2.1. Kinetic MHD

We begin with the equations of kinetic MHD (KMHD), as derived by Kulsrud (Reference Kulsrud and Rosenbluth1964, Reference Kulsrud, Galeev and Sudan1983). KMHD is a hybrid fluid-kinetic theory, appropriate for scales $k^{-1}$ larger than the ion gyroradius ( $k{\it\rho}_{i}\ll 1$ ) and frequencies ${\it\omega}$ smaller than the ion gyrofrequency ( ${\it\omega}\ll {\it\Omega}_{i}$ ) in a weakly collisional ( ${\it\rho}_{i}\ll {\it\lambda}_{mfp}$ ) plasma. The equations for species $s$ are the continuity equation

(2.1) $$\begin{eqnarray}\left(\frac{\partial }{\partial t}+\boldsymbol{u}_{s}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)n_{s}=-n_{s}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{s},\end{eqnarray}$$

the momentum equation

(2.2) $$\begin{eqnarray}m_{s}n_{s}\left(\frac{\partial }{\partial t}+\boldsymbol{u}_{s}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)\boldsymbol{u}_{s}=-\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\unicode[STIX]{x1D64B}_{s}+q_{s}n_{s}\left(\boldsymbol{E}+\frac{\boldsymbol{u}_{s}}{c}\times \boldsymbol{B}\right)+\boldsymbol{F}_{s},\end{eqnarray}$$

and the drift-kinetic equation

(2.3) $$\begin{eqnarray}\frac{\text{D}f_{s}}{\text{D}t}+\frac{\text{D}\ln B}{\text{D}t}\frac{w_{\bot }}{2}\frac{\partial f_{s}}{\partial w_{\bot }}+\left(\frac{q_{s}E_{\Vert }}{m_{s}}+\frac{w_{\bot }^{2}}{2}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\hat{\boldsymbol{b}}-\frac{\text{D}\boldsymbol{u}_{\bot s}}{\text{D}t}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\right)\frac{\partial f_{s}}{\partial v_{\Vert }}=\left(\frac{\partial f_{s}}{\partial t}\right)_{c},\end{eqnarray}$$

where $\text{D}/\text{D}t\doteq \partial /\partial t+\boldsymbol{u}_{\bot s}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}$ . Our notation is standard: $m_{s}$ , $q_{s}~(=Z_{s}e)$ , $n_{s}$ , $\boldsymbol{u}_{s}$ , and $\unicode[STIX]{x1D64B}_{s}$ are, respectively, the mass, charge, number density, mean velocity, and pressure tensor of species $s$ ; $\boldsymbol{B}$ is the magnetic field; and $\boldsymbol{E}$ is the electric field. The rate of change in the momentum of species $s$ due to inter-species collisions, $\boldsymbol{F}_{s}$ , is obtained by taking the first velocity moment of the collision term $(\partial f_{s}/\partial t)_{c}$ . The distribution function of species $s$ , $f_{s}=f_{s}(v_{\Vert },w_{\bot })$ , is written in terms of the velocity-space variables $v_{\Vert }$ and $w_{\bot }$ , which are measured parallel and perpendicular, respectively, to the magnetic field direction, $\hat{\boldsymbol{b}}\doteq \boldsymbol{B}/B$ . As in Kulsrud’s original formulation, we use as the perpendicular kinetic variable $\boldsymbol{w}_{\bot }=\boldsymbol{v}_{\bot }-\boldsymbol{u}_{\bot s}$ , the perpendicular velocity peculiar to the mean perpendicular flow of species $s$ . The parallel component of (2.2) is redundant, as it may be straightforwardly obtained by taking the first parallel velocity moment of the drift-kinetic equation (2.3).

With Larmor gyrations ordered out of these equations, $f_{s}$ is independent of gyrophase. As a result, the pressure tensor is diagonal in a coordinate system defined by the parallel and perpendicular directions with respect to the magnetic field:

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D64B}_{s}=p_{\bot s}(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})+p_{\Vert s}\hat{\boldsymbol{b}}\hat{\boldsymbol{b}},\end{eqnarray}$$

where $\unicode[STIX]{x1D644}$ is the unit dyadic, and

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle p_{\Vert s}\doteq n_{s}T_{\Vert s}=\int \text{d}^{3}\boldsymbol{v}\,m_{s}(v_{\Vert }-u_{\Vert s})^{2}f_{s}, & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle p_{\bot s}\doteq n_{s}T_{\bot s}=\int \text{d}^{3}\boldsymbol{v}\,m_{s}\frac{w_{\bot }^{2}}{2}f_{s} & \displaystyle\end{eqnarray}$$

$s$

$(m_{e}/m_{i})^{1/2}$

(2.7) $$\begin{eqnarray}\mathop{\sum }_{s}q_{s}n_{s}=0,\end{eqnarray}$$

it is straightforward to show that the parallel component of the electric field satisfies

(2.8) $$\begin{eqnarray}E_{\Vert }=-\hat{\boldsymbol{b}}\boldsymbol{\cdot }\left(\frac{\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\unicode[STIX]{x1D64B}_{e}}{en_{e}}\right)=-\frac{1}{en_{e}}[\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}p_{\Vert e}-(p_{\bot e}-p_{\Vert e})\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\hat{\boldsymbol{b}}]\end{eqnarray}$$

to leading order.Footnote ² Equation (2.8) expresses parallel pressure balance for the (effectively massless) electron fluid, taking into account the forcing-out of large-pitch-angle particles (those with ${\it\alpha}\doteq \cos ^{-1}(v_{\Vert }/v)\sim {\rm\pi}/2$ ) from regions of increased magnetic-field strength. Indeed, in the absence of collisions, (2.3) guarantees that each particle’s adiabatic invariant ${\it\mu}_{s}\doteq m_{s}w_{\bot }^{2}/2B$ is identically preserved (i.e.  $f_{s}$ remains constant along the phase-space trajectory $\text{D}{\it\mu}_{s}/\text{D}t=0$ ). The second term in parentheses on the left-hand side of (2.3) is then straightforwardly interpreted as the mirror force, $(m_{s}w_{\bot }^{2}/2)\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=-{\it\mu}_{s}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}B$ , which, by (2.8), is compensated in the electron fluid by the parallel electric force, $-eE_{\Vert }$ , and by the divergence of the parallel electron pressure, $-(1/n_{e})\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(p_{\Vert e}\hat{\boldsymbol{b}})$ .

2.2. Particle drifts and centre-of-mass variables

The same ordering as one uses to obtain a gyrophase-independent $f_{s}$ also guarantees that all species drift perpendicular to the magnetic field with identical velocities. Indeed, the lowest-order contribution to the perpendicular electric field is $\boldsymbol{E}_{\bot }=-(\boldsymbol{u}_{s}/c)\times \boldsymbol{B}$ , so that $\boldsymbol{u}_{\bot s}=\boldsymbol{u}_{\bot }=c\boldsymbol{E}\times \boldsymbol{B}/B^{2}$ . With this borne in mind, we interpret the $\text{D}/\text{D}t$ operator in (2.3) as measuring the rate of change of a quantity in a Lagrangian frame that is transported parallel to the magnetic field at velocity $v_{\Vert }$ and drifts perpendicular to the magnetic field at the $\boldsymbol{E}\times \boldsymbol{B}$ velocity.

It then follows that the mean drift of any species relative to the centre-of-mass velocity $\boldsymbol{u}\doteq \sum _{s}m_{s}n_{s}\boldsymbol{u}_{s}/\!\sum _{s}m_{s}n_{s}$ must be in the parallel direction, viz., $\boldsymbol{u}_{s}=\boldsymbol{u}+u_{\Vert s}^{\prime }\hat{\boldsymbol{b}}$ with

(2.9) $$\begin{eqnarray}u_{\Vert s}^{\prime }=\frac{1}{n_{s}}\int \text{d}^{3}\boldsymbol{v}\,(v_{\Vert }-u_{\Vert })f_{s}.\end{eqnarray}$$

In centre-of-mass variables with ${\it\rho}\doteq \sum _{s}m_{s}n_{s}$ , (2.1) and (2.2) become

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\partial }{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right){\it\rho}=-{\it\rho}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}\left(\frac{\partial }{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)\boldsymbol{u}=-\boldsymbol{{\rm\nabla}}\left(p_{\bot }+\frac{B^{2}}{8{\rm\pi}}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}\left(p_{\bot }-p_{\Vert }-\mathop{\sum }_{s}m_{s}n_{s}u_{\Vert s}^{\prime 2}+\frac{B^{2}}{4{\rm\pi}}\right)\right], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

$p_{\bot }\doteq \sum _{s}p_{\bot s}$

$p_{\Vert }\doteq \sum _{s}p_{\Vert s}$

(2.12) $$\begin{eqnarray}\frac{\partial \boldsymbol{B}}{\partial t}=-c\boldsymbol{{\rm\nabla}}\times \boldsymbol{E}=\boldsymbol{{\rm\nabla}}\times (\boldsymbol{u}\times \boldsymbol{B}),\end{eqnarray}$$

i.e. the magnetic flux is frozen into a frame moving perpendicular to the magnetic field at the velocity $\boldsymbol{u}_{\bot }$ .

2.3. Reduced ordering and dimensionless parameters

We proceed by separating all fields into equilibrium values plus fluctuations: $f_{s}=f_{0s}+{\it\delta}f_{s}$ , $\boldsymbol{B}=B_{0}\hat{\boldsymbol{z}}+{\it\delta}\boldsymbol{B}$ , $n_{s}=n_{0s}+{\it\delta}n_{s}$ , $p_{\bot s}=p_{\bot 0s}+{\it\delta}p_{\bot s}$ , $p_{\Vert s}=p_{\Vert 0s}+{\it\delta}p_{\Vert s}$ , and $u_{\Vert s}^{\prime }=u_{\Vert 0s}^{\prime }+{\it\delta}u_{\Vert s}^{\prime }$ . The mean magnetic field $B_{0}\hat{\boldsymbol{z}}$ and the equilibrium distribution function $f_{0s}$ are both taken to be spatially uniform (the slab limit). The asymptotic ordering

(2.13) $$\begin{eqnarray}\frac{k_{\Vert }}{k_{\bot }}\sim \frac{{\it\delta}f_{s}}{f_{0s}}\sim \frac{u_{\bot }}{v_{A}}\sim \frac{{\it\delta}B_{\bot }}{B_{0}}\sim \frac{u_{\Vert }}{v_{A}}\sim \frac{{\it\delta}B_{\Vert }}{B_{0}}\sim \frac{{\it\delta}n_{s}}{n_{0s}}\sim \frac{{\it\delta}p_{\bot s}}{p_{\bot 0s}}\sim \frac{{\it\delta}p_{\Vert s}}{p_{\Vert 0s}}\sim \frac{{\it\delta}u_{\Vert s}^{\prime }}{u_{\Vert 0s}^{\prime }}\sim {\it\epsilon},\end{eqnarray}$$

where

(2.14) $$\begin{eqnarray}v_{A}\doteq \frac{B_{0}}{\sqrt{4{\rm\pi}{\it\rho}_{0}}}\end{eqnarray}$$

is the Alfvén speed, is then applied to ‘reduce’ the equations so that they describe the evolution of anisotropic ( $k_{\Vert }\ll k_{\bot }$ ) fluctuations whose parallel Alfvén time scale and perpendicular nonlinear time scale are of the same order, $k_{\Vert }v_{A}\sim k_{\bot }u_{\bot }$ – the so-called critical-balance conjecture (Goldreich & Sridhar Reference Goldreich and Sridhar1995), used here as an ordering assumption (S09). Such spatial anisotropy is both measured directly in the solar wind (e.g. Bieber, Wanner & Matthaeus Reference Bieber, Wanner and Matthaeus1996; Horbury, Forman & Oughton Reference Horbury, Forman and Oughton2008; Podesta Reference Podesta2009; Wicks et al. Reference Wicks, Horbury, Chen and Schekochihin2010; Chen et al. Reference Chen, Mallet, Yousef, Schekochihin and Horbury2011) and observed in numerical simulations of Alfvénic turbulence (e.g. Shebalin, Matthaeus & Montgomery Reference Shebalin, Matthaeus and Montgomery1983; Oughton, Priest & Matthaeus Reference Oughton, Priest and Matthaeus1994; Cho & Vishniac Reference Cho and Vishniac2000; Maron & Goldreich Reference Maron and Goldreich2001). The perpendicular perturbations are taken to be Alfvénic ( ${\it\delta}B_{\bot }/B_{0}\sim u_{\bot }/v_{A}$ ) and the compressive perturbations ( ${\it\delta}u_{\Vert }$ , ${\it\delta}B_{\Vert }$ , ${\it\delta}n_{s}$ , ${\it\delta}p_{\bot s}$ , ${\it\delta}p_{\Vert s}$ , ${\it\delta}u_{\Vert s}^{\prime }$ ) are ordered comparable to the Alfvénic ones, with the parallel and perpendicular thermal speeds of species $s$ ,

(2.15a,b ) $$\begin{eqnarray}v_{th\Vert s}\doteq \sqrt{\frac{2T_{\Vert 0s}}{m_{s}}}\quad \text{and}\quad v_{th\bot s}\doteq \sqrt{\frac{2T_{\bot 0s}}{m_{s}}},\end{eqnarray}$$

respectively, ordered comparable to the Alfvén speed (i.e.  ${\it\beta}_{\Vert s}$ , ${\it\beta}_{\bot s}$ defined in (2.21) ordered unity). We further assume that the characteristic frequency of the fluctuations satisfies ${\it\omega}\sim k_{\Vert }v_{A}\sim v_{th\Vert i}/L$ . This means that fast magnetosonic modes, for which ${\it\omega}\sim k_{\bot }v_{A}$ , are ordered out of our equations. Fast-wave fluctuations are rarely seen in the solar wind (Howes et al. Reference Howes, Bale, Klein, Chen, Salem and TenBarge2012); observations of turbulence in the solar wind confirm that it is primarily Alfvénic (e.g. Belcher & Davis Reference Belcher and Davis1971) and that its compressive component is substantially pressure-balanced (Burlaga et al. Reference Burlaga, Scudder, Klein and Isenberg1990; Roberts Reference Roberts1990; Marsch & Tu Reference Marsch and Tu1993; Bavassano, Pietropaolo & Bruno Reference Bavassano, Pietropaolo and Bruno2004; see (2.26)). We expect the same to hold true in the intracluster medium, where observationally inferred turbulent velocities are convincingly subsonic (e.g. Sanders & Fabian Reference Sanders and Fabian2013; Zhuravleva et al. Reference Zhuravleva, Churazov, Schekochihin, Allen, Arévalo, Fabian, Forman, Sanders, Simionescu, Sunyaev, Vikhlinin and Werner2014). Indeed, our reduced ordering is consistent with a small sonic Mach number, $\mathit{Ma}=u_{\bot }/v_{thi}\sim {\it\epsilon}(v_{A}/v_{thi})\ll 1$ .

