2.3.1 Gyrotropy of the background distribution function

Under the ordering (2.11), the largest term in the Vlasov–Landau equation (2.1) corresponds to Larmor motion of the mean distribution about the uniform guide field:

(2.17) $$\begin{eqnarray}-\unicode[STIX]{x1D6FA}_{s}\,\hat{\boldsymbol{z}}\boldsymbol{\cdot }\left(\boldsymbol{v}\times \frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\boldsymbol{v}}\right)=0.\end{eqnarray}$$

This directional bias allows us to set up a local Cartesian coordinate system and decompose the particle velocity in terms of the parallel velocity $v_{\Vert }$ , the perpendicular velocity $v_{\bot }$ , and the gyrophase angle $\unicode[STIX]{x1D717}$ ,

(2.18) $$\begin{eqnarray}\boldsymbol{v}=v_{\Vert }\hat{\boldsymbol{z}}+v_{\bot }(\cos \unicode[STIX]{x1D717}\,\hat{\boldsymbol{x}}+\sin \unicode[STIX]{x1D717}\,\hat{\boldsymbol{y}}).\end{eqnarray}$$

Equation (2.17) then takes on the simple form

(2.19) $$\begin{eqnarray}-\unicode[STIX]{x1D6FA}_{s}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}=0,\end{eqnarray}$$

which states that the mean distribution function is gyrotropic (independent of the gyrophase):

(2.20) $$\begin{eqnarray}f_{0s}=f_{0s}(v_{\Vert },v_{\bot },t).\end{eqnarray}$$

All velocity-space derivatives of $f_{0s}$ that enter (2.1) are thus with respect to $v_{\Vert }$ and $v_{\bot }$ , viz.

(2.21) $$\begin{eqnarray}\frac{q_{s}}{m_{s}}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\boldsymbol{v}}=-(v_{\Vert }-u_{\Vert 0s}^{\prime })\hat{\boldsymbol{z}}\,\frac{q_{s}f_{0s}^{\Vert }}{T_{\Vert 0s}}-\boldsymbol{v}_{\bot }\,\frac{q_{s}f_{0s}^{\bot }}{T_{\bot 0s}},\end{eqnarray}$$

where

(2.22a,b ) $$\begin{eqnarray}f_{0s}^{\Vert }\doteq -v_{\text{th}\Vert s}^{2}\,\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}(v_{\Vert }-u_{\Vert 0s}^{\prime })^{2}}\quad \text{and}\quad f_{0s}^{\bot }\doteq -v_{\text{th}\bot s}^{2}\,\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}v_{\bot }^{2}}\end{eqnarray}$$

are dimensionless derivatives of a species’ mean distribution function with respect to the square of the parallel velocity (peculiar to the species drift velocity) and the perpendicular velocity, respectively. Their weighted difference,

(2.23) $$\begin{eqnarray}\,\mathfrak{D}f_{0s}\doteq \frac{T_{\bot 0s}}{T_{\Vert 0s}}f_{0s}^{\Vert }-f_{0s}^{\bot },\end{eqnarray}$$

measures the velocity-space anisotropy of the mean distribution function. For a bi-Maxwellian distribution (2.9), $f_{0s}^{\Vert }=f_{0s}^{\bot }=f_{0s}$ and

(2.24) $$\begin{eqnarray}\,\mathfrak{D}f_{0s}=\left(\frac{T_{\bot 0s}}{T_{\Vert 0s}}-1\right)f_{0s}\doteq \unicode[STIX]{x1D6E5}_{s}f_{0s},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}_{s}$ is the temperature (or, equivalently, pressure) anisotropy of the mean distribution function of species $s$ .

2.3.2 Boltzmann response

At $\mathit{O}(\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}_{s}f_{0s})$ , we learn from (2.1) that the first-order distribution function $\unicode[STIX]{x1D6FF}f_{1s}$ may be split into two parts. The first of these is the so-called adiabatic (or ‘Boltzmann’) response,

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}f_{1s,\text{Boltz}}=-\frac{q_{s}\unicode[STIX]{x1D711}_{s}^{\prime }}{T_{\Vert 0s}}f_{0s}^{\Vert }+\frac{q_{s}}{T_{\bot 0s}}\left(\unicode[STIX]{x1D711}-\frac{v_{\Vert }A_{\Vert }}{c}\right)\,\mathfrak{D}f_{0s},\end{eqnarray}$$

where

(2.26) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{s}^{\prime }\doteq \unicode[STIX]{x1D711}-\frac{u_{\Vert 0s}^{\prime }A_{\Vert }}{c}\end{eqnarray}$$

is the fluctuating electrostatic potential in the frame of the parallel-drifting species $s$ . This part of $\unicode[STIX]{x1D6FF}f_{1s}$ represents the (leading-order) evolution of $f_{0s}$ under the influence of the perturbed electromagnetic fields. To see this, we first introduce the total particle energy in the parallel-drifting frame,

(2.27) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D700}}_{s}=\unicode[STIX]{x1D700}_{0s}+\unicode[STIX]{x1D700}_{1s}\doteq {\textstyle \frac{1}{2}}m_{s}|\boldsymbol{v}-u_{\Vert 0s}^{\prime }\,\hat{\boldsymbol{z}}|^{2}+q_{s}\unicode[STIX]{x1D711}_{s}^{\prime }\end{eqnarray}$$

and the (gyrophase-dependent part of the) first adiabatic invariant,

(2.28) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D707}}_{s}=\unicode[STIX]{x1D707}_{0s}+\unicode[STIX]{x1D707}_{1s}\doteq \frac{m_{s}v_{\bot }^{2}}{2B_{0}}+\frac{q_{s}}{B_{0}}\left(\unicode[STIX]{x1D711}-\frac{v_{\Vert }A_{\Vert }}{c}\right),\end{eqnarray}$$

both written out to first order in the fluctuation amplitudes (e.g. Kruskal Reference Kruskal1958; Hastie, Taylor & Haas Reference Hastie, Taylor and Haas1967; Taylor Reference Taylor1967; Catto, Tang & Baldwin Reference Catto, Tang and Baldwin1981; Parra Reference Parra2013). It is then straightforward to show by using

(2.29a,b ) $$\begin{eqnarray}f_{0s}^{\Vert }=-T_{\Vert 0s}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{0s}}\quad \text{and}\quad f_{0s}^{\bot }=-T_{\bot 0s}\left(\frac{1}{B_{0}}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{0s}}+\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{0s}}\right)\end{eqnarray}$$

(see § I–C.4) that the sum of the mean distribution function and the Boltzmann response is simply

(2.30) $$\begin{eqnarray}f_{0s}(v_{\Vert },v_{\bot })+\unicode[STIX]{x1D6FF}f_{1s,\text{Boltz}}=f_{0s}(\overline{\unicode[STIX]{x1D700}}_{s},\overline{\unicode[STIX]{x1D707}}_{s})+\mathit{O}(\unicode[STIX]{x1D716}^{2}).\end{eqnarray}$$

In other words, the Boltzmann response does not change the form of the mean distribution function if the latter is written as a function of the constants of the motion (calculated sufficiently accurately).

2.3.3 Gyrokinetic response

The second part of $\unicode[STIX]{x1D6FF}f_{1s}$ , which we denote by $h_{s}$ , represents the response of rings of charge to the fluctuating fields, and is thus referred to as the gyrokinetic response. It satisfies

(2.31) $$\begin{eqnarray}\boldsymbol{v}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }h_{s}-\unicode[STIX]{x1D6FA}_{s}\frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}\bigg|_{\boldsymbol{r}}=-\unicode[STIX]{x1D6FA}_{s}\frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}\bigg|_{\,\!\boldsymbol{R}_{s}}=0,\end{eqnarray}$$

where we have transformed the $\unicode[STIX]{x1D717}$ derivative taken at constant position $\boldsymbol{r}$ to one taken at constant guiding centre

(2.32) $$\begin{eqnarray}\boldsymbol{R}_{s}=\boldsymbol{r}+\frac{\boldsymbol{v}\times \hat{\boldsymbol{z}}}{\unicode[STIX]{x1D6FA}_{s}}.\end{eqnarray}$$

Thus, $h_{s}$ is independent of the gyrophase angle at constant guiding centre $\boldsymbol{R}_{s}$ (but not at constant position $\boldsymbol{r}$ ):

(2.33) $$\begin{eqnarray}h_{s}=h_{s}(t,\boldsymbol{R}_{s},v_{\Vert },v_{\bot }).\end{eqnarray}$$

2.3.4 Gyrokinetic equation

At $\mathit{O}(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FA}_{s}f_{0s})$ , we find from (2.1) that the gyrokinetic response evolves via the gyrokinetic equation

(2.34) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}t}+v_{\Vert }\frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}z}+\frac{c}{B_{0}}\{\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}},h_{s}\} & = & \displaystyle \frac{q_{s}f_{0s}^{\Vert }}{T_{\Vert 0s}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+u_{\Vert 0s}^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right)\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{q_{s}\,\mathfrak{D}f_{0s}}{T_{\bot 0s}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+v_{\Vert }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right)\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}},\end{eqnarray}$$

where

(2.35) $$\begin{eqnarray}\unicode[STIX]{x1D712}\doteq \unicode[STIX]{x1D711}-\frac{v_{\Vert }A_{\Vert }}{c}-\frac{\boldsymbol{v}_{\bot }\boldsymbol{\cdot }\boldsymbol{A}_{\bot }}{c}\end{eqnarray}$$

is the gyrokinetic potential and

(2.36) $$\begin{eqnarray}\langle \unicode[STIX]{x1D712}(t,\boldsymbol{r},\boldsymbol{v})\rangle _{\!\boldsymbol{R}_{s}}\doteq \frac{1}{2\unicode[STIX]{x03C0}}\oint \text{d}\unicode[STIX]{x1D717}\,\unicode[STIX]{x1D712}\left(t,\boldsymbol{R}_{s}-\frac{\boldsymbol{v}\times \hat{\boldsymbol{z}}}{\unicode[STIX]{x1D6FA}_{s}},\boldsymbol{v}\right)\end{eqnarray}$$

denotes the ring average of $\unicode[STIX]{x1D712}$ at fixed guiding centre $\boldsymbol{R}_{s}$ . The Poisson bracket

(2.37) $$\begin{eqnarray}\{\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}},h_{s}\}\doteq \hat{\boldsymbol{z}}\boldsymbol{\cdot }\left(\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}}{\unicode[STIX]{x2202}\boldsymbol{R}_{s}}\times \frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}\boldsymbol{R}_{s}}\right)\end{eqnarray}$$

represents the nonlinear interaction between the gyrocentre rings and the ring-averaged electromagnetic fields.

The gyrokinetic equation (2.34) can also be written in the following, perhaps more physically illuminating, form:

(2.38) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}t}+\langle \dot{\boldsymbol{R}}_{s}\rangle _{\!\boldsymbol{R}_{s}}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}h_{s}}{\unicode[STIX]{x2202}\boldsymbol{R}_{s}}=-\langle \dot{\overline{\unicode[STIX]{x1D700}}}_{s}\rangle _{\!\boldsymbol{R}_{s}}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D700}}_{s}}-\langle \dot{\overline{\unicode[STIX]{x1D707}}}_{s}\rangle _{\!\boldsymbol{R}_{s}}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D707}}_{s}},\end{eqnarray}$$

where

(2.39) $$\begin{eqnarray}\langle \dot{\boldsymbol{R}}_{s}\rangle _{\!\boldsymbol{R}_{s}}=v_{\Vert }\hat{\boldsymbol{z}}-\frac{c}{B_{0}}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}}{\unicode[STIX]{x2202}\boldsymbol{R}_{s}}\times \hat{\boldsymbol{z}}\end{eqnarray}$$

is the ring velocity,

(2.40) $$\begin{eqnarray}\langle \dot{\overline{\unicode[STIX]{x1D700}}}_{s}\rangle _{\!\boldsymbol{R}_{s}}=q_{s}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+u_{\Vert 0s}^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right)\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}\end{eqnarray}$$

is the ring-averaged rate of change of the particle energy (2.27), and

(2.41) $$\begin{eqnarray}\langle \dot{\overline{\unicode[STIX]{x1D707}}}_{s}\rangle _{\!\boldsymbol{R}_{s}}=\frac{q_{s}}{B_{0}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+v_{\Vert }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right)\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}\end{eqnarray}$$

is the ring-averaged rate of change of the (gyrophase-dependent part of the) first adiabatic invariant (2.28). The right-hand side of (2.38) represents the effect of collisionless work done on the rings by the fields (the wave–ring interaction). Written in this way, (2.34) is simply the ring-averaged Vlasov equation,

(2.42) $$\begin{eqnarray}\langle {\dot{f}}_{s}(t,\boldsymbol{r},\overline{\unicode[STIX]{x1D700}}_{s},\overline{\unicode[STIX]{x1D707}}_{s})\rangle _{\!\boldsymbol{R}_{s}}=0\end{eqnarray}$$

to lowest order in $\unicode[STIX]{x1D716}$ .

