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Collisional whistler instability and electron temperature staircase in inhomogeneous plasma

N. A. Lopez, A. F. A. Bott, A. A. Schekochihin

TL;DR

This work addresses heat-transport regulation in high-$\beta$ collisional plasmas by deriving a phase-space (Wigner–Moyal) description of the collisional whistler instability from Braginskii-like electron MHD in a 1D slab. It shows that background gradients introduce gradient-damping terms in the growth rate, stabilizing nonuniform temperature profiles and driving globally marginally stable temperature staircases along magnetic field lines, especially at large $\beta_0$ and moderate $\mathcal{M}$. The analysis reveals back-reaction effects: frictional cooling can occur in early stages, and Ettingshausen-driven heat-flux suppression is possible under specific alignment of intensity and temperature gradients. Although strong heat-flux quenching in high-$\beta$ plasmas is unlikely within the fluid model used, the results demonstrate a robust mechanism by which whistler waves regulate transport even in the collisional limit, with implications for cold-fronts in clusters and inertial confinement fusion. The framework lays the groundwork for future work incorporating nonlinearities, kinetic effects, and more realistic geometries.

Abstract

High-beta magnetized plasmas often exhibit anomalously structured temperature profiles, as seen from galaxy cluster observations and recent experiments. It is well known that when such plasmas are collisionless, temperature gradients along the magnetic field can excite whistler waves that efficiently scatter electrons to limit their heat transport. Only recently has it been shown that parallel temperature gradients can excite whistler waves also in collisional plasmas. Here we develop a Wigner--Moyal theory for the collisional whistler instability starting from Braginskii-like fluid equations in a slab geometry. This formalism is necessary because, for a large region in parameter space, the fastest-growing whistler waves have wavelengths comparable to the background temperature gradients. We find additional damping terms in the expression for the instability growth rate involving inhomogeneous Nernst advection and resistivity. They (i) enable whistler waves to re-arrange the electron temperature profile via growth, propagation, and subsequent dissipation, and (ii) allow non-constant temperature profiles to exist stably. For high-beta plasmas, the marginally stable solutions take the form of a temperature staircase along the magnetic field lines. The electron heat flux can also be suppressed by the Ettingshausen effect when the whistler intensity profile is sufficiently peaked and oriented opposite the background temperature gradient. This mechanism allows cold fronts without magnetic draping, might reduce parallel heat losses in inertial fusion experiments, and generally demonstrates that whistler waves can regulate transport even in the collisional limit.

Collisional whistler instability and electron temperature staircase in inhomogeneous plasma

TL;DR

This work addresses heat-transport regulation in high- collisional plasmas by deriving a phase-space (Wigner–Moyal) description of the collisional whistler instability from Braginskii-like electron MHD in a 1D slab. It shows that background gradients introduce gradient-damping terms in the growth rate, stabilizing nonuniform temperature profiles and driving globally marginally stable temperature staircases along magnetic field lines, especially at large and moderate . The analysis reveals back-reaction effects: frictional cooling can occur in early stages, and Ettingshausen-driven heat-flux suppression is possible under specific alignment of intensity and temperature gradients. Although strong heat-flux quenching in high- plasmas is unlikely within the fluid model used, the results demonstrate a robust mechanism by which whistler waves regulate transport even in the collisional limit, with implications for cold-fronts in clusters and inertial confinement fusion. The framework lays the groundwork for future work incorporating nonlinearities, kinetic effects, and more realistic geometries.

Abstract

High-beta magnetized plasmas often exhibit anomalously structured temperature profiles, as seen from galaxy cluster observations and recent experiments. It is well known that when such plasmas are collisionless, temperature gradients along the magnetic field can excite whistler waves that efficiently scatter electrons to limit their heat transport. Only recently has it been shown that parallel temperature gradients can excite whistler waves also in collisional plasmas. Here we develop a Wigner--Moyal theory for the collisional whistler instability starting from Braginskii-like fluid equations in a slab geometry. This formalism is necessary because, for a large region in parameter space, the fastest-growing whistler waves have wavelengths comparable to the background temperature gradients. We find additional damping terms in the expression for the instability growth rate involving inhomogeneous Nernst advection and resistivity. They (i) enable whistler waves to re-arrange the electron temperature profile via growth, propagation, and subsequent dissipation, and (ii) allow non-constant temperature profiles to exist stably. For high-beta plasmas, the marginally stable solutions take the form of a temperature staircase along the magnetic field lines. The electron heat flux can also be suppressed by the Ettingshausen effect when the whistler intensity profile is sufficiently peaked and oriented opposite the background temperature gradient. This mechanism allows cold fronts without magnetic draping, might reduce parallel heat losses in inertial fusion experiments, and generally demonstrates that whistler waves can regulate transport even in the collisional limit.
Paper Structure (37 sections, 158 equations, 15 figures)

This paper contains 37 sections, 158 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Summary
  3. Governing fluid equations
  4. Extended electron MHD equations in slab geometry
  5. Eigenbasis projection
  6. Chapman--Enskog friction coefficients
  7. Resulting equations for Chapman--Enskog friction forces
  8. Dispersion relation, growth rate, and breakdown of the short-wavelength approximation
  9. Derivation via Wigner--Moyal phase-space formulation
  10. Insufficiency of short-wavelength approximation
  11. Linear stability condition
  12. Dynamical relevance of collisional whistler instability
  13. Global marginally stable temperature profiles
  14. General theory
  15. Magnetization staircases as a general class of solutions
  16. ...and 22 more sections

Figures (15)

  • Figure 1: Plots of the normalized group-velocity dispersion $\widetilde{G_d} = \beta_0G_d/v_t r_L$ [see (\ref{['eq:group']})] for a linear temperature profile (\ref{['eq:linT']}) (left) and a Gaussian temperature profile (\ref{['eq:gaussT']}) (right). All normalization quantities are defined with respect to $T_0$, and $\mathcal{M}_0 = \mathcal{M}(T_0)$.
  • Figure 2: Same as Fig. \ref{['fig:group']} but for the normalized resistivity $\widetilde{\eta} = \beta_0\eta/v_t r_L$ [see (\ref{['eq:resist']})].
  • Figure 3: Same as Fig. \ref{['fig:group']} but for the normalized cross-gradient Nernst velocity $\widetilde{u_\gamma} = Lu_\gamma/r_L v_t$ [see (\ref{['eq:growthV']})].
  • Figure 4: Same as Fig. \ref{['fig:group']} but for the normalized Nernst velocity $\widetilde{v_N} = Lv_N/r_L v_t$ [see (\ref{['eq:nerstV']})].
  • Figure 5: Region of parameter space where a geometrical-optics description of the collisional whistler instability is valid (green, $k_z L_T \gg 1$), questionable (yellow lined region, $k_z L_T \gtrsim 1$), and not valid (red crossed region, $k_z L_T < 1$). The regions are determined by the expression for $k_z L_T$ given by (\ref{['eq:WKBvalid']}) using the transport coefficients of Lopez24a.
  • ...and 10 more figures

Theorems & Definitions (1)

  • Conjecture