The density fluctuations of the various species are related to one another via quasineutrality (2.7):

(2.16) $$\begin{eqnarray}\mathop{\sum }_{s}c_{s}\frac{{\it\delta}n_{s}}{n_{0s}}=\mathop{\sum }_{s}\frac{c_{s}}{n_{0s}}\int \text{d}^{3}\boldsymbol{v}\,{\it\delta}f_{s}=0,\end{eqnarray}$$

where $c_{s}\doteq Z_{s}n_{0s}/n_{0e}$ is the charge-weighted ratio of number densities; note that $c_{e}=-1$ and $\sum _{s}c_{s}=0$ . The perturbed pressures are calculated via

(2.17a,b ) $$\begin{eqnarray}{\it\delta}p_{\Vert s}=\int \text{d}^{3}\boldsymbol{v}\,m_{s}(v_{\Vert }-u_{\Vert 0s}^{\prime })^{2}\,{\it\delta}f_{s}\quad \text{and}\quad {\it\delta}p_{\bot s}=\int \text{d}^{3}\boldsymbol{v}\,m_{s}\frac{w_{\bot }^{2}}{2}\,{\it\delta}f_{s}.\end{eqnarray}$$

Perturbed parallel drifts may be obtained directly from taking first moments of the perturbed distribution functions,

(2.18) $$\begin{eqnarray}{\it\delta}u_{\Vert s}^{\prime }=\frac{1}{n_{0s}}\int \text{d}^{3}\boldsymbol{v}\,(v_{\Vert }-u_{\Vert 0s}^{\prime })\,{\it\delta}f_{s}-u_{\Vert }\quad \text{with }u_{\Vert }=\frac{\displaystyle \mathop{\sum }_{s}m_{s}\int \text{d}^{3}\boldsymbol{v}\,v_{\Vert }{\it\delta}f_{s}}{\displaystyle \mathop{\sum }_{s}m_{s}n_{0s}},\end{eqnarray}$$

rather than indirectly via the momentum equations (2.2) and (2.11). In other words, they are not independent quantities.

The resulting set of equations has a number of dimensionless free parameters:

(2.19a,b ) $$\begin{eqnarray}{\it\tau}_{\Vert s}\doteq \frac{T_{\Vert 0s}}{T_{\Vert 0e}}\quad \text{and}\quad {\it\tau}_{\bot s}\doteq \frac{T_{\bot 0s}}{T_{\bot 0e}}\end{eqnarray}$$

are the ratios of the parallel and perpendicular temperatures of species $s$ to their respective electron temperatures (note that ${\it\tau}_{\Vert e}={\it\tau}_{\bot e}=1$ );

(2.20) $$\begin{eqnarray}{\it\Delta}_{s}\doteq \frac{p_{\bot 0s}}{p_{\Vert 0s}}-1\end{eqnarray}$$

is the dimensionless pressure anisotropy of species $s$ ; and

(2.21a,b ) $$\begin{eqnarray}{\it\beta}_{\Vert s}\doteq \frac{8{\rm\pi}p_{\Vert 0s}}{B_{0}^{2}}\quad \text{and}\quad {\it\beta}_{\bot s}\doteq \frac{8{\rm\pi}p_{\bot 0s}}{B_{0}^{2}}\end{eqnarray}$$

are the ratios of the parallel and perpendicular pressures of species $s$ to the magnetic pressure. We use the shorthand ${\it\beta}_{\Vert }\doteq \sum _{s}{\it\beta}_{\Vert s}$ and ${\it\beta}_{\bot }\doteq \sum _{s}{\it\beta}_{\bot s}$ . All of these quantities, evaluated in the equilibrium state, are taken to be order unity in the ${\it\epsilon}$ expansion (subsidiary limits in high and low ${\it\beta}_{\Vert }$ , for example, can be taken after the ${\it\epsilon}$ expansion is done: see § 4.4). Likewise, we order $u_{\Vert 0s}^{\prime }\sim v_{th\Vert i}$ for all species $s$ . This precludes equilibrium parallel drifts in the electron fluid from entering into our equations, because $v_{\Vert }\sim v_{th\Vert e}\sim \sqrt{m_{i}/m_{e}}v_{th\Vert i}\gg v_{th\Vert i}$ for the electrons (i.e. the random thermal motions of electrons are characterised by speeds much in excess of the ion thermal speed and, therefore, any parallel drifts in the equilibrium state).

We will also make frequent use of the following compact notation:

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

which are dimensionless derivatives of a species’ equilibrium distribution with respect to the square of the parallel velocity (peculiar to the species equilibrium drift velocity) and perpendicular velocity (peculiar to the $\boldsymbol{E}\times \boldsymbol{B}$ drift velocity), respectively; and

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

which measures the velocity-space anisotropy of that distribution. These definitions are, of course, only useful insofar as particle collisions and scatterings are unable to maintain a Maxwellian distribution in the equilibrium state, for which $f_{0s}^{\Vert }=f_{0s}^{\bot }=f_{0s}$ and $\,\mathfrak{D}f_{0s}=0$ . To allow for this to be the case, we order the collision frequency ${\it\nu}_{ii}\ll {\it\epsilon}{\it\omega}$ . This means that collisional relaxation of $f_{0s}$ towards a Maxwellian distribution occurs at higher orders than will be treated in this paper, as does the heating of the background plasma due to collisional smoothing of the (secularly increasing) fine-scale structure in velocity space. For our purposes, the background equilibrium is thus stationary in time.

While most applications to non-Maxwellian space and astrophysical plasmas make use of the bi-Maxwellian distribution function,

(2.24) $$\begin{eqnarray}f_{bi\text{-}M,s}(v_{\Vert },w_{\bot })\doteq \frac{n_{0s}}{\sqrt{{\rm\pi}}v_{th\Vert s}}\exp \left[-\frac{(v_{\Vert }-u_{\Vert 0s}^{\prime })^{2}}{v_{th\Vert s}^{2}}\right]\frac{1}{{\rm\pi}v_{th\bot s}^{2}}\exp \left(-\frac{w_{\bot }^{2}}{v_{th\bot s}^{2}}\right),\end{eqnarray}$$

to describe the equilibrium distribution function of the plasma (for which $f_{0s}^{\Vert }=f_{0s}^{\bot }=f_{0s}$ and $\,\mathfrak{D}f_{0s}={\it\Delta}_{s}\,f_{0s}$ ), we keep our derivation of KRMHD general with respect to the form of $f_{0s}(v_{\Vert },w_{\bot })$ .Footnote ² That being said, because of its widespread use by the space and astrophysical communities, we will often refer back to (2.24) to present useful particular cases of our more general results. To ease application of our theory to the solar wind, in appendix D we also specialise our equations for the case of a bi-kappa equilibrium distribution function.

Finally, we caution that there are some situations in weakly collisional magnetised astrophysical plasmas for which the KRMHD ordering (2.13) is inapplicable. One such situation is when the equilibrium pressure anisotropy takes on values beyond the firehose stability boundary (3.2), since $k_{\Vert }$ is not much smaller than $k_{\bot }$ for the fastest-growing firehose modes. Likewise, beyond the mirror stability boundary (B 14), the nonlinear evolution of the mirror instability involves the trapping of particles, a feature not accounted for in the KRMHD ordering (note however that the expected wavevector anisotropy $k_{\Vert }/k_{\bot }\ll 1$ of these modes is captured). That being said, the majority of the solar wind does lie within these stability boundaries (e.g. Bale et al. Reference Bale, Kasper, Howes, Quataert, Salem and Sundkvist2009), and so our theory should be appropriate for describing Alfvénic turbulence in that part of parameter space. A second limitation of the KRMHD ordering is that there are times and places in both the solar wind and the magnetosheath where ${\it\delta}B/B_{0}$ is not small (e.g. Alexandrova, Lacombe & Mangeney Reference Alexandrova, Lacombe and Mangeney2008), usually at the outer scale. However, as the free energy stored in the turbulent fluctuations cascades to smaller scales (in the inertial range and beyond), the fluctuations become smaller and more anisotropic. If anything, these fluctuations tend towards the KRMHD limit (or, more generally, the gyrokinetic limit: see appendix C).

2.4. Alfvénic fluctuations

To zeroth order in ${\it\epsilon}$ , (2.10) becomes $\boldsymbol{{\rm\nabla}}_{\!\bot }\,\boldsymbol{\cdot }\,\boldsymbol{u}_{\bot }=0$ . Likewise, the divergence-free constraint on the magnetic field, $\boldsymbol{{\rm\nabla}}\,\boldsymbol{\cdot }\,\boldsymbol{B}=0$ , becomes $\boldsymbol{{\rm\nabla}}_{\!\bot }\,\boldsymbol{\cdot }\,{\it\delta}\boldsymbol{B}_{\bot }=0$ . These simplifications allow the Alfvénic fluctuations to be expressed in terms of scalar stream (flux) functions:

(2.25a,b ) $$\begin{eqnarray}\boldsymbol{u}_{\bot }=\hat{\boldsymbol{z}}\times \boldsymbol{{\rm\nabla}}_{\!\bot }{\it\Phi}\quad \text{and}\quad \frac{{\it\delta}\boldsymbol{B}_{\bot }}{\sqrt{4{\rm\pi}{\it\rho}_{0}}}=\hat{\boldsymbol{z}}\times \boldsymbol{{\rm\nabla}}_{\!\bot }{\it\Psi}.\end{eqnarray}$$

We substitute these expressions into (2.11) and (2.12) and examine the result order by order in ${\it\epsilon}$ .

To lowest order, (2.11) and (2.12) become, respectively,

(2.26) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}_{\!\bot }\left(\mathop{\sum }_{s}{\it\delta}p_{\bot s}+\frac{B_{0}^{2}}{4{\rm\pi}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)=0, & \displaystyle\end{eqnarray}$$

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial }{\partial t}{\it\Psi}+\{{\it\Phi},{\it\Psi}\}=v_{A}\frac{\partial }{\partial z}{\it\Phi}, & \displaystyle\end{eqnarray}$$

(2.28) $$\begin{eqnarray}\{{\it\Phi},{\it\Psi}\}\doteq \,\hat{\boldsymbol{z}}\boldsymbol{\cdot }(\boldsymbol{{\rm\nabla}}_{\!\bot }{\it\Phi}\times \boldsymbol{{\rm\nabla}}_{\!\bot }{\it\Psi}).\end{eqnarray}$$

The first of these equations expresses perpendicular force balance, a result which will aid our description of the compressive fluctuations in § 2.5. The second equation is identical to the induction equation in standard RMHD. At the next order, we derive an evolution equation for the stream function. Its simplest form is obtained by taking the $z$ -component of the curl of the force equation (2.11) – the vorticity equation – which gives

(2.29) $$\begin{eqnarray}\frac{\partial }{\partial t}{\rm\nabla}_{\bot }^{2}{\it\Phi}+\{{\it\Phi},{\rm\nabla}_{\bot }^{2}{\it\Phi}\}=\left[1+\mathop{\sum }_{s}\frac{{\it\beta}_{\Vert s}}{2}\left({\it\Delta}_{s}-\frac{2u_{\Vert 0s}^{\prime 2}}{v_{th\Vert s}^{2}}\right)\right]\left(v_{A}\frac{\partial }{\partial z}{\rm\nabla}_{\bot }^{2}{\it\Psi}+\{{\it\Psi},{\rm\nabla}_{\bot }^{2}{\it\Psi}\}\right).\end{eqnarray}$$

For an isotropic equilibrium pressure ( $p_{\bot 0s}=p_{\Vert 0s},{\it\Delta}_{s}=0$ ) and no equilibrium parallel drifts ( $u_{\Vert 0s}^{\prime }=0$ ), this reduces to the standard RMHD momentum equation.