It is a manifestly good idea in much of what follows to absorb the final term of (2.34) (and, likewise, of (2.38)), into $h_{s}$ by writing the latter in terms of the velocity-space coordinates $(v_{\Vert },\unicode[STIX]{x1D707}_{s})$ , where

(2.43) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{s}\doteq \overline{\unicode[STIX]{x1D707}}_{s}-\frac{q_{s}}{B_{0}}\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}\end{eqnarray}$$

is the full adiabatic invariant, viz., $\dot{\unicode[STIX]{x1D707}}_{s}\sim \mathit{O}(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D714}T_{\bot 0s}/B_{0})$ . At long wavelengths satisfying $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ ,

(2.44) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{s}\simeq \frac{m_{s}|\boldsymbol{v}-\boldsymbol{u}_{\bot }-\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}|^{2}}{2B}\doteq \frac{m_{s}w_{\bot }^{2}}{2B},\end{eqnarray}$$

which is simply the magnetic moment of a particle in a magnetic field of strength $B$ drifting across said field at the $\boldsymbol{E}\times \boldsymbol{B}$ velocity,

(2.45) $$\begin{eqnarray}\boldsymbol{u}_{\bot }=\frac{c}{B_{0}}\hat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D711}(\boldsymbol{r}).\end{eqnarray}$$

Then, introducingFootnote ⁴

(2.46a ) $$\begin{eqnarray}\displaystyle \widetilde{h}_{s}(v_{\Vert },\unicode[STIX]{x1D707}_{s}) & \doteq & \displaystyle h_{s}(v_{\Vert },v_{\bot })+\frac{q_{s}\,\mathfrak{D}f_{0s}}{T_{\bot 0s}}\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}\end{eqnarray}$$

(2.46b ) $$\begin{eqnarray}\displaystyle \text{} & = & \displaystyle h_{s}(\unicode[STIX]{x1D700}_{0s},\unicode[STIX]{x1D707}_{0s})+\frac{q_{s}}{B_{0}}\frac{\unicode[STIX]{x2202}f_{0s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{0s}}\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}},\end{eqnarray}$$

(2.47) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{h}_{s}}{\unicode[STIX]{x2202}t}+\langle \dot{\boldsymbol{R}}_{s}\rangle _{\!\boldsymbol{R}_{s}}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\widetilde{h}_{s}}{\unicode[STIX]{x2202}\boldsymbol{R}_{s}}=\frac{q_{s}f_{0s}^{\Vert }}{T_{\Vert 0s}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+u_{\Vert 0s}^{\prime }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right)\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{s}}.\end{eqnarray}$$

This form of the gyrokinetic equation is particularly well suited for deriving the gyrokinetic invariants (§ 4). Its right-hand side represents the collisionless work done on the rings by the fields in a frame comoving with the parallel drift velocity of species $s$ .

It will also prove useful in what follows to modify the energy variable $\overline{\unicode[STIX]{x1D700}}_{s}$ to obtain

(2.48) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{s}\doteq \overline{\unicode[STIX]{x1D700}}_{s}-q_{s}\langle \unicode[STIX]{x1D711}_{s}^{\prime }\rangle _{\!\boldsymbol{ R}_{s}},\end{eqnarray}$$

which is the kinetic energy of the particle as measured in the frame moving with the $u_{\Vert 0s}^{\prime }$ and $\boldsymbol{E}\times \boldsymbol{B}$ drifts (e.g. Parra Reference Parra2013); indeed,

(2.49) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{s}\simeq {\textstyle \frac{1}{2}}m_{s}(v_{\Vert }-u_{\Vert 0s}^{\prime })^{2}+{\textstyle \frac{1}{2}}m_{s}w_{\bot }^{2}\end{eqnarray}$$

at long wavelengths. If the mean distribution function is expressed in terms of these new velocity-space variables, viz.  $f_{s}=\widetilde{f}_{0s}(\unicode[STIX]{x1D700}_{s},\unicode[STIX]{x1D707}_{s})+\unicode[STIX]{x1D6FF}\widetilde{f}_{s}$ , then the perturbed distribution function $\unicode[STIX]{x1D6FF}\widetilde{f}_{s}$ becomes (see (I–C52))

(2.50a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\widetilde{f}_{s}(\unicode[STIX]{x1D700}_{s},\unicode[STIX]{x1D707}_{s}) & = & \displaystyle \unicode[STIX]{x1D6FF}f_{1s}-(\unicode[STIX]{x1D6FF}f_{1s,\text{Boltz}}-\langle \unicode[STIX]{x1D6FF}f_{1s,\text{Boltz}}\rangle _{\!\boldsymbol{R}_{s}})-\frac{q_{s}\,\mathfrak{D}f_{0s}}{T_{\bot 0s}}\left\langle \frac{\boldsymbol{v}_{\bot }\boldsymbol{\cdot }\boldsymbol{A}_{\bot }}{c}\right\rangle _{~\!\boldsymbol{R}_{s}}\end{eqnarray}$$

(2.50b ) $$\begin{eqnarray}\displaystyle \text{} & = & \displaystyle \widetilde{h}_{s}(v_{\Vert },\unicode[STIX]{x1D707}_{s})-\frac{q_{s}f_{0s}^{\Vert }}{T_{\Vert 0s}}\langle \unicode[STIX]{x1D711}_{s}^{\prime }\rangle _{\!\boldsymbol{R}_{s}}.\end{eqnarray}$$

$k_{\bot }\unicode[STIX]{x1D70C}_{s}\sim 1$

2.3.5 Field equations

The equations governing the electromagnetic potentials are most easily obtained by substituting the decomposition

(2.51) $$\begin{eqnarray}f_{s}=f_{0s}(v_{\Vert },v_{\bot })+\unicode[STIX]{x1D6FF}f_{1s,\text{Boltz}}+h_{s}(t,\boldsymbol{R}_{s},v_{\Vert },v_{\bot })+\cdots\end{eqnarray}$$

into the leading-order expansions of the quasineutrality constraint (2.3) and Ampère’s law (2.4). The result is (see § I–C.3)

(2.52) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\mathop{\sum }_{s}q_{s}\left[\int \text{d}^{3}\boldsymbol{v}\,\langle h_{s}\rangle _{\boldsymbol{r}}-\frac{q_{s}n_{0s}}{T_{\bot 0s}}\left(C_{0s}^{\bot }\unicode[STIX]{x1D711}-C_{1s}^{\bot }\frac{u_{\Vert 0s}^{\prime }A_{\Vert }}{c}\right)\right], & \displaystyle\end{eqnarray}$$

(2.53) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=-\frac{4\unicode[STIX]{x03C0}}{c}\mathop{\sum }_{s}q_{s}\left[\int \text{d}^{3}\boldsymbol{v}\,v_{\Vert }\langle h_{s}\rangle _{\boldsymbol{r}}-\frac{q_{s}n_{0s}v_{\text{th}\Vert s}}{2T_{\bot 0s}}\left(C_{1s}^{\bot }\unicode[STIX]{x1D711}\frac{2u_{\Vert 0s}^{\prime }}{v_{\text{th}\Vert s}}+\widetilde{\unicode[STIX]{x1D6E5}}_{s}\frac{v_{\text{th}\Vert s}A_{\Vert }}{c}\right)\right],\qquad & \displaystyle\end{eqnarray}$$

(2.54) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6FF}B_{\Vert }=-\frac{4\unicode[STIX]{x03C0}}{c}\hat{\boldsymbol{z}}\boldsymbol{\cdot }\left[\unicode[STIX]{x1D735}_{\bot }\times \mathop{\sum }_{s}q_{s}\int \text{d}^{3}\boldsymbol{v}\,\langle \boldsymbol{v}h_{s}\rangle _{\boldsymbol{r}}\right], & \displaystyle\end{eqnarray}$$

where

(2.55) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6E5}}_{s}\doteq \frac{T_{\bot 0s}}{T_{\Vert 0s}}-C_{2s}^{\bot }\left(1+\frac{2u_{\Vert 0s}^{\prime 2}}{v_{\text{th}\Vert s}^{2}}\right)\end{eqnarray}$$

is the temperature anisotropy of species $s$ augmented by the parallel ram pressure from background parallel drifts, $C_{\ell s}^{\bot }$ are parallel moments of the perpendicular-differentiated mean distribution function (all of which equate to unity for a drifting bi-Maxwellian distribution; see appendix A), and

(2.56) $$\begin{eqnarray}\langle h_{s}(t,\boldsymbol{R}_{s},v_{\Vert },v_{\bot })\rangle _{\boldsymbol{r}}\doteq \frac{1}{2\unicode[STIX]{x03C0}}\oint \text{d}\unicode[STIX]{x1D717}\,h_{s}\left(t,\boldsymbol{r}+\frac{\boldsymbol{v}\times \hat{\boldsymbol{z}}}{\unicode[STIX]{x1D6FA}_{s}},v_{\Vert },v_{\bot }\right)\end{eqnarray}$$

denotes the gyro-average of $h_{s}$ at fixed $\boldsymbol{r}$ . Together with the gyrokinetic equation (2.34), the field equations (2.52)–(2.54) constitute a closed system that describes the evolution of a gyrokinetic plasma with non-Maxwellian $f_{0s}$ and parallel interspecies drifts.

This completes our abbreviated review of the material derived in Paper I on the gyrokinetic framework for homogeneous, non-Maxwellian plasmas. We now proceed to analyse the linear and nonlinear behaviour of the perturbations governed by this system of equations.

3.6.1 KRMHD limit: $Z_{i}m_{e}/m_{i}$ , $k_{\bot }\unicode[STIX]{x1D70C}_{i}\rightarrow 0$

In the limit where $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ and the electron–ion mass ratio $Z_{i}m_{e}/m_{i}$ are both asymptotically small, one should recover the linear theory for bi-Maxwellian KRMHD (cf. § I–4.4 and § 3.5). In this limit, ${\mathcal{B}}\simeq \unicode[STIX]{x1D6FC}_{i}$ , ${\mathcal{E}}\simeq -(3/2)\unicode[STIX]{x1D6FC}_{i}$ , and the dispersion relation (3.27) becomes

(3.28) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D6FC}_{\ast }}{\overline{\unicode[STIX]{x1D714}}^{2}}-\unicode[STIX]{x1D6FC}_{i}\right)\left(\frac{2{\mathcal{A}}}{\unicode[STIX]{x1D6FD}_{\bot i}}-{\mathcal{A}}{\mathcal{D}}+{\mathcal{C}}^{2}\right)=0.\end{eqnarray}$$

Setting the first factor of (3.28) to zero and simplifying $\unicode[STIX]{x1D6FC}_{\ast }\simeq \unicode[STIX]{x1D6FC}_{i}[1+(\unicode[STIX]{x1D6FD}_{\Vert i}/2)\unicode[STIX]{x1D6E5}_{i}+(\unicode[STIX]{x1D6FD}_{\Vert e}/2)\unicode[STIX]{x1D6E5}_{e}]$ (see (3.26f )), we obtain the dispersion relation for undamped Alfvén waves modified by the ion and electron pressure anisotropies:

(3.29) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm k_{\Vert }v_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert i}}{2}\unicode[STIX]{x1D6E5}_{i}+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}}.\end{eqnarray}$$

Again, when

(3.30) $$\begin{eqnarray}1+\frac{\unicode[STIX]{x1D6FD}_{\Vert i}}{2}\unicode[STIX]{x1D6E5}_{i}+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}<0,\end{eqnarray}$$

the effective Alfvén speed becomes imaginary and the firehose instability results (cf. (3.21)).

Setting the second factor of (3.28) to zero, and using the leading-order expressions for ${\mathcal{C}}\simeq (T_{\bot 0i}/T_{\Vert 0i})[1+\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})-\unicode[STIX]{x1D70F}_{\Vert i}/\unicode[STIX]{x1D70F}_{\bot i}]$ and ${\mathcal{D}}\simeq 2{\mathcal{C}}+2(1+Z_{i}/\unicode[STIX]{x1D70F}_{\bot i})\unicode[STIX]{x1D6E5}_{e}$ , we obtain (after some straightforward but tedious algebra) the dispersion relation for the compressive fluctuations (i.e. those with density and magnetic-field-strength fluctuations),

(3.31) $$\begin{eqnarray}[1+\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})-\unicode[STIX]{x1D6EC}^{+}][1+\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})-\unicode[STIX]{x1D6EC}^{-}]=0,\end{eqnarray}$$

where

(3.32) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}^{\pm }=-\frac{\unicode[STIX]{x1D70F}_{\Vert i}}{Z_{i}}+\frac{T_{\Vert 0i}}{T_{\bot 0i}}\frac{\unicode[STIX]{x1D70D}_{i}}{\unicode[STIX]{x1D6FD}_{\bot i}}\pm \sqrt{\left(\frac{\unicode[STIX]{x1D70F}_{\Vert i}}{\unicode[STIX]{x1D70F}_{\bot i}}+\frac{\unicode[STIX]{x1D70F}_{\Vert i}}{Z_{i}}\right)^{2}+\left(\frac{T_{\Vert 0i}}{T_{\bot 0i}}\frac{\unicode[STIX]{x1D70D}_{i}}{\unicode[STIX]{x1D6FD}_{\bot i}}\right)^{2}},\end{eqnarray}$$

and

(3.33) $$\begin{eqnarray}\unicode[STIX]{x1D70D}_{i}\doteq 1-\unicode[STIX]{x1D6FD}_{\bot }\unicode[STIX]{x1D6E5}_{e}.\end{eqnarray}$$

