Whistler-regulated Magnetohydrodynamics: Transport Equations for Electron Thermal Conduction in the High-β Intracluster Medium of Galaxy Clusters

We begin with an equation for the distribution function f(x, V_∥, V_⊥, t), where x is the space variable parallel to the ambient magnetic field B ₀ and V_∥ and V_⊥ are the velocities parallel and perpendicular to B ₀. Since the whistler waves in a high β_e system have k ρ_e ∼ 1 so that the whistler wave frequency is well below the electron cyclotron frequency (see Equation (1)), the electrons remain gyrotropic. The kinetic equation for electrons becomes

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}f+{v}_{\parallel }{{\rm{\nabla }}}_{\parallel }f-\displaystyle \frac{e}{{m}_{e}}{E}_{\parallel }\displaystyle \frac{\partial }{\partial {v}_{\parallel }}f=C\left(f\right),\end{eqnarray} \tag{ 17 }$$

where a parallel electric field E_∥ has been included to maintain zero net current. The zero current condition is valid even in a system with complex 3D magnetic fields as long as the kinetic scales are ordered out of the dynamics (Arnold et al. 2019; Drake et al. 2019). The collision operator includes scattering by both classical electron–ion collisions and whistler waves

$$\begin{eqnarray}&&C(f)={C}_{{ei}}(f)+{C}_{w}(f)\end{eqnarray} \tag{ 18 }$$

with

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{C}_{{ei}} & = & \displaystyle \frac{1}{2}{\nu }_{{ei}}(| {\boldsymbol{V}}-{\boldsymbol{U}}| )\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\\ & & \cdot \left(\Space{0ex}{2.25ex}{0ex}{\boldsymbol{I}}| {\boldsymbol{V}}-{\boldsymbol{U}}{| }^{2}-({\boldsymbol{V}}-{\boldsymbol{U}})({\boldsymbol{V}}-{\boldsymbol{U}})\Space{0ex}{2.25ex}{0ex}\right)\cdot \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\end{array}\end{array}\end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray}\begin{array}{c}\begin{array}{c}\begin{array}{rcl}{C}_{w} & = & \displaystyle \frac{1}{2}{\nu }_{w}\left(| {\boldsymbol{V}}-{\boldsymbol{U}}-{{\boldsymbol{V}}}_{w}| \right)\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\cdot \left(\Space{0ex}{2.25ex}{0ex}{\boldsymbol{I}}| {\boldsymbol{V}}-{\boldsymbol{U}}-{{\boldsymbol{V}}}_{w}{| }^{2}\right.\\ & & \left.-({\boldsymbol{V}}-{\boldsymbol{U}}-{{\boldsymbol{V}}}_{w})({\boldsymbol{V}}-{\boldsymbol{U}}-{{\boldsymbol{V}}}_{w})\Space{0ex}{2.25ex}{0ex}\right)\cdot \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 20 }$$

describing the respective types of scattering. To simplify the derivation, we include here only whistlers propagating in the positive direction, which is valid for a locally negative temperature gradient. The generalization to counter-propagating whistlers is straightforward and the results are presented in the Appendix. The collision operators in Equations (19) and (20) describe scattering only in pitch angle—in the frame moving with the net parallel plasma drift U in the case of C_ei and with respect to the whistler wave frame U + V_w in the case of whistler waves. Both scattering rates have energy dependences with ν_ei ∝ V⁻³ and ν_w ∝ V^γ . It will follow from the solutions to Equation (17) that the energy dependence of ν_w is crucial for maintaining a net whistler-driven advective heat flux while at the same time maintaining zero net current. An important assumption in the model for whistler scattering is that the scattering preserves energy in the whistler wave frame, which follows from the fact that, for long-wavelength oscillations in that frame, the electric field vanishes. An additional assumption, however, is that ν_w is independent of pitch angle. Such a model is supported by PIC simulations of heat-flux-driven whistlers in which the particles were isotropized in pitch angle. On the other hand, the amplitudes of whistlers in a real system are likely to saturate at much lower values than in the simulations because the temperature gradient scale lengths in real systems are far larger. As discussed in Section 2, the saturation amplitude of the whistler fluctuations ε_w scales as nT ρ_e /L ≪ nT. Quasilinear models of electron scattering based on an assumed spectrum of waves yield scattering rates that depend on the pitch angle through ζ = V_∥/V. However, we argue that the spectrum of heat-flux-driven waves will evolve so that the scattering rate is insensitive to ζ since any pileup of the distribution function with pitch angle would lead to local growth of whistlers which would enhance the local scattering. For this reason, the whistler scattering rate is taken to be independent of pitch angle.

Equation (17) is solved to second order in the parameter discussed at the end of Section 1. Fundamental to the ordering is the assumption that the collisional mean free path is short compared with the parallel system scale length L and that both the mean drift speed U and the whistler phase speed V_w are small compared with the thermal speed so that the collision operators as well as f can be expanded in powers of ∼ V_w /V_te ∼ U/V_te . Thus, we begin by writing Equation (17) to lowest order,

$$\begin{eqnarray}&&0={C}_{0}({f}_{0}),\end{eqnarray} \tag{ 21 }$$

with

$$\begin{eqnarray}&&{C}_{0}=\displaystyle \frac{1}{2}({\nu }_{{ei}}+{\nu }_{w})\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\cdot \left({\boldsymbol{I}}{V}^{2}-{\boldsymbol{V}}{\boldsymbol{V}}\right)\cdot \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}},\end{eqnarray} \tag{ 22 }$$

whose solution is a Maxwellian with zero drift,

$$\begin{eqnarray}&&{f}_{0}={n}_{0}{\left(\displaystyle \frac{{m}_{e}}{2\pi {T}_{0}}\right)}^{3/2}{e}^{-{m}_{e}{V}^{2}/2{T}_{0}}.\end{eqnarray} \tag{ 23 }$$

Technically, C₀ only scatters in pitch angle so f₀ could be written as any function of V². However, we take f₀ to be a Maxwellian. To first order, the derivatives on the left side of Equation (17) need to be included,

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}{f}_{0}+{V}_{\parallel }{{\rm{\nabla }}}_{\parallel }{f}_{0}-\displaystyle \frac{e}{{m}_{e}}{E}_{\parallel }\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{V}_{\parallel }}{f}_{0}\\ \,\,\,=\,{C}_{0}({f}_{1})+{C}_{1}({f}_{0}),\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 24 }$$

where

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{C}_{1} & = & \displaystyle \frac{1}{2}{\nu }_{w}(V)\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\cdot \left(\Space{0ex}{2.25ex}{0ex}-2{\boldsymbol{I}}{\boldsymbol{V}}\cdot ({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\right.\\ & & \left.+{\boldsymbol{V}}({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})+({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w}){\boldsymbol{V}}\Space{0ex}{2.50ex}{0ex}\right)\cdot \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\\ & & -\hspace{0.53ex}\displaystyle \frac{{\rm{\partial }}{\nu }_{w}}{{\rm{\partial }}{V}^{2}}{\boldsymbol{V}}\cdot ({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\cdot ({\boldsymbol{I}}{V}^{2}-{\boldsymbol{V}}{\boldsymbol{V}})\cdot \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\\ & & +\hspace{0.50ex}({\nu }_{w}\to {\nu }_{{ei}},{{\boldsymbol{V}}}_{w}\to 0).\end{array}\end{array}\end{eqnarray} \tag{ 25 }$$

Both C₀ and C₁ preserve the number density and the kinetic energy when averaged over velocity and the mean drift of f₀ is zero so

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{0}}{\partial t}=0,\,\,\,\displaystyle \frac{3}{2}\displaystyle \frac{\partial {P}_{0}}{\partial t}=0\end{eqnarray} \tag{ 26 }$$

with P₀ = n₀ T₀. Thus, the time derivative of f₀ in Equation (24) can be neglected. The various derivatives acting on f₀ (from ∇_∥, ∂/∂ V and C₀) can be readily evaluated. The inversion of C₀(f₁) is simplified by noting that f₁ = f₁(x, V, ζ, t) so that

$$\begin{eqnarray}&&{C}_{0}({f}_{1})=\displaystyle \frac{\nu }{2}\displaystyle \frac{\partial }{\partial \zeta }(1-{\zeta }^{2})\displaystyle \frac{\partial }{\partial \zeta }{f}_{1},\end{eqnarray} \tag{ 27 }$$

which is then readily inverted to yield the solution for f₁,

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{f}_{1} & = & -\displaystyle \frac{1}{\nu }\displaystyle \frac{V\zeta {f}_{0}}{{T}_{0}}\left[\displaystyle \frac{1}{{n}_{0}}{{\rm{\nabla }}}_{\parallel }{P}_{0}+\left(\displaystyle \frac{{m}_{e}{V}^{2}}{2{T}_{0}}-\displaystyle \frac{5}{2}\right){{\rm{\nabla }}}_{\parallel }{T}_{0}\right.\\ & & \left.+{{eE}}_{\parallel }-{m}_{e}\left(\Space{0ex}{2.3ex}{0ex}{\nu }_{w}({V}_{w}+U)+{\nu }_{{ei}}U\Space{0ex}{2.3ex}{0ex}\right)\Space{0ex}{3.0ex}{0ex}\right].\end{array}\end{array}\end{eqnarray} \tag{ 28 }$$