The nonlinearities in (2.27) and (2.29) involving the magnetic field imply that Alfvénic fluctuations propagate along the locally deformed magnetic field rather than the uniform equilibrium field, and so the parallel and perpendicular directions do not strictly lie along the Cartesian axes defined by the guide field. Indeed, by introducing the Lagrangian operators

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\doteq \frac{\partial }{\partial t}+\boldsymbol{u}_{\bot }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{\!\bot }=\frac{\partial }{\partial t}+\{{\it\Phi},\dots \}, & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\doteq \frac{\partial }{\partial z}+\frac{{\it\delta}\boldsymbol{B}_{\bot }}{B_{0}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{\!\bot }=\frac{\partial }{\partial z}+\frac{1}{v_{A}}\{{\it\Psi},\dots \}, & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\Psi}}{\partial t}=v_{A}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\Phi}, & \displaystyle\end{eqnarray}$$

(2.33) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}{\rm\nabla}_{\bot }^{2}{\it\Phi}=v_{A}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\frac{v_{A\ast }^{2}}{v_{A}^{2}}{\rm\nabla}_{\bot }^{2}{\it\Psi}, & \displaystyle\end{eqnarray}$$

(2.34) $$\begin{eqnarray}v_{A\ast }\doteq v_{A}\left[1+\mathop{\sum }_{s}\frac{{\it\beta}_{\Vert s}}{2}\left({\it\Delta}_{s}-\frac{2u_{\Vert 0s}^{\prime 2}}{v_{th\Vert s}^{2}}\right)\right]^{1/2}.\end{eqnarray}$$

In systems where $p_{\bot 0}-p_{\Vert 0}-\sum _{s}m_{s}n_{0s}u_{\Vert 0s}^{\prime 2}<0$ , the speed at which deformations in the magnetic field are propagated is effectively reduced by the excess parallel pressure, which undermines the restoring force exerted by the tension of the magnetic-field lines. When $p_{\bot 0}-p_{\Vert 0}-\sum _{s}m_{s}n_{0s}u_{\Vert 0s}^{\prime 2}=-B_{0}^{2}/4{\rm\pi}$ , the magnetic tension is exactly balanced by the anisotropy in the distribution function and the plasma does not respond to (perpendicular) magnetic perturbations. Parallel drifts in the equilibrium distribution functions of different species make this criterion easier to satisfy by supplementing the parallel thermal pressure. For values of $p_{\bot 0}-p_{\Vert 0}$ below this threshold, the effective Alfvén speed becomes imaginary and the stream and flux functions acquire a ${\rm\pi}/2$ relative phase shift. The plasma then becomes firehose unstable, an effect that we discuss in § 3.1. For now we caution that, if the equilibrium pressure anisotropy and parallel drifts make $v_{A\ast }/v_{A}$ as small as ${\it\epsilon}$ , the reduced ordering (2.13) is broken and the KRMHD theory, as derived here, becomes inapplicable.

Equations (2.32) and (2.33) form a closed set, and so the Alfvén-wave inertial-range cascade is completely decoupled from all other (compressive) types of perturbations (the Alfvénic cascade is further discussed in § 3). While this result is usually derived starting from the collisional MHD limit, we have shown that the same holds true even for a collisionless plasma (as in S09) with arbitrary gyrotropic equilibrium distribution function.

2.5. Compressive fluctuations

2.5.1. Parallel electric field

To obtain the equations describing the density ( ${\it\delta}n_{e}$ ) and magnetic-field-strength ( ${\it\delta}B_{\Vert }$ ) fluctuations, we return to the drift-kinetic equation (2.3). Applying the reduced ordering (2.13) and neglecting collisions, we have for the electron species

(2.35) $$\begin{eqnarray}\left(\frac{\text{d}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)\left({\it\delta}f_{e}-\frac{{\it\delta}B_{\Vert }}{B_{0}}\frac{w_{\bot }^{2}}{v_{th\bot e}^{2}}f_{0e}^{\bot }\right)+(v_{\Vert }-u_{\Vert 0e}^{\prime })\left(\frac{eE_{\Vert }}{T_{\Vert 0e}}+\frac{w_{\bot }^{2}}{v_{th\Vert e}^{2}}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)f_{0e}^{\Vert }=0.\end{eqnarray}$$

Further expanding (2.35) in the small parameter $(m_{e}/m_{i})^{1/2}$ removes the $\text{d}/\text{d}t$ term and the equilibrium electron drift ( $u_{\Vert 0e}^{\prime }$ ) to lowest order. Dividing the result by $v_{\Vert }$ and employing the $\,\mathfrak{D}$ notation (2.23) to group terms, we find

(2.36) $$\begin{eqnarray}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left({\it\delta}f_{e}+\frac{w_{\bot }^{2}}{v_{th\bot e}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\,\mathfrak{D}f_{0e}\right)+\frac{eE_{\Vert }}{T_{\Vert 0e}}\,f_{0e}^{\Vert }=0.\end{eqnarray}$$

Integrating over the velocity space leads to an expression for the parallel electric field,

(2.37) $$\begin{eqnarray}E_{\Vert }=-\frac{1}{C_{0e}^{\Vert }}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\,\frac{T_{\Vert 0e}}{e}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right),\end{eqnarray}$$

where we use the following notation: for integer $\ell$ ,

(2.38) $$\begin{eqnarray}\displaystyle C_{\ell e}^{\Vert } & \doteq & \displaystyle \frac{1}{n_{0e}}\int \text{d}^{3}\boldsymbol{v}\,\frac{1}{\ell !}\left(\frac{w_{\bot }}{v_{th\bot e}}\right)^{2\ell }f_{0e}^{\Vert }\nonumber\\ \displaystyle & = & \displaystyle 1~\text{for a bi-Maxwellian}\end{eqnarray}$$

are dimensionless coefficients related to perpendicular moments of the parallel-differentiated equilibrium electron distribution function, and

(2.39) $$\begin{eqnarray}{\it\Delta}_{\ell e}\doteq C_{\ell e}^{\Vert }\frac{p_{\bot 0e}}{p_{\Vert 0e}}-1\end{eqnarray}$$

is the dimensionless pressure anisotropy (cf. (2.20)) of the electrons weighted by those coefficients. For isotropic electrons, the parallel electric field is entirely related to fluctuations in the electron (and therefore ion) density; the corresponding (first) term in (2.37) ultimately leads to the Landau damping of ion acoustic waves. When the equilibrium electron pressure is anisotropic, fluctuations in magnetic-field strength also contribute to the parallel electric field; this second term enforces quasineutrality in the face of preferential exclusion of large-pitch-angle electrons from regions of enhanced field strength.

2.5.2. Pressure perturbations

With knowledge of the parallel electric field (2.37), we can rewrite the perturbed electron distribution asFootnote ²

(2.40) $$\begin{eqnarray}{\it\delta}f_{e}=\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)f_{0e}^{\Vert }-\frac{w_{\bot }^{2}}{v_{th\bot e}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\,\mathfrak{D}f_{0e}\end{eqnarray}$$

and compute the perturbed parallel and perpendicular electron pressures by taking the appropriate second moments (cf. (2.17)):

(2.41) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}p_{\Vert e}}{p_{\Vert 0e}}=\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)-{\it\Delta}_{e}\frac{{\it\delta}B_{\Vert }}{B_{0}}, & \displaystyle\end{eqnarray}$$

(2.42) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}p_{\bot e}}{p_{\bot 0e}}=\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)-2{\it\Delta}_{2e}\frac{{\it\delta}B_{\Vert }}{B_{0}}. & \displaystyle\end{eqnarray}$$

Note that, if the equilibrium distribution function of electrons is isotropic, no electron pressure anisotropy can be generated by the fluctuations and the electron fluid remains isothermal along magnetic-field lines:

(2.43) $$\begin{eqnarray}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\delta}T_{e}=0.\end{eqnarray}$$

The latter occurs physically by rapid electron conduction along field lines. Deviations from isothermality in non-Maxwellian plasmas arise when electrons conserve their adiabatic invariant ${\it\mu}_{e}$ in the presence of field-line compressions and rarefactions. Indeed, by taking the third velocity moments of (2.40), we see that the parallel flows of parallel and perpendicular electron heat, respectively

(2.44a,b ) $$\begin{eqnarray}Q_{\Vert e}\doteq \int \text{d}^{3}\boldsymbol{v}\,m_{e}v_{\Vert }^{3}\,{\it\delta}f_{e}\quad \text{and}\quad Q_{\bot e}\doteq \int \text{d}^{3}\boldsymbol{v}\,m_{e}v_{\Vert }\frac{w_{\bot }^{2}}{2}\,{\it\delta}f_{e},\end{eqnarray}$$

satisfy $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}Q_{\Vert e}=\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}Q_{\bot e}=0$ . For bi-Maxwellian electrons, this translates into

(2.45a,b ) $$\begin{eqnarray}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\delta}T_{\Vert e}=0\quad \text{and}\quad \hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}({\it\delta}T_{\bot e}+{\it\Delta}_{e}\langle {\it\mu}_{e}\rangle {\it\delta}B_{\Vert })=0,\end{eqnarray}$$

where $\langle {\it\mu}_{e}\rangle =m_{e}v_{th\bot e}^{2}/2B_{0}$ is the lowest-order contribution to the mean adiabatic invariant of the electrons. Equation (2.45) states that, while the parallel temperature of the electrons remains constant along field lines, the perpendicular temperature cannot do so without violating ${\it\mu}_{e}$ conservation (cf. equations 39–40 of Snyder, Hammett & Dorland Reference Snyder, Hammett and Dorland1997).

2.5.3. Reduced drift-kinetic equation

It is often computationally convenient as well as physically illuminating to replace the perturbed distribution function ${\it\delta}f_{s}$ by the function

(2.46) $$\begin{eqnarray}g_{s}\doteq {\it\delta}f_{s}-\frac{w_{\bot }^{2}}{v_{th\bot s}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}f_{0s}^{\bot },\end{eqnarray}$$

which is the perturbed distribution function if $f_{0s}$ is taken to be a function of the exact adiabatic invariant ${\it\mu}_{s}=mw_{\bot }^{2}/2B$ (rather than of $mw_{\bot }^{2}/2B_{0}$ ); to wit,

(2.47) $$\begin{eqnarray}\displaystyle f_{s}-f_{0s}(v_{\Vert },{\it\mu}_{s}) & = & \displaystyle f_{s}-f_{0s}\left(v_{\Vert },\frac{m_{s}w_{\bot }^{2}}{2B_{0}}-\frac{m_{s}w_{\bot }^{2}}{2B_{0}}\left(1-\frac{B_{0}}{B}\right)\right)\nonumber\\ \displaystyle & \simeq & \displaystyle f_{s}-f_{0s}(v_{\Vert },w_{\bot })+\frac{w_{\bot }}{2}\frac{{\it\delta}B_{\Vert }}{B_{0}}\frac{\partial f_{0s}}{\partial w_{\bot }}\nonumber\\ \displaystyle & = & \displaystyle {\it\delta}f_{s}-\frac{w_{\bot }^{2}}{v_{th\bot s}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}f_{0s}^{\bot }=g_{s}.\end{eqnarray}$$

Indeed, using (2.40) in (2.46), we find that

(2.48) $$\begin{eqnarray}g_{e}=\left[\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)-\frac{w_{\bot }^{2}}{v_{th\Vert e}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right]f_{0e}^{\Vert }\end{eqnarray}$$

does not contain any derivatives of the equilibrium distribution function with respect to $w_{\bot }$ . We will see that the same holds true for $g_{i}$ , whose evolution equation we now derive.

The evolution equation for the perturbed ion distribution function is obtained by applying the reduced ordering (2.13) to (2.3) with $s=i$ . Using (2.37) for the parallel electric field, the reduced kinetic equation for the ions may be written in a compact form analogous to equation (145) of S09:

(2.49) $$\begin{eqnarray}\left(\frac{\text{d}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)g_{i}+(v_{\Vert }-u_{\Vert 0i}^{\prime })\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left[\frac{1}{C_{0e}^{\Vert }}\frac{Z_{i}}{{\it\tau}_{\Vert i}}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+\frac{w_{\bot }^{2}}{v_{th\Vert i}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right]f_{0i}^{\Vert }=0.\end{eqnarray}$$

Note that this equation does not contain any derivatives of the equilibrium distribution function with respect to $w_{\bot }$ . In terms of $g_{i}$ , (2.16), (2.18), and (2.26) become

(2.50) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}n_{e}}{n_{0e}}-\frac{{\it\delta}B_{\Vert }}{B_{0}}=\mathop{\sum }_{i}\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,g_{i}, & \displaystyle\end{eqnarray}$$

(2.51) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\Vert }+\mathop{\sum }_{i}c_{i}{\it\delta}u_{\Vert i}^{\prime }=\mathop{\sum }_{i}\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,(v_{\Vert }-u_{\Vert 0i}^{\prime })\,g_{i}, & \displaystyle\end{eqnarray}$$

(2.52) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+2\left(\mathop{\sum }_{i}c_{i}\frac{{\it\tau}_{\bot i}}{Z_{i}}+\frac{1}{{\it\beta}_{\bot e}}-{\it\Delta}_{2e}\right)\!\frac{{\it\delta}B_{\Vert }}{B_{0}}=-\mathop{\sum }_{i}\frac{{\it\tau}_{\bot i}}{Z_{i}}\frac{c_{i}}{n_{0i}}\!\int \!\text{d}^{3}\boldsymbol{v}\frac{w_{\bot }^{2}}{v_{th\bot i}^{2}}g_{i},\hspace{-6.0pt} & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which reduce to equations (146)–(148) of S09 for a single-ion-species Maxwellian plasma.

Equations (2.49)–(2.52) evolve the ion distribution function $g_{i}$ ; the ‘slow-wave quantities’ $u_{\Vert }$ , ${\it\delta}u_{\Vert i}^{\prime }$ , and ${\it\delta}B_{\Vert }$ ; and the density fluctuations ${\it\delta}n_{e}$ . All nonlinearities are contained in the $\text{d}/\text{d}t$ and $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}$ Lagrangian operators, which include the Alfvénic quantities ${\it\Phi}$ and ${\it\Psi}$ ; these are determined separately and independently by (2.27) and (2.29). Nonlinear scattering/mixing of slow waves and the entropy mode by the Alfvénic perturbations takes the form of passive advection of the distribution function $g_{i}$ . In other words, even when the equilibrium distribution function is non-Maxwellian and there are parallel drifts between the various species, the compressive fluctuations are passively transported by the Alfvénic fluctuations, a result that we have, thus, generalised from MHD (Lithwick & Goldreich Reference Lithwick and Goldreich2001) and Maxwellian KRMHD (S09). The passive cascades of compressive fluctuations, as well as their kinetic damping and susceptibility to mirror instability in a pressure-anisotropic plasma, are further discussed in § 4.