Equation (3.31) indicates two compressive branches of solutions, a ‘ $+$ ’ branch (corresponding to the compressive-wave eigenvector $\boldsymbol{G}^{+}$ ) and a ‘ $-$ ’ branch (corresponding to the other compressive-wave eigenvector $\boldsymbol{G}^{-}$ ), which are discussed in §§ I–4.3 and I–4.4 (see (I–4.20a,b) in particular) and consist of linear combinations of density and magnetic-field-strength fluctuations. An important limit of (3.31) is obtained for $\unicode[STIX]{x1D6FD}_{\bot s}\sim \unicode[STIX]{x1D6FD}_{\Vert s}\sim 1/\unicode[STIX]{x1D6E5}_{s}\gg 1$ , in which the ‘ $+$ ’ compressive branch, consisting primarily of magnetic-field-strength fluctuations, is collisionlessly damped at a rate

(3.34) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}\doteq -\text{i}\unicode[STIX]{x1D714}=-\frac{|k_{\Vert }|v_{A}}{\sqrt{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}_{\Vert i}}}\frac{T_{\Vert 0i}^{2}}{T_{\bot 0i}^{2}}\left(1-\displaystyle \mathop{\sum }_{s}\unicode[STIX]{x1D6FD}_{\bot s}\unicode[STIX]{x1D6E5}_{s}\right).\end{eqnarray}$$

(In this limit, the ‘ $-$ ’ branch consists mainly of density fluctuations and is strongly damped with $\text{Im}(\unicode[STIX]{x1D709}_{i})\sim 1$ .) Equation (3.34) captures the effect of pressure anisotropy on the Barnes (Reference Barnes1966) damping of slow modes (in the limit $k_{\Vert }/k_{\bot }\ll 1$ ). The damping is due to Landau-resonant particles interacting with the mirror force associated with the magnetic compressions in the wave. When

(3.35) $$\begin{eqnarray}1-\unicode[STIX]{x1D6FD}_{\bot i}\unicode[STIX]{x1D6E5}_{i}-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}<0,\end{eqnarray}$$

the proportional increase (for $\unicode[STIX]{x1D6E5}_{s}>0$ ) in the number of large-pitch-angle particles in the magnetic troughs ( $\unicode[STIX]{x1D6FF}B_{\Vert }<0$ ) of the slow mode results in more perpendicular pressure than can be stably balanced by the magnetic pressure. The result is the mirror instability (see, e.g. Southwood & Kivelson Reference Southwood and Kivelson1993). A more general criterion for the long-wavelength mirror instability in a single-ion-species plasma is given by (I–B14) – see also (3.15); for an electron–ion bi-Maxwellian plasma with arbitrary $\unicode[STIX]{x1D6FD}_{\bot s}$ and $\unicode[STIX]{x1D6FD}_{\Vert s}$ , it becomes

(3.36) $$\begin{eqnarray}1-\unicode[STIX]{x1D6FD}_{\bot i}\unicode[STIX]{x1D6E5}_{i}-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}<-\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\frac{(\unicode[STIX]{x1D6E5}_{i}-\unicode[STIX]{x1D6E5}_{e})^{2}}{1+Z_{i}/\unicode[STIX]{x1D70F}_{\Vert i}},\end{eqnarray}$$

a more restrictive condition than (3.35). The difference between the right-hand sides of (3.35) and (3.36) is due to the stabilizing effect of the parallel electric field, which is small when $\unicode[STIX]{x1D6FD}_{\bot s}\sim \unicode[STIX]{x1D6FD}_{\Vert s}\sim 1/\unicode[STIX]{x1D6E5}_{s}\gg 1$ . We refer the reader to § I–4.4.2 for further analysis and discussion.

3.6.2 KAW limit: $k_{\bot }\unicode[STIX]{x1D70C}_{e}\ll 1\ll k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , $\overline{\unicode[STIX]{x1D714}}\sim \mathit{O}(\unicode[STIX]{x1D6FC}_{i}^{1/2})$

In the limit $k_{\bot }\unicode[STIX]{x1D70C}_{e}\ll 1\ll k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , we have $\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D6FC}_{i})$ , $\unicode[STIX]{x1D6E4}_{1}(\unicode[STIX]{x1D6FC}_{i})\rightarrow 0$ for the ions and $\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D6FC}_{e})\simeq \unicode[STIX]{x1D6E4}_{1}(\unicode[STIX]{x1D6FC}_{e})\simeq 1$ for the electrons, whence ${\mathcal{B}}\simeq 1$ and ${\mathcal{E}}\simeq -1$ in (3.27). We also drop the electron plasma dispersion functions to lowest order in $k_{\bot }\unicode[STIX]{x1D70C}_{e}$ , a simplification that will be justified a posteriori. The gyrokinetic dispersion relation (3.27) becomes

(3.37) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D6FC}_{\ast }{\mathcal{A}}}{\overline{\unicode[STIX]{x1D714}}^{2}}-{\mathcal{A}}+1\right)\left(\frac{2{\mathcal{A}}}{\unicode[STIX]{x1D6FD}_{\bot i}}-{\mathcal{A}}{\mathcal{D}}+{\mathcal{C}}^{2}\right)=(-{\mathcal{A}}+{\mathcal{C}})^{2}.\end{eqnarray}$$

With $\unicode[STIX]{x1D6FD}_{\bot s}\sim \unicode[STIX]{x1D6FD}_{\Vert s}\sim \unicode[STIX]{x1D6E5}_{s}^{-1}\sim 1$ and $\overline{\unicode[STIX]{x1D714}}\sim \mathit{O}(\unicode[STIX]{x1D6FC}_{i}^{1/2})$ , we have ${\mathcal{A}}\simeq 1+T_{\bot 0i}/Z_{i}T_{\Vert 0e}$ , ${\mathcal{C}}\simeq -\unicode[STIX]{x1D6E5}_{e}$ , and ${\mathcal{D}}\simeq 2(Z_{i}/\unicode[STIX]{x1D70F}_{\bot i})\unicode[STIX]{x1D6E5}_{e}$ .Footnote ⁸ The solutions are then

(3.38) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm \frac{k_{\Vert }v_{A}k_{\bot }\unicode[STIX]{x1D70C}_{i}}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}+2/(1+Z_{i}T_{\Vert 0e}/T_{\bot 0i})-2{\mathcal{K}}}}\,\sqrt{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)},\end{eqnarray}$$

where

(3.39) $$\begin{eqnarray}{\mathcal{K}}\doteq \frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\frac{\unicode[STIX]{x1D6E5}_{e}^{2}}{1+Z_{i}T_{\Vert 0e}/T_{\bot 0i}}>0\end{eqnarray}$$

is a stabilizing term due to the parallel electric field at sub-ion-Larmor scales (cf. the right-hand side of (3.36)). Equation (3.38) is a generalization of the standard KAW dispersion relation (e.g. Kingsep, Chukbar & Yan’kov Reference Kingsep, Chukbar and Yan’kov1990),

(3.40) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm \frac{k_{\Vert }v_{A}k_{\bot }\unicode[STIX]{x1D70C}_{i}}{\sqrt{\unicode[STIX]{x1D6FD}_{i}+2/(1+Z_{i}T_{0e}/T_{0i})}},\end{eqnarray}$$

for bi-Maxwellian plasmas. Note that, for this solution, $\unicode[STIX]{x1D709}_{e}\sim \mathit{O}(k_{\bot }\unicode[STIX]{x1D70C}_{e})\ll 1$ , as promised. The KAW dispersion relation for a bi-kappa ion–electron plasma is given by (B 19).

The equations governing the corresponding sub-ion-scale fluctuations in the electron density, the parallel flow velocity and the magnetic-field strength are obtained from the gyrokinetic field equations (2.52)–(2.54) after expanding in $k_{\bot }\unicode[STIX]{x1D70C}_{e}\ll 1\ll k_{\bot }\unicode[STIX]{x1D70C}_{i}$ . The resulting equations are equivalent to (I–C88)–(I–C90) with all the $\unicode[STIX]{x1D6E4}_{\ell m}(\unicode[STIX]{x1D6FC}_{i})$ coefficients set to zero, viz.,

(3.41a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}} & = & \displaystyle -\frac{Z_{i}e\unicode[STIX]{x1D711}}{T_{\bot 0i}}=-\frac{2}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\frac{\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x1D70C}_{i}v_{A}},\end{eqnarray}$$

(3.41b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}u_{\Vert e} & = & \displaystyle \frac{c}{4\unicode[STIX]{x03C0}en_{0e}}\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=-\frac{\unicode[STIX]{x1D70C}_{i}}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9},\end{eqnarray}$$

(3.41c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}u_{\Vert i} & = & \displaystyle -\frac{c}{4\unicode[STIX]{x03C0}Z_{i}en_{0i}}\unicode[STIX]{x1D6FD}_{\Vert i}\unicode[STIX]{x1D6E5}_{i}\frac{A_{\Vert }}{\unicode[STIX]{x1D70C}_{i}^{2}}=\frac{\unicode[STIX]{x1D70C}_{i}}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\unicode[STIX]{x1D6FD}_{\Vert i}\unicode[STIX]{x1D6E5}_{i}\frac{\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D70C}_{i}^{2}},\end{eqnarray}$$

(3.41d ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{\bot i}}{2}\left(1+\frac{Z_{i}}{\unicode[STIX]{x1D70F}_{\bot i}}\right)\left(1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)^{-1}\frac{Z_{i}e\unicode[STIX]{x1D711}}{T_{\bot 0i}}\nonumber\\ \displaystyle \text{} & = & \displaystyle \sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}\left(1+\frac{Z_{i}}{\unicode[STIX]{x1D70F}_{\bot i}}\right)\left(1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)^{-1}\frac{\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x1D70C}_{i}v_{A}},\end{eqnarray}$$

$\unicode[STIX]{x1D6F7}$

$\unicode[STIX]{x1D6F9}$

(3.42a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D711}=\frac{B_{0}}{c}\unicode[STIX]{x1D6F7}\quad \text{and}\quad A_{\Vert }=-\frac{B_{0}}{v_{A}}\unicode[STIX]{x1D6F9}.\end{eqnarray}$$

Equations (3.41a–d ) are to be compared with equations (221)–(223) of S09. They reflect the fact that, for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , the ion response is effectively Boltzmann (see (2.25)), with the gyrokinetic response $\widetilde{h}_{i}$ contributing nothing either to the fields or to the flows. Note that the parallel ion flow velocity (3.41c ) is ${\sim}(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{-2}\ll 1$ smaller than the corresponding pressure-anisotropic terms in the parallel electron flow velocity (3.41b ), and thus contributes almost nothing to the parallel current.

There are several things to note about the KAW dispersion relation (3.38). First, KAWs in a bi-Maxwellian plasma are subject to both the mirror and firehose instability thresholds, whose geometric mean appears as the final term in (3.38), repeated here:

(3.43)

This makes sense, as Alfvénic and compressive fluctuations are coupled in the KAW by finite-Larmor-radius effects. Indeed, the eigenfunctions corresponding to the frequencies (3.38) are (cf. (231) of S09)

(3.44) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{\boldsymbol{k}}^{\pm } & = & \displaystyle \sqrt{\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)\left[2+\unicode[STIX]{x1D6FD}_{\bot i}\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)-\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}^{2}\right]}~\frac{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})^{1/2}}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}\frac{\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}}{\unicode[STIX]{x1D70C}_{i}}\nonumber\\ \displaystyle & & \displaystyle \mp \sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}k_{\bot }}\unicode[STIX]{x1D6F9}_{\boldsymbol{k}}.\end{eqnarray}$$

We see that the factor $1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2$ , related to the firehose threshold, is associated with the Alfvénic fluctuation $\unicode[STIX]{x1D6FF}B_{\bot \boldsymbol{k}}\propto k_{\bot }\unicode[STIX]{x1D6F9}_{\boldsymbol{k}}$ ; the factor $1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}}$ , related to the mirror threshold, is associated with the compressive fluctuation $\unicode[STIX]{x1D6FF}B_{\Vert \boldsymbol{k}}\propto \unicode[STIX]{x1D6F7}_{\boldsymbol{k}}/\unicode[STIX]{x1D70C}_{i}$ .Footnote ⁹ The KAW eigenfunctions $\unicode[STIX]{x1D6E9}_{\boldsymbol{k}}^{\pm }$ combine both effects.