The first-order electron drift velocity U₁ can then be evaluated by multiplying f₁ by V_∥ = V ζ and integrating over velocity. The result is Ohm's law,

$$\begin{eqnarray}\begin{array}{rcl}0 & = & {m}_{e}{\nu }_{e}({U}_{1}-U)=-{\alpha }_{1}\left(\displaystyle \frac{1}{{n}_{0}}{{\rm{\nabla }}}_{\parallel }{P}_{0}+{{eE}}_{\parallel }\right)\\ & & -{\alpha }_{2}{{\rm{\nabla }}}_{\parallel }{T}_{0}+{\alpha }_{3}{m}_{e}{\nu }_{{we}}{V}_{w},\end{array}\end{eqnarray} \tag{ 29 }$$

where we have invoked the zero current condition to require U₁ = U. This equation then determines E_∥. The dimensionless parameters α₁, α₂, and α₃ are given in Table 1. The collision rates in Equation (29) are evaluated at the electron thermal speed, ${\nu }_{e}={\nu }_{{we}}+{\nu }_{{ei}}^{e}$ with ν_we = ν_w (V_te ) and ${\nu }_{{ei}}^{e}={\nu }_{{ei}}\left({V}_{{te}}\right)$. The term proportional to ∇_∥ T₀ in Equation (29) is the thermal force and is only nonzero when the velocity dependence of the scattering rate is included. The first-order electron heat flux Q₁ can also be evaluated by multiplying f₁ by m_e V² V_∥/2 and averaging over velocity. After eliminating E_∥ using Equation (29), the result for Q₁ is given by

$$\begin{eqnarray}&&{Q}_{1}=\left(\displaystyle \frac{5}{2}U+{\alpha }_{10}\displaystyle \frac{{\nu }_{{we}}}{{\nu }_{e}}{V}_{w}\right){n}_{0}{T}_{0}-{\kappa }_{e}{{\rm{\nabla }}}_{\parallel }T,\end{eqnarray} \tag{ 30 }$$

with κ_e given in Equation (6). When the whistler scattering rate is small, the heat flux reduces to the classical form, which includes advection as well as parallel thermal conduction.

To obtain the evolution equations for the electron density and temperature, it is necessary to write Equation (17) to second order. It takes the form

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}{f}_{1}+{V}_{\parallel }{{\rm{\nabla }}}_{\parallel }{f}_{1}-\displaystyle \frac{e}{{m}_{e}}{E}_{\parallel }\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{V}_{\parallel }}{f}_{1}\\ \,\,=\,{C}_{0}({f}_{2})+{C}_{1}({f}_{1})+{C}_{2}({f}_{0}),\end{array}\end{array}\end{eqnarray} \tag{ 31 }$$

where C₀ and C₁ are given in Equations (22) and (25). There are terms from the second-order collision operator C₂ that arise from the velocity dependence of the collision rates ν_ei and ν_w . However, these terms all cancel when computing the continuity and energy moments of Equation (31) so the minimal required expression for C₂ is given by

$$\begin{eqnarray}\begin{array}{c}\begin{array}{c}\begin{array}{rcl}{C}_{2} & = & \displaystyle \frac{1}{2}{\nu }_{w}(V)\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\cdot \left(\Space{0ex}{2.25ex}{0ex}{\boldsymbol{I}}{\left(U+{V}_{w}\right)}^{2}\right.\\ & & \left.+({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\Space{0ex}{2.25ex}{0ex}\right)\cdot \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{\boldsymbol{V}}}\\ & & +({\nu }_{w}\to {\nu }_{{ei}},{{\boldsymbol{V}}}_{w}\to 0).\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 32 }$$

The evolution equation for the plasma density is obtained by integrating Equation (31) over the velocity. All of the collision terms on the right side of the equation as well as the E_∥ term integrate to zero, leaving the continuity equation as written in Equation (3). Note that it is not necessary to evaluate f₂ to obtain the density evolution equation since C₀(f₂) integrates to zero. The equation for the evolution of the electron energy is obtained by multiplying Equation (31) by m_e V²/2 and integrating over the velocity. As in the continuity equation, f₂ drops out since in a stationary frame where C₀ is evaluated the scattering does not change the particle energy. The electron energy evolution equation takes the form presented in Equation (4).

As discussed in Section 3, the temperature gradient drive of the whistlers is only nonzero if the whistler scattering rate is energy dependent. The energy dependence of the whistler-driven scattering rate was not evaluated in PIC simulations of heat flux driven whistlers (Roberg-Clark et al. 2016, 2018a; Komarov et al. 2018) but the bounce frequency of electrons trapped in large-amplitude whistlers as seen in the simulations increases with the particle energy (Karimabadi et al. 1990). However, in real systems in which the temperature gradient scale length is large, the whistler wave amplitude will be small (see the discussion in Section 3) and with a broader spectrum of waves than documented in the PIC simulations. The quasilinear scattering rate for an assumed spectrum of low-frequency Alfvén waves has been calculated previously (Lee 1982; Schlickeiser 1989; Schlickeiser & Miller 1998). The velocity dependence of the scattering rate takes the form of a power law V^γ where γ = q − 1 and where the wave energy spectrum was assumed to have a power-law form k^−q . Since the whistler waves of interest in electron scattering are also sub-cyclotron, these earlier quasilinear results also apply to the electrons' response to whistlers. The spectrum of whistler waves in PIC simulations of heat-flux-driven whistlers fell off steeply (Roberg-Clark et al. 2018a) but the limited spectral range of the simulations likely impacted the results. The cascade of electron–MHD turbulence has also been explored (Biskamp et al. 1999) and yielded power-law spectra that scaled like k^−7/3 so that γ ∼ 4/3. Thus, we take the whistler scattering rate to take the quasilinear form with a power-law dependence on velocity:

$$\begin{eqnarray}&&{\nu }_{w}=0.1{{\rm{\Omega }}}_{e}\displaystyle \frac{{\varepsilon }_{w}}{{\varepsilon }_{B}}{\left(V/{V}_{{te}}\right)}^{\gamma },\end{eqnarray} \tag{ 33 }$$

with γ ∼ 4/3. The numerical factor of 0.1 in Equation (33) is based on the results of PIC simulations (Roberg-Clark et al. 2018a) rather than a detailed quasilinear calculation. However, as shown in Section 2, while this factor impacts the rate of growth of whistlers, it does not control their saturation and the associated saturated value of ν_we . Since the timescale for whistler growth is short compared with dynamical timescales, which vary as L/V_w , the details of the whistler growth rate are not important. More important is the onset criterion for whistler growth which, as shown in Section 2, is independent of the factors appearing in Equation (33).

Finally, we present a derivation of Equation (9), which describes the whistler energy transport. The extraction and heating terms follow from the transfer of energy between the electrons and whistlers through the scattering process. The calculation that leads to the advection terms on the left side of the equation, which are important in describing the propagation of the whistlers and evaluating their interaction with the bulk fluid, is presented here. However, to simplify the equations we limit the calculation to a simple system with straight magnetic field lines. The more general case is presented in the Appendix. We start with Faraday's law for the perturbed whistler magnetic field,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}\delta {{\boldsymbol{B}}}_{w}+c{\rm{\nabla }}\times \delta {{\boldsymbol{E}}}_{w}=0,\end{eqnarray} \tag{ 34 }$$

where δ B _w and δ E _w are the transverse magnetic and electric field of the whistler wave. Although the scattering of electrons by the whistlers requires that the waves be oblique with respect to the ambient magnetic field, here we explore only the transport along the ambient magnetic field and therefore consider a simple 1D model of the wave dynamics. Since the full 3D MHD equations will be displayed in the Appendix, the convection velocity U of the MHD fluid is taken as a general vector. Taking the dot product of Equation (34) with δB _w and completing some vector algebra, we obtain an equation for the wave energy ε_w = ∣δB _w ∣²/8π:

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}{\varepsilon }_{w}+\displaystyle \frac{c}{4\pi }{\boldsymbol{\nabla }}\cdot (\delta {{\boldsymbol{E}}}_{w}\times \delta {{\boldsymbol{B}}}_{w})+\delta {{\boldsymbol{J}}}_{w}\cdot \delta {{\boldsymbol{E}}}_{w}\\ \,=\,-{V}_{w}\left({\alpha }_{10}\displaystyle \frac{{\nu }_{we}}{{\nu }_{e}}n{{\rm{\nabla }}}_{\parallel }T+{F}_{w}\right),\end{array}\end{array}\end{eqnarray} \tag{ 35 }$$

where we have used Ampère's law without the displacement current and have included the scattering terms on the right side. To evaluate δJ _w · δE _w , we first note that δJ _w = − neδ v _w . An equation for the whistler electric field E _w is obtained from the linearized electron equation of motion with no inertia:

$$\begin{eqnarray}&&0={\boldsymbol{\delta }}{E}_{w}+\displaystyle \frac{1}{c}\delta {{\boldsymbol{V}}}_{w}\times {\boldsymbol{B}}+\displaystyle \frac{1}{c}{\boldsymbol{U}}\times \delta {{\boldsymbol{B}}}_{w},\end{eqnarray} \tag{ 36 }$$

where the electron streaming velocity U is equal to that of the ions because of the current in an MHD system is zero unless the Hall terms in Ohm's law are retained (Arnold et al. 2019; Drake et al. 2019). In the Poynting flux the whistler electric field δE _w takes the form

$$\begin{eqnarray}&&\delta {{\boldsymbol{E}}}_{w}=-\displaystyle \frac{1}{c}({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\times \delta {{\boldsymbol{B}}}_{w}.\end{eqnarray} \tag{ 37 }$$

This is equivalent to transforming from the whistler wave frame, where the electric field is zero, into the laboratory frame. The Poynting flux takes the form

$$\begin{eqnarray}&&{{\boldsymbol{S}}}_{w}=\displaystyle \frac{c}{4\pi }\delta {\boldsymbol{E}}\times \delta {\boldsymbol{B}}=\left(2({U}_{\parallel }+{V}_{w}){\boldsymbol{b}}+{{\boldsymbol{U}}}_{\perp }\right){\varepsilon }_{w}.\end{eqnarray} \tag{ 38 }$$

Using Equation (36) to calculate δE _W , the heating term can be written as δJ · δE _w = U · F _pw where F _pw is the whistler ponderomotive force acting on the electrons, which is given by

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{{pw}}=-\displaystyle \frac{{ne}}{c}\delta {{\boldsymbol{V}}}_{w}\times \delta {{\boldsymbol{B}}}_{w}.\end{eqnarray} \tag{ 39 }$$

The right side is easily evaluated using Ampère's law for δ V _w and carrying out some vector algebra. The resulting expression for the ponderomotive force is

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{{pw}}=-{\boldsymbol{b}}{\boldsymbol{b}}\cdot {\boldsymbol{\nabla }}{\varepsilon }_{w}.\end{eqnarray} \tag{ 40 }$$

Inserting F _pw and S _w into Equation (35) yields the transport equation for ε_w given in Equation (9). The interaction between the whistler radiation pressure and the ions through the last term on the left side of the whistler transport equation requires a corresponding term in the ion momentum equation. This arises when the individual electron and ion momentum equations are added to produce the one-fluid momentum equation. For completeness the full set of MHD equations along with the whistler-constrained electron energy transport equations are presented in the Appendix. The results in the Appendix generalize the ponderomotive force to the case when magnetic fields have curvature.

Further constraints on electron thermal transport from whistler scattering will make it even less likely that heat conduction from the outer regions of the ICM to the cool core can limit the radiative collapse at the largest scales. On the other hand, an important question is whether the local dynamics of radiative instabilities that are likely to develop within the cool core will be altered compared with the classical model. In the case of an isobaric perturbation about an equilibrium in which radiative cooling is balanced by some local heat source, the suppression of thermal transport relative to the classical Spitzer value will decrease the Field length (Field 1965), thereby extending the domain of thermal instability down to smaller spatial scales. Further, the significant consequences of whistler-limited transport can be more clearly identified by considering a state in which radiative cooling is balanced by a nonzero divergence of the heat flux. In the simplest 1D model of an isobaric condensation, the heat flux in the classical model scales as T^7/2/L with L the ambient temperature scale length. To maintain a constant heat flux into the condensing region where the temperature is dropping requires the local temperature gradient to increase or L to decrease. The reduction in L pushes the system toward the whistler unstable domain. Specifically, the heat flux Q scales as

$$\begin{eqnarray}&&Q\sim \displaystyle \frac{{L}_{c}}{L}{{nT}}_{e}{V}_{{te}}\end{eqnarray} \tag{ 47 }$$

so the whistler instability criterion can be written as

$$\begin{eqnarray}&&\displaystyle \frac{L}{{\beta }_{e}{L}_{c}}\sim \displaystyle \frac{{{nT}}_{e}{V}_{{te}}}{{\beta }_{e}Q}\sim \displaystyle \frac{{B}^{2}}{8\pi Q}{V}_{{te}}\lt 1.\end{eqnarray} \tag{ 48 }$$

Thus, in a 1D system where B is a constant, the threshold for whistler onset will be crossed when the temperature is low enough. At that point the scaling of the heat flux will change from T^7/2/L to ${V}_{w}{{nT}}_{e}\sim {T}_{e}^{1/2}$. Thus, the dynamics of condensation is likely be substantially changed in light of the new transport model.

The temperature gradients across cold fronts can be quite large and, depending on the strength of the locally draped magnetic field, are likely to drive whistlers that in turn can control transport and the structure of these fronts. The equations developed here provide a fully self-consistent framework for exploring the dynamics and structure of cold fronts with no constraints on the possible breakdown of the classical model of thermal transport.

A pure hydrodynamic and gravitationally stratified plasma is unstable to perturbations that cause convective motions if the entropy is a decreasing function of radius, i.e., if $\partial \mathrm{ln}K/\partial \mathrm{ln}r\lt 0$ (Schwarzschild 1906), where K = P ρ^−Γ is the entropic function labeling an adiabatic curve, ρ is the mass density, and Γ = 5/3 is the ratio of specific heats. The entropy profiles of all observed galaxy clusters are increasing functions of radius, which should render them convectively stable (Cavagnolo et al. 2008). If a fluid element in a stably stratified atmosphere is adiabatically displaced in radius from its equilibrium position by δ r, it experiences a buoyant restoring force per unit volume (Ruszkowski & Oh 2010)

$$\begin{eqnarray}&&{F}_{\mathrm{adiab}}=\rho \ddot{r}=-\displaystyle \frac{\rho g}{{\rm{\Gamma }}}\displaystyle \frac{\partial \mathrm{ln}K}{\partial r}\delta r,\end{eqnarray} \tag{ 49 }$$

where g is the gravitational acceleration, that causes oscillations around its equilibrium position at the classical Brunt–Väisälä frequency N where

$$\begin{eqnarray}&&{N}^{2}=\displaystyle \frac{g}{{\rm{\Gamma }}r}\displaystyle \frac{\partial \mathrm{ln}K}{\partial \mathrm{ln}r}.\end{eqnarray} \tag{ 50 }$$

More fundamentally, the Brunt–Väisälä frequency is the limiting frequency of linear internal gravity waves (which have frequency ${\omega }_{\mathrm{gw}}^{2}={k}_{h}^{2}{N}^{2}/| {k}^{2}|$ where k_h is the component of the wavenumber orthogonal to the gravity gradient), so real N translates into linear stability of the atmosphere. However, if an external turbulent driving force is larger than the buoyant restoring force then it can overcome the stable equilibrium and induce mixing.

This paradigm experiences a fundamental change in the weakly collisional, magnetized plasma of a galaxy cluster because it changes the response of the plasma to perturbations. Provided there is a radial temperature gradient, ∂T_e /∂r ≠ 0, anisotropic thermal conduction along the mean magnetic field causes the ICM to be almost always buoyantly unstable regardless of the sign of the temperature and entropy gradients. If ∂T_e /∂r > 0 (which applies to the central cooling regions in cool core clusters) the ICM is unstable to the HBI (Quataert 2008) provided that there are regions where the magnetic field is mostly radially aligned. If ∂T_e /∂r < 0 (which applies to all clusters on large scales and is a consequence of hydrostatic rearrangement in the presence of the universal dark matter potential) the ICM is unstable to the MTI (Balbus 2000) provided that there are regions where the magnetic field is mostly horizontally aligned. Using a linear stability analysis (Kunz 2011) and nonlinear simulations (Kunz et al. 2012) it was shown that anisotropic viscosity (i.e., Braginskii pressure anisotropy) affects the MTI and HBI growth rates by stifling the convergence/divergence of magnetic field lines, but cannot suppress it. When field geometries are chosen to stabilize the HBI/MTI, the system is still subject to related overstabilities (Balbus & Reynolds 2010). In consequence, any perturbation will be convectively unstable and cause instant mixing of the thermal plasma because it lacks a restoring force that provides an energy penalty (Sharma et al. 2009) which facilitates advection and turbulent mixing of AGN-injected energy (Kannan et al. 2017). In the saturated state of these anisotropic transport instabilities, the buoyant restoring force is altered to $| {F}_{\mathrm{cond}}| \sim \rho g(\partial \mathrm{ln}{T}_{e}/\partial r)\delta r$, and replaces the restoring force of Equation (49) that is based on the classic Schwarzschild criterion (Sharma et al. 2009).

