3.1 Notation: Alfvénic fields

The electric and magnetic fields are described by the scalar potential $\unicode[STIX]{x1D719}$ and vector potential $\boldsymbol{A}$ . It is convenient to introduce dimensionless versions of $\unicode[STIX]{x1D719}$ and of the component of $\boldsymbol{A}$ parallel to the equilibrium magnetic field $\boldsymbol{B}_{0}=B_{0}\hat{\boldsymbol{z}}$ :

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D711}=\frac{Ze\unicode[STIX]{x1D719}}{T_{i}}=\frac{2\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x1D70C}_{i}v_{\text{th}i}},\quad {\mathcal{A}}=\frac{A_{\Vert }}{\unicode[STIX]{x1D70C}_{i}B_{0}}=-\frac{\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D70C}_{i}v_{\text{A}}},\end{eqnarray}$$

where $-e$ is the electron charge, $Ze$ the ion charge, $T_{i}$ the ion equilibrium temperature, $v_{\text{th}i}=\sqrt{2T_{i}/m_{i}}$ the ion thermal speed, $m_{i}$ the ion mass, $\unicode[STIX]{x1D70C}_{i}=v_{\text{th}i}/\unicode[STIX]{x1D6FA}_{i}$ the ion Larmor radius, $\unicode[STIX]{x1D6FA}_{i}=ZeB_{0}/m_{i}c$ the ion-Larmor frequency, $c$ the speed of light, $v_{\text{A}}=B_{0}/\sqrt{4\unicode[STIX]{x03C0}m_{i}n_{i}}$ the Alfvén speed and $n_{i}$ the equilibrium ion density. In what follows, we shall drop the ion species index everywhere except for some iconic quantities (e.g. $\unicode[STIX]{x1D70C}_{i}$ ) or where there is a possibility of ambiguity (e.g. $T_{i}$ versus $T_{e}$ ).

In the above, we have also introduced the stream function $\unicode[STIX]{x1D6F7}$ ( $=c\unicode[STIX]{x1D719}/B_{0}$ ) of the $\boldsymbol{E}\times \boldsymbol{B}$ flow associated with $\unicode[STIX]{x1D719}$ and the flux function $\unicode[STIX]{x1D6F9}$ giving (in velocity units) the magnetic-field perturbation perpendicular to $\boldsymbol{B}_{0}$ :

(3.2) $$\begin{eqnarray}\boldsymbol{u}_{\bot }=\hat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F7},\quad \boldsymbol{b}_{\bot }=\hat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}.\end{eqnarray}$$

Physically these perturbations are Alfvén waves (AW). In the inertial range of magnetised plasma turbulence, they decouple from all other modes (the fast modes, which are ordered out in the GK approximation, and the slow, or compressive, modes) and satisfy the ‘reduced MHD’ equations (RMHD, first derived by Kadomtsev & Pogutse (Reference Kadomtsev and Pogutse1974) and Strauss (Reference Strauss1976); for a GK derivation, see S09–§ 5.3):

(3.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}t}=v_{\text{A}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6F7},\quad \frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}=v_{\text{A}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}.\end{eqnarray}$$

Here the nonlinearities are hidden in the convective time derivative and in the spatial derivative along the perturbed field lines:

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\{\unicode[STIX]{x1D6F7},\ldots \}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{t h }i}}{2}\{\unicode[STIX]{x1D711},\ldots \}, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}_{\Vert }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}+\frac{\boldsymbol{b}_{\bot }}{v_{\text{A}}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}+\frac{1}{v_{\text{A}}}\{\unicode[STIX]{x1D6F9},\ldots \}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}-\unicode[STIX]{x1D70C}_{i}\{{\mathcal{A}},\ldots \}, & \displaystyle\end{eqnarray}$$

where $\{\,f,g\}=(\unicode[STIX]{x2202}_{x}f)(\unicode[STIX]{x2202}_{y}g)-(\unicode[STIX]{x2202}_{x}g)(\unicode[STIX]{x2202}_{y}f)$ . These derivatives will appear ubiquitously in what follows.