Secondly, the ion pressure anisotropy does not appear in (3.38). Physically, this is because the ion response is essentially Boltzmann (2.25), which produces an isothermal pressure response

(3.45a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}p_{\bot i}=T_{\bot 0i}\unicode[STIX]{x1D6FF}n_{i}\quad \text{and}\quad \unicode[STIX]{x1D6FF}p_{\Vert i}=T_{\Vert 0i}\unicode[STIX]{x1D6FF}n_{i}.\end{eqnarray}$$

By contrast, the electron pressure response is (see (I–2.45a,b))

(3.45c,d ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}p_{\bot e}=T_{\bot 0e}\unicode[STIX]{x1D6FF}n_{e}-p_{\bot 0e}\unicode[STIX]{x1D6E5}_{e}\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}\quad \text{and}\quad \unicode[STIX]{x1D6FF}p_{\Vert e}=T_{\Vert 0e}\unicode[STIX]{x1D6FF}n_{e},\end{eqnarray}$$

so a magnetic-field-strength perturbation produces a perpendicular electron temperature perturbation proportional to the electron pressure anisotropy. This difference arises because, at scales satisfying $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , the ions do not ‘see’ the magnetic-field-strength fluctuation, which varies rapidly along the ion gyro-orbit and is thus ring-averaged away. In this situation, the ions have no reason to adjust their perpendicular pressure according to the changes in the magnetic-field strength. Put differently, because $\unicode[STIX]{x1D707}$ conservation for the ions holds with respect to the magnetic-field strength averaged over the particle orbit (see (2.43)), fluctuations in field strength ( $\unicode[STIX]{x1D6FF}B_{\Vert }$ ) as seen by the ions are reduced by a factor of $2J_{1}(a_{i})/a_{i}\sim a_{i}^{-3/2}\ll 1$ . For the electrons, on the other hand, we have $2J_{1}(a_{e})/a_{e}\simeq 1$ , so $\unicode[STIX]{x1D6FF}p_{\bot e}$ fully adjusts to the fluctuating field strength (see the second term in (3.45c,d ), as well as § I–2.5.2).

The ion pressure anisotropy is also absent from the firehose factor $1+(\unicode[STIX]{x1D6FD}_{\Vert e}/2)\unicode[STIX]{x1D6E5}_{e}$ in (3.38) and (3.44) for a similar reason: pressure-anisotropy corrections to the effective tension in the field lines stem from the $\hat{\boldsymbol{b}}\unicode[STIX]{x1D6FF}\hat{\boldsymbol{b}}(p_{\bot 0s}-p_{\Vert 0s})$ term in the perturbed magnetized pressure tensor, which is only relevant if species $s$ can ‘see’ the field fluctuation $\unicode[STIX]{x1D6FF}\hat{\boldsymbol{b}}$ . This result explains why sub-ion-Larmor cascades of KAWs were observed in the hybrid-kinetic simulations of Kunz et al. (Reference Kunz, Schekochihin and Stone2014) and Kunz et al. (Reference Kunz, Stone and Quataert2016), despite the larger scales being driven mirror unstable by positive ion pressure anisotropy: in those calculations, the electrons were assumed to be pressure isotropic and so KAWs were stable.

Thirdly, the KAW in the gyrokinetic limit satisfies perpendicular pressure balance:

(3.45) $$\begin{eqnarray}\frac{B_{0}\unicode[STIX]{x1D6FF}B_{\Vert }}{4\unicode[STIX]{x03C0}}+\unicode[STIX]{x1D6FF}p_{\bot e}+\unicode[STIX]{x1D6FF}p_{\bot i}=p_{\bot 0i}\left(1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\frac{2}{\unicode[STIX]{x1D6FD}_{\bot i}}\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}+p_{\bot 0i}\left(1+\frac{Z_{i}}{\unicode[STIX]{x1D70F}_{\bot i}}\right)\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}=0,\end{eqnarray}$$

which follows from combining (3.40a,d ) and (3.45a,b ). This equation states that an increase in number density must be accompanied by a decrease in the magnetic-field strength, the amount of this decrease depending upon the factor $1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2$ . If $\unicode[STIX]{x1D6E5}_{e}>0$ , then the magnetic-field lines must inflate further in order to maintain perpendicular pressure balance as large-pitch-angle particles are squeezed into the magnetic troughs. When the concentration of these particles leads to more perpendicular pressure than can be stably balanced by the magnetic pressure, the troughs must grow deeper to compensate. In the long-wavelength limit, the pressure-balanced slow mode then goes unstable to the mirror instability. In the short-wavelength limit, the KAW goes unstable for the same reason.

Finally, there is obviously something amiss about (3.41d ) and (3.44) when $\unicode[STIX]{x1D6E5}_{e}=2/\unicode[STIX]{x1D6FD}_{\bot e}$ . In this case, $\unicode[STIX]{x1D6FF}B_{\Vert }\neq 0$ even though $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D6FF}n_{e}=0$ . The perpendicular pressure balance (3.45) is then achieved by balancing the perturbed magnetic pressure $B_{0}\unicode[STIX]{x1D6FF}B_{\Vert }/4\unicode[STIX]{x03C0}$ with the adiabatic electron response $\unicode[STIX]{x1D6FF}p_{\bot e}=p_{\bot 0e}(\unicode[STIX]{x1D6FF}T_{\bot e}/T_{\bot 0e})=-p_{\bot 0e}\unicode[STIX]{x1D6E5}_{e}\unicode[STIX]{x1D6FF}B_{\Vert }/B_{0}$ . The KAW then consists only of magnetic-field fluctuations, with polarization

(3.46) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot \boldsymbol{k}}}{\unicode[STIX]{x1D6FF}B_{\Vert \boldsymbol{k}}}=\pm \hat{\boldsymbol{z}}\times \frac{\boldsymbol{k}_{\bot }}{k_{\bot }}\left(2+\frac{2}{\unicode[STIX]{x1D6FD}_{\bot e}}\right)^{-1/2}\end{eqnarray}$$

and growth/decay rate

(3.47) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\pm k_{\Vert }v_{A}k_{\bot }d_{i}\left[\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\right]^{1/2}=\pm k_{\Vert }v_{A}k_{\bot }d_{i}\,\frac{(2+2/\unicode[STIX]{x1D6FD}_{\bot e})^{1/2}}{1+2/\unicode[STIX]{x1D6FD}_{\bot e}}.\end{eqnarray}$$

The instability is allowed because perpendicular pressure balance is maintained for any $\unicode[STIX]{x1D6FF}B_{\Vert }$ so long as $\unicode[STIX]{x1D6E5}_{e}=2/\unicode[STIX]{x1D6FD}_{\bot e}$ .

Generally speaking, the location of the wavenumber transition from Alfvén waves to KAWs during a turbulent cascade (at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ for a Maxwellian plasma) is a function of the electron pressure anisotropy. This dependence may be tested by looking for a shift in the ion-Larmor-scale spectral break in measurements of Alfvénic turbulence in the non-Maxwellian solar wind and in simulations of gyrokinetic turbulence in bi-Maxwellian plasmas (using our theory). Furthermore, in a Maxwellian, high- $\unicode[STIX]{x1D6FD}$ plasma, the dispersion relation exhibits a sharp frequency jump at the Alfvén-wave–KAW transition (see figure 8c of S09). This jump is accompanied by very strong ion-Landau damping. It is easy to imagine from the above discussion that the electron pressure anisotropy, by affecting the rate of collisionless damping, would thus play an important role in determining the fraction of wave energy that is damped on the ions versus cascaded down to electron scales. This is manifest in numerical solutions to (3.27), which we present in § 3.6.4.

3.6.3 Slow waves in the sub-ion-Larmor range: $k_{\bot }\unicode[STIX]{x1D70C}_{e}\ll 1\ll k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , $\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})\sim \mathit{O}(\unicode[STIX]{x1D6FC}_{i}^{3/2})$

The approximate values of ${\mathcal{A}}$ , ${\mathcal{C}}$ , and ${\mathcal{D}}$ derived in § 3.6.2 made use of the asymptotic forms $\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D6FC}_{i})\sim 1/\sqrt{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{i}}$ and $\unicode[STIX]{x1D6E4}_{1}(\unicode[STIX]{x1D6FC}_{i})\sim 1/\sqrt{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{i}^{3}}$ for $\unicode[STIX]{x1D6FC}_{i}\gg 1$ . Terms proportional to these asymptotically small $\unicode[STIX]{x1D6E4}_{m}$ coefficients were dropped, assuming that all multiplicative factors of them were of order unity. However, solutions of the dispersion relation (3.27) exist with $\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})\sim \mathit{O}(\unicode[STIX]{x1D6FC}_{i}^{3/2})$ , and so some of these terms are non-negligible and must be retained. In this case, we find the leading-order expressions

(3.48) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}\sim \frac{\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})}{\sqrt{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{i}}}\frac{T_{\bot 0i}}{T_{\Vert 0i}}, & \displaystyle\end{eqnarray}$$

(3.49) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{C}}\sim \frac{\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})}{\sqrt{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{i}^{3}}}\frac{T_{\bot 0i}}{T_{\Vert 0i}}-\left[\unicode[STIX]{x1D6E5}_{e}+\unicode[STIX]{x1D709}_{e}Z_{\text{M}}(\unicode[STIX]{x1D709}_{e})\frac{T_{\bot 0e}}{T_{\Vert 0e}}\right], & \displaystyle\end{eqnarray}$$

(3.50) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}\sim \frac{\unicode[STIX]{x1D709}_{i}Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})}{\sqrt{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{i}^{3}}}\frac{T_{\bot 0i}}{T_{\Vert 0i}}+\frac{2Z_{i}}{\unicode[STIX]{x1D70F}_{\bot i}}\left[\unicode[STIX]{x1D6E5}_{e}+\unicode[STIX]{x1D709}_{e}Z_{\text{M}}(\unicode[STIX]{x1D709}_{e})\frac{T_{\bot 0e}}{T_{\Vert 0e}}\right]. & \displaystyle\end{eqnarray}$$

With ${\mathcal{B}}\simeq 1$ and ${\mathcal{E}}\simeq -1$ , the dispersion relation (3.27) becomes

(3.51) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D6FC}_{\ast }}{\overline{\unicode[STIX]{x1D714}}^{2}}-1\right)\left(\frac{2}{\unicode[STIX]{x1D6FD}_{\bot i}}-{\mathcal{D}}\right)\simeq 1.\end{eqnarray}$$

In general, each of the terms in this equation is $\mathit{O}(1)$ and a numerical solution is required. However, when $\unicode[STIX]{x1D6FD}_{\bot s}\sim \unicode[STIX]{x1D6FD}_{\Vert s}\sim \unicode[STIX]{x1D6E5}_{s}^{-1}\gg 1$ , a simple analytic solution can be obtained. In this limit, the dispersion relation (3.51) becomes ${\mathcal{D}}\simeq 1$ . Writing $\unicode[STIX]{x1D709}_{i}=-\text{i}\widetilde{\unicode[STIX]{x1D709}}_{i}$ and using $Z_{\text{M}}(\unicode[STIX]{x1D709}_{i})\simeq 2\text{i}\sqrt{\unicode[STIX]{x03C0}}\exp (\widetilde{\unicode[STIX]{x1D709}}_{i}^{2})$ for $\widetilde{\unicode[STIX]{x1D709}}_{i}\gg 1$ , the solution is $\widetilde{\unicode[STIX]{x1D709}}_{i}\sim \sqrt{3\ln |k_{\bot }\unicode[STIX]{x1D70C}_{i}|}$ up to logarithmically small corrections, or

(3.52) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}\sim -|k_{\Vert }|v_{A}\sqrt{3\unicode[STIX]{x1D6FD}_{\Vert i}\ln |k_{\bot }\unicode[STIX]{x1D70C}_{i}|}.\end{eqnarray}$$

This is an aperiodic, Barnes-damped slow mode in the sub-ion-Larmor range. Electron contributions to this solution are small, and the pressure anisotropy of either species makes no appearance to leading order. These features can be seen in the numerical solutions, which we now present.

Figure 1.

Real and imaginary parts of the frequency $\unicode[STIX]{x1D714}=\text{Re}(\unicode[STIX]{x1D714})+\text{i}\unicode[STIX]{x1D6FE}$ on the Alfvénic branch as functions of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ for $\unicode[STIX]{x1D6FD}_{\Vert i}=\unicode[STIX]{x1D6FD}_{\Vert e}=1$ , 10 and 100. (Recall that $\unicode[STIX]{x1D70C}_{i}\doteq v_{\text{th}\bot i}/\unicode[STIX]{x1D6FA}_{i}\propto \sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}$ , and so the abscissa is a function of pressure anisotropy.) Equal ion and electron equilibrium pressure anisotropies $\unicode[STIX]{x1D6E5}_{s}$ are equally spaced from negative (blue) to positive (red) values, the hues of the lines darkening towards larger absolute values (as indicated by the colour bars). The solution for a Maxwellian equilibrium, $\unicode[STIX]{x1D6E5}_{s}=0$ , is shown as the black line. Stable (unstable) modes are denoted by solid (dotted) lines. See § 3.6.4 for discussion.

3.6.4 Numerical solutions for an electron–ion bi-Maxwellian plasma

In this section, the dispersion relation (3.27) is solved numerically. We group the resulting solutions into ‘Alfvénic’ and ‘compressive’ ones, based on whether their long-wavelength limit returns the Alfvén-wave solution (3.29) or the compressive solution (3.31), respectively. These two branches are coupled at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gtrsim 1$ by finite-Larmor-radius effects.

Figure 1 displays the real part, $\text{Re}(\unicode[STIX]{x1D714})$ , and the absolute value of the imaginary part, $|\unicode[STIX]{x1D6FE}|\doteq |\text{Im}(\unicode[STIX]{x1D714})|$ , of the complex frequency $\unicode[STIX]{x1D714}$ (normalized to $k_{\Vert }v_{A}$ ) describing waves on the Alfvénic branch as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , for varying $\unicode[STIX]{x1D6FD}_{\Vert i}$ and pressure anisotropy $\unicode[STIX]{x1D6E5}_{i}$ . We take $\unicode[STIX]{x1D6FD}_{\Vert e}=\unicode[STIX]{x1D6FD}_{\Vert i}$ and $\unicode[STIX]{x1D6E5}_{e}=\unicode[STIX]{x1D6E5}_{i}$ (i.e.  $T_{\Vert 0e}=T_{\Vert 0i}$ ), so that the equal-temperature ions and electrons both contribute to the total pressure anisotropy of the equilibrium plasma. The solid black lines correspond to $\unicode[STIX]{x1D6E5}_{s}=0$ , i.e. to a Maxwellian equilibrium, and match those presented in figure 4 of Howes et al. (Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006) and figure 8 of S09. The red (blue) lines correspond to positive (negative) equilibrium pressure anisotropy, their hues darkening towards larger absolute values (as indicated by the accompanying colour bars).