Kinetic plasma physics and finite Larmor-radius effects can however significantly impact those HBIs/MTIs, which have been found using a magnetofluid (i.e., Braginskii) description. While at long wavelengths (the "drift-kinetic" limit), a kinetic analysis reinforces the MTI and its Alfvénic counterpart, at sub-ion-Larmor scales, there is an overstability driven by the electron-temperature gradient of kinetic-Alfvén drift waves whose growth rate is even larger than the standard MTI (Xu & Kunz 2016). While the effective heat conductivity can locally be suppressed by the ion-scale mirror instability (Komarov et al. 2016; Riquelme et al. 2016), recent work found that the nonlinear saturation of the MTI is not significantly modified (Berlok et al. 2021). Given that the predicted reduction in parallel thermal transport can be significant in our picture of whistler-mediated thermal conduction, an important question is whether this reduction might affect these large-scale fluid MTIs and HBIs. Formally, the current treatments of MTIs and HBIs explicitly assume diffusive parallel heat transport and so would need to be reworked with our modified scheme that highlights the importance of advective transport. However, as a first approach, it is possible to argue physically. The MTIs/HBIs rely on three properties: (i) the presence of a radial temperature gradient, ∂T_e /∂r ≠ 0, (ii) the large ratio of electron thermal transport along to that across the ambient magnetic field and (iii) a short conduction time in comparison to the buoyancy and advection timescales on the length scales considered. ¹⁰

The perpendicular thermal transport arising from the whistler instability has not been explored in detail. The invariance of the canonical momentum in the symmetry direction of a 2D PIC model constrains particle motion perpendicular to B so that the exploration of perpendicular transport requires the PIC modeling to be carried out in a more computationally challenging 3D system. On the other hand, based on the basic characteristics of the heat-flux-driven whistler, we can estimate the perpendicular transport. The characteristic transverse scale of the whistler turbulence is the electron Larmor radius ρ_e so that a reasonable estimate of the cross-field diffusion is

$$\begin{eqnarray}&&{D}_{\perp }\sim {\rho }_{e}^{2}{\nu }_{{we}}.\end{eqnarray} \tag{ 51 }$$

The parallel transport is given by

$$\begin{eqnarray}&&{D}_{\parallel }\sim \displaystyle \frac{{V}_{{te}}^{2}}{{\nu }_{{we}}}.\end{eqnarray} \tag{ 52 }$$

Thus, the ratio R_trans of the parallel to perpendicular transport is given by

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{trans}} & = & \displaystyle \frac{{D}_{\parallel }}{{D}_{\perp }}\\ & & \sim {\left(\displaystyle \frac{L}{\beta {\rho }_{e}}\right)}^{2}\sim 3\times {10}^{26}{\left(\displaystyle \frac{L}{30\,\mathrm{kpc}}\right)}^{2}{\left(\displaystyle \frac{{T}_{e}}{{10}^{7}\,{\rm{K}}}\right)}^{-1}{\left(\displaystyle \frac{B}{1\,\mu {\rm{G}}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 53 }$$

where we have used the saturated whistler scattering rate ν_we = β_e V_te /L given in Equation (14). Thus, the anisotropy ratio remains extreme, in spite of the factor β_e in the denominator of Equation (53).

While whistler wave scattering does not qualitatively change the anisotropic character of heat transport and hence the physical basis for the MTIs and HBIs, the scale ℓ that can go unstable is modified. Assuming a hydrostatic atmosphere, these instabilities only grow at interesting rates provided the characteristic timescale at which conduction acts on a given perturbation is much shorter than the buoyancy timescale, which we identify with the inverse Brunt–Väisälä frequency, N, and obtain the condition

$$\begin{eqnarray}&&{\tau }_{{\rm{cond}}}=\displaystyle \frac{\ell }{{V}_{w}}=\displaystyle \frac{\ell {\beta }_{e}}{{V}_{{te}}}\ll {N}^{-1}={\left(\displaystyle \frac{g}{{\rm{\Gamma }}r}\displaystyle \frac{{\rm{\partial }}{\rm{ln}}K}{{\rm{\partial }}{\rm{ln}}r}\right)}^{-1/2}.\end{eqnarray} \tag{ 54 }$$

This timescale ordering is the basis for the excitation of the MTIs and HBIs and ensures quasi-isothermality along a given field line (Quataert 2008). Because tangled magnetic field lines have a significant portion of azimuthal components not aligned with the vertical buoyancy direction, this adds furthermore to the large separation of timescales. We assume a Navarro–Frenk–White dark matter density profile (Navarro et al. 1997) which is characterized by a scale radius, r_s , and a mass density at the scale radius, ρ_s , so that the gravitational acceleration in the inner regions at radii r < r_s is given by

$$\begin{eqnarray}&&{g}_{0}=\displaystyle \frac{{GM}(\lt r)}{{r}^{2}}=2\pi G{\rho }_{s}{r}_{s}=\mathrm{const}.,\end{eqnarray} \tag{ 55 }$$

where G is Newton's constant. We furthermore assume an exponentially stratified atmosphere where the pressure scale height is given by $h={V}_{{tp}}^{2}/{g}_{0}$ (where V_tp is the isothermal sound speed) so that the critical length scale for instability is

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}\ell & \ll & \displaystyle \frac{{V}_{{te}}}{{\beta }_{e}}{N}^{-1}=\sqrt{\displaystyle \frac{{\rm{\Gamma }}}{{\rm{\Gamma }}-1}}\displaystyle \frac{{V}_{{te}}\,{V}_{{tp}}}{{\beta }_{e}{g}_{0}}\\ & & \sim 20\left(\displaystyle \frac{\sqrt{{T}_{e}{T}_{p}}}{{10}^{7}\,{\rm{K}}}\right){\left(\displaystyle \frac{{\beta }_{e}}{100}\right)}^{-1}{\left(\displaystyle \frac{{g}_{0}}{{10}^{-8}{\rm{cm}}{{\rm{s}}}^{-2}}\right)}^{-1}\,{\rm{kpc}}.\end{array}\end{array}\end{eqnarray} \tag{ 56 }$$

On larger length scales, conduction is slower than buoyancy and the system transitions to obey the classical Schwarzschild criterion for convective instability, $\partial \mathrm{ln}K/\partial \mathrm{ln}r\lt 0$. However, on scales smaller than ℓ, there is no restoring force associated with the entropy gradient and passive scalars such as metals should get easily mixed with the surrounding ICM as explained above. This picture is predicated upon the formal applicability of the MTIs and HBIs but, for realistic magnetic field strengths in clusters with β_e ∼ 100, magnetic tension provides an additional restoring force and can suppress a significant fraction of HBI-unstable modes, thus either impairing or completely inhibiting the HBIs on scales smaller than ∼50–70 kpc, depending on the unknown magnetic field coherence length and the fraction of volume partitioned with these intermittent strong magnetic fields (Yang & Reynolds 2016). These considerations strongly constrain the applicability of the MTIs and HBIs and call for a careful assessment as to whether they are excited in linear theory at all in the presence of whistler-mediated thermal conduction.

If there are bulk flows in the ICM, the condition of Equation (56) is modified and we require that the conductive timescale be much shorter than the advective timescale for the MTIs and HBIs to be excited, which can be rewritten into a condition for the advection velocity,

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcll}U & \ll & {V}_{w}= & \displaystyle \frac{{V}_{{te}}}{{\beta }_{e}}\sim 100{\left(\displaystyle \frac{{T}_{e}}{{10}^{7}{\rm{K}}}\right)}^{1/2}{\left(\displaystyle \frac{{\beta }_{e}}{100}\right)}^{-1}\,{\rm{km}}\,{{\rm{s}}}^{-1}.\end{array}\end{array}\end{eqnarray} \tag{ 57 }$$

Both conditions of Equations (56) and (57) are constraining and may severely limit the applicability of the MTI and HBI in galaxy clusters.

Intermittent activity of bipolar AGN jets at the centers of cool core clusters is expected to generate sound waves in the ICM. Concentric ripples in the X-ray emissivity have been detected in the Perseus (Fabian et al. 2003) and Virgo clusters (Forman et al. 2005) and have been interpreted as sound waves generated by central supermassive black holes. These arcs have characteristic widths (or wavelengths) λ of order ∼1 to 10 kpc and are seen up to several tens of kiloparsecs away from cluster centers, suggesting that the waves propagate over substantial fractions of cool core radii before completely dissipating. The recent investigation of Bambic & Reynolds (2019) showed that up to 25%–30% of the energy injected as fast jets by the AGN can end up as sound waves, highlighting the relevance of this physics to the question of AGN feedback.