3.2 Gyrokinetic equation

Our starting point is standard, slab, Maxwellian gyrokinetics (see the derivation in H06 or a summary in S09–§ 3). In it, the ion distribution function is represented as

(3.6) $$\begin{eqnarray}f=F_{0}+\unicode[STIX]{x1D6FF}f,\quad \unicode[STIX]{x1D6FF}f=-\unicode[STIX]{x1D711}(\boldsymbol{r})F_{0}+h(\boldsymbol{R}),\quad \boldsymbol{R}=\boldsymbol{r}+\unicode[STIX]{x1D746},\quad \unicode[STIX]{x1D746}=\frac{\boldsymbol{v}_{\bot }\times \hat{\boldsymbol{z}}}{\unicode[STIX]{x1D6FA}},\end{eqnarray}$$

where $\boldsymbol{R}$ is the GK spatial coordinate (centre of Larmor ring), whereas $\boldsymbol{r}$ is the usual spatial coordinate (position of the particle). Then the GK equation for the evolution of $h$  is

(3.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}+v_{\Vert }\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}z}+\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{th}}}{2}\left\{\langle \unicode[STIX]{x1D712}\rangle _{\boldsymbol{R}},h\right\}=\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D712}\rangle _{\boldsymbol{R}}}{\unicode[STIX]{x2202}t}F_{0}+C[h]+\frac{2v_{\Vert }\langle a_{\text{ext}}\rangle _{\boldsymbol{R}}}{v_{\text{th}}^{2}}F_{0}.\end{eqnarray}$$

Here the GK potential gyroaveraged at constant $\boldsymbol{R}$ (an operation denoted by angle brackets) is

(3.8) $$\begin{eqnarray}\langle \unicode[STIX]{x1D712}\rangle _{\boldsymbol{R}}=\frac{Ze}{T_{i}}\left\langle \unicode[STIX]{x1D719}-\frac{\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{A}}{c}\right\rangle _{\boldsymbol{R}}={\hat{J}}_{0}\unicode[STIX]{x1D711}-2\hat{v}_{\Vert }{\hat{J}}_{0}{\mathcal{A}}+\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}B}{B},\end{eqnarray}$$

where $\hat{v}_{\Vert }=v_{\Vert }/v_{\text{th}}$ , $\hat{v}_{\bot }=v_{\bot }/v_{\text{th}}$ , $\unicode[STIX]{x1D6FF}B$ is the perturbation of the magnetic field along itself (related to $\boldsymbol{A}_{\bot }$ ), which is also the perturbation of the field’s strength; the gyroaveraging Bessel operators are defined in terms of their Fourier space representations:

(3.9) $$\begin{eqnarray}{\hat{J}}_{0}\leftrightarrow J_{0}(a)=1-\frac{a^{2}}{4}+\cdots \,,\quad {\hat{J}}_{1}\leftrightarrow \frac{2J_{1}(a)}{a}=1-\frac{a^{2}}{8}+\cdots \,,\quad a=\frac{k_{\bot }v_{\bot }}{\unicode[STIX]{x1D6FA}}=\hat{v}_{\bot }k_{\bot }\unicode[STIX]{x1D70C}_{i}.\end{eqnarray}$$

Obviously, $a^{2}\leftrightarrow -\hat{v}_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}\unicode[STIX]{x1D6FB}_{\bot }^{2}=-\hat{v}_{\bot }^{2}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}$ , where we denote $\hat{\unicode[STIX]{x1D735}}_{\bot }=\unicode[STIX]{x1D70C}_{i}\unicode[STIX]{x1D735}_{\bot }$ . We shall use the ${\hat{J}}$ notation (Kunz et al. Reference Kunz, Abel, Klein and Schekochihin2018) interchangeably with $\langle \ldots \rangle _{\boldsymbol{R}}$ (or with $\langle \ldots \rangle _{\boldsymbol{r}}$ , the gyroaverage of an $\boldsymbol{R}$ -dependent quantity at constant $\boldsymbol{r}$ ), as proves convenient.