For all values of $\unicode[STIX]{x1D6FD}_{\Vert i}$ and $\unicode[STIX]{x1D6E5}_{s}$ , the analytic solution (3.29) accurately describes the Alfvén waves (and the firehose-unstable modes) at long wavelengths, with negative (positive) pressure anisotropies decreasing (increasing) the effective Alfvén speed and, thus, the real part of the frequency. (Solutions for $\unicode[STIX]{x1D6FD}_{\Vert i}\ll 1$ are not presented, as very extreme pressure anisotropies are required to appreciably modify the waves in this case.) The weak damping affecting these pressure-anisotropy-modified Alfvén waves is obtained analytically in appendix B.1 for $\unicode[STIX]{x1D6FD}_{\Vert i}\gg 1$ with $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim \mathit{O}(\unicode[STIX]{x1D6FD}_{\Vert i}^{-1/4})$ , $\overline{\unicode[STIX]{x1D714}}\sim \mathit{O}(1)$ (see (B 4)):

(3.53) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6FE}}\simeq -\frac{9}{16}\frac{k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}}{2}\sqrt{\frac{\unicode[STIX]{x1D6FD}_{\Vert i}}{\unicode[STIX]{x03C0}}},\end{eqnarray}$$

which is independent (to leading order) of the pressure anisotropy (as it indeed is in the figure). For $\unicode[STIX]{x1D6FD}_{\Vert i}^{-1/4}\ll k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ (see (B 5a )), or $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim \mathit{O}(1)$ , $\overline{\unicode[STIX]{x1D714}}\sim \mathit{O}(\unicode[STIX]{x1D6FD}_{\Vert i}^{-1/2})$ (see (B 8b )), we have

(3.54) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6FE}}\simeq -\frac{8}{9}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert i}}{2}\unicode[STIX]{x1D6E5}_{i}+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\left(\frac{k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}}{2}\right)^{-1}\sqrt{\frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FD}_{\Vert i}}}\frac{T_{\bot 0i}}{T_{\Vert 0i}},\end{eqnarray}$$

which captures well the decrease of $|\overline{\unicode[STIX]{x1D6FE}}|$ with $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ and its dependence on the pressure anisotropy that are seen in the $\unicode[STIX]{x1D6FD}_{\Vert i}=10$ and $100$ plots near $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ . For $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , the damping rate increases again with $\unicode[STIX]{x1D6FC}_{i}$ (see (B 9)), this time with an explicit dependence on the electron pressure anisotropy:

(3.55) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6FE}}\simeq -\frac{k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}}{2}\sqrt{\frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FD}_{\Vert i}}}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\frac{T_{\bot 0e}^{2}}{T_{\Vert 0e}^{2}}\frac{T_{\Vert 0i}}{T_{\bot 0i}}\left(\frac{Z_{i}}{\unicode[STIX]{x1D70F}_{\Vert i}}\frac{Z_{i}m_{e}}{m_{i}}\right)^{1/2}.\end{eqnarray}$$

Positive anisotropy $\unicode[STIX]{x1D6E5}_{e}>0$ increases the rate of collisionless damping by effectively increasing the Alfvén speed and thereby brings the wave frequency (see (B 9)) closer to the ion-Landau resonance; this is manifest in the $\unicode[STIX]{x1D6FD}_{\Vert i}=10$ and $100$ panels of figure 1. Negative electron pressure anisotropy, on the other hand, drives these KAWs firehose unstable for $\unicode[STIX]{x1D6E5}_{e}<-2/\unicode[STIX]{x1D6FD}_{\Vert e}$ . Note that there exist Alfvénic fluctuations that are firehose unstable at long wavelengths, which become stable at short wavelengths (the blue lines that are absent at low $k_{\bot }$ but appear at high $k_{\bot }$ in the $\text{Re}(\unicode[STIX]{x1D714})$ panel). This is because the ion contribution to the total pressure anisotropy is only relevant at long wavelengths. Both the analytic solutions (cf. (3.29) and (3.38)) and the numerical solutions shown in figure 2 demonstrate this point.

Figure 2.

Imaginary part of the frequency $\unicode[STIX]{x1D714}=\text{Re}(\unicode[STIX]{x1D714})+\text{i}\unicode[STIX]{x1D6FE}$ on the Alfvénic branch as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ for $\unicode[STIX]{x1D6FD}_{\Vert i}=\unicode[STIX]{x1D6FD}_{\Vert e}=100$ . Either (a) the equilibrium ion distribution is taken to be isotropic, $\unicode[STIX]{x1D6E5}_{i}=0$ , or (b) the equilibrium electron distribution is taken to be isotropic, $\unicode[STIX]{x1D6E5}_{e}=0$ . Stable (unstable) modes are denoted by solid (dotted) lines. Sub-ion-Larmor fluctuations are primarily affected by electron pressure anisotropy, while long-wavelength fluctuations and the transition at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ are affected by both. See § 3.6.4 for discussion.

Figure 3.

Ratio of damping/growth rate $|\unicode[STIX]{x1D6FE}|$ and $\text{Re}(\unicode[STIX]{x1D714})$ on the Alfvénic branch, partitioned into contributions from damping on ions ( $|\unicode[STIX]{x1D6FE}_{i}|$ ; a,c,e) and electrons ( $|\unicode[STIX]{x1D6FE}_{e}|$ ; b,d,f), as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ for $\unicode[STIX]{x1D6FD}_{\Vert i}=\unicode[STIX]{x1D6FD}_{\Vert e}=1$ , $10$ , and $100$ . Equal ion and electron equilibrium pressure anisotropies $\unicode[STIX]{x1D6E5}_{s}$ are varied from negative (blue) to positive (red) values, with $\unicode[STIX]{x1D6E5}_{s}=0$ indicated by the black line. Stable (unstable) modes are denoted by solid (dotted) lines. See § 3.6.4 for discussion.

In § 3.6.2, we speculated that the location of the wavenumber transition from Alfvén waves to KAWs and the consequent ion-Larmor-scale spectral break that is routinely observed in kinetic turbulence (e.g. Leamon et al. Reference Leamon, Smith, Ness, Matthaeus and Wong1998, Reference Leamon, Smith, Ness and Wong1999; Bale et al. Reference Bale, Kellogg, Mozer, Horbury and Reme2005; Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Markovskii, Vasquez & Smith Reference Markovskii, Vasquez and Smith2008; Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Chen et al. Reference Chen, Leung, Boldyrev, Maruca and Bale2014) should be a function of the pressure anisotropy. This conjecture is supported especially by the $\unicode[STIX]{x1D6FD}_{\Vert i}=10$ and $100$ plots, which display a sharp frequency jump near $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ at a location dependent upon the pressure anisotropy. This dependence can be estimated by equating the asymptotic damping rates (3.53) or (3.54) to the real frequency (3.29); this yields

(3.56) $$\begin{eqnarray}(k_{\bot }\unicode[STIX]{x1D70C}_{i})_{\text{break}}\propto \left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert i}}{2}\unicode[STIX]{x1D6E5}_{i}+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)^{1/4}.\end{eqnarray}$$

Positive (negative) anisotropies are thus predicted to shift the ion-Larmor-scale spectral break towards larger (smaller) perpendicular wavenumbers. This frequency jump is seen to be accompanied by very strong ion Landau damping ( $|\overline{\unicode[STIX]{x1D6FE}}|\sim 1$ ), with order-of-magnitude variations in the damping rate depending upon $\unicode[STIX]{x1D6E5}_{s}$ . Note further that the width of the $\text{Re}(\unicode[STIX]{x1D714})=0$ gap at large $\unicode[STIX]{x1D6FD}_{\Vert i}$ near $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ , evident in the $\unicode[STIX]{x1D6FD}_{\Vert i}=100$ plot, increases with $\unicode[STIX]{x1D6E5}_{s}>0$ . (Figure 2 shows that both the ion and electron pressure anisotropies affect the strength of the collisionless damping and the location of the break point.)

Because the amount of Landau damping at these scales is likely to be related to how much energy ultimately goes into heating the ions or the electrons, pressure anisotropy ought to be considered alongside $\unicode[STIX]{x1D6FD}_{i}$ and $T_{0i}/T_{0e}$ when assessing the efficiency of ion heating. This consideration may be important for developing a quantitative and predictive theory of collisionless radiatively inefficient accretion flows, for which the efficiency of ion heating is a key unknown (e.g. Quataert & Gruzinov Reference Quataert and Gruzinov1999; Howes Reference Howes2010) and in which pressure anisotropy is predicted to govern much of the nonlinear dynamics and thermodynamics (e.g. Sharma et al. Reference Sharma, Hammett, Quataert and Stone2006, Reference Sharma, Quataert, Hammett and Stone2007; Kunz et al. Reference Kunz, Stone and Quataert2016). We investigate this possibility quantitatively using a simple cascade model in § 4.5. Here we simply examine which species (ions or electrons) contributes the most to the Landau damping of the Alfvénic fluctuations at different scales. In figure 3, the ratio of the absolute value of the imaginary part and the real part of the frequency, separated into contributions from damping on the ions (a,c,e) and on the electrons (b,d,f), are shown versus $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , $\unicode[STIX]{x1D6FD}_{\Vert i}$ , and $\unicode[STIX]{x1D6E5}_{s}$ . (See § 4.5 and, in particular, (4.58) for how this partitioning of $\unicode[STIX]{x1D6FE}$ into $\unicode[STIX]{x1D6FE}_{i}$ and $\unicode[STIX]{x1D6FE}_{e}$ is computed.) In all cases, the collisionless damping on the ions is stronger than that on the electrons for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\lesssim 2$ , whereas the collisionless damping on the electrons at sub-ion-Larmor scales is dominant. While the latter is much stronger than the former, one would anticipate in a turbulent cascade that the energy carried by the sub-ion-Larmor fluctuations would be much smaller than that carried by the long-wavelength Alfvénic fluctuations, and so the damping rates on both the ions and the electrons (and the impact of pressure anisotropy on them) are likely important for calculating ion versus electron heating (see § 4.5).

Figure 4.

Imaginary part of the frequency $\unicode[STIX]{x1D714}=\text{Re}(\unicode[STIX]{x1D714})+\text{i}\unicode[STIX]{x1D6FE}$ on the compressive branch as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ for $\unicode[STIX]{x1D6FD}_{\Vert i}=\unicode[STIX]{x1D6FD}_{\Vert e}=1$ and $100$ . Equal ion and electron equilibrium pressure anisotropies $\unicode[STIX]{x1D6E5}_{s}$ are varied from negative (blue) to positive (red) values, with $\unicode[STIX]{x1D6E5}_{s}=0$ indicated by the black line. Stable (unstable) modes are denoted by solid (dotted) lines. See § 3.6.4 for discussion.

Figure 5.

Imaginary part of the frequency $\unicode[STIX]{x1D714}=\text{Re}(\unicode[STIX]{x1D714})+\text{i}\unicode[STIX]{x1D6FE}$ on the compressive branch as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ for $\unicode[STIX]{x1D6FD}_{\Vert i}=\unicode[STIX]{x1D6FD}_{\Vert e}=10$ and either (a) $\unicode[STIX]{x1D6E5}_{i}=\unicode[STIX]{x1D6E5}_{e}$ or (b) $\unicode[STIX]{x1D6E5}_{i}=0$ . Pressure anisotropies are varied from negative (blue) to positive (red) values, the hues of the lines darkening towards larger absolute values (as indicated by the colour bars). The solution for a Maxwellian equilibrium, $\unicode[STIX]{x1D6E5}_{s}=0$ , is indicated by the black line. Stable (unstable) modes are denoted by solid (dotted) lines. Ion pressure anisotropy only affects the behaviour of the modes at long wavelengths, $k_{\bot }\unicode[STIX]{x1D70C}_{i}\lesssim 1$ . See § 3.6.4 for discussion.

Figures 4 and 5 display the imaginary part of the frequency describing waves on the compressive branch as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , for varying plasma $\unicode[STIX]{x1D6FD}_{\Vert i}$ and pressure anisotropy $\unicode[STIX]{x1D6E5}_{s}$ . These modes always have $\text{Re}(\unicode[STIX]{x1D714})=0$ and thus are aperiodic. As in figure 1, the solid black lines correspond to $\unicode[STIX]{x1D6E5}_{s}=0$ , while the red (blue) lines correspond to positive (negative) equilibrium pressure anisotropy. In figure 4, we take $\unicode[STIX]{x1D6FD}_{\Vert e}=\unicode[STIX]{x1D6FD}_{\Vert i}$ and $\unicode[STIX]{x1D6E5}_{e}=\unicode[STIX]{x1D6E5}_{i}$ , so that the equal-temperature ions and electrons both contribute to the total pressure anisotropy of the equilibrium plasma. By comparing solutions obtained using $\unicode[STIX]{x1D6E5}_{i}=\unicode[STIX]{x1D6E5}_{e}$ with those obtained using $\unicode[STIX]{x1D6E5}_{i}=0$ , figure 5 demonstrates that it is only the pressure anisotropy of the electrons that affects the sub-ion-Larmor range (as predicted in § 3.6.3).