2.6. Summary

The reduced theory derived here evolves $4+N_{ion}$ unknown functions: ${\it\Phi}$ , ${\it\Psi}$ , ${\it\delta}B_{\Vert }$ , ${\it\delta}n_{e}$ , and $g_{i}$ for each of the $N_{ion}$ different ionic species. The stream and flux functions, ${\it\Phi}$ and ${\it\Psi}$ respectively, are related to the fluid quantities (perpendicular velocity and magnetic-field perturbations) via (2.25). They satisfy a closed set of equations, (2.27)–(2.29), which describe the decoupled cascade of Alfvénic fluctuations whose phase speed is modified by pressure anisotropy and inter-species parallel drifts. In the collisional limit, they revert to the standard equations of RMHD. The density and magnetic-field-strength fluctuations (the ‘compressive’ fluctuations, or the slow waves and the entropy mode in the collisional limit) require a kinetic description in terms of the ion distribution function $g_{i}$ , which is evolved by the kinetic equation (2.49). This kinetic equation itself contains ${\it\delta}n_{e}$ and ${\it\delta}B_{\Vert }$ , which are, in turn, calculated by taking velocity-space integrals of $g_{i}$ via (2.50) and (2.52). The nonlinear evolution of $g_{i}$ , ${\it\delta}B_{\Vert }$ , and ${\it\delta}n_{e}$ is due solely to passive advection of $g_{i}$ by the Alfvénic turbulence, which mixes ${\it\delta}n_{e}$ and ${\it\delta}B_{\Vert }$ in the direction transverse to the magnetic field. In the Lagrangian frame associated with the Alfvénic fluctuations, the compressive fluctuations obey a one-dimensional linear equation, which may be solved independently of the Alfvénic turbulence.

Here we summarise our new set of equations:

(2.53a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\Psi}}{\partial t}=v_{A}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\Phi}, & \displaystyle\end{eqnarray}$$

(2.53b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}{\rm\nabla}_{\bot }^{2}{\it\Phi}=v_{A}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\frac{v_{A\ast }^{2}}{v_{A}^{2}}{\rm\nabla}_{\bot }^{2}{\it\Psi}, & \displaystyle\end{eqnarray}$$

(2.53c ) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\text{d}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)g_{i}+(v_{\Vert }-u_{\Vert 0i}^{\prime })\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left[\frac{1}{C_{0e}^{\Vert }}\frac{Z_{i}}{{\it\tau}_{\Vert i}}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+\frac{w_{\bot }^{2}}{v_{th\Vert i}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right]f_{0i}^{\Vert }=0; & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.53d ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{{\it\delta}n_{e}}{n_{0e}}=-\mathop{\sum }_{i}[\dots ]^{-1}\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,\left[\frac{w_{\bot }^{2}}{v_{th\bot i}^{2}}-2c_{i}\left(\mathop{\sum }_{i^{\prime }}\frac{c_{i^{\prime }}{\it\tau}_{\bot i^{\prime }}Z_{i}}{c_{i}{\it\tau}_{\bot i}Z_{i^{\prime }}}+\frac{1}{{\it\beta}_{\bot i}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \;\;\qquad \qquad \qquad \qquad \qquad \qquad \left.+\,\frac{Z_{i}}{{\it\tau}_{\bot i}}\left(2{\it\Delta}_{2e}-\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}{\it\Delta}_{1e}\right)\right]g_{i},\end{eqnarray}$$

(2.53e ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}B_{\Vert }}{B_{0}}=-\mathop{\sum }_{i}[\dots ]^{-1}\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,\left(\frac{w_{\bot }^{2}}{v_{th\bot i}^{2}}+\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}\frac{Z_{i}}{{\it\tau}_{\bot i}}\right)g_{i}, & \displaystyle\end{eqnarray}$$

(2.54) $$\begin{eqnarray}[\dots ]\doteq \frac{Z_{i}}{{\it\tau}_{\bot i}}\left(\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}-2{\it\Delta}_{2e}+\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}{\it\Delta}_{1e}\right)+2c_{i}\left(\mathop{\sum }_{i^{\prime }}\frac{c_{i^{\prime }}{\it\tau}_{\bot i^{\prime }}Z_{i}}{c_{i}{\it\tau}_{\bot i}Z_{i^{\prime }}}+\frac{1}{{\it\beta}_{\bot i}}\right).\end{eqnarray}$$

These equations reduce to (155)–(159) of S09 when the equilibrium distribution function is Maxwellian and only one ionic species is present (for which $v_{A\ast }=v_{A}$ , $u_{\Vert 0i}^{\prime }=0$ , $C_{\ell e}^{\Vert }=1$ , ${\it\tau}_{\Vert s}={\it\tau}_{\bot s}$ , ${\it\Delta}_{\ell e}=0$ , $v_{th\Vert s}=v_{th\bot s}$ , ${\it\beta}_{\Vert s}={\it\beta}_{\bot s}$ , and $f_{0s}^{\Vert }=f_{0s}^{\bot }=f_{0s}$ ).

It should be noted that (2.53) are ideal, in that they are ignorant of any physics capable of dissipating the large gradients in phase space that will inevitably be produced as the turbulence cascades to smaller and smaller scales. While some of this fine-scale structure is regularised by finite-Larmor-radius effects, which are included in the non-Maxwellian gyrokinetic theory derived in appendix C, numerical implementation of (2.53) requires the addition of finite collisionality and resistivity.

3.1. Linear theory of Alfvénic fluctuations: Alfvén waves and firehose instability

The linear theory of Alfvénic fluctuations in KRMHD can be readily obtained by dropping the nonlinear terms in (2.53a ) and (2.53b ) and adopting the solutions ${\it\Phi}$ , ${\it\Psi}\sim \exp (-\text{i}{\it\omega}t+\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r})$ . The resulting dispersion relation is simply

(3.1) $$\begin{eqnarray}{\it\omega}=\pm k_{\Vert }v_{A\ast },\end{eqnarray}$$

with eigenvectors satisfying ${\it\Phi}=\mp {\it\Psi}(v_{A\ast }/v_{A})$ . When

(3.2) $$\begin{eqnarray}p_{\bot 0}-p_{\Vert 0}-\mathop{\sum }_{s}m_{s}n_{0s}u_{\Vert 0s}^{\prime 2}<-\frac{B_{0}^{2}}{4{\rm\pi}},\end{eqnarray}$$

the phase speed of the Alfvén wave becomes imaginary and the firehose instability results (Rosenbluth Reference Rosenbluth1956; Chandrasekhar, Kaufman & Watson Reference Chandrasekhar, Kaufman and Watson1958; Parker Reference Parker1958; Vedenov & Sagdeev Reference Vedenov and Sagdeev1958). Physically, negative pressure anisotropies and/or parallel drifts reduce the elasticity of the magnetic-field lines, undermining the supplied restoring force necessary to propagate the wave. In KRMHD, the fastest growth occurs at arbitrarily small parallel scales, with no small-scale regularisation accessible within the long-wavelength approximation, $k{\it\rho}_{i}\ll 1$ , in which (2.2) is derived. To obtain the fastest growing mode, finite-Larmor-radius effects must be taken into account (cf. Schekochihin et al. Reference Schekochihin, Cowley, Rincon and Rosin2010). Direct calculation from the hot-plasma dispersion relation yields $k_{\Vert }{\it\rho}_{i}\sim |{\it\beta}_{\bot }/{\it\beta}_{\Vert }-1+2/{\it\beta}_{\Vert }|^{1/2}$ for the parallel ( $k_{\bot }=0$ ) firehose (Kennel & Sagdeev Reference Kennel and Sagdeev1967; Davidson & Völk Reference Davidson and Völk1968) and $k{\it\rho}_{i}\sim 1$ for the oblique firehose with $k_{\bot }\neq 0$ (Yoon, Wu & de Assis Reference Yoon, Wu and de Assis1993; Hellinger & Matsumoto Reference Hellinger and Matsumoto2000).

3.2. Elsasser fields and Alfvén ratio

The effective Alfvén speed can be used to cast (2.32) and (2.33) in a symmetric form via the introduction of the generalised Elsasser potentials,

(3.3) $$\begin{eqnarray}{\it\zeta}^{\pm }\doteq {\it\Phi}\pm \frac{v_{A\ast }}{v_{A}}{\it\Psi},\end{eqnarray}$$

and the corresponding Elsasser fields

(3.4) $$\begin{eqnarray}\boldsymbol{z}^{\pm }\doteq \hat{\boldsymbol{z}}\times \boldsymbol{{\rm\nabla}}_{\!\bot }{\it\zeta}^{\pm }=\boldsymbol{u}_{\bot }\pm \frac{v_{A\ast }}{v_{A}}\frac{{\it\delta}\boldsymbol{B}_{\bot }}{\sqrt{4{\rm\pi}{\it\rho}_{0}}}.\end{eqnarray}$$

The latter are a straightforward generalisation of the standard Elsasser (Reference Elsasser1950) variables to non-Maxwellian equilibria, for which $v_{A\ast }\neq v_{A}$ . Combining (2.27) and (2.29), one can show that the Elsasser potentials satisfy

(3.5) $$\begin{eqnarray}\left(\frac{\partial }{\partial t}\mp v_{A\ast }\frac{\partial }{\partial z}\right){\rm\nabla}_{\bot }^{2}{\it\zeta}^{\pm }=-\frac{1}{2}\left(\left\{{\it\zeta}^{+},{\rm\nabla}_{\bot }^{2}{\it\zeta}^{-}\right\}+\left\{{\it\zeta}^{-},{\rm\nabla}_{\bot }^{2}{\it\zeta}^{+}\right\}\mp {\rm\nabla}_{\bot }^{2}\left\{{\it\zeta}^{+},{\it\zeta}^{-}\right\}\right)\!.\end{eqnarray}$$

Thus, the standard result that nonlinear interactions (‘scatterings’) of Alfvénic fluctuations occur only between counter-propagating fluctuations (Kraichnan Reference Kraichnan1965) holds true for general (gyrotropic) distribution functions. What is modified by the non-Maxwellian nature of the distribution function is the amount of (perpendicular) magnetic fluctuations that comprise each of the Elsasser potentials. As $v_{A\ast }\rightarrow 0$ , the magnetic fluctuations fail to propagate and the distinction between ${\it\zeta}^{+}$ and ${\it\zeta}^{-}$ is no longer meaningful. Indeed, the very idea of critical balance that underpins the RMHD ordering $k_{\Vert }v_{A}\sim k_{\bot }u_{\bot }$ is based upon a causality argument: fluctuations cannot be correlated over a distance larger than that over which an Alfvén wave propagates in a nonlinear interaction time. Significantly reducing the signal speed, with $v_{A\ast }/v_{A}\sim {\it\epsilon}$ or smaller, interferes with this argument and breaks the reduced ordering used in this paper. This is what will happen if the firehose threshold is approached. On the unstable side of the threshold, the firehose fluctuations that emerge are not anisotropic in the same way that Alfvénic, or more generally gyrokinetic, fluctuations are: in fact, they have $k_{\Vert }\sim k_{\bot }$ (Yoon et al. Reference Yoon, Wu and de Assis1993; Hellinger & Matsumoto Reference Hellinger and Matsumoto2000; Kunz, Schekochihin & Stone Reference Kunz, Schekochihin and Stone2014). This is why the considerations in this paper do not describe the turbulence on the unstable side of the firehose threshold.

The fact that the Alfvén ratio

(3.6) $$\begin{eqnarray}r_{A}\doteq \left|\frac{{\it\Phi}}{{\it\Psi}}\right|^{2}=1+\mathop{\sum }_{s}\frac{{\it\beta}_{\Vert s}}{2}\left({\it\Delta}_{s}-\frac{2u_{\Vert 0s}^{\prime 2}}{v_{th\Vert s}^{2}}\right)\end{eqnarray}$$

depends on the anisotropy inherent to the distribution function (cf. Barnes Reference Barnes, Parker, Kennel and Lanzerotti1979) becomes testable in the solar wind, where measurements find that the energy in magnetic-field fluctuations exceeds the energy in the velocity fluctuations, $r_{A}<1$ (e.g. Belcher & Davis Reference Belcher and Davis1971; Matthaeus & Goldstein Reference Matthaeus and Goldstein1982; Bruno, Bavassano & Villante Reference Bruno, Bavassano and Villante1985; Roberts et al. Reference Roberts, Klein, Goldstein and Matthaeus1987; Tu, Marsch & Thieme Reference Tu, Marsch and Thieme1989; Marsch & Tu Reference Marsch and Tu1990; Grappin, Velli & Mangeney Reference Grappin, Velli and Mangeney1991; Goldstein, Neugebauer & Smith Reference Goldstein, Neugebauer and Smith1995; Bavassano, Pietropaolo & Bruno Reference Bavassano, Pietropaolo and Bruno1998; Podesta, Roberts & Goldstein Reference Podesta, Roberts and Goldstein2007; Salem et al. Reference Salem, Mangeney, Bale and Veltri2009; Perri & Balogh Reference Perri and Balogh2010; Chen et al. Reference Chen, Mallet, Yousef, Schekochihin and Horbury2011; Borovsky Reference Borovsky2012). This result is often interpreted in terms of ‘residual energy’, ${\it\sigma}_{r}\doteq (r_{A}-1)/(r_{A}+1)$ , the difference between the energy in velocity and magnetic-field fluctuations that is believed to be an inherent feature of the turbulence itself (Pouquet, Frisch & Leorat Reference Pouquet, Frisch and Leorat1976; see Chen et al. Reference Chen, Bale, Salem and Maruca2013 and Wicks et al. Reference Wicks, Roberts, Mallet, Schekochihin, Horbury and Chen2013 for brief reviews of the relevant literature and contemporary analyses). While the observed scale-dependent component of the residual energy is likely to be intrinsic – recent theory predicts a $\propto k_{\bot }^{-2}$ residual-energy spectrum for both balanced and moderately imbalanced strong turbulence (e.g. Müller & Grappin Reference Müller and Grappin2005; Boldyrev et al. Reference Boldyrev, Perez, Borovsky and Podesta2011; Boldyrev, Perez & Wang Reference Boldyrev, Perez, Wang, Pogorelov, Font, Audit and Zank2012) – the constant component can (at least partially) be attributed to non-MHD corrections to the Alfvén speed due to pressure anisotropies and parallel drifts. Indeed, this interpretation is supported by (3.6), and one may thus construct a more appropriate Alfvén ratio by weighting the flux function by $v_{A\ast }/v_{A}$ (Belcher & Davis Reference Belcher and Davis1971; Matthaeus & Goldstein Reference Matthaeus and Goldstein1982). This was the route followed by Chen et al. (Reference Chen, Bale, Salem and Maruca2013), who found that such a ‘kinetic normalisation’, equivalent to using

(3.7) $$\begin{eqnarray}r_{A\ast }\doteq \left|\frac{v_{A}}{v_{A\ast }}\frac{{\it\Phi}}{{\it\Psi}}\right|^{2}\end{eqnarray}$$

instead of $r_{A}$ , yields fluctuations that are closer to equipartition, with a mean residual energy of ${\it\sigma}_{r\ast }\doteq (r_{A\ast }-1)/(r_{A\ast }+1)=-0.19$ and a mean Alfvén ratio of $r_{A\ast }=0.71$ (rather than ${\it\sigma}_{r}=-0.43$ and $r_{A}=0.40$ using (3.6)). Using the appropriate kinetic normalisation is thus essential when measuring quantities like the residual energy in plasmas with anisotropic distribution functions, such as the solar wind.