The last term in (3.7) represents energy injection by means of an external parallel acceleration $a_{\text{ext}}$ . This will be a convenient model of the excitation of compressive perturbations for further calculations dealing with free-energy budgets. Finally, the collision operator $C[h]$ contains both the ion–ion and ion–electron collisions, but the latter are negligible in the mass-ratio expansion adopted below.

3.3 Isothermal electron fluid

We supplement the ion GK equation (3.7) with two fluid equations arising from the isothermal approximation for electrons, which is a result of expansion in the electron–ion mass ratio and holds at $k_{\bot }\unicode[STIX]{x1D70C}_{e}\ll 1$ ,Footnote ⁷ and with field equations that follow from quasineutrality and Ampère’s law in the same approximation (this system of equations was derived in S09–§ 4, implemented numerically by Kawazura & Barnes (Reference Kawazura and Barnes2018) and simulated to some useful effect by Kawazura et al. (Reference Kawazura, Barnes and Schekochihin2019)):

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{A}}}{\unicode[STIX]{x2202}t}+\frac{v_{\text{t h }}}{2}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D711}=\frac{v_{\text{t h }}}{2}\unicode[STIX]{x1D735}_{\Vert }\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}n}{n}+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}{\mathcal{A}}, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{\unicode[STIX]{x1D6FF}n}{n}-\frac{\unicode[STIX]{x1D6FF}B}{B}\right)+\unicode[STIX]{x1D735}_{\Vert }u_{\Vert e}=-\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{t h }}}{2}\left\{\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}n}{n},\frac{\unicode[STIX]{x1D6FF}B}{B}\right\}, & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}n}{n}=-\unicode[STIX]{x1D711}+\overline{{\hat{J}}_{0}h}, & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{\Vert e}}{v_{\text{t h }}}=\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+\overline{\hat{v}_{\Vert }{\hat{J}}_{0}h}+{\mathcal{J}}_{\text{ext}}, & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}B}{B}=\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)\unicode[STIX]{x1D711}-\frac{Z}{\unicode[STIX]{x1D70F}}\overline{{\hat{J}}_{0}h}-\overline{\hat{v}_{\bot }^{2}{\hat{J}}_{1}h}. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ , $\unicode[STIX]{x1D6FF}n/n$ is the relative electron density perturbation (which is the same as the ion one, by quasineutrality), $u_{\Vert e}$ the parallel electron flow velocity, and overlines denote velocity integrals: $\overline{(\ldots )}=(1/n_{i})\int \text{d}^{3}\boldsymbol{v}(\ldots )$ ; note that the integrals are taken at constant $\boldsymbol{r}$ and so $\boldsymbol{R}$ -dependent quantities under them must be gyroaveraged at constant $\boldsymbol{r}$ , hence the appearance of the ${\hat{J}}_{0}$ and ${\hat{J}}_{1}$ operators. In the above system, (3.10) is the parallel component of Ohm’s law (electron’s momentum equation), (3.11) is the electron continuity equation, (3.12) is the statement of quasineutrality, (3.13) and (3.14) are the parallel and perpendicular components, respectively, of Ampère’s law (the perpendicular one is equivalent to the statement of perpendicular pressure balance; see S09–§ 3.3).

On the right-hand side of (3.13), we have added a forcing term for the Alfvénic perturbations, which can be viewed as arising from a model external (non-dimensionalised) ‘AW-antenna’ current ${\mathcal{J}}_{\text{ext}}\equiv j_{\Vert \text{ext}}/en_{e}v_{\text{th}}$ (e.g. TenBarge et al. Reference TenBarge, Howes, Dorland and Hammett2014). On the right-hand side of (3.10), we have added a resistive term ( $\unicode[STIX]{x1D702}$ is the magnetic diffusivity) to represent dissipation of energy into electron heat and to allow flux unfreezing at small scales (an important concern: see Eyink Reference Eyink2015, Reference Eyink2018; Boozer Reference Boozer2018). Formally, this effect is outside the mass-ratio ordering that gave us the hybrid equations introduced above and would have to be brought in alongside electron inertia and electron-collisional effects (see S09–§ 7.12 and Zocco & Schekochihin Reference Zocco and Schekochihin2011), but we can treat the resistive term as a representative for all of that as far as magnetic reconnection and free-energy thermalisation on electrons are concerned.Footnote ⁸ This is reasonable because the precise details of how the energy is removed from the system should not matter, so long as it happens at scales smaller than the ion Larmor scale and does not introduce artificial coupling between ions and electrons.Footnote ⁹ These features – forcing and resistivity – will be useful in working out free-energy budgets.