At long wavelengths ( $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ ), the damping/growth rate follows closely the ‘ $+$ ’ branch obtained from the dispersion relation (3.31), as it is the more weakly damped of the two branches.Footnote ¹⁰ This is the branch that goes over to the aperiodic Barnes-damped/mirror-unstable slow mode for $\unicode[STIX]{x1D6FD}_{\bot s},\unicode[STIX]{x1D6FD}_{\Vert s}\gg 1$ (see (3.34)). As such, it is unstable if (3.36) is satisfied; the figures show this. At $k_{\bot }\unicode[STIX]{x1D70C}\sim 1$ , the character of the mode changes abruptly to become either strongly damped at a rate independent of the pressure anisotropy (see (3.52) for an approximation solution) or, if the pressure anisotropy is sufficiently positive, mirror unstable at a rate given approximately by (3.38). Note that, in the latter case, the compressive solution is coupled to the Alfvénic solution, and the unstable mode is a KAW.

At electron-Larmor scales ( $k_{\bot }\unicode[STIX]{x1D70C}_{e}\gtrsim 1$ ), both branches of the dispersion relation – Alfvénic and compressive – are strongly damped via Landau resonance on the electrons.

This completes our discussion of the linear gyrokinetic theory. We now make use of the physics contained in these results to explain how pressure anisotropy affects the nonlinear turbulent cascade of free energy to small scales in phase space.

4.3.1 ERMHD and the KAW cascade

Using (3.41d ) to relate the magnetic-field-strength fluctuation $\unicode[STIX]{x1D6FF}B_{\Vert }$ in the sub-ion-Larmor range to the electrostatic potential $\unicode[STIX]{x1D711}$ , writing the energy in the perpendicular magnetic field $\unicode[STIX]{x1D6FF}B_{\bot }^{2}$ as $|\unicode[STIX]{x1D735}_{\bot }A_{\Vert }|^{2}$ , and re-introducing the stream and flux functions via (3.42), the KAW invariant that appears in (4.18) may be written as

(4.19a ) $$\begin{eqnarray}\displaystyle W_{\text{KAW}} & = & \displaystyle \int \text{d}^{3}\boldsymbol{r}\,\frac{m_{i}n_{0i}}{2}\left\{\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}|^{2}\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}}}{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2)^{2}}\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)\!\!\left[2+\unicode[STIX]{x1D6FD}_{\bot i}\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)-\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}^{2}\right]\frac{\unicode[STIX]{x1D6F7}^{2}}{\unicode[STIX]{x1D70C}_{i}^{2}}\right\}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.19b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \int \text{d}^{3}\boldsymbol{r}\,\frac{m_{i}n_{0i}}{4}(|\unicode[STIX]{x1D6E9}^{+}|^{2}+|\unicode[STIX]{x1D6E9}^{-}|^{2}),\end{eqnarray}$$

$+$

$-$

$\unicode[STIX]{x1D6E9}^{+}$

$\unicode[STIX]{x1D6E9}^{-}$

$\unicode[STIX]{x1D6E9}^{+}$

$\unicode[STIX]{x1D6E9}^{-}$

$k_{\bot }$

We now show that the form of $W_{\text{KAW}}$ in (4.19) can be similarly obtained from the equations describing nonlinear KAWs – so-called electron reduced magnetohydrodynamics (ERMHD) – and thus is the ‘fluid energy’ conserved during the nonlinear cascade of KAWs to small spatial scales.

In appendix C.8 of Paper I, we derived, via a mass-ratio expansion, nonlinear equations describing the electron kinetics for $k_{\bot }\unicode[STIX]{x1D70C}_{e}\sim k_{\bot }\unicode[STIX]{x1D70C}_{i}(m_{e}/m_{i})^{1/2}\ll 1$ . For the purposes of the present paper, the two most important electron equations are those specifying the parallel electric field (see (I–C72))

(4.20) $$\begin{eqnarray}E_{\Vert }=-\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}-\frac{1}{c}\frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}=-\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\frac{T_{\Vert 0e}}{e}\left(\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}+\unicode[STIX]{x1D6E5}_{e}\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}\right)\end{eqnarray}$$

and what amounts to a reduced electron continuity equation (see (I–C78) and the accompanying discussion in § I–C.8.3):

(4.21) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\left(\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}-\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}\right)+\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}u_{\Vert e}^{(1)}+\frac{cT_{\bot 0e}}{eB_{0}}\left\{\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}},\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}\right\}=0,\end{eqnarray}$$

where

(4.22) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\doteq \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\frac{c}{B_{0}}\{\unicode[STIX]{x1D711},\ldots \!\}\end{eqnarray}$$

is the Lagrangian time derivative measured in a frame transported at the $\boldsymbol{E}\times \boldsymbol{B}$ drift velocity, $\boldsymbol{u}_{\bot }$ (defined by (2.45)), and

(4.23) $$\begin{eqnarray}u_{\Vert e}^{(1)}\doteq \frac{1}{n_{0e}}\int \text{d}^{3}\boldsymbol{v}\,v_{\Vert }g_{e}^{(1)}=\frac{c}{4\unicode[STIX]{x03C0}en_{0e}}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }\end{eqnarray}$$

is the parallel electron flow velocity associated with the $\mathit{O}(\sqrt{m_{e}/m_{i}})$ piece of $g_{e}$ , denoted $g_{e}^{(1)}$ in Paper I (see § I–C.8).Footnote ¹⁴

Using (3.41a ) and (3.41d ) for the electron density fluctuation and the magnetic-field-strength fluctuation in the sub-ion-Larmor-scale range, and introducing $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D6F9}$ via (3.42), we find that equations (4.20) and (4.21) become, respectively,

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}t}=v_{A}\left\{1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\left[1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\left(\frac{1+\unicode[STIX]{x1D70F}_{\bot i}/Z_{i}}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}\right)\right]\right\}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}, & \displaystyle\end{eqnarray}$$

(4.25) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}t}\left[2+\unicode[STIX]{x1D6FD}_{\bot i}\left(\frac{1+Z_{i}/\unicode[STIX]{x1D70F}_{\bot i}}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}\right)\right]=-v_{A}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}_{i}^{2}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}. & \displaystyle\end{eqnarray}$$

These equations generalize the linear theory of KAWs (§ 3.6.2) to the nonlinear regime (cf. equations (226)–(227) of S09, to which (4.24)–(4.25) reduce in the pressure-isotropic limit). Introducing the perturbed magnetic-field vector

(4.26) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}\boldsymbol{B}}{B_{0}}=\frac{1}{v_{A}}\hat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}+\hat{\boldsymbol{z}}\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FF}B_{\Vert }$ given by (3.41d ), equations (4.24) and (4.25) can be recast as two coupled evolution equations for the perpendicular and parallel components of the perturbed magnetic field, respectively.

We now show that (4.24) and (4.25) conserve the energy $W_{\text{KAW}}$ given by (4.19). First, apply $\unicode[STIX]{x1D735}_{\bot }$ to (4.24) and dot the result with $(1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2)\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}$ ; then integrate over space and use integration by parts to obtain

(4.27) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{r}\,\frac{1}{2}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}|^{2}=-\int \text{d}^{3}\boldsymbol{r}\,v_{A}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\nonumber\\ \displaystyle & & \displaystyle \quad \times \left\{1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\left[1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\left(\frac{1+\unicode[STIX]{x1D70F}_{\bot i}/Z_{i}}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}\right)\right]\right\}(\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9})\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}.\end{eqnarray}$$

Next, we multiply (4.25) by $(1+Z_{i}T_{\Vert 0e}/T_{\bot 0i})(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2)(\unicode[STIX]{x1D6F7}/\unicode[STIX]{x1D70C}_{i}^{2})$ , integrate over space and use integration by parts to find

(4.28) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-5.0pt}\frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{r}\,\frac{1}{2}\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)\left[2+\unicode[STIX]{x1D6FD}_{\bot i}\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)-\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}^{2}\right]\frac{\unicode[STIX]{x1D6F7}^{2}}{\unicode[STIX]{x1D70C}_{i}^{2}}\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\quad =\int \text{d}^{3}\boldsymbol{r}\,v_{A}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\left(1+\frac{Z_{i}T_{\Vert 0e}}{T_{\bot 0i}}\right)\left(1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)(\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7})\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}.\end{eqnarray}$$

Further multiplying (4.28) by a factor of $(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})/(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2)^{2}$ and adding it to (4.27) cancels their right-hand sides and gives

(4.29) $$\begin{eqnarray}\frac{\text{d}W_{\text{KAW}}}{\text{d}t}=0,\end{eqnarray}$$

with the KAW invariant $W_{\text{KAW}}$ identical to that given by (4.19).

It is clear from the ERMHD equations (4.24) and (4.25), as well as from the form of the quadratic quantity they conserve (4.19), that electron pressure anisotropy influences the ability of the plasma to support certain magnetic-field perturbations. As the firehose threshold is approached from the stable side (i.e.  $1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2\rightarrow 0^{+}$ ), the energetic cost for a KAW to bend the magnetic-field lines is reduced. Likewise, as the mirror threshold is approached from the stable side (i.e.  $1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}}\rightarrow 0^{+}$ ), the energetic cost for a KAW to compress/rarefy the magnetic-field lines is reduced. Beyond these thresholds, it becomes energetically profitable to grow these fluctuations. (Again, note that these instabilities themselves fall outside of the gyrokinetic ordering employed here.) Equation (4.29) thus establishes a thermodynamic connection between the linear and nonlinear stability of KAWs.

Before proceeding to investigate the ion-entropy cascade, in which $W_{\widetilde{h}_{i}}$ is independently conserved, we briefly remark on the case of $\unicode[STIX]{x1D6E5}_{e}=2/\unicode[STIX]{x1D6FD}_{\bot e}$ , for which the nonlinear ERMHD equations (4.24) and (4.25) are ill posed. Instead, the appropriate nonlinear equations are

(4.30) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{v_{A}d_{i}}\frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}B_{\Vert }=\frac{1}{1+2/\unicode[STIX]{x1D6FD}_{\bot e}}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}B_{\Vert }, & \displaystyle\end{eqnarray}$$

(4.31) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{v_{A}d_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}B_{\Vert }}{\unicode[STIX]{x2202}t}=\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{2+2/\unicode[STIX]{x1D6FD}_{\bot e}}{1+2/\unicode[STIX]{x1D6FD}_{\bot e}}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }, & \displaystyle\end{eqnarray}$$

which of course return the eigenvectors (3.46) and growth/decay rates (3.47) in the linear regime. The corresponding conserved energy is

(4.32) $$\begin{eqnarray}W_{\text{KAW}}=\int \text{d}^{3}\boldsymbol{r}\,\Bigg[\left(2+\frac{2}{\unicode[STIX]{x1D6FD}_{\bot e}}\right)\frac{\unicode[STIX]{x1D6FF}B_{\bot }^{2}}{8\unicode[STIX]{x03C0}}-\frac{\unicode[STIX]{x1D6FF}B_{\Vert }^{2}}{8\unicode[STIX]{x03C0}}\Bigg]\left(1+\frac{2}{\unicode[STIX]{x1D6FD}_{\bot e}}\right)^{-1},\end{eqnarray}$$

which, as with (4.19), may be written as the sum of the energies of the ‘ $+$ ’ and ‘ $-$ ’ linear KAW eigenmodes found in (3.46). Note that, if $\unicode[STIX]{x1D6FF}B_{\Vert }^{2}$ grows or decays, so too must $\unicode[STIX]{x1D6FF}B_{\bot }^{2}$ grow or decay proportionally to keep $W_{\text{KAW}}$ constant. As explained after (3.47), the resulting instability occurs because perpendicular pressure balance is maintained for any $\unicode[STIX]{x1D6FF}B_{\Vert }$ so long as $\unicode[STIX]{x1D6E5}_{e}=2/\unicode[STIX]{x1D6FD}_{\bot e}$ .

4.3.2 Ion-entropy cascade

The other piece of the gyrokinetic invariant, $W_{\widetilde{h}_{i}}$ (see (4.18)), represents the cascade of ion-entropy fluctuations to small scales in phase space (velocity and position) via nonlinear phase mixing in the gyrocentre space (see § 7.9 of S09 for further details). It may be obtained directly from the gyrokinetic equation (2.47) as follows.

