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Nonlinear Frequency Shifts due to Phase Coherent Interactions in Incompressible Hall MHD Turbulence

Erik C. Hansen, Prerana Sharma, Swadesh M. Mahajan

Abstract

Turbulence in the magnetized plasma is well understood to be the consequence of wave interactions. When the Hall effect is added to the minimum magnetohydrodynamics (MHD), the MHD waves become dispersive and different nonlinear interactions are expected. The emergent turbulent state will thus be expected to be different. For incompressible Hall MHD we develop a reduced model for wave-wave interactions concentrating on those processes that will lead to phase coherent modifications to the linear dispersion of a given wave. We show that these special interactions provide an amplitude-dependent contribution to the linear dispersion relation, which yields nonlinear frequency shifts. The resonance-driven frequency shifts are dominant and add damping or growth to the linear dispersion. The damping/growth rates represent the nonlinear time scales for energy redistribution and can be used in conjunction with a conjecture like the "critical balance" to estimate the energy spectral content.

Nonlinear Frequency Shifts due to Phase Coherent Interactions in Incompressible Hall MHD Turbulence

Abstract

Turbulence in the magnetized plasma is well understood to be the consequence of wave interactions. When the Hall effect is added to the minimum magnetohydrodynamics (MHD), the MHD waves become dispersive and different nonlinear interactions are expected. The emergent turbulent state will thus be expected to be different. For incompressible Hall MHD we develop a reduced model for wave-wave interactions concentrating on those processes that will lead to phase coherent modifications to the linear dispersion of a given wave. We show that these special interactions provide an amplitude-dependent contribution to the linear dispersion relation, which yields nonlinear frequency shifts. The resonance-driven frequency shifts are dominant and add damping or growth to the linear dispersion. The damping/growth rates represent the nonlinear time scales for energy redistribution and can be used in conjunction with a conjecture like the "critical balance" to estimate the energy spectral content.
Paper Structure (13 sections, 96 equations, 4 figures)

This paper contains 13 sections, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Linear Theory of Incompressible Hall MHD
  3. Hall MHD Coupling Coefficients
  4. Implicit Nonlinear Dispersion Relation
  5. The Frequency Integral
  6. Example Nonlinear Frequency Calculations
  7. $\mathbf{n}\approx \mathbf{m}$
  8. Leading Order Broad Spectrum
  9. Summary
  10. Reduced Form for Interaction Kernel
  11. Integration of a Circular Spectrum
  12. Leading Order Expansion of $g_{nm}$ and $h_{nm}$
  13. General Integration of the Broad Frequency Spectrum

Figures (4)

  • Figure 1: Contours of coupling coefficients of incompressible Hall MHD as a function of $\mathbf{n} = y\, \mathbf{m} + x\, \mathbf{m}\times\hat{z}$.
  • Figure 2: Contours of $g_{nm}$ and $h_{nm}$ corresponding to Figures \ref{['fig:gmn2']}, \ref{['fig:hmn2']}, and \ref{['fig:hmn23']}, respectively.
  • Figure 3: Frequency shift due to wavenumbers near $\mathbf{m}$. We observe what appears to be power law behavior for $\abs{\mathbf{m}}$ large and small, with the frequency shift scaling as $\abs{\mathbf{m}}_\perp^{4}$ for small $\abs{\mathbf{m}}$ and $\abs{\mathbf{m}}_\perp ^6$ for large $\abs{\mathbf{m}}$.
  • Figure 4: Integration domain in $\abs{\mathbf{n}}$ and $\abs{\mathbf{m}-\mathbf{n}}$ coordinates for small $\abs{\mathbf m}$. We show the regions where $\abs{\mathbf n}$ and $\abs{\mathbf m -\mathbf n}$ are considered large and small, corresponding to where $\Sigma \equiv \abs{\mathbf n} + \abs{\mathbf m -\mathbf n} = 2$.