The above argument suggests that sound waves could be a promising agent responsible for converting the mechanical energy of the AGN to the thermal energy of the ICM. Appealing features of this mode of heating are that the waves provide a natural mechanism to: (i) quickly deliver AGN energy to the ICM (i.e., on sound crossing timescales that are typically shorter than radiative cooling timescales), and (ii) distribute the energy over a large fraction of the cool core volume (rather than dissipating the energy close to the AGN jet axis). These are desirable features of the model because they can help to explain why cool cores remain globally thermally stable over times comparable to the Hubble time. Furthermore, ICM heating via sound wave thermalization is a gentle process involving very subsonic velocity fluctuations. This is a particularly appealing feature of this mode of heating given recent Hitomi observations (Hitomi Collaboration et al. 2016; Fabian et al. 2017) that suggest that the ICM in the Perseus cluster is very calm.

Tapping of sound wave energy can occur via a number of mechanisms. Hydrodynamical simulations of AGN outbursts invoking Spitzer ion viscosity demonstrated that sound wave dissipation can offset radiative cooling losses and that the waves can propagate significant distances away from cluster centers (Ruszkowski et al. 2004a, 2004b). However, the central cool core regions tend to be somewhat overheated in this model. This problem is further exacerbated by the fact that classical Spitzer conduction is ${\left({m}_{p}/{m}_{e}\right)}^{1/2}$ more effective than Spitzer viscosity in dissipating waves. This suggests that transport processes need to be substantially suppressed in the ICM in order to eliminate tension with the observations. The necessary level of suppression was quantified using linear theory by Fabian et al. (2005), and this work was further extended by Zweibel et al. (2018), who included the self-limiting nature of dissipation by electron thermal conduction, electron–ion non-equilibration effects, and provided estimates of kinetic effects by comparing to a semi-collisionless theory.

One of the main limitations of the above investigations was that the level of transport was quantified by specifying ad hoc Spitzer suppression factors. Our model allows one to relax these assumptions and it can be applied to make physically motivated predictions for the evolution of the dissipating sound waves. In particular, our model self-consistently bridges the transition from the collisional to whistler-mediated transport regimes. Properly accounting for this transition may prove crucial for our understanding of the thermalization of the AGN-induced sound waves—while the sound wave dissipation at the very centers of cool cores may occur via collisional processes, collisionless processes are likely to dominate over a wide range of distances away from cool core centers where L_c β > λ. For example, the Coulomb mean free path is L_c ∼ 2 × 10⁻³ kpc and ∼ 4 × 10⁻² kpc at the centers of the Virgo and Perseus clusters, respectively; and at the distance of ∼50 kpc from the center, the corresponding values are L_c ∼ 0.4 and ∼0.2 kpc (see, e.g., Zhuravleva et al. 2014 for density and temperature profiles in these clusters). Given typical plasma β ∼ 10² in the ICM and sound wave λ ∼ 1 to 10 kpc (e.g., Fabian et al. 2006), both the collisional and whistler regimes are expected to play a role in sound wave dissipation and propagation. Furthermore, the suppression of conduction expected in the whistler-dominated scattering regime may offer a natural explanation for the observed large propagation lengths of sound waves in the ICM. Interestingly, Kunz et al. (2020) demonstrated that suppression of collisionless Landau damping of ion acoustic waves is expected when the relative wave amplitudes exceed 2/β. Such waves could then be self-sustaining and propagate over large distances in a manner resembling sound wave propagation in a weakly collisional ICM.

An investigation of the consequences of whistler-mediated transport for the evolution of the AGN-induced waves represents an interesting future research direction, and we intend to report on the results of this investigation in future publications.

The solar wind is diffuse plasma flowing at high Mach number outward in the heliosphere from the Sun. In situ satellite observations have produced enormous amounts of data on its properties and the Parker Solar Probe mission will facilitate measurements as close as 10R_⊙ with R_⊙ the solar radius. The electron temperature falls slowly with radial distance, from around 30 eV at 35R_⊙ to 10 eV at 120 R_⊙ (Moncuquet et al. 2020). The collisionality of the solar wind depends on the plasma density and varies over a wide range, including a transition from collisional to collisionless behavior as the collisional mean free path L_c varies from smaller than to larger than the scale length of the electron temperature gradient L. Measurements from the large Wind spacecraft data set revealed that the electron heat flux rolls over to a value below the collisionless heat flux Q₀ as L_c /L increases and the ambient plasma becomes more collisionless (Bale et al. 2013). While β_e of the solar wind is nominally of order unity around 1 au, this value is actually highly variable. The Wind data set revealed more than 12k measurements of β_e in the range from 5 to 100. An important conclusion from this data set in light of the threshold for whistler growth in Equation (48) is that the value of L_c /L above which the electron heat flux rolled over to a constant value decreased with higher β_e . An important development was the confirmation from the Wind (Tong et al. 2018) and ARTEMIS (Tong et al. 2019) data sets that the ratio of Q/Q₀ scales like ${\beta }_{e}^{-1}$ at high β_e , consistent with whistler-limited heat flux. Further, the measured amplitude of whistler waves increased both with heat flux and with β_e (Tong et al. 2019), consistent with the heat flux as the whistler drive mechanism.

While the general features of the whistler wave activity and the associated heat flux measurements support the idea that heat-flux-driven whistlers play a role in limiting electron heat flux in the solar wind, significant uncertainties remain. The ARTEMIS observations have been interpreted as "quasi-parallel" whistler waves (Tong et al. 2019). The PIC simulations as well as analytic analysis, however, have established that parallel whistlers are not capable of significantly limiting electron heat flux since the electrons carrying the dominant heat flux do not resonate with parallel whistlers (Roberg-Clark et al. 2016; Komarov et al. 2018). On the other hand, the magnetic field measurements are limited to the spin plane of the spacecraft so no direct measurements of the direction of the wavevectors of the whistlers was possible. The amplitude of measured whistlers in the solar wind was small, around 2% of the ambient magnetic field, leading to concern that the whistler waves were too small in amplitude to limit electron thermal transport. On the other hand, the saturated level of fluctuations given below Equation (14), ε_w ∼10(ρ_e /L)nT, is very small for the realistic values of ρ_e /L of the solar wind. For T_e ∼ 10 eV and B₀ ∼ 10⁻⁴ G, ρ_e ∼ 1 km. Taking a temperature scale length of around 100R_⊙ ∼ 10⁵ km and β_e ∼ 10, the predicted whistler fluctuation level δ B/B₀ ∼ 1%, which is in the range of the observations. The scaling $Q/{Q}_{0}\sim {\beta }_{e}^{-1}$ at high β_e is now firmly established (Tong et al. 2018, 2019). There has been no mechanism other than heat-flux-limited whistlers proposed to explain this scaling.

The more recent measurements from the Parker Solar Probe mission have further established the role of whistlers in limiting the heat flux in the solar wind and specifically scattering the field-aligned electron strahl, which has energies in the range of 100 eV–1 keV, into the more isotropic halo distribution (Agapitov et al. 2020; Cattell et al. 2021a, 2021b). The presence of large-amplitude whistlers was correlated with local regions of increased plasma β (Agapitov et al. 2020; Cattell et al. 2021b). The presence of whistlers also correlated well with thresholds of fan instability, which is an oblique whistler driven by the n = −1 resonance (Vasko et al. 2019; Cattell et al. 2021b). Further, the pitch angle width of the measured strahl electrons was also linked to the presence or absence of large-amplitude whistlers, with the pitch angle width increasing with the strength of whistler wave activity, establishing that strahl scattering is caused by resonant interactions with whistlers.

For the sake of transparency of this derivation, we suppress the source terms in the whistler wave equation, namely the whistler wave growth and loss terms (due to drag and second-order Fermi acceleration). Moreover, in this derivation we only consider whistler waves that move in one direction and generalize the result at the end of this section to also account for backward-moving whistlers.

The energy equation for electromagnetic fields is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\partial }}({{\boldsymbol{E}}}^{2}/2+{{\boldsymbol{B}}}^{2}/2)}{{\rm{\partial }}t}+c{\boldsymbol{\nabla }}\cdot ({\boldsymbol{E}}\times {\boldsymbol{B}})+{\boldsymbol{J}}\cdot {\boldsymbol{E}}=0,\end{eqnarray} \tag{ A1 }$$

where the electric field in Hall-MHD (which is the appropriate limit) is given by

$$\begin{eqnarray}&&{\boldsymbol{E}}+\displaystyle \frac{[{\boldsymbol{U}}-{\boldsymbol{J}}/({ne})]\times {\boldsymbol{B}}}{c}={\bf{0}}.\end{eqnarray} \tag{ A2 }$$

The Hall term, which is proportional to J /(ne), is assumed to be zero in standard MHD. This term does not do any work on the electromagnetic field as can be shown:

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{J}}\cdot {\boldsymbol{E}} & = & -{\boldsymbol{J}}\cdot \displaystyle \frac{[{\boldsymbol{U}}-{\boldsymbol{J}}/({ne})]\times {\boldsymbol{B}}}{c}\\ & = & -{\boldsymbol{J}}\cdot \displaystyle \frac{{\boldsymbol{U}}\times {\boldsymbol{B}}}{c}={\boldsymbol{U}}\cdot \displaystyle \frac{{\boldsymbol{J}}\times {\boldsymbol{B}}}{c}\end{array}\end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray}&&=\,{\boldsymbol{U}}\cdot \left[({\boldsymbol{\nabla }}\times {\boldsymbol{B}})\times {\boldsymbol{B}}\right]=-{\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}\right)\right],\end{eqnarray} \tag{ A4 }$$

where we use vector identities in the last step and assume a divergence-free magnetic field.