3.3. Alfvén-wave invariants

Another standard result – that interactions between ‘ $+$ ’ and ‘ $-$ ’ waves occur without exchanging energy – can also be shown to hold true in general. Multiplying (3.5) by ${\it\rho}_{0}{\it\zeta}^{\pm }$ and integrating the result over space, we find

(3.8) $$\begin{eqnarray}\frac{\text{d}W_{AW}^{\pm }}{\text{d}t}=0,\end{eqnarray}$$

where

(3.9) $$\begin{eqnarray}W_{AW}^{\pm }\doteq \frac{1}{2}\int \text{d}^{3}\boldsymbol{r}\,{\it\rho}_{0}|\boldsymbol{{\rm\nabla}}_{\!\bot }{\it\zeta}^{\pm }|^{2}\end{eqnarray}$$

are the independently conserved (free) energies of the forward- and backward-propagating Alfvénic fluctuations, respectively. Their sum,

(3.10) $$\begin{eqnarray}\displaystyle W_{AW} & \doteq & \displaystyle W_{AW}^{+}+W_{AW}^{-}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\int \text{d}^{3}\boldsymbol{r}\,{\it\rho}_{0}(|\boldsymbol{{\rm\nabla}}_{\!\bot }{\it\zeta}^{+}|^{2}+|\boldsymbol{{\rm\nabla}}_{\!\bot }{\it\zeta}^{-}|^{2})\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\int \text{d}^{3}\boldsymbol{r}\,{\it\rho}_{0}\left(|\boldsymbol{{\rm\nabla}}_{\!\bot }{\it\Phi}|^{2}+\frac{v_{A\ast }^{2}}{v_{A}^{2}}|\boldsymbol{{\rm\nabla}}_{\!\bot }{\it\Psi}|^{2}\right)\nonumber\\ \displaystyle & = & \displaystyle \int \text{d}^{3}\boldsymbol{r}\left\{\frac{{\it\rho}_{0}u_{\bot }^{2}}{2}+\left[\,1+\mathop{\sum }_{s}\frac{{\it\beta}_{\Vert s}}{2}\left({\it\Delta}_{s}-\frac{2u_{\Vert 0s}^{\prime 2}}{v_{th\Vert s}^{2}}\right)\right]\frac{{\it\delta}B_{\bot }^{2}}{8{\rm\pi}}\right\},\end{eqnarray}$$

is, of course, also conserved. If $v_{A\ast }^{2}$ (equivalently, the expression in the square brackets in (3.10)) is positive – i.e. if the plasma is firehose-stable – $W_{AW}$ is a positive definite quantity measuring the total kinetic and potential energy stored in the Alfvénic fluctuations. To interpret (3.10) when $v_{A\ast }^{2}$ is driven negative by the pressure anisotropy and parallel drifts, we separate the various terms in the conservation law for $W_{AW}$ as follows:

(3.11) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{r}\left(\frac{{\it\rho}_{0}u_{\bot }^{2}}{2}+\frac{{\it\delta}B_{\bot }^{2}}{8{\rm\pi}}\right)=\mathop{\sum }_{s}\frac{n_{0s}T_{\Vert 0s}}{2}\int \text{d}^{3}\boldsymbol{r}\,\left|{\it\Delta}_{s}-\frac{2u_{\Vert 0s}^{\prime 2}}{v_{th\Vert s}^{2}}\right|\frac{\partial }{\partial t}\frac{{\it\delta}B_{\bot }^{2}}{B_{0}^{2}}.\end{eqnarray}$$

We then see that the terms on the right-hand side of this equation constitute a source for the kinetic and magnetic fluctuations on the left-hand side. Indeed, recent work on the nonlinear evolution of the firehose instability (e.g. Schekochihin et al. Reference Schekochihin, Cowley, Kulsrud, Rosin and Heinemann2008b ; Rosin et al. Reference Rosin, Schekochihin, Rincon and Cowley2011; Kunz et al. Reference Kunz, Schekochihin and Stone2014) has shown that the rate of relaxation of the pressure anisotropy ${\it\Delta}_{s}$ is related to $\partial {\it\delta}B_{\bot }^{2}/\partial t$ , and so one can, heuristically, interpret the right-hand side of (3.11) as a velocity-space source of free energy multiplied by the rate at which fluctuations act to remove that source of free energy. In this case, the Alfvénic invariant (3.10) is minimised by growing fluctuations.

Another way to interpret (3.10) for a non-Maxwellian plasma is as follows. How close the equilibrium pressure-anisotropic distribution is to the firehose threshold has the effect of weighting the (free) energy associated with perpendicular magnetic perturbations. As the threshold is approached, bending field lines becomes energetically less demanding (dynamically, the negative-pressure-anisotropy stress cancels the tension force: see (2.11)). As the threshold is crossed, $W_{AW}$ is no longer positive definite and so can be conserved even if perturbations grow – which is indeed what happens.

In § 5.1, we show that $W_{AW}$ is part of a generalised free-energy invariant conserved and cascaded to small scales in phase space by the plasma turbulence.

4.1. Compressive invariant

In § 2.5.3, we derived the evolution and constraint equations for the perturbed ion distribution function $g_{i}$ , whose moments describe compressive fluctuations in a pressure-anisotropic plasma. The nonlinear evolution of these fluctuations is due solely to passive advection of $g_{i}$ by the Alfvénic turbulence, which mixes ${\it\delta}n_{e}$ and ${\it\delta}B_{\Vert }$ in the direction transverse to the local magnetic field. (Mathematically, this is a statement that, in the Lagrangian frame associated with the Alfvénic component of the turbulence, (2.53c ) is linear.) During this mixing, the compressive fluctuations satisfy an important conservation law, which we now derive.

Multiplying (2.49) by $({\it\tau}_{\Vert i}/Z_{i})(c_{i}/n_{0i})(g_{i}/f_{0i}^{\Vert })$ and integrating over the phase space, we find that

(4.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{r}\,\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,\frac{g_{i}^{2}}{2f_{0i}^{\Vert }}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\int \text{d}^{3}\boldsymbol{r}\,\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,(v_{\Vert }-u_{\Vert 0i}^{\prime })g_{i}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left[\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{w_{\bot }^{2}}{v_{th\Vert i}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right]=0.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

On the other hand, multiplying (2.49) by the term in the square brackets in (4.1), integrating the result over phase space, and performing integration by parts gives

(4.2) $$\begin{eqnarray}\displaystyle & & \displaystyle \int \text{d}^{3}\boldsymbol{r}\int \text{d}^{3}\boldsymbol{v}\,\frac{\text{d}g_{i}}{\text{d}t}\left[\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{w_{\bot }^{2}}{v_{th\Vert i}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =\int \text{d}^{3}\boldsymbol{r}\int \text{d}^{3}\boldsymbol{v}\,v_{\Vert }g_{i}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left[\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{w_{\bot }^{2}}{v_{th\Vert i}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right].\end{eqnarray}$$

Using this expression in (4.1), summing over ion species, and using (2.50) and (2.52) to eliminate the resulting velocity-space integrals of $g_{i}$ produces the following conservation law:

(4.3) $$\begin{eqnarray}\displaystyle \frac{\text{d}W_{compr}}{\text{d}t} & = & \displaystyle \int \text{d}^{3}\boldsymbol{r}\,\mathop{\sum }_{i}u_{\Vert 0i}^{\prime }\left[T_{\Vert 0i}{\it\delta}n_{i}\frac{Z_{i}}{{\it\tau}_{\Vert i}}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\Delta}_{1e}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)+{\it\delta}p_{\bot i}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right]\nonumber\\ \displaystyle & = & \displaystyle -\int \text{d}^{3}\boldsymbol{r}\,u_{\Vert 0i}^{\prime }\left(Z_{i}e{\it\delta}n_{i}E_{\Vert }-{\it\delta}p_{\bot i}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\right),\end{eqnarray}$$

where

(4.4) $$\begin{eqnarray}\displaystyle W_{compr} & \doteq & \displaystyle \frac{n_{0e}T_{\Vert 0e}}{2}\int \text{d}^{3}\boldsymbol{r}\left\{\mathop{\sum }_{i}\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{c_{i}}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,\,\frac{g_{i}^{2}}{f_{0i}^{\Vert }}+\frac{1}{C_{0e}^{\Vert }}\left(\frac{{\it\delta}n_{e}}{n_{0e}}-\frac{{\it\delta}B_{\Vert }}{B_{0}}\right)^{2}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.\frac{p_{\bot 0e}}{p_{\Vert 0e}}\left[\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}-2{\it\Delta}_{2e}+\frac{C_{1e}^{\Vert }}{C_{0e}^{\Vert }}{\it\Delta}_{1e}+2\left(\mathop{\sum }_{i}c_{i}\frac{{\it\tau}_{\bot i}}{Z_{i}}+\frac{1}{{\it\beta}_{\bot e}}\right)\right]\frac{{\it\delta}B_{\Vert }^{2}}{B_{0}^{2}}\right\}.\end{eqnarray}$$

In the absence of interspecies drifts, $W_{compr}$ is an invariant conserved by (2.49)–(2.52). The simpler version of $W_{compr}$ that is conserved for the pressure-isotropic case ((201) of S09) is related to the perturbed entropy ${\it\delta}S_{s}$ of the system (cf. Krommes & Hu Reference Krommes and Hu1994; Sugama et al. Reference Sugama, Okamoto, Horton and Wakatani1996; Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Plunk, Quataert and Tatsuno2008a ). With interspecies parallel drifts, the right-hand side of (4.3) constitutes a source or sink for this quantity. It is the work done by the fluctuating parallel electric field (2.37) and by magnetic-mirror forces acting on the interspecies drifts, and represents the exchange of free energy between these drifts and the compressive fluctuations. (See § B.2 for a specific example of this physics, where we explicitly demonstrate that free energy can flow into or out of the interspecies drifts depending upon whether the system is unstable to an ion-acoustic, i.e. streaming, instability.)

Adding and subtracting the phase-space integral of $T_{\Vert 0e}g_{e}^{2}/2f_{0e}^{\Vert }$ to the right-hand side of (4.4), with $g_{e}$ given by (2.48), the compressive invariant may be rewritten in the following compact form:

(4.5) $$\begin{eqnarray}W_{compr}=\int \text{d}^{3}\boldsymbol{r}\left[\mathop{\sum }_{s}\int \text{d}^{3}\boldsymbol{v}\,\frac{T_{\Vert 0s}g_{s}^{2}}{2f_{0s}^{\Vert }}-(1+{\it\beta}_{\bot })\frac{{\it\delta}B_{\Vert }^{2}}{8{\rm\pi}}\right].\end{eqnarray}$$

We now make a transformation analogous to that made in equation (149) of S09. Defining

(4.6) $$\begin{eqnarray}{\it\delta}\widetilde{f}_{s}\doteq g_{s}+\frac{w_{\bot }^{2}}{v_{th\Vert s}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}f_{0s}^{\Vert }={\it\delta}f_{s}+\frac{w_{\bot }^{2}}{v_{th\bot s}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\,\mathfrak{D}f_{0s}\end{eqnarray}$$

and using (2.48) and (2.52) to eliminate the resulting velocity-space integrals over $g_{e}$ and $g_{i}$ , respectively, we can rewrite (4.5) in a particularly useful form:

(4.7) $$\begin{eqnarray}W_{compr}=\int \text{d}^{3}\boldsymbol{r}\left[\mathop{\sum }_{s}\int \text{d}^{3}\boldsymbol{v}\,\frac{T_{\Vert 0s}{\it\delta}\widetilde{f}_{s}^{\,2}}{2f_{0s}^{\Vert }}+\left(1-\mathop{\sum }_{s}{\it\beta}_{\bot s}{\it\Delta}_{2s}\right)\frac{{\it\delta}B_{\Vert }^{2}}{8{\rm\pi}}\right],\end{eqnarray}$$

where

(4.8) $$\begin{eqnarray}{\it\Delta}_{2s}=\left(\frac{1}{n_{0s}}\int \text{d}^{3}\boldsymbol{v}\,\frac{1}{2}\frac{w_{\bot }^{4}}{v_{th\bot s}^{4}}f_{0s}^{\Vert }\right)\frac{p_{\bot 0s}}{p_{\Vert 0s}}-1\end{eqnarray}$$

is the extension of ${\it\Delta}_{2e}$ to arbitrary species $s$ (cf. (2.39)). This form of $W_{compr}$ parallels the final expression in (3.10) for the Alfvénic invariant: we have a quantity that has one interpretation if $1-\sum _{s}{\it\beta}_{\bot s}{\it\Delta}_{2s}>0$ , namely that it is the generalised energy of the compressive fluctuations, and another if $1-\sum _{s}{\it\beta}_{\bot s}{\it\Delta}_{2s}<0$ , in which case the final term in (4.7) becomes a free-energy source for the mirror instability, for which the expression multiplying ${\it\delta}B_{\Vert }^{2}/8{\rm\pi}$ in (4.7) is related to the stability parameter (see § 4.4.2 and § B.3); neglecting interspecies drifts,