In the sub-ion-Larmor range, the ring-averaged gyrokinetic potential $\langle \unicode[STIX]{x1D712}\rangle _{\!\boldsymbol{R}_{i}}$ is dominated by the contribution from the electrostatic potential:

(4.33) $$\begin{eqnarray}\displaystyle \frac{Z_{i}e}{T_{\bot 0i}}\langle \unicode[STIX]{x1D712}_{\boldsymbol{k}}\rangle _{\!\boldsymbol{R}_{i}} & = & \displaystyle \frac{2}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\left\{\left[J_{0}(a_{i})+\frac{v_{\bot }^{2}}{v_{A}^{2}}\frac{J_{1}(a_{i})}{a_{i}}\frac{1+Z_{i}/\unicode[STIX]{x1D70F}_{\bot i}}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}\right]\frac{\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}}{\unicode[STIX]{x1D70C}_{i}v_{A}}+J_{0}(a_{i})\frac{v_{\Vert }}{v_{A}}\frac{\unicode[STIX]{x1D6F9}_{\boldsymbol{k}}}{\unicode[STIX]{x1D70C}_{i}v_{A}}\right\}\nonumber\\ \displaystyle & \simeq & \displaystyle \frac{2}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\sqrt{\frac{2}{\unicode[STIX]{x03C0}a_{i}}}\cos \Bigl(a_{i}-\frac{\unicode[STIX]{x03C0}}{4}\Bigr)\frac{\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}}{\unicode[STIX]{x1D70C}_{i}v_{A}}\quad \text{ for }a_{i}\doteq k_{\bot }\unicode[STIX]{x1D70C}_{i}\frac{v_{\bot }}{v_{\text{th}\bot i}}\gg 1,\end{eqnarray}$$

because of the KAW scaling $\unicode[STIX]{x1D6F9}_{\boldsymbol{k}}\sim \unicode[STIX]{x1D6F7}_{\boldsymbol{k}}/k_{\bot }\unicode[STIX]{x1D70C}_{i}$ (see (3.44)).Footnote ¹⁵ The gyrokinetic equation (2.47) in the sub-ion-Larmor range is then

(4.34) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{h}_{i}}{\unicode[STIX]{x2202}t}+v_{\Vert }\frac{\unicode[STIX]{x2202}\widetilde{h}_{i}}{\unicode[STIX]{x2202}z}+\{\langle \unicode[STIX]{x1D6F7}\rangle _{\!\boldsymbol{R}_{i}},\widetilde{h}_{i}\}=\frac{Z_{i}eB_{0}}{cT_{\Vert 0i}}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6F7}\rangle _{\!\boldsymbol{R}_{i}}}{\unicode[STIX]{x2202}t}f_{0i}.\end{eqnarray}$$

This equation states that $\widetilde{h}_{i}$ is linearly phase mixed by parallel-streaming ions, nonlinearly phase mixed in the gyrocentre space by ring-averaged $\boldsymbol{E}\times \boldsymbol{B}$ flows (i.e. KAW turbulence), and sourced by wave–particle interactions, which become asymptotically small as $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ . Multiplying (4.34) by $T_{\Vert 0i}\widetilde{h}_{i}/f_{0i}$ and integrating over the phase space, we obtain

(4.35) $$\begin{eqnarray}\frac{\text{d}W_{\widetilde{h}_{i}}}{\text{d}t}\doteq \frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{v}\int \text{d}^{3}\boldsymbol{R}_{i}\,\frac{T_{\Vert 0i}\widetilde{h}_{i}^{2}}{2f_{0i}}=\frac{Z_{i}eB_{0}}{c}\int \text{d}^{3}\boldsymbol{v}\int \text{d}^{3}\boldsymbol{R}_{i}\,\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6F7}\rangle _{\!\boldsymbol{R}_{i}}}{\unicode[STIX]{x2202}t}\widetilde{h}_{i}.\end{eqnarray}$$

Deep in the sub-ion-Larmor range, the right-hand side of (4.35) representing the collisionless damping at the ion gyroscale is small and the total variance of $\widetilde{h}_{i}$ is conserved (as was shown also by (4.18)). The ion-entropy fluctuations are transferred across this scale range by means of a cascade, in which particles with like gyrocentre positions but with different $v_{\bot }$ (and so different gyroradii) sample spatially decorrelated electromagnetic fluctuations and thus $\hat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}\langle \unicode[STIX]{x1D6F7}\rangle _{\!\boldsymbol{R}_{i}}$ drift with different velocities.

Equation (4.35) makes clear that neither ion nor electron pressure anisotropy (in a bi-Maxwellian plasma) affects the ion-entropy cascade itself. Instead, by shifting the precise $k$ -space location of the frequency break that occurs near $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ (see § 3.6.4), pressure anisotropy could adjust the values of $\unicode[STIX]{x1D6F7}$ and $\widetilde{h}_{i}$ at the end of the inertial range, where the ion-entropy cascade is sourced. This may, in turn, alter the fraction of free energy that enters the entropy cascade rather than the KAW cascade. Since the entropy cascade is what ultimately carries the fraction of the energy diverted from the inertial-range electromagnetic fluctuations by the collisionless damping (wave–particle interaction) to collisional scales, the ion and electron pressure anisotropies are likely to affect the ratio of ion to electron heating.

4.4.1 Inertial-range scaling relations

We first treat the long-wavelength inertial range $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ (Paper I) by considering Alfvénic fluctuations with perpendicular scale $\unicode[STIX]{x1D706}$ and parallel scale $\ell _{\unicode[STIX]{x1D706}}\gg \unicode[STIX]{x1D706}$ . The critical balance condition at scale $\unicode[STIX]{x1D706}$ is given by

(4.36) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D706}}\sim \frac{v_{A\ast }}{\ell _{\unicode[STIX]{x1D706}}}\sim \frac{u_{\bot \unicode[STIX]{x1D706}}}{\unicode[STIX]{x1D706}},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D706}}$ is the typical frequency of the fluctuations at scale $\unicode[STIX]{x1D706}$ and $v_{A\ast }$ includes the effect of temperature anisotropy on the propagation speed of linear Alfvén waves (see (3.20)). The cascade time is then $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}\sim \ell _{\unicode[STIX]{x1D706}}/v_{A\ast }\sim \unicode[STIX]{x1D706}/u_{\bot \unicode[STIX]{x1D706}}$ , whence

(4.37a,b ) $$\begin{eqnarray}\frac{u_{\bot \unicode[STIX]{x1D706}}^{2}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}}\sim \unicode[STIX]{x1D700}=\text{ const. }\,\Longrightarrow \,\,\,u_{\bot \unicode[STIX]{x1D706}}\sim (\unicode[STIX]{x1D700}\unicode[STIX]{x1D706})^{1/3}\quad \text{and}\quad \ell _{\unicode[STIX]{x1D706}}\sim \ell _{0}^{1/3}\unicode[STIX]{x1D706}^{2/3},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}$ is the (spatially constant) average energy per unit time per unit volume that the system dissipates and $\ell _{0}\doteq v_{A\ast }^{3}/\unicode[STIX]{x1D700}$ . These scaling relations state that the kinetic-energy spectrum is ${\sim}k_{\bot }^{-5/3}$ and that the spatial anisotropy of the turbulent fluctuations increases at smaller scales. For a given rate of energy injection, background temperature anisotropies do not affect the fluctuation amplitude at each scale, but do affect the spatial anisotropy at each scale by requiring larger (smaller) $\ell _{\unicode[STIX]{x1D706}}$ for positive (negative) temperature anisotropies in order to maintain critical balance.

The compressive fluctuations are passively advected by the Alfvénic component of the turbulence (Lithwick & Goldreich Reference Lithwick and Goldreich2001; S09), a feature that carries over unaltered to the pressure-anisotropic case (§ I–2.6). If these compressive fluctuations are weakly damped, they share the Alfvénic $k_{\bot }^{-5/3}$ spectrum. In particular, this is true for the fluctuating pressure anisotropies produced by these modes, which may be obtained from the KRMHD equations (I–2.41), (I–2.42), (I–2.49) and (I–2.52). The fluctuating electron pressure anisotropy is given by

(4.38) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}p_{\bot e}-\unicode[STIX]{x1D6FF}p_{\Vert e}}{p_{\Vert 0e}}=\unicode[STIX]{x1D6E5}_{e}\left(\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}-\frac{T_{\bot 0e}}{T_{\Vert 0e}}\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}\right),\end{eqnarray}$$

which indicates that no scale-dependent electron pressure anisotropy will be generated by the compressive fluctuations if no electron pressure anisotropy exists in the background state. Since electron pressure anisotropy affects the sub-ion-scale dynamics, this is worth noting. On the other hand, the fluctuating ion perpendicular pressure satisfies perpendicular pressure balance and is thus related to the density and magnetic-field-strength fluctuations as follows (see (I–2.52) with (I–2.46) substituted for $g_{i}$ ):

(4.39) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}p_{\bot i}}{p_{\bot 0i}}=-\frac{Z_{i}}{\unicode[STIX]{x1D70F}_{\bot i}}\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}-\frac{2}{\unicode[STIX]{x1D6FD}_{\bot i}}\left(1-\frac{\unicode[STIX]{x1D6FD}_{\bot e}}{2}\unicode[STIX]{x1D6E5}_{e}\right)\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}.\end{eqnarray}$$

The fluctuating ion parallel pressure cannot be written in terms of $\unicode[STIX]{x1D6FF}n_{e}$ and $\unicode[STIX]{x1D6FF}B_{\Vert }$ without further assumptions; its evolution is given by

(4.40) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\left(\frac{\unicode[STIX]{x1D6FF}p_{\Vert i}}{p_{\Vert 0i}}-\frac{\unicode[STIX]{x1D6FF}B_{\Vert }}{B_{0}}\right)+\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\frac{Q_{\Vert i}}{p_{\Vert 0i}}=0,\end{eqnarray}$$

where $Q_{\Vert i}\doteq \int \text{d}^{3}\boldsymbol{v}\,m_{i}v_{\Vert }^{3}\unicode[STIX]{x1D6FF}f_{i}$ is the parallel flow of parallel ion heat. Whether the parallel ion heat flow is negligible ( $\unicode[STIX]{x1D714}\gg |k_{\Vert }|c_{s}$ ) or dominant ( $\unicode[STIX]{x1D714}\ll |k_{\Vert }|c_{s}$ ), scale-dependent ion pressure anisotropy will be generated by the fluctuations even when $\unicode[STIX]{x1D6E5}_{i}=0$ .

If the strength of the collisionless damping of these fluctuations determines whether a cascade of compressive fluctuations can occur, then the background pressure anisotropy will affect the amount of $\unicode[STIX]{x1D6FF}B_{\Vert }$ and $\unicode[STIX]{x1D6FF}n_{e}$ , and consequently of $\unicode[STIX]{x1D6FF}p_{\bot s}$ and $\unicode[STIX]{x1D6FF}p_{\Vert s}$ , in this cascade (see § I–4.4.2 and equation I–4.32a,b); for example, positive $\unicode[STIX]{x1D6E5}_{s}$ reduces the rate of Barnes damping of magnetic-field-strength fluctuations in a high- $\unicode[STIX]{x1D6FD}$ plasma (see (3.34)). If compressive fluctuations are strongly damped, or if no compressive fluctuations are excited at long wavelengths in the first place, then the temperature anisotropy would instead be driven adiabatically by the Alfvénic fluctuations (e.g. Squire et al. Reference Squire, Quataert and Schekochihin2016, Reference Squire, Kunz, Quataert and Schekochihin2017):

(4.41) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D6FF}p_{\bot s}-\unicode[STIX]{x1D6FF}p_{\Vert s}}{p_{\Vert 0s}}\right)_{\!\unicode[STIX]{x1D706}}\propto \biggl|\frac{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot \unicode[STIX]{x1D706}}}{B_{0}}\biggr|^{2}\propto \unicode[STIX]{x1D706}^{2/3},\end{eqnarray}$$

which translates into a $k_{\bot }^{-7/3}$ power spectrum of pressure anisotropy. To the extent one can differentiate between $-5/3$ and $-7/3$ , the source of the scale-dependent pressure anisotropy could thus be inferred. While it would be interesting to investigate how such scale-dependent pressure anisotropies affect the Alfvén-wave cascade, this physics, as well as the scaling relation (4.41), lie outside of the gyrokinetic ordering (2.11).

Before moving on to the kinetic range, we note that modern theories of strong inertial-range MHD turbulence have appeared to converge on a perpendicular spectral slope of $-3/2$ , rather than the Goldreich–Sridhar $-5/3$ (see Schekochihin (Reference Schekochihin2017) for a biased but up-to-date review), the former scaling law having its origins in the phenomenon of dynamical alignment proposed by Boldyrev (Reference Boldyrev2006). It is possible to argue that the discrepancy between $-5/3$ and $-3/2$ is fundamentally an intermittency effect (Chandran, Schekochihin & Mallet Reference Chandran, Schekochihin and Mallet2015; Mallet & Schekochihin Reference Mallet and Schekochihin2017); the specific scaling laws (4.37) and (4.41) do not account for intermittency. In any case, the point of this exposition is not tied to whether the turbulence follows one power law or the other, but rather is that the Alfvénic and compressive fluctuations in the inertial range can independently drive scale-dependent pressure anisotropies, whose scaling laws say something about the dominant physics and the possible role of pressure anisotropy in influencing the efficacy of collisionless damping.