We are interested in the energy transport of small-amplitude whistler waves. To this end, we perturb the energy equation in Equation (A1) assuming small deviations of the electric and magnetic fields from its mean values. Carrying out a perturbation analysis and neglecting the energy density of the electric field in the energy equation yields

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\partial }}\delta {{\boldsymbol{B}}}^{2}/2}{{\rm{\partial }}t}+c{\boldsymbol{\nabla }}\cdot (\delta {\boldsymbol{E}}\times \delta {\boldsymbol{B}})+\delta {\boldsymbol{J}}\cdot \delta {\boldsymbol{E}}=0,\end{eqnarray} \tag{ A5 }$$

and the perturbed electric field for whistler waves is given by

$$\begin{eqnarray}&&\delta {\boldsymbol{E}}+\displaystyle \frac{({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\times \delta {\boldsymbol{B}}}{c}={\bf{0}},\end{eqnarray} \tag{ A6 }$$

where V _w is the whistler wave velocity. Inserting the perturbed electric field into the divergence terms and expanding the double-cross product yields

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}\delta {{\boldsymbol{B}}}^{2}/2}{{\rm{\partial }}t}-{\boldsymbol{\nabla }}\cdot \left[({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\cdot \delta {\boldsymbol{B}}\delta {\boldsymbol{B}}-\delta {{\boldsymbol{B}}}^{2}({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\right]\\ \,\,+\,\delta {\boldsymbol{J}}\cdot \delta {\boldsymbol{E}}=0.\end{array}\end{array}\end{eqnarray} \tag{ A7 }$$

The magnetic field tensor is given by

$$\begin{eqnarray}\delta {\boldsymbol{B}}\delta {\boldsymbol{B}}=\left[\begin{array}{ccc}\delta {B}_{x}\delta {B}_{x} & \delta {B}_{y}\delta {B}_{x} & 0\\ \delta {B}_{x}\delta {B}_{y} & \delta {B}_{y}\delta {B}_{y} & 0\\ 0 & 0 & 0\end{array}\right]\end{eqnarray} \tag{ A8 }$$

where we assumed that the whistler waves are propagating quasi-parallel to the mean magnetic field and evaluated the tensor in a coordinate system where this mean magnetic field is locally aligned with the z-axis. In general the whistler wave has some phase. As we have no a priori information about this phase, we take an average of the magnetic field tensor over all possible phases and assume an equal probability for all phases. This results in

$$\begin{eqnarray}\left\langle \delta {\boldsymbol{B}}\delta {\boldsymbol{B}}\right\rangle =\left[\begin{array}{ccc}\delta {{\boldsymbol{B}}}^{2}/2 & 0 & 0\\ 0 & \delta {{\boldsymbol{B}}}^{2}/2 & 0\\ 0 & 0 & 0\end{array}\right]=\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}({\bf{1}}-{\boldsymbol{b}}{\boldsymbol{b}}),\end{eqnarray} \tag{ A9 }$$

where 〈〉 denotes the ensemble average, $\delta {B}_{x}=| \delta {\boldsymbol{B}}| \cos (\phi )$, $\delta {B}_{y}=\pm | \delta {\boldsymbol{B}}| \sin (\phi )$, b = B /∣ B ∣ is the unit vector along B , and we applied the ergodic theorem so that we effectively replace ergodic averaging by phase averaging. ¹¹ Taking the ensemble average of Equation (A7) and inserting the result of Equation (A9) yields

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}\delta {{\boldsymbol{B}}}^{2}/2}{{\rm{\partial }}t}-{\boldsymbol{\nabla }}\cdot \left[\left(({\bf{1}}-{\boldsymbol{b}}{\boldsymbol{b}})\cdot ({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}-\delta {{\boldsymbol{B}}}^{2}({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})\right)\right]\\ \,\,+\, \langle \delta {\boldsymbol{J}}\cdot \delta {\boldsymbol{E}} \rangle =0.\end{array}\end{array}\end{array}\end{eqnarray} \tag{ A10 }$$

After noting that (1 − b b ) · V _w = 0 we get

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}\delta {{\boldsymbol{B}}}^{2}/2}{{\rm{\partial }}t}+{\boldsymbol{\nabla }}\cdot \left[\delta {{\boldsymbol{B}}}^{2}({\boldsymbol{U}}+{{\boldsymbol{V}}}_{w})-({\bf{1}}-{\boldsymbol{b}}{\boldsymbol{b}})\cdot {\boldsymbol{U}}\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}\right]\\ \,\,+\, \langle \delta {\boldsymbol{J}}\cdot \delta {\boldsymbol{E}} \rangle =0.\end{array}\end{array}\end{eqnarray} \tag{ A11 }$$

The work done on the perturbed electromagnetic field is

$$\begin{eqnarray}&&\delta {\boldsymbol{J}}\cdot \delta {\boldsymbol{E}}=-{\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-\delta {\boldsymbol{B}}\delta {\boldsymbol{B}}\right)\right],\end{eqnarray} \tag{ A12 }$$

or, after taking the ensemble average (using Equation (A9)),

$$\begin{eqnarray} &&\langle \delta {\boldsymbol{J}}\cdot \delta {\boldsymbol{E}} \rangle =-{\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}\right)\right].\end{eqnarray} \tag{ A13 }$$

Inserting this expression into Equation (A11) yields

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}\delta {{\boldsymbol{B}}}^{2}/2}{{\rm{\partial }}t}+{\boldsymbol{\nabla }}\cdot \left[\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\boldsymbol{U}}+\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}\cdot {\boldsymbol{U}}+\delta {{\boldsymbol{B}}}^{2}{{\boldsymbol{V}}}_{w}\right]\\ \,=\,{\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{1}{2}\delta {{\boldsymbol{B}}}^{2}{\boldsymbol{b}}{\boldsymbol{b}}\right)\right].\end{array}\end{array}\end{eqnarray} \tag{ A14 }$$

Defining the energy density and pressure tensor of whistler waves, respectively,

$$\begin{eqnarray}&&{\varepsilon }_{w}=\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2},\ {\rm{and}}\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}&&{{\bf\mathsf{P}}}_{w}=\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}\end{eqnarray} \tag{ A16 }$$

Equation (A14) takes the simpler form:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\partial }}{\varepsilon }_{w}}{{\rm{\partial }}t}+{\boldsymbol{\nabla }}\cdot \left[\left({\varepsilon }_{w}{\bf{1}}+{{\boldsymbol{P}}}_{w})\cdot {\boldsymbol{U}}+2{\varepsilon }_{w}{{\boldsymbol{V}}}_{w}\right.\right]={\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot {{\boldsymbol{P}}}_{w}\right].\end{eqnarray} \tag{ A17 }$$

In order to generalize this result to forward- and backward-propagating whistlers, we define the corresponding energy densities, ${\varepsilon }_{w}^{\pm }$, and wave pressure tensor, ${{\bf\mathsf{P}}}_{w}^{\pm }$, add the whistler source terms to the right side, and obtain ¹²

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\varepsilon }_{w}^{\pm }}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[({\varepsilon }_{w}^{\pm }{\bf{1}}+{{\bf\mathsf{P}}}_{w}^{\pm })\cdot {\boldsymbol{U}}\pm 2{\varepsilon }_{w}^{\pm }{{\boldsymbol{V}}}_{w}\right]\\ \ =\ {\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot {{\bf\mathsf{P}}}_{w}^{\pm }\right]+{G}_{w}^{\pm }-{H}_{w}^{\pm }.\end{array}\end{eqnarray} \tag{ A19 }$$

The superscript ± denotes waves propagating down and up the temperature gradient, respectively. The whistler energy density and pressure tensor are given by

$$\begin{eqnarray}&&{\varepsilon }_{w}^{\pm }=\displaystyle \frac{\delta {{\boldsymbol{B}}}_{\pm }^{2}}{2},\,{\rm{and}}\end{eqnarray} \tag{ A20 }$$

$$\begin{eqnarray}&&{{\bf\mathsf{P}}}_{w}^{\pm }=\displaystyle \frac{\delta {{\boldsymbol{B}}}_{\pm }^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}.\end{eqnarray} \tag{ A21 }$$