(4.9) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{r}\left(\mathop{\sum }_{s}\int \text{d}^{3}\boldsymbol{v}\,\frac{T_{\Vert 0s}{\it\delta}\widetilde{f}_{s}^{\,2}}{2f_{0s}^{\Vert }}+\frac{{\it\delta}B_{\Vert }^{2}}{8{\rm\pi}}\right)=\mathop{\sum }_{s}n_{0s}T_{\bot 0s}\int \text{d}^{3}\boldsymbol{r}\,{\it\Delta}_{2s}\frac{\partial }{\partial t}\frac{{\it\delta}B_{\Vert }^{2}}{B_{0}^{2}}.\end{eqnarray}$$

In the unstable case, one can interpret $\partial {\it\delta}B_{\Vert }^{2}/\partial t$ as the rate of relaxation of the pressure anisotropy as the mirror fluctuations grow (see e.g. Schekochihin et al. Reference Schekochihin, Cowley, Kulsrud, Rosin and Heinemann2008b ; Kunz et al. Reference Kunz, Schekochihin and Stone2014; Riquelme, Quataert & Verscharen Reference Riquelme, Quataert and Verscharen2014; Rincon, Schekochihin & Cowley Reference Rincon, Schekochihin and Cowley2015). In this case, $W_{compr}$ is minimised by growing fluctuations.

Another interpretation of what happens when the stability threshold is approached and crossed is analogous to the one we offered at the end of § 3.3 for a similar situation concerning the Alfvénic fluctuations. Within the mirror stability boundary, $W_{compr}$ is a positive definite conserved free-energy-like quantity. As the system gets closer to the mirror threshold, it becomes energetically ‘cheaper’ to produce magnetic compressions or rarefactions ( ${\it\delta}B_{\Vert }$ ) – dynamically, this is due to the fact that the effect of positive $p_{\bot 0}-p_{\Vert 0}$ is to reduce the magnetic pressure response (cf. Southwood & Kivelson Reference Southwood and Kivelson1993). Once the threshold is crossed, $W_{compr}$ is no longer positive definite and its conservation is compatible with the growth of ${\it\delta}B_{\Vert }$ and ${\it\delta}\widetilde{f}_{s}$ (the mirror instability).

The astute reader will recognise that the factor multiplying ${\it\delta}B_{\Vert }^{2}$ in the free-energy invariant, namely $1-\sum _{s}{\it\beta}_{\bot s}{\it\Delta}_{2s}$ , is not the exact mirror stability parameter, (B 14). While the latter reduces to the former in the case of very high ${\it\beta}$ or of cold electrons (for which the right-hand side of (B 14) vanishes), in general there is a stabilising term due to the interaction of linearly resonant particles with the parallel electric field (i.e. Landau damping). This physics is contained inside the first term in the compressive invariant, proportional to ${\it\delta}\widetilde{f}_{s}^{2}$ . In order to see this, and to make better sense of the structure of $W_{compr}$ , we must understand the physical meaning of ${\it\delta}\widetilde{f}_{s}$ .

4.2. Meaning of ${\it\delta}\widetilde{f}_{s}$ : $(v_{\Vert },w_{\bot })$ versus $({\it\varepsilon}_{s},{\it\mu}_{s})$ coordinates

Our decision to write $f_{0s}$ as a function of $v_{\Vert }$ and $w_{\bot }$ , while analytically convenient, is not the most natural choice for interpreting the compressive invariant $W_{compr}$ . Instead, let us introduce the kinetic energy and adiabatic invariant of a particle of species $s$ , given respectively by

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\varepsilon}_{s}\doteq {\textstyle \frac{1}{2}}m_{s}(v_{\Vert }-u_{\Vert 0s}^{\prime })^{2}+{\textstyle \frac{1}{2}}m_{s}w_{\bot }^{2}, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\mu}_{s}\doteq \frac{m_{s}w_{\bot }^{2}}{2B}, & \displaystyle\end{eqnarray}$$

${\it\varepsilon}_{s}$

${\it\mu}_{s}$

(4.12) $$\begin{eqnarray}\frac{\text{D}f_{s}}{\text{D}t}+\left[m_{s}(v_{\Vert }-u_{\Vert 0s}^{\prime })\left(\frac{q_{s}}{m_{s}}E_{\Vert }-\frac{\text{D}\boldsymbol{u}_{\bot }}{\text{D}t}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\right)+{\it\mu}_{s}\left(\frac{\text{d}B}{\text{d}t}+u_{\Vert 0s}^{\prime }\frac{\partial B}{\partial z}\right)\right]\frac{\partial f_{s}}{\partial {\it\varepsilon}_{s}}=\left(\frac{\partial f_{s}}{\partial t}\right)_{c},\end{eqnarray}$$

where $v_{\Vert }-u_{\Vert 0s}^{\prime }=\pm \sqrt{2m_{s}({\it\varepsilon}_{s}-{\it\mu}_{s}B)}$ . This is perhaps the most transparent form of KMHD: particles parallel-stream, $\boldsymbol{E}\times \boldsymbol{B}$ -drift, conserve ${\it\mu}_{s}$ , and change their kinetic energy by interacting with a parallel electric field $E_{\Vert }$ and/or a changing magnetic field (mirroring) in the frame of the equilibrium species drift; the term $-(\text{D}\boldsymbol{u}_{\bot }/\text{D}t)\boldsymbol{\cdot }\hat{\boldsymbol{b}}=\boldsymbol{u}_{\bot }\boldsymbol{\cdot }(\text{D}\hat{\boldsymbol{b}}/\text{D}t)$ is an inertial term having to do with the fact that the direction of the magnetic-field line changes as the particle streams along it and so the plane of the $\boldsymbol{E}\times \boldsymbol{B}$ drift tilts.

When compared to the formulation of KMHD in $(v_{\Vert },w_{\bot })$ coordinates (2.3), this formulation makes clear that there is a more general class of equilibrium solutions in a collisionless drift-kinetic plasma than those satisfying $f_{s}=f_{0s}(v_{\Vert },w_{\bot })$ , namely, $f_{s}=\widetilde{f}_{0s}({\it\varepsilon}_{s},{\it\mu}_{s})$ . Thus, what is referred to as the ‘equilibrium state’ may in fact contain inhomogeneous fluctuations (e.g. Alfvén waves), so long as the energy of each particle is conserved. As a result, there are pieces of ${\it\delta}f_{s}$ in the $(v_{\Vert },w_{\bot })$ formulation that may be absorbed into the equilibrium distribution function when $f_{0s}$ is taken to be a function ${\it\varepsilon}_{s}$ and ${\it\mu}_{s}$ (e.g. fluctuations in magnetic-field strength that do not violate ${\it\mu}_{s}$ -conservation). To see that this is the case, we relate the two formulations via

(4.13) $$\begin{eqnarray}\displaystyle f_{0s}\left(\left(v_{\Vert }-u_{\Vert 0s}^{\prime }\right)^{2},w_{\bot }^{2}\right) & = & \displaystyle f_{0s}\left(\frac{2{\it\varepsilon}_{s}}{m_{s}}-\frac{2{\it\mu}_{s}B}{m_{s}},\frac{2{\it\mu}_{s}B}{m_{s}}\right)\nonumber\\ \displaystyle & \simeq & \displaystyle f_{0s}\left(\frac{2{\it\varepsilon}_{s}}{m_{s}}-\frac{2{\it\mu}_{s}B_{0}}{m_{s}},\frac{2{\it\mu}_{s}B_{0}}{m_{s}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2{\it\mu}_{s}B_{0}}{m_{s}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\left[-\frac{\partial f_{0s}}{\partial (v_{\Vert }-u_{\Vert 0s}^{\prime })^{2}}+\frac{\partial f_{0s}}{\partial w_{\bot }^{2}}\right]\nonumber\\ \displaystyle & = & \displaystyle \widetilde{f}_{0s}({\it\varepsilon}_{s},{\it\mu}_{s})+\frac{w_{\bot }^{2}}{v_{th\bot s}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\,\mathfrak{D}f_{0s}.\end{eqnarray}$$

Comparing (4.6) and (4.13), it then becomes clear that the perturbed distribution function appearing in the compressive invariant $W_{compr}$ (see (4.7)) satisfies

(4.14) $$\begin{eqnarray}{\it\delta}\widetilde{f}_{s}\doteq {\it\delta}f_{s}+\frac{w_{\bot }^{2}}{v_{th\bot s}^{2}}\frac{{\it\delta}B_{\Vert }}{B_{0}}\,\mathfrak{D}f_{0s}={\it\delta}f_{s}+f_{0s}-\widetilde{f}_{0s}=f_{s}-\widetilde{f}_{0s},\end{eqnarray}$$

i.e. it is the perturbed distribution function if $f_{0s}$ is taken to be a function of $({\it\varepsilon}_{s},{\it\mu}_{s})$ instead of $(v_{\Vert },w_{\bot })$ .

The meaning of the first entropy-like term in the compressive invariant $W_{compr}$ (see (4.7)) is then readily apparent: it is the non-Alfvénic piece of the distribution function that represents changes in the kinetic energy of the particles due to interactions with the compressive fluctuations. In it are contributions from Landau-resonant particles, whose energy is changed by the parallel electric and magnetic-mirror forces in such a way as to facilitate Landau (§ B.2) and Barnes (§ B.3) damping of ion-acoustic waves and slow modes. As long as the plasma stays within the mirror and streaming stability boundaries, the compressive invariant is positive definite.

In the next four sections (§§ 4.3–4.6), we show that the conservation of $W_{compr}$ represents a turbulent cascade of compressive fluctuations from large to small scales in phase space.

4.3. Parallel kinetics: two decoupled collisionless cascades

We begin by noting that, under our collisionless ordering, the $w_{\bot }$ dependence in (2.53c ) can be integrated out. We introduce two auxiliary functions, $G_{n}(v_{\Vert })$ and $G_{B}(v_{\Vert })$ , defined so that

(4.15a,b ) $$\begin{eqnarray}\frac{{\it\delta}n_{e}}{n_{0e}}=\int \text{d}v_{\Vert }\,G_{n}\quad \text{and}\quad \frac{{\it\delta}B_{\Vert }}{B_{0}}=\int \text{d}v_{\Vert }\,G_{B},\end{eqnarray}$$

i.e. they contain all the $w_{\bot }$ integrals and species summations in the right linear combination as per (2.53d ) and (2.53e ). Then (2.53c ) reduces to the following two coupled one-dimensional kinetic equations (cf. (179) and (180) of S09):

(4.16a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}G_{n}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}G_{n}=\mathop{\sum }_{i}(v_{\Vert }-u_{\Vert 0i}^{\prime })F_{0i}^{\Vert }(v_{\Vert })\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left[{\it\lambda}_{i}^{nn}(v_{\Vert })\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\lambda}_{i}^{nB}(v_{\Vert })\frac{{\it\delta}B_{\Vert }}{B_{0}}\right],\qquad & \displaystyle\end{eqnarray}$$

(4.16b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}G_{B}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}G_{B}=\mathop{\sum }_{i}(v_{\Vert }-u_{\Vert 0i}^{\prime })F_{0i}^{\Vert }(v_{\Vert })\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left[{\it\lambda}_{i}^{Bn}(v_{\Vert })\frac{{\it\delta}n_{e}}{n_{0e}}+{\it\lambda}_{i}^{BB}(v_{\Vert })\frac{{\it\delta}B_{\Vert }}{B_{0}}\right],\qquad & \displaystyle\end{eqnarray}$$

$v_{\Vert }$

${\it\lambda}_{i}$

(4.17) $$\begin{eqnarray}F_{\ell i}^{\Vert }(v_{\Vert })\doteq \frac{2{\rm\pi}}{n_{0i}}\int _{0}^{\infty }\text{d}w_{\bot }w_{\bot }\,\frac{1}{\ell !}\left(\frac{w_{\bot }}{v_{th\bot i}}\right)^{2\ell }f_{0i}^{\Vert }(v_{\Vert },w_{\bot })\end{eqnarray}$$

for integer $\ell$ . This coupled system of integral equations, compactly expressed as

(4.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(\frac{\text{d}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)\left[\begin{array}{@{}c@{}}G_{n}(v_{\Vert })\\ G_{B}(v_{\Vert })\end{array}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathop{\sum }_{i}(v_{\Vert }-u_{\Vert 0i}^{\prime })F_{0i}^{\Vert }(v_{\Vert })\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\int _{-\infty }^{\infty }\text{d}v_{\Vert }^{\prime }\left[\begin{array}{@{}cc@{}}{\it\lambda}_{i}^{nn}(v_{\Vert }) & {\it\lambda}_{i}^{nB}(v_{\Vert })\\ {\it\lambda}_{i}^{Bn}(v_{\Vert }) & {\it\lambda}_{i}^{BB}(v_{\Vert })\end{array}\right]\left[\begin{array}{@{}c@{}}G_{n}(v_{\Vert }^{\prime })\\ G_{B}(v_{\Vert }^{\prime })\end{array}\right],\end{eqnarray}$$

has a simple physical interpretation. Kinetic fluctuations in each of the $i^{\prime }=1\dots \,N_{ion}$ ionic species collectively excite compressive fluctuations in the magnetic-field strength (via perpendicular pressure balance), the electron density (via quasineutrality), and thereby the electric field (via parallel pressure balance in the massless electron fluid; see (2.37)). These electromagnetic fields in turn feed back upon the kinetic fluctuations exhibited by each individual ionic species, in a way that is dictated by the various ${\it\lambda}_{i}$ coefficients. This system of equations describing the evolution of the compressive fluctuations and their interactions with the particles is closed, signalling the fact that the compressive dynamics proceeds independently from that of the Alfvénic fluctuations.