4.4.2 Kinetic-range scaling laws

In the sub-ion-Larmor kinetic range $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , a nonlinear cascade of KAWs can be set up. To obtain scalings associated with it, we follow S09 in assuming that the KAW fluctuations satisfy $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D706}}^{+}\sim \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D706}}^{-}$ (see (3.44)), or

(4.42) $$\begin{eqnarray}\frac{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})^{1/2}}\,\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}}\sim \sqrt{\frac{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}{1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2}}\,\frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D70C}_{i}}\,\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D706}}\end{eqnarray}$$

(for the purposes of these scaling arguments, we set $Z_{i}T_{\Vert 0e}/T_{\bot 0i}\sim 1$ , but retain the $\unicode[STIX]{x1D6FD}_{i}$ and $\unicode[STIX]{x1D6FD}_{e}\unicode[STIX]{x1D6E5}_{e}$ dependences). Using (4.42), the constant-flux assumption becomes

(4.43) $$\begin{eqnarray}\frac{(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}}/\unicode[STIX]{x1D706})^{2}}{\unicode[STIX]{x1D70F}_{\text{KAW}\unicode[STIX]{x1D706}}}\sim \frac{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}{1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2}\frac{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}}}{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2)^{2}}\,\frac{(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D706}}/\unicode[STIX]{x1D70C}_{i})^{2}}{\unicode[STIX]{x1D70F}_{\text{KAW}\unicode[STIX]{x1D706}}}\sim \unicode[STIX]{x1D700}_{\text{KAW}}=\text{const.},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{\text{KAW}\unicode[STIX]{x1D706}}$ is the cascade time and $\unicode[STIX]{x1D700}_{\text{KAW}}$ is the energy flux of KAWs. The latter is the fraction of the total energy flux $\unicode[STIX]{x1D700}$ that is converted into the KAW cascade at the ion gyroscale (which itself is likely a function of the background pressure anisotropy). The cascade time follows from examining the nonlinear terms proportional to $v_{A}(\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot }/B_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }\sim \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}}/\unicode[STIX]{x1D706}^{2}$ in (4.24) and (4.25). Assuming the turbulence is critically balanced, this time is comparable to the inverse of the KAW frequency (3.38):Footnote ¹⁶

(4.44) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\text{KAW}\unicode[STIX]{x1D706}}\sim \frac{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}}}\frac{\unicode[STIX]{x1D706}^{2}}{\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D706}}}\sim \sqrt{\frac{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})(1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2)}}\,\frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D70C}_{i}}\frac{\ell _{\unicode[STIX]{x1D706}}}{v_{A}}.\end{eqnarray}$$

Combining (4.43) and (4.44) leads to the following scaling relations for KAW turbulence:

(4.45) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D706}}\sim \left(\frac{\unicode[STIX]{x1D700}_{\text{KAW}}}{\unicode[STIX]{x1D700}}\right)^{1/3}\ell _{0}^{-1/3}\unicode[STIX]{x1D70C}_{i}^{2/3}\unicode[STIX]{x1D706}^{2/3}\,\frac{v_{A\ast }(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2)}{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})^{2/3}}\left(\frac{1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2}{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}\right)^{1/3}, & \displaystyle\end{eqnarray}$$

(4.46) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}}\sim \left(\frac{\unicode[STIX]{x1D700}_{\text{KAW}}}{\unicode[STIX]{x1D700}}\right)^{1/3}\ell _{0}^{-1/3}\unicode[STIX]{x1D70C}_{i}^{-1/3}\unicode[STIX]{x1D706}^{5/3}\,\frac{v_{A\ast }}{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})^{1/6}}\left(\frac{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}{1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2}\right)^{1/6}, & \displaystyle\end{eqnarray}$$

(4.47) $$\begin{eqnarray}\displaystyle & \displaystyle \ell _{\unicode[STIX]{x1D706}}\sim \left(\frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D700}_{\text{KAW}}}\right)^{1/3}\ell _{0}^{1/3}\unicode[STIX]{x1D70C}_{i}^{1/3}\unicode[STIX]{x1D706}^{1/3}\,\frac{v_{A}}{v_{A\ast }}\left[\frac{(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})(1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2)}{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}\right]^{1/6}. & \displaystyle\end{eqnarray}$$

The corresponding $k_{\bot }^{-7/3}$ spectrum of magnetic energy is identical to that found in the isotropic-Maxwellian case, but with fluctuation amplitudes that depend on the electron anisotropy and its distance from the firehose and mirror instability thresholds. KAWs also generate scale-dependent ion and electron pressure anisotropy, which may be calculated from the relations (3.40a,d ) and (3.45a,b ):

(4.48) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D6FF}p_{\bot i}-\unicode[STIX]{x1D6FF}p_{\Vert i}}{p_{\Vert 0i}}\right)_{\!\unicode[STIX]{x1D706}} & = & \displaystyle \unicode[STIX]{x1D6E5}_{i}\frac{\unicode[STIX]{x1D6FF}n_{i\unicode[STIX]{x1D706}}}{n_{0i}}=-\unicode[STIX]{x1D6E5}_{i}\frac{2}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\frac{\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D706}}}{\unicode[STIX]{x1D70C}_{i}v_{A}}\propto \unicode[STIX]{x1D6E5}_{i}\,\unicode[STIX]{x1D706}^{2/3},\end{eqnarray}$$

(4.49) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D6FF}p_{\bot e}-\unicode[STIX]{x1D6FF}p_{\Vert e}}{p_{\Vert 0e}}\right)_{\!\unicode[STIX]{x1D706}} & = & \displaystyle \unicode[STIX]{x1D6E5}_{e}\left(\frac{\unicode[STIX]{x1D6FF}n_{e\unicode[STIX]{x1D706}}}{n_{0e}}-\frac{T_{\bot 0e}}{T_{\Vert 0e}}\frac{\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D706}}}{B_{0}}\right)\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D6E5}_{e}\frac{2}{\sqrt{\unicode[STIX]{x1D6FD}_{\bot i}}}\left(1+\frac{T_{\bot 0e}}{T_{\Vert 0e}}\frac{\unicode[STIX]{x1D6FD}_{\bot i}}{2}\frac{1+Z_{i}/\unicode[STIX]{x1D70F}_{\bot i}}{1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}/2}\right)\frac{\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D706}}}{\unicode[STIX]{x1D70C}_{i}v_{A}}\propto \unicode[STIX]{x1D6E5}_{e}\,\unicode[STIX]{x1D706}^{2/3}.\qquad\end{eqnarray}$$

Interestingly, KAWs do not produce scale-dependent ion (electron) pressure anisotropy if no ion (electron) anisotropy exists in the background state. This is because the KAW pressure response is isothermal when $\unicode[STIX]{x1D6E5}_{i,e}=0$ (see (3.44)).

Once again, these specific scaling laws are probably not accurate, and the coefficients accompanying them are probably not measurable. (For example, Boldyrev & Perez (Reference Boldyrev and Perez2012) appeal to intermittency corrections to predict a steeper spectrum of $-8/3$ , in closer agreement with solar-wind observations that yield slopes between $-3.1$ and $-2.5$ with a mode of $-2.8$ (e.g. Sahraoui et al. Reference Sahraoui, Huang, Belmont, Goldstein, Rétino, Robert and De Patoul2013).) But here we have a clear demonstration of three salient features: (i) that background pressure anisotropy is required to excite scale-dependent pressure anisotropy in the KAW range (at least within the gyrokinetic theory); (ii) that the relative amplitudes of the magnetic-field-strength and density fluctuations depend on the electron pressure anisotropy and not the ion one; and (iii) that the nonlinear KAW cascade time at any given scale becomes larger the closer the plasma is to the mirror/firehose thresholds (if critical balance continues to hold).

This last point becomes particularly important for the ion-entropy cascade, in which $W_{\widetilde{h}_{i}}$ is conserved. Following similar reasoning, a constant flux of entropy implies

(4.50) $$\begin{eqnarray}\frac{v_{\text{th}\bot i}^{6}v_{\text{th}\Vert i}^{2}}{n_{0i}^{2}}\frac{\widetilde{h}_{i\unicode[STIX]{x1D706}}^{2}}{\unicode[STIX]{x1D70F}_{h\unicode[STIX]{x1D706}}}\sim \unicode[STIX]{x1D700}_{h}=\text{const.},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{h}$ is the entropy flux proportional to the fraction of the total energy flux $\unicode[STIX]{x1D700}$ that is converted into the entropy cascade at the ion gyroscale (again, which itself is a function of the background pressure anisotropy). Following the arguments presented in § 7.9.2 of S09, the nonlinear time is obtained by weighting $\unicode[STIX]{x1D70F}_{\text{KAW}\unicode[STIX]{x1D706}}$ (see (4.44)) by a factor of $(\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D706})^{1/2}$ , due to the ring averaging in the gyrokinetic nonlinearity (the third term in (4.34)). An additional factor of $(\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D706})^{1/2}$ accounts for the random-walk accumulation of changes to $\widetilde{h}_{i\unicode[STIX]{x1D706}}$ during one cascade time – an argument analogous to that used to describe weak turbulence of Alfvén waves (Kraichnan Reference Kraichnan1965; see equation (255) of S09). Thus,

(4.51) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{h\unicode[STIX]{x1D706}}\sim \frac{\unicode[STIX]{x1D70C}_{i}}{\unicode[STIX]{x1D706}}\,\unicode[STIX]{x1D70F}_{\text{KAW}\unicode[STIX]{x1D706}}\sim \left(\frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D700}_{\text{KAW}}}\right)^{1/3}\frac{\ell _{0}^{1/3}\unicode[STIX]{x1D70C}_{i}^{1/3}\unicode[STIX]{x1D706}^{1/3}}{v_{A\ast }}\left[\frac{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}{(1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2)(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})}\right]^{1/3}.\end{eqnarray}$$

Substituting this into (4.50) provides the scaling

(4.52) $$\begin{eqnarray}\widetilde{h}_{i\unicode[STIX]{x1D706}}\sim \frac{n_{0i}}{v_{\text{th}\bot i}^{3}}\left(\frac{\unicode[STIX]{x1D700}_{h}}{\unicode[STIX]{x1D700}}\right)^{1/2}\left(\frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D700}_{\text{KAW}}}\right)^{1/6}\frac{\ell _{0}^{-1/3}\unicode[STIX]{x1D70C}_{i}^{1/6}\unicode[STIX]{x1D706}^{1/6}}{\sqrt{\unicode[STIX]{x1D6FD}_{\Vert i}}}\frac{v_{A\ast }}{v_{A}}\left[\frac{1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}}}{(1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2)(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})}\right]^{1/6},\end{eqnarray}$$

which corresponds to a $k_{\bot }^{-4/3}$ entropy spectrum.

Again, the point is not so much the $-4/3$ ion-entropy spectrum, which is unchanged from the prediction of isotropic-pressure gyrokinetics (S09), but that the time over which $\widetilde{h}_{i}$ is decorrelated due to mixing by ring-averaged $\boldsymbol{E}\times \boldsymbol{B}$ flows is a function of the electron pressure anisotropy. One instance in which this dependence may become important concerns the ratio of parallel (i.e. linear) versus perpendicular (i.e. nonlinear) phase mixing. If we assume that the parallel decorrelation scale of $\widetilde{h}_{i}$ is inherited from the KAW scaling (4.47), then after one cascade time $\unicode[STIX]{x1D70F}_{h\unicode[STIX]{x1D706}}$ (see (4.51)) $\widetilde{h}_{i}$ becomes decorrelated on the parallel velocity scales

(4.53) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}v_{\Vert }}{v_{\text{th}\Vert i}}\sim \frac{\ell _{\unicode[STIX]{x1D706}}}{v_{\text{th}\Vert i}\unicode[STIX]{x1D70F}_{h\unicode[STIX]{x1D706}}}\sim \sqrt{\frac{(1+\unicode[STIX]{x1D6FD}_{\Vert e}\unicode[STIX]{x1D6E5}_{e}/2)(1-\unicode[STIX]{x1D6FD}_{\bot e}\unicode[STIX]{x1D6E5}_{e}+{\mathcal{K}})}{\unicode[STIX]{x1D6FD}_{\Vert i}(1+\unicode[STIX]{x1D6FD}_{\bot i}-2{\mathcal{K}})}}.\end{eqnarray}$$

In S09, it was argued that within the ion-entropy cascade the parallel phase mixing can be ignored relative to the perpendicular phase mixing. This was done by comparing the estimate $\unicode[STIX]{x1D6FF}v_{\Vert }/v_{\text{th}i}\sim 1/\sqrt{\unicode[STIX]{x1D6FD}_{i}(1+\unicode[STIX]{x1D6FD}_{i})}\sim 1$ after one cascade time (i.e. (4.53) with $\unicode[STIX]{x1D6E5}_{e}=0$ ) to the smallest perpendicular-velocity-space interval between spatially decorrelated electromagnetic fluctuations, $\unicode[STIX]{x1D6FF}v_{\bot }/v_{\text{th}i}\sim (k_{\bot }\unicode[STIX]{x1D70C}_{i})^{-1}$ with $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ given by its value at the collisional cutoff. Recent gyrokinetic simulations of KAW turbulence that demonstrate predominantly perpendicular collisional heating of the ions (Navarro et al. Reference Navarro, Teaca, Told, Groselj, Crandall and Jenko2016) support this argument. But in a pressure-anisotropic plasma, the entropy cascade slows down near the firehose and mirror thresholds (see (4.51)). The parallel phase mixing could then compete with the perpendicular phase mixing, since there is more time for small scales in $v_{\Vert }$ to be secularly produced and ultimately activate collisional dissipation. This idea can be tested using high-resolution gyrokinetic simulations of KAW turbulence in a pressure-anisotropic plasma.