The whistler growth term, ${G}_{w}^{\pm }$, and the whistler loss term, ${H}_{w}^{\pm }$, are given by

$$\begin{eqnarray}&&{G}_{w}^{\pm }=\pm {V}_{\mathrm{st}}^{\pm }{{nk}}_{{\rm{B}}}{\boldsymbol{b}}\cdot {\boldsymbol{\nabla }}T,\end{eqnarray} \tag{ A22 }$$

$$\begin{eqnarray}&&{H}_{w}^{\pm }={m}_{e}{{nV}}_{w}^{2}\left({\alpha }_{11}{\nu }_{{we}}^{\pm }+{\alpha }_{13}\displaystyle \frac{{\nu }_{{we}}^{+}{\nu }_{{we}}^{-}}{{\nu }_{e}}\right),\end{eqnarray} \tag{ A23 }$$

where k_B is Boltzmann's constant. The first term in the whistler loss term ${H}_{w}^{\pm }$ is associated with the wave drag on the electrons that causes them to be heated and includes contributions from whistlers propagating in both directions. The term proportional to the product of the scattering rates of bi-direction whistlers corresponds to second-order Fermi acceleration. The generalized electron streaming velocities ${V}_{\mathrm{st}}^{\pm }$ with the forward- and backward-propagating whistler waves are defined as

$$\begin{eqnarray}&&{V}_{\mathrm{st}}^{\pm }={V}_{w}{\alpha }_{10}\displaystyle \frac{{\nu }_{{we}}^{\pm }}{{\nu }_{e}},\end{eqnarray} \tag{ A24 }$$

where V_w = V _w · b = V_te /β_e is the non-directional whistler phase speed.

Analogously, it is possible to perturb the momentum equation to account for the forces exerted by the whistler waves. We start with the momentum equation for the composite ion–electron fluid, which is given by

$$\begin{eqnarray}&&\begin{array}{l}\rho \displaystyle \frac{d{\boldsymbol{U}}}{{dt}}+{\boldsymbol{\nabla }}{P}_{\mathrm{th}}=\displaystyle \frac{{\boldsymbol{J}}\times {\boldsymbol{B}}}{c}=({\boldsymbol{\nabla }}\times {\boldsymbol{B}})\times {\boldsymbol{B}}\\ \quad =\ -{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}\right),\end{array}\end{eqnarray} \tag{ A25 }$$

where d/dt = ∂/∂t + U · ∇ is the Lagrangian derivative, P_th = P_e + P_i is the total (ion plus electron) pressure, and we used J = c∇ × B (neglecting the displacement current in the Hall-MHD approximation). Introducing the perturbations in the magnetic field while neglecting linear contributions due to ensemble averaging yields

$$\begin{eqnarray}&&\rho \displaystyle \frac{d{\boldsymbol{U}}}{{dt}}+{\boldsymbol{\nabla }}\cdot \left({P}_{\mathrm{th}}{\bf{1}}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}+\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-\langle \delta {\boldsymbol{B}}\delta {\boldsymbol{B}}\rangle \right)={\bf{0}},\end{eqnarray} \tag{ A26 }$$

which can be simplified using Equation (A9) to yield

$$\begin{eqnarray}&&\rho \displaystyle \frac{d{\boldsymbol{U}}}{{dt}}+{\boldsymbol{\nabla }}\cdot \left({P}_{\mathrm{th}}{\bf{1}}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}+\displaystyle \frac{\delta {{\boldsymbol{B}}}^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}\right)={\bf{0}}.\end{eqnarray} \tag{ A27 }$$

Using the definitions for the wave energy and wave pressure while accounting for the two wave types and inserting the continuity equation yields the final conservative form of the momentum equation in the presence of whistler waves:

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho {\boldsymbol{U}}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left(\rho {\boldsymbol{U}}{\boldsymbol{U}}+{P}_{\mathrm{th}}{\bf{1}}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}+{{\bf\mathsf{P}}}_{w}^{+}+{{\bf\mathsf{P}}}_{w}^{-}\right)={\bf{0}}.\end{eqnarray} \tag{ A28 }$$

Multiplying this equation by U and using the continuity equation yields the kinetic energy equation

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\varepsilon }_{\mathrm{kin}}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left({\varepsilon }_{\mathrm{kin}}{\boldsymbol{U}}\right)\\ \ =\ -{\boldsymbol{U}}\cdot \left\{{\boldsymbol{\nabla }}\cdot \left[\left({P}_{\mathrm{th}}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}\right){\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}+{{\bf\mathsf{P}}}_{w}^{+}+{{\bf\mathsf{P}}}_{w}^{-}\right]\right\}\end{array}\end{eqnarray} \tag{ A29 }$$

where ε_kin = ρ U ²/2.

The magnetic induction law is given by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\partial }}{\boldsymbol{B}}}{{\rm{\partial }}t}-{\boldsymbol{\nabla }}\cdot ({\boldsymbol{B}}{\boldsymbol{U}}-{\boldsymbol{U}}{\boldsymbol{B}})={\bf{0}}.\end{eqnarray} \tag{ A30 }$$

Multiplying this equation by B yields the equation for the magnetic energy

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\displaystyle \frac{{\rm{\partial }}{\varepsilon }_{B}}{{\rm{\partial }}t}+{\boldsymbol{\nabla }}\cdot \left[{\boldsymbol{U}}\left({\varepsilon }_{B}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}\right)-({\boldsymbol{U}}\cdot {\boldsymbol{B}}){\boldsymbol{B}}\right]\\ \,=\,{\boldsymbol{U}}\cdot \left\{{\boldsymbol{\nabla }}\cdot \left[\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}{\bf{1}}-{\boldsymbol{B}}{\boldsymbol{B}}\right]\right\},\end{array}\end{array}\end{eqnarray} \tag{ A31 }$$

where ε_B = B ²/2.

The ion and electron energy equations are

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\varepsilon }_{i}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[{\boldsymbol{U}}({\varepsilon }_{i}+{P}_{i})\right]={\boldsymbol{U}}\cdot {\boldsymbol{\nabla }}{P}_{i}-\displaystyle \frac{{\varepsilon }_{i}-{\varepsilon }_{e}}{{\tau }_{\mathrm{eq}}},\end{eqnarray} \tag{ A32 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\varepsilon }_{e}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[{\boldsymbol{U}}({\varepsilon }_{e}+{P}_{e})+{{\boldsymbol{Q}}}_{e,\mathrm{st}}+{{\boldsymbol{Q}}}_{e,\mathrm{dif}}\right]\\ \ =\ {\boldsymbol{U}}\cdot {\boldsymbol{\nabla }}{P}_{e}-\displaystyle \frac{{\varepsilon }_{e}-{\varepsilon }_{i}}{{\tau }_{\mathrm{eq}}}+\sum _{\pm }({H}_{w}^{\pm }-{G}_{w}^{\pm })\end{array}\end{eqnarray} \tag{ A33 }$$

where ε_i = 3/2nk_B T_i and ε_e = 3/2nk_B T_e are the energy densities of ions and electrons, respectively, and the electron streaming and diffusion fluxes are given by

$$\begin{eqnarray}&&{{\boldsymbol{Q}}}_{e,\mathrm{st}}=({V}_{\mathrm{st}}^{+}-{V}_{\mathrm{st}}^{-}){P}_{e}{\boldsymbol{b}},\end{eqnarray} \tag{ A34 }$$

$$\begin{eqnarray}&&{{\boldsymbol{Q}}}_{e,\mathrm{dif}}=-{\kappa }_{e}{\boldsymbol{b}}{\boldsymbol{b}}\cdot {\boldsymbol{\nabla }}{T}_{e}.\end{eqnarray} \tag{ A35 }$$

While the ion energy equation is standard, the equation for the electron energy density is the same as given in Equation (4) but generalized for the general fluid velocity U and the whistler streaming velocities ${V}_{\mathrm{st}}^{\pm }$. The energy equilibration time between electrons and ions is denoted by τ_eq, which is typically longer by $\sqrt{{m}_{i}/{m}_{e}}$ than the classical electron–ion scattering time. The whistler waves can also transfer energy between electrons and ions but the detailed scaling behavior for this transfer has not been established (Roberg-Clark et al. 2018b). Viscous terms could also be included in the momentum and ion pressure equations. The parallel conductivity κ_e is

$$\begin{eqnarray}&&{\kappa }_{e}={\alpha }_{12}\displaystyle \frac{{{nk}}_{{\rm{B}}}{T}_{e}}{{m}_{e}{\nu }_{e}}.\end{eqnarray} \tag{ A36 }$$

The total scattering rate ${\nu }_{e}={\nu }_{{ei}}^{e}+{\nu }_{{we}}^{+}+{\nu }_{{we}}^{-}$ is the sum of ${\nu }_{{ei}}^{e}$, the classical electron–ion scattering rate, and the scattering rates ${\nu }_{{we}}^{\pm }$ from the forward- and backward-propagating whistlers, all evaluated at the electron thermal velocity V_te ,

$$\begin{eqnarray}&&{\nu }_{{ei}}^{e}=\displaystyle \frac{4\pi {e}^{4}n{\boldsymbol{\Lambda }}}{{m}_{e}^{2}{V}_{{te}}^{3}},\,\,{\nu }_{{we}}^{\pm }=0.1{{\rm{\Omega }}}_{e}\displaystyle \frac{{\varepsilon }_{w}^{\pm }}{{\varepsilon }_{B}},\end{eqnarray} \tag{ A37 }$$

where ${\varepsilon }_{w}^{\pm }$ and ε_B = B²/8π are the whistler and magnetic energy densities and Λ is the Coulomb logarithm.