Because each ion species responds to the compressive fluctuations in a different way ( ${\it\lambda}_{i}\neq {\it\lambda}_{i^{\prime }}$ for $i\neq i^{\prime }$ ), (4.18) cannot be diagonalised in general. Moreover, the mode structure excited in the parallel-velocity space by these fluctuations will generally be different to the mode structure present in the fluctuations themselves (in general, ${\it\lambda}(v_{\Vert })\neq {\it\lambda}(v_{\Vert }^{\prime })$ for $v_{\Vert }\neq v_{\Vert }^{\prime }$ ), and so this system can only be diagonalised if the ${\it\lambda}_{i}$ coefficients are independent of $v_{\Vert }$ . Both of these requirements for diagonalising (4.18) are guaranteed only for a plasma consisting of a single species of bi-Maxwellian ions, for which $F_{\ell i}^{\Vert }(v_{\Vert })=F_{M}(v_{\Vert })\doteq (1/\sqrt{{\rm\pi}}v_{th\Vert i})\exp (-v_{\Vert }^{2}/v_{th\Vert i}^{2})$ for all integer $\ell$ . Then, assuming a bi-Maxwellian plasma and proceeding with the diagonalisation, we find that (4.18) is equivalent to

(4.19) $$\begin{eqnarray}\frac{\text{d}G^{\pm }}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}G^{\pm }=\frac{v_{\Vert }F_{M}(v_{\Vert })}{{\it\Lambda}^{\pm }}\,\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\int _{-\infty }^{\infty }\text{d}v_{\Vert }^{\prime }\,G^{\pm }(v_{\Vert }^{\prime }),\end{eqnarray}$$

where

(4.20a,b ) $$\begin{eqnarray}G^{+}=G_{B}+\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{1}{{\it\sigma}_{i}}\left(1+\frac{Z_{i}}{{\it\tau}_{\bot i}}\right)G_{n}\quad \text{and}\quad G^{-}=G_{n}+\frac{{\it\varpi}_{i}}{{\it\sigma}_{i}}\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{2}{{\it\beta}_{\bot i}}\,G_{B}\end{eqnarray}$$

are the eigenvectors,

(4.21) $$\begin{eqnarray}{\it\Lambda}^{\pm }=-\frac{{\it\tau}_{\Vert i}}{Z_{i}}+\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{{\it\varsigma}_{i}}{{\it\beta}_{\bot i}}\pm \sqrt{\left(\frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right)^{2}+\left(\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{{\it\varsigma}_{i}}{{\it\beta}_{\bot i}}\right)^{2}}\end{eqnarray}$$

are the (inverses of) the corresponding eigenvalues, and we have defined

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}_{i}\doteq \frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}+\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{1}{{\it\beta}_{\bot i}}+\sqrt{\left(\frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right)^{2}+\left(\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{{\it\varsigma}_{i}}{{\it\beta}_{\bot i}}\right)^{2}}, & \displaystyle\end{eqnarray}$$

(4.23) $$\begin{eqnarray}\displaystyle & {\it\varsigma}_{i}\doteq 1-{\it\beta}_{\bot }{\it\Delta}_{e}, & \displaystyle\end{eqnarray}$$

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\varpi}_{i}\doteq 1+\frac{p_{\Vert 0e}}{p_{\bot 0i}}{\it\Delta}_{e}\left(1-\frac{1}{2}{\it\beta}_{\bot }{\it\Delta}_{e}\right), & \displaystyle\end{eqnarray}$$

Equation (4.19), which reduces to equation (181) of S09 for a Maxwellian plasma, describes two decoupled kinetic cascades. These are the subject of § 4.5, in which we derive the two collisionless invariants associated with $G^{+}$ and $G^{-}$ . But first, we specialise $G^{\pm }$ and ${\it\Lambda}^{\pm }$ for application to two different astrophysical systems representing two distinct parameter regimes and determine what their values imply for the evolution of compressive fluctuations in the linear regime.

4.4. Linear theory: collisionless damping and mirror instability

To develop the linear theory for a two-component bi-Maxwellian plasma,Footnote ² we Fourier-transform (4.19) in time ( $\partial /\partial t\rightarrow -\text{i}{\it\omega}$ ) and space ( $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\rightarrow \text{i}k_{\Vert }$ ), divide both sides by $-\text{i}({\it\omega}-k_{\Vert }v_{\Vert })$ , and integrate over the parallel velocity. Dividing both sides of the resulting equation by $\int \!\text{d}v_{\Vert }G^{\pm }$ and using $\int \text{d}v_{\Vert }\,F_{M}(v_{\Vert })=1$ , we obtain the following dispersion relation:

(4.25) $$\begin{eqnarray}{\it\Lambda}^{\pm }-1=\frac{{\it\omega}}{k_{\Vert }}\int \text{d}v_{\Vert }\,\frac{F_{M}(v_{\Vert })}{v_{\Vert }-{\it\omega}/k_{\Vert }}={\it\xi}_{i}Z_{M}({\it\xi}_{i}),\end{eqnarray}$$

where ${\it\xi}_{i}\doteq {\it\omega}/k_{\Vert }v_{th\Vert i}$ is the dimensionless phase speed and the (Maxwellian) plasma dispersion function (Fried & Conte Reference Fried and Conte1961)

(4.26) $$\begin{eqnarray}Z_{M}({\it\xi})\doteq \frac{1}{\sqrt{{\rm\pi}}}\int _{-\infty }^{\infty }\text{d}x\,\frac{\text{e}^{-x^{2}}}{x-{\it\xi}},\end{eqnarray}$$

the integration being performed along the Landau contour. Formally, (4.25) has an infinite number of solutions, most of which are strongly damped with damping rates $\text{Im}({\it\xi}_{i})\sim 1$ . A few of the more interesting solutions may be obtained analytically in the low- and high-beta limits.

4.4.1. Low-beta limit: solar corona

When ${\it\beta}_{s}\ll 1\sim {\it\Delta}_{s}$ , (4.20) and (4.21) giveFootnote ²

(4.27a,b ) $$\begin{eqnarray}{\it\Lambda}^{-}-1\simeq -\left(1+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right),\quad G^{-}\simeq G_{n}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\left(1+{\it\Delta}_{i}+{\it\Delta}_{e}\,\frac{Z_{i}}{{\it\tau}_{\Vert i}}\right)G_{B},\end{eqnarray}$$

(4.28a,b ) $$\begin{eqnarray}\displaystyle {\it\Lambda}^{+}-1\simeq \frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{2}{{\it\beta}_{\bot i}},\quad G^{+}\simeq G_{B} & & \displaystyle\end{eqnarray}$$

For the ‘ $-$ ’ branch, we have $\text{Im}({\it\xi}_{i})\sim 1$ , and so

(4.29) $$\begin{eqnarray}{\it\omega}\sim -\text{i}|k_{\Vert }|v_{A}\sqrt{{\it\beta}_{\Vert i}},\end{eqnarray}$$

which is much smaller than the Alfvénic cascade rate $k_{\Vert }v_{A}$ . For the ‘ $+$ ’ branch, consisting predominantly of fluctuations in magnetic-field strength,

(4.30) $$\begin{eqnarray}{\it\omega}\sim -\text{i}|k_{\Vert }|v_{A}\sqrt{{\it\beta}_{\Vert i}\left|\ln \frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{2}{{\it\beta}_{\bot i}}\right|},\end{eqnarray}$$

up to logarithmically small corrections. This damping rate is slightly greater than that of the ‘ $-$ ’ branch, though still much smaller than the Alfvénic cascade rate. Compressive fluctuations in a low-beta plasma are therefore weakly damped. Pressure anisotropies do not affect this conclusion.

4.4.2. High-beta limit: intracluster medium

When ${\it\beta}_{s}\sim 1/{\it\Delta}_{s}\gg 1$ , we have

(4.31a,b ) $$\begin{eqnarray}{\it\Lambda}^{-}-1\simeq -2\left(\frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right),\quad G^{-}\simeq G_{n},\end{eqnarray}$$

(4.32a,b ) $$\begin{eqnarray}{\it\Lambda}^{+}-1\simeq \frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{1}{{\it\beta}_{\bot i}}\left(1-\mathop{\sum }_{s}{\it\beta}_{\bot s}{\it\Delta}_{s}\right),\quad G^{+}\simeq G_{B}+\frac{1}{2}\frac{Z_{i}}{{\it\tau}_{\Vert i}}\frac{p_{\Vert 0i}}{p_{\bot 0i}}\,G_{n}.\end{eqnarray}$$

The ‘ $-$ ’ branch corresponds to the density fluctuations and is strongly damped with $\text{Im}({\it\xi}_{i})\sim 1$ . In contrast, the damping rate of the ‘ $+$ ’ branch is small: it can be obtained by expanding $Z_{M}({\it\xi}_{i})=\text{i}\sqrt{{\rm\pi}}+O({\it\xi}_{i})$ , which gives

(4.33) $$\begin{eqnarray}{\it\gamma}\doteq -\text{i}{\it\omega}=-\frac{|k_{\Vert }|v_{A}}{\sqrt{{\rm\pi}{\it\beta}_{\Vert i}}}\frac{p_{\Vert 0i}^{2}}{p_{\bot 0i}^{2}}\left(1-\mathop{\sum }_{s}{\it\beta}_{\bot s}{\it\Delta}_{s}\right).\end{eqnarray}$$

This expression generalises (in the limit $k_{\Vert }/k_{\bot }\ll 1$ ) for bi-Maxwellian distribution functions what is known in astrophysics as the Barnes (Reference Barnes1966) damping and in plasma physics as transit-time damping (Stix Reference Stix1962). Particles that are almost at rest with respect to the slow wave (i.e. ‘Landau resonant’ with ${\it\omega}\sim k_{\Vert }v_{\Vert }$ ) are subject to the action of the mirror force associated with the magnetic compressions in the wave. Since, for a monotonically decreasing distribution function ( $f_{0i}^{\Vert }>0$ ), there are more particles with $v_{\Vert }<{\it\omega}/k_{\Vert }$ than with $v_{\Vert }>{\it\omega}/k_{\Vert }$ , the energy exchange between resonant particles and the wave leads to a net gain (loss) of energy by the particles (wave). Put differently, the only way to maintain perpendicular pressure balance for a slow wave is to increase the energy of the resonant particles (referred to as ‘betatron acceleration’, due to the last term in square brackets in (4.12)) at the expense of the wave energy. The result is wave damping.

Because $G_{n}$ is strongly damped, the fluctuations that are damped at the rate (4.33) are predominantly of the magnetic-field strength. For large ion beta, the damping rate is a small fraction ${\sim}1/\sqrt{{\it\beta}_{\Vert i}}$ of the Alfvénic cascade rate, becoming even smaller in a plasma exhibiting positive pressure anisotropy. This reduction is due to the proportional increase (for ${\it\Delta}_{s}>0$ ) in the number of large-pitch-angle particles in the magnetic troughs ( ${\it\delta}B_{\Vert }<0$ ). This inflates the field lines (in order to maintain perpendicular pressure balance), thereby (partially) offsetting the damping of the field-strength fluctuations. If 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. This process runs away as the resonant particles in the deepening troughs lose energy at a rate ${\it\mu}\partial B/\partial t\sim {\it\mu}{\it\gamma}{\it\delta}B_{\Vert }$ (‘betatron deceleration’). This is the mirror instability (e.g. Southwood & Kivelson Reference Southwood and Kivelson1993). It is the inevitable outcome of trying to maintain perpendicular pressure balance in the midst of an effectively negative magnetic pressure (see appendix B and, in particular, (B 6)).

4.5. Collisionless invariants and phase mixing

The linear theory elucidated, we now return to the principal result of § 4.3: that, for a plasma with a single species of bi-Maxwellian ions, the compressive fluctuations can be decomposed into two decoupled kinetic cascades. Here we show that these two cascades independently obey their own conservation laws.