3.4 Free-energy budget

The $\unicode[STIX]{x1D6FF}f$ gyrokinetics conserves (except for explicit sources and sinks) a quadratic norm of the perturbations, known as the free energy (see S09–§ 3.4 and references therein),

(3.15) $$\begin{eqnarray}W=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left(\mathop{\sum }_{s}\int \text{d}^{3}\boldsymbol{v}\frac{T_{s}\unicode[STIX]{x1D6FF}f_{s}^{2}}{2F_{0s}}+\frac{|\unicode[STIX]{x1D6FF}\boldsymbol{B}|^{2}}{8\unicode[STIX]{x03C0}}\right),\end{eqnarray}$$

where $V$ is the volume of the plasma. Here the perturbed ion distribution function is given by (3.6), the perturbed electron distribution function under the mass-ratio expansion is $\unicode[STIX]{x1D6FF}f_{e}=(\unicode[STIX]{x1D6FF}n/n)F_{0e}$ (see S09–§ 4.4), and so, in our notation,

(3.16) $$\begin{eqnarray}W=\frac{v_{\text{th}}^{2}}{4}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\frac{\overline{\langle h^{2}\rangle _{\boldsymbol{r}}}}{F_{0}}-\unicode[STIX]{x1D711}^{2}-2\unicode[STIX]{x1D711}\frac{\unicode[STIX]{x1D6FF}n}{n}+\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}n^{2}}{n^{2}}+\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\left(|\hat{\unicode[STIX]{x1D735}}_{\bot }{\mathcal{A}}|^{2}+\frac{\unicode[STIX]{x1D6FF}B^{2}}{B^{2}}\right)\right],\end{eqnarray}$$

where we have dropped the prefactor of $m_{i}n_{i}$ .

Since $\int \text{d}^{3}\boldsymbol{r}\overline{\langle h^{2}\rangle _{\boldsymbol{r}}/F_{0}}=\overline{\int \text{d}^{3}\boldsymbol{R}\,h^{2}/F_{0}}$ , we may derive the evolution equation for $W$ by multiplying (3.7) by $h/F_{0}$ , integrating over the entire GK phase space and using (3.10)–(3.14) opportunely. The result is

(3.17) $$\begin{eqnarray}\frac{\text{d}W}{\text{d}t}=\unicode[STIX]{x1D700}_{\text{AW}}+\unicode[STIX]{x1D700}_{\text{compr}}-Q_{i}-Q_{e},\end{eqnarray}$$

where the sources are the injection rates of the Alfvénic and compressive perturbations,

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}_{\text{AW}}=v_{\text{th}}^{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\frac{\unicode[STIX]{x2202}{\mathcal{A}}}{\unicode[STIX]{x2202}t}{\mathcal{J}}_{\text{ext}}=-\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{j}_{\text{ext}}, & \displaystyle\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}_{\text{compr}}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,a_{\text{ext}}\overline{v_{\Vert }{\hat{J}}_{0}h}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,a_{\text{ext}}u_{\Vert i}, & \displaystyle\end{eqnarray}$$

and the sinks are the ion- and electron-heating ratesFootnote ¹⁰

(3.20) $$\begin{eqnarray}\displaystyle \displaystyle Q_{i} & = & \displaystyle -\frac{v_{\text{t h }}^{2}}{2}\overline{\int \frac{\text{d}^{3}\boldsymbol{R}}{V}\frac{hC[h]}{F_{0}}}\geqslant 0,\end{eqnarray}$$