The total electron heat flux Q _e = U (ε_e + P_e ) + Q _e,st + Q _e,dif is composed of the flux due to advection of electron enthalpy, the effective streaming flux due to electron–whistler wave scattering, as well as the diffusive flux. In Equation (A33), the forward- and backward-propagating whistlers try to carry the electron energy in opposite directions along B . The wave with the dominant scattering rate wins out and as a result the direction of advection can change sign with the ambient parallel temperature gradient.

For completeness, here we summarize the complete set of equations for whistler MHD that constitute a complete description of plasma dynamics in the high β ICM (to which a description of anisotropic viscosity can be added):

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\boldsymbol{\nabla }}\cdot (\rho {\boldsymbol{U}})=0,\end{eqnarray} \tag{ A38 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial \rho {\boldsymbol{U}}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[\rho {\boldsymbol{U}}{\boldsymbol{U}}+\left({P}_{\mathrm{th}}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}\right){\bf{1}}\right.\\ \left.-\ {\boldsymbol{B}}{\boldsymbol{B}}+{{\bf\mathsf{P}}}_{w}^{+}+{{\bf\mathsf{P}}}_{w}^{-}\Space{0ex}{3.5ex}{0ex}\right]={\bf{0}},\end{array}\end{eqnarray} \tag{ A39 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}+{\boldsymbol{\nabla }}\cdot ({\boldsymbol{B}}{\boldsymbol{U}}-{\boldsymbol{U}}{\boldsymbol{B}})={\bf{0}},\end{eqnarray} \tag{ A40 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\varepsilon }_{i}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[{\boldsymbol{U}}({\varepsilon }_{i}+{P}_{i})\right]={\boldsymbol{U}}\cdot {\boldsymbol{\nabla }}{P}_{i}-\displaystyle \frac{{\varepsilon }_{i}-{\varepsilon }_{e}}{{\tau }_{\mathrm{eq}}},\end{eqnarray} \tag{ A41 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\varepsilon }_{e}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[{\boldsymbol{U}}({\varepsilon }_{e}+{P}_{e})+{{\boldsymbol{Q}}}_{e,\mathrm{st}}+{{\boldsymbol{Q}}}_{e,\mathrm{dif}}\right]\\ \ =\ {\boldsymbol{U}}\cdot {\boldsymbol{\nabla }}{P}_{e}-\displaystyle \frac{{\varepsilon }_{e}-{\varepsilon }_{i}}{{\tau }_{\mathrm{eq}}}+\sum _{\pm }({H}_{w}^{\pm }-{G}_{w}^{\pm }),\end{array}\end{eqnarray} \tag{ A42 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\varepsilon }_{w}^{\pm }}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[({\varepsilon }_{w}^{\pm }{\bf{1}}+{{\bf\mathsf{P}}}_{w}^{\pm })\cdot {\boldsymbol{U}}\pm 2{\varepsilon }_{w}^{\pm }{{\boldsymbol{V}}}_{w}\right]\\ \ =\ {\boldsymbol{U}}\cdot \left[{\boldsymbol{\nabla }}\cdot {{\bf\mathsf{P}}}_{w}^{\pm }\right]+{G}_{w}^{\pm }-{H}_{w}^{\pm }\end{array}\end{eqnarray} \tag{ A43 }$$

where P_th = P_i + P_e and the equations of state are given by

$$\begin{eqnarray}&&{P}_{i}={{nk}}_{{\rm{B}}}{T}_{i}=({\rm{\Gamma }}-1){\varepsilon }_{i},\end{eqnarray} \tag{ A44 }$$

$$\begin{eqnarray}&&{P}_{e}={{nk}}_{{\rm{B}}}{T}_{e}=({\rm{\Gamma }}-1){\varepsilon }_{e},\end{eqnarray} \tag{ A45 }$$

where Γ = 5/3. The whistler growth term, ${G}_{w}^{\pm }$, and the whistler loss terms, ${H}_{w}^{\pm }$, (due to drag and second-order Fermi acceleration) are given by

$$\begin{eqnarray}&&{G}_{w}^{\pm }=\pm {V}_{\mathrm{st}}^{\pm }{{nk}}_{{\rm{B}}}{\boldsymbol{b}}\cdot {\boldsymbol{\nabla }}{T}_{e},\end{eqnarray} \tag{ A46 }$$

$$\begin{eqnarray}&&{H}_{w}^{\pm }={m}_{e}{{nV}}_{w}^{2}\left({\alpha }_{11}{\nu }_{{we}}^{\pm }+{\alpha }_{13}\displaystyle \frac{{\nu }_{{we}}^{+}{\nu }_{{we}}^{-}}{{\nu }_{e}}\right).\end{eqnarray} \tag{ A47 }$$

The whistler energy density and pressure tensor are given by

$$\begin{eqnarray}&&{\varepsilon }_{w}^{\pm }=\displaystyle \frac{\delta {{\boldsymbol{B}}}_{\pm }^{2}}{2},\end{eqnarray} \tag{ A48 }$$

$$\begin{eqnarray}&&{{\bf\mathsf{P}}}_{w}^{\pm }=\displaystyle \frac{\delta {{\boldsymbol{B}}}_{\pm }^{2}}{2}{\boldsymbol{b}}{\boldsymbol{b}}.\end{eqnarray} \tag{ A49 }$$

The parameters α_i with various subscripts in Equations(A42)–(A47) are given in Table 1. As shown in the table these parameters have simple analytic forms in the limit of large or small classical collisions but no simple analytic form for arbitrary ${\nu }_{{we}}/{\nu }_{{ei}}^{e}$. The parameter γ in the table controls the dependence of ν_w on velocity, ${\nu }_{w}={\nu }_{{we}}{\left(V/{V}_{{te}}\right)}^{\gamma }$, where, as discussed in Section 3, our best estimate is that γ = 4/3. The values of the parameters α_i in the two collisionality limits have been explicitly evaluated for γ = 4/3. A connection formula for the two collisionality limits could be constructed in a numerical implementation of the transport equations. Further, as mentioned previously the thermal conduction in Equation (A36) does not reduce to the Spitzer value because we have discarded electron–electron collisions. The correct Spitzer value is obtained by setting α₁₂ to 4.25 rather than the value given in Table 1.

Finally, we note that in a large-scale system in which the transport timescale L/V_w is long compared the electron–ion energy exchange time τ_eq, Equations (A41) and (A41) can be combined into a single energy equation for electrons and ions.

Adding up Equations (A19), (A29), (A31), (A32), and (A33), we obtain total energy conservation

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {\varepsilon }_{\mathrm{tot}}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left[{\boldsymbol{U}}\cdot ({\varepsilon }_{\mathrm{tot}}{\bf{1}}+{{\bf\mathsf{P}}}_{\mathrm{tot}})-({\boldsymbol{U}}\cdot {\boldsymbol{B}}){\boldsymbol{B}}\right.\\ \ \ \left.+\ 2{{\boldsymbol{V}}}_{w}({\varepsilon }_{w}^{+}-{\varepsilon }_{w}^{-})+{{\boldsymbol{Q}}}_{e,\mathrm{st}}+{{\boldsymbol{Q}}}_{e,\mathrm{dif}}\right]=0\end{array}\end{eqnarray} \tag{ A50 }$$

where

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{tot}}={\varepsilon }_{i}+{\varepsilon }_{e}+{\varepsilon }_{\mathrm{kin}}+{\varepsilon }_{B}+{\varepsilon }_{w}^{+}+{\varepsilon }_{w}^{-},\,{\rm{and}}\end{eqnarray} \tag{ A51 }$$

$$\begin{eqnarray}&&{{\bf\mathsf{P}}}_{\mathrm{tot}}=\left({P}_{i}+{P}_{e}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}\right){\bf{1}}+{{\bf\mathsf{P}}}_{w}^{+}+{{\bf\mathsf{P}}}_{w}^{-}\end{eqnarray} \tag{ A52 }$$

are the total energy density and the pressure tensor, respectively.