If we multiply (4.19) by $G^{\pm }(v_{\Vert })/F_{M}(v_{\Vert })$ , integrate over space and parallel velocity, and perform integration by parts on the right-hand side, we find that

(4.34) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int \text{d}^{3}\boldsymbol{r}\int \text{d}v_{\Vert }\,\frac{(G^{\pm })^{2}}{2F_{M}(v_{\Vert })}=-\frac{1}{{\it\Lambda}^{\pm }}\int \text{d}^{3}\boldsymbol{r}\left(\int \text{d}v_{\Vert }\,G^{\pm }\right)\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\int \text{d}v_{\Vert }\,v_{\Vert }G^{\pm }.\end{eqnarray}$$

On the other hand, integrating (4.19) over parallel velocity gives

(4.35) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int \text{d}v_{\Vert }\,G^{\pm }=-\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\int \text{d}v_{\Vert }\,v_{\Vert }G^{\pm },\end{eqnarray}$$

because $\int \!\text{d}v_{\Vert }\,v_{\Vert }F_{M}(v_{\Vert })=0$ . Using this to replace the final term on the right-hand side of (4.34), we find

(4.36) $$\begin{eqnarray}\frac{\text{d}W_{compr}^{\pm }}{\text{d}t}=0,\end{eqnarray}$$

where the two invariants are

(4.37) $$\begin{eqnarray}W_{compr}^{\pm }=\frac{n_{0i}T_{\Vert 0i}}{2}\int \text{d}^{3}\boldsymbol{r}\left[\int \text{d}v_{\Vert }\,\frac{(G^{\pm })^{2}}{F_{M}(v_{\Vert })}-\frac{1}{{\it\Lambda}^{\pm }}\left(\int \text{d}v_{\Vert }\,G^{\pm }\right)^{2}\right].\end{eqnarray}$$

For these invariants to be conserved, the first term in the square bracket must grow to compensate for the decay of the second due to collisionless damping. To see how this arrangement proceeds, it is useful to split

(4.38) $$\begin{eqnarray}G^{\pm }=\widetilde{G}^{\pm }+F_{M}(v_{\Vert })\int \text{d}v_{\Vert }\,G^{\pm },\end{eqnarray}$$

noting that $\int \!\text{d}v_{\Vert }\,\widetilde{G}^{\pm }=0$ since $\int \text{d}v_{\Vert }\,F_{M}(v_{\Vert })=1$ . Then

(4.39) $$\begin{eqnarray}W_{compr}^{\pm }=\frac{n_{0i}T_{\Vert 0i}}{2}\int \text{d}^{3}\boldsymbol{r}\left[\int \text{d}v_{\Vert }\,\frac{(\widetilde{G}^{\pm })^{2}}{F_{M}(v_{\Vert })}+\left(1-\frac{1}{{\it\Lambda}^{\pm }}\right)\left(\int \text{d}v_{\Vert }\,G^{\pm }\right)^{2}\right].\end{eqnarray}$$

For a Maxwellian equilibrium distribution function, the two invariants $W_{compr}^{\pm }$ are guaranteed to be positive definite because ${\it\Lambda}^{+}>1$ and ${\it\Lambda}^{-}<0$ (see (4.21)). Collisionless damping leads to exponential decay of the density and magnetic-field-strength fluctuations, or, equivalently, of $\int \!\text{d}v_{\Vert }\,G^{\pm }$ , while conserving $W_{compr}^{\pm }$ . This means that damping is a redistribution of the conserved quantity $W_{compr}^{\pm }$ : the first term grows to compensate for the decay of the second. This is a manifestation of linear parallel phase mixing (Landau Reference Landau1946; Hammett, Dorland & Perkins Reference Hammett, Dorland and Perkins1992; Krommes & Hu Reference Krommes and Hu1994; Krommes Reference Krommes1999; Watanabe & Sugama Reference Watanabe and Sugama2004): free energy passes from the low (density) moment of the distribution function to higher moments (contained in $\widetilde{G}$ ). As time goes on, the latter part of the solution becomes increasingly oscillatory in $v_{\Vert }$ ( $\widetilde{G}^{\pm }\propto \text{e}^{-\text{i}k_{\Vert }v_{\Vert }t}$ , the so-called ballistic response), representing the development of finer structure in the parallel-velocity space (see § 6.2.4 of S09 for further discussion).

What is novel for anisotropic distribution functions is what happens near the mirror threshold ${\it\Lambda}^{+}=1$ (cf. (4.32)). When the perpendicular pressure is greater than the parallel pressure, the Barnes damping rate is reduced (4.33), and so the free energy is transferred from the electromagnetic fluctuations to the ballistic kinetic fluctuations at a slower rate. As we explained in our discussion (4.7), it is easier to compress magnetic-field lines ( ${\it\delta}B_{\Vert }\neq 0$ ) when $p_{\bot 0}>p_{\Vert 0}$ , so the energetic cost of perturbing the magnetic-field strength is relatively small; this is the physical origin of the small factor $(1-1/{\it\Lambda}^{+})$ multiplying the second term in (4.39). In other words, as the mirror threshold is approached, the generation of fine-scale structure in velocity space by phase mixing is achieved by damping increasingly large magnetic-field-strength fluctuations at an ever decreasing rate. When ${\it\Lambda}^{+}<1$ and the plasma is driven mirror-unstable, the second term in (4.39) becomes increasingly negative as the magnetic-field-strength fluctuations grow. Consequently, the first term in (4.39) must grow increasingly positive to compensate, which corresponds to the production of fine-scale structure in velocity space. This structure is caused by the fraction of particles that are linearly resonant with (and nonlinearly trapped by) the unstable mirror mode (e.g. Southwood & Kivelson Reference Southwood and Kivelson1993).

4.6. $W_{compr}$ revisited: compressive phase-space cascade

We now show that, for a bi-Maxwellian plasma with a single ion species, the compressive invariant $W_{compr}$ obtained in § 4.1 incorporates the two invariants derived in § 4.5. We begin by expressing the density and magnetic-field-strength fluctuations in terms of $G^{\pm }$ :

(4.40) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}n_{e}}{n_{0e}}=\frac{1}{{\it\kappa}_{i}}\left({\it\sigma}_{i}\int \text{d}v_{\Vert }\,G^{-}-{\it\varpi}_{i}\frac{{\it\tau}_{\Vert i}}{Z_{i}}\frac{2}{{\it\beta}_{\bot i}}\int \text{d}v_{\Vert }\,G^{+}\right), & \displaystyle\end{eqnarray}$$

(4.41) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}B_{\Vert }}{B_{0}}=\frac{1}{{\it\kappa}_{i}}\left[\,{\it\sigma}_{i}\int \text{d}v_{\Vert }\,G^{+}-\frac{p_{\Vert 0i}}{p_{\bot 0i}}\left(1+\frac{Z_{i}}{{\it\tau}_{\bot i}}\right)\int \text{d}v_{\Vert }\,G^{-}\,\right], & \displaystyle\end{eqnarray}$$

${\it\sigma}_{i}$

${\it\varpi}_{i}$

(4.42) $$\begin{eqnarray}{\it\kappa}_{i}\doteq 2\,\sqrt{\left(\frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right)^{2}+\left(\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{{\it\varsigma}_{i}}{{\it\beta}_{\bot i}}\right)^{2}}.\end{eqnarray}$$

(There is a factor of 2 missing in the definition of ${\it\kappa}$ in S09, their equation (204). This error affects their equation (213).) To express $g_{i}$ in terms of $G^{\pm }$ , we decompose $g_{i}$ as follows (cf. § 6.2.5 of S09):

(4.43a,b ) $$\begin{eqnarray}g_{i}=\frac{n_{0i}}{{\rm\pi}v_{th\bot i}^{2}}\text{e}^{-x}\,{\hat{g}}(x,v_{\Vert }),\quad {\hat{g}}(x,v_{\Vert })\doteq \mathop{\sum }_{\ell =0}^{\infty }L_{\ell }(x)G_{\ell }(v_{\Vert }),\end{eqnarray}$$

where $x=w_{\bot }^{2}/v_{th\bot i}^{2}$ and the Laguerre polynomials $L_{\ell }(x)=(\text{e}^{x}/\ell !)(\text{d}^{\ell }/\text{d}x^{\ell })\,x^{\ell }\text{e}^{-x}$ . Because Laguerre polynomials are orthogonal, we have

(4.44) $$\begin{eqnarray}\frac{1}{n_{0i}}\int \text{d}^{3}\boldsymbol{v}\,\frac{g_{i}^{2}}{2f_{0i}}=\mathop{\sum }_{\ell =0}^{\infty }\int \text{d}v_{\Vert }\,\frac{G_{\ell }^{2}}{2F_{M}(v_{\Vert })},\end{eqnarray}$$

where the expansion coefficients are determined via the Laguerre transform:

(4.45) $$\begin{eqnarray}G_{\ell }(v_{\Vert })=\int _{0}^{\infty }\text{d}x\,\text{e}^{-x}\,L_{\ell }(x){\hat{g}}(x,v_{\Vert }).\end{eqnarray}$$

Since $L_{0}=1$ and $L_{1}=1-x$ , (2.50) and (2.52) can be used to write the $\ell =0$ and $\ell =1$ expansion coefficients as linear combinations of $G_{n}$ and $G_{B}$ . Using (4.20) and (4.21) to replace $G_{n}$ and $G_{B}$ by suitable combinations of ${\it\Lambda}^{+}G^{+}$ and ${\it\Lambda}^{-}G^{-}$ , we find after some straightforward but tedious algebra that

(4.46) $$\begin{eqnarray}\displaystyle G_{0} & \!=\! & \displaystyle -\frac{1}{{\it\kappa}}\left\{\left[\,{\it\sigma}\,\frac{p_{\bot 0i}}{p_{\Vert 0i}}\left(1+\frac{p_{\Vert 0e}}{p_{\bot 0i}}{\it\Delta}_{e}\right)-{\it\varpi}_{i}\frac{2}{{\it\beta}_{\bot i}}\right]{\it\Lambda}^{+}G^{+}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{Z_{i}}{{\it\tau}_{\Vert i}}\left[\,{\it\sigma}-\left(\frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right)\left(1+\frac{p_{\Vert 0e}}{p_{\bot 0i}}{\it\Delta}_{e}\right)\right]{\it\Lambda}^{-}G^{-}\right\},\end{eqnarray}$$

(4.47) $$\begin{eqnarray}\displaystyle G_{1} & \!=\! & \displaystyle \frac{1}{{\it\kappa}}\left[\,{\it\sigma}\,\frac{p_{\bot 0i}}{p_{\Vert 0i}}\,{\it\Lambda}^{+}G^{+}-\left(1+\frac{Z_{i}}{{\it\tau}_{\bot i}}\right){\it\Lambda}^{-}G^{-}\,\right].\end{eqnarray}$$

(4.48) $$\begin{eqnarray}\frac{\text{d}G_{\ell }}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}G_{\ell }=0,\quad \ell >1.\end{eqnarray}$$

Thus, the distribution function can be explicitly written in terms of $G^{\pm }$ :

(4.49) $$\begin{eqnarray}g_{i}=\left[G_{0}(v_{\Vert })+\left(1-\frac{w_{\bot }^{2}}{v_{th\bot i}^{2}}\right)G_{1}(v_{\Vert })\right]\frac{n_{0i}}{{\rm\pi}v_{th\bot i}^{2}}\text{e}^{-w_{\bot }^{2}/v_{th\bot i}^{2}}+\widetilde{g}_{i},\end{eqnarray}$$

where $\widetilde{g}_{i}$ comprises all $G_{\ell }$ with $\ell >1$ . In other words, $\widetilde{g}_{i}$ is a passively mixed, undamped, ballistic-type mode that contributes to neither density nor magnetic-field strength:

(4.50a−c ) $$\begin{eqnarray}\frac{\text{d}\widetilde{g}_{i}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\widetilde{g}_{i}=0,\quad \int \text{d}^{3}\boldsymbol{v}\,\widetilde{g}_{i}=0,\quad \int \text{d}^{3}\boldsymbol{v}\,\frac{w_{\bot }^{2}}{v_{th\bot i}^{2}}\widetilde{g}_{i}=0;\end{eqnarray}$$

it is the homogeneous solution of (2.53c ).

Equipped with expressions for $G_{0}$ (4.46), $G_{1}$ (4.47), ${\it\delta}n_{e}/n_{0e}$ (4.40), and ${\it\delta}B_{\Vert }/B_{0}$ (4.41) written in terms of $G^{\pm }$ , we now substitute these into our expression for the compressive invariant (4.5) with (4.44) to find that

(4.51) $$\begin{eqnarray}\displaystyle W_{compr} & = & \displaystyle \int \text{d}^{3}\boldsymbol{r}\int \text{d}^{3}\boldsymbol{v}\,\frac{T_{\Vert 0i}\widetilde{g}_{i}^{\,2}}{2f_{0i}}+\frac{p_{\bot 0i}^{2}}{p_{\Vert 0i}^{2}}\left\{\vphantom{\left[\left(1+\frac{p_{\Vert 0e}}{p_{\bot 0i}}{\it\Delta}_{e}\right)^{2}-1\right]}1+\frac{2}{{\it\kappa}_{i}}\left(\frac{{\it\tau}_{\Vert i}}{{\it\tau}_{\bot i}}+\frac{{\it\tau}_{\Vert i}}{Z_{i}}\right)\left(1+\frac{p_{\Vert 0e}}{p_{\bot 0i}}{\it\Delta}_{e}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\,\frac{1}{2}\left[\left(1+\frac{p_{\Vert 0e}}{p_{\bot 0i}}{\it\Delta}_{e}\right)^{2}-1\right]\left(1-\frac{2}{{\it\kappa}_{i}}\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{{\it\varsigma}_{i}}{{\it\beta}_{\bot i}}\right)\right\}({\it\Lambda}^{+})^{2}\,W_{compr}^{+}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2}\frac{Z_{i}^{2}}{{\it\tau}_{\Vert i}^{2}}\left(1+\frac{2}{{\it\kappa}_{i}}\frac{p_{\Vert 0i}}{p_{\bot 0i}}\frac{{\it\varsigma}_{i}}{{\it\beta}_{\bot i}}\right)({\it\Lambda}^{-})^{2}\,W_{compr}^{-}.\end{eqnarray}$$

Thus, for a bi-Maxwellian plasma, the generalised invariant for compressive fluctuations splits into three independently cascading parts: $W_{compr}^{\pm }$ associated with the density and magnetic-field-strength fluctuations and a purely kinetic part given by the first term in (4.51):

(4.52) $$\begin{eqnarray}W_{\widetilde{g}_{i}}\doteq \int \text{d}^{3}\boldsymbol{r}\int \text{d}^{3}\boldsymbol{v}\,\frac{T_{\Vert 0i}\widetilde{g}_{i}^{\,2}}{2f_{0i}}.\end{eqnarray}$$

All three cascade channels lead to small perpendicular spatial scales via passive mixing by the Alfvénic turbulence and to small scales in $v_{\Vert }$ via the linear parallel phase mixing, the rates of mixing being functions of the velocity-space anisotropy of the equilibrium distribution function.