbulence. Dependence of the cross-helicity evolution on the large-scale velocity structures, such as velocity shear and flow expansion, is also discussed.

A Simple Model for Reflection-Driven Spectral Evolution of Turbulence in the Corona and Inner Heliosphere

Dissipation of turbulence is generally considered as a prime candidate for the heating and acceleration of so- lar wind from the corona throughout the heliosphere, due to radially evolving dissipation processes and/or cascading of energy to dissipation scales. In order to model the latter consistently, we must consider the evolution of cross-helicity when modeling the strength of the cascade, as the non-linear interactions causing it require counterstreaming waves. In this paper, we present a simple model of non-WKB wave reflection from large-scale gradients, and apply the result- ing cross-helicity to evaluate the evolution of the turbulence spectrum up to 0.3 AU by using a phenomenological cascade model. The study is restricted to a cascade in perpendicular direction. We study the ability of this spectral flux to heat the solar wind, and its dependence on the frequency of the excited waves. We find that the cascade advances fast in the low corona, and has a strong frequency dependence. The heating rate is high close to the coronal base, but decreases very quickly. In the corona, the heating rate varies by an or- der of magnitude in the frequency range of 10 6 to 10 3 Hz, suggesting that the shape and range of the frequency spec- trum has important implications on the solar wind modeling.

Initial condition sensitivity of global quantities in magnetohydrodynamic turbulence

The effect of subtle changes in initial conditions on the evolution of global quantities in two-dimensional magnetohydrodynamic (MHD) turbulence is studied. It is found that a change in the initial phases of complex Fourier modes of the Elsässer variables, while keeping the initial values of total energy, cross helicity, and Alfvén ratio unchanged, has a significant effect on the evolution of cross helicity. On the contrary, the total energy and Alfvén ratio are insensitive to the initial phases. The simulations are based on direct numerical simulation using the pseudospectral method.

Beat, modulational, and decay instabilities of a circularly polarized Alfvén wave

A circularly polarized low‐frequency electromagnetic pump wave propagating along an ambient magnetic field is known to be unstable to the growth of several parallel‐propagating parametric instabilities. If ion‐cyclotron effects are retained in a two‐fluid description, the dispersion relation is a sixth‐order polynomial. We present a series of new analytical approximations to this dispersion relation. We emphasize new results for the beat instability that occurs as an interaction of the forward propagating upper sideband with the backward propagating lower sideband. The nature of the beat instability depends on β = (υsound/υA)² and on the sense of polarization of the pump wave. The beat and decay instabilities can occur together if the pump is left‐handed (i.e., ion resonant) and if β ≲ 1, but they cannot occur together if the pump is right‐handed. For a left‐handed pump the beat mode is the only instability if β ≳ 1. If the pump is right‐handed and β ≳ 1, then the beat instability exists only when the pump amplitude exceeds a threshold value, and the beat will be the only instability if the pump amplitude is large enough to stabilize the modulational instability. If the pump is left‐handed and β ≲ 1, then the beat mode is stabilized when the pump amplitude becomes sufficiently large. The beat instability primarily produces a forward propagating transverse wave in the upper sideband. Thus if β ≳ 1, the instabilities considered here do not produce the backward propagating waves which are thought to affect turbulence and the evolution of cross helicity in the solar wind. New analytical results are presented also for the decay and modulational instabilities when β ≈ 1.

Evidence for a “background” spectrum of solar wind turbulence in the inner heliosphere

This statistical analysis of power spectra of the fluctuating Elsässer variables, δZ± = δV±δVA, covers 3408 hours of interplanetary magnetic field and plasma observations made by experiments on the Helios 1 and 2 spacecraft during the years 1975, 1976, and 1980. The averaged spectra of δZ− in the range 2×10−8 km−1 < k* < 4×10−7 km−1 (k* = k/2π, k is the wave number) are found to show little variation with heliocentric distance, solar wind velocity, and solar activity. The resulting averaged power spectrum of δZ− obeys a power law with a slope of −5/3. We suggest that it could be thought of a persistent “background” spectrum. The averaged spectrum of δZ+, however, is flatter and of higher amplitude than the invariant spectrum of δZ−, and it appears to evolve radially toward that of δZ−. The concurrent decrease of the normalized cross‐helicity with increasing heliocentric distance (e.g., Roberts et al., 1987a, b) may therefore mainly result from a decrease of δZ+ and not from an increase of δZ−. This finding places an observational constraint on theoretical models for the evolution of cross‐helicity.