(3.21) $$\begin{eqnarray}\displaystyle \displaystyle Q_{e} & = & \displaystyle \unicode[STIX]{x1D702}\frac{v_{\text{t h }}^{2}}{\unicode[STIX]{x1D6FD}_{i}\unicode[STIX]{x1D70C}_{i}^{2}}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,|\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}|^{2}=\unicode[STIX]{x1D702}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\frac{|\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }|^{2}}{4\unicode[STIX]{x03C0}m_{i}n_{i}}\geqslant 0.\end{eqnarray}$$

We have restored dimensions these expressions to make their physical meaning more transparent. Note that in (3.18), the final expression – the work done by the electric field against the external current – is obtained by noticing that there is a perpendicular current associated with $j_{\Vert \text{ext}}$ , which is small in the GK expansion (because, to avoid injecting charges, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{j}_{\text{ext}}=0$ ) and so, to lowest order in $k_{\Vert }/k_{\bot }$ ,

(3.22) $$\begin{eqnarray}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\frac{1}{c}\frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}j_{\Vert \text{ext}}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\frac{1}{c}\frac{\unicode[STIX]{x2202}\boldsymbol{A}}{\unicode[STIX]{x2202}t}\boldsymbol{\cdot }\boldsymbol{j}_{\text{ext}}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left(\frac{1}{c}\frac{\unicode[STIX]{x2202}\boldsymbol{A}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right)\boldsymbol{\cdot }\boldsymbol{j}_{\text{ext}},\end{eqnarray}$$

with the expression for $-\boldsymbol{E}$ able to be completed with $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ under the integral because $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{j}_{\text{ext}}=0$ .

In steady state, equation (3.17) is the overall free-energy budget, which says that the total injection is equal to the total dissipation. The main purpose of this paper is to work out more restrictive energy budgets that constrain $Q_{i}$ and $Q_{e}$ separately.

3.5 Separating Alfvénic perturbations

We now rearrange the perturbed distribution function in a way that has the effect of separating the Alfvénic part of the distribution function from its ‘compressive’ part:Footnote ¹¹

(3.23) $$\begin{eqnarray}h=\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}F_{0}+g={\hat{J}}_{0}\unicode[STIX]{x1D711}F_{0}+g\quad \Rightarrow \quad \unicode[STIX]{x1D6FF}f=(\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}-\unicode[STIX]{x1D711})F_{0}+g,\quad g=\langle \unicode[STIX]{x1D6FF}f\rangle _{\boldsymbol{R}}.\end{eqnarray}$$

The field equations (3.12)–(3.14) become

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}n}{n}=-(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}+\overline{{\hat{J}}_{0}g}, & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{\Vert e}}{v_{\text{t h }}}=\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+\overline{\hat{v}_{\Vert }{\hat{J}}_{0}g}+{\mathcal{J}}_{\text{ext}}, & \displaystyle\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}B}{B}=-\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}n}{n}+(1-\hat{\unicode[STIX]{x1D6E4}}_{1})\unicode[STIX]{x1D711}-\overline{\hat{v}_{\bot }^{2}{\hat{J}}_{1}g}, & \displaystyle\end{eqnarray}$$

where two more Bessel operators have arisen:

(3.27) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6E4}}_{0}\leftrightarrow \overline{J_{0}^{2}(a)F_{0}}=I_{0}(\unicode[STIX]{x1D6FC})\text{e}^{-\unicode[STIX]{x1D6FC}}=1-\unicode[STIX]{x1D6FC}+\cdots \,,\quad \unicode[STIX]{x1D6FC}=\frac{k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}}{2}\leftrightarrow -\frac{1}{2}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2},\end{eqnarray}$$

(3.28) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6E4}}_{1}\leftrightarrow \overline{\hat{v}_{\bot }^{2}\frac{2J_{1}(a)J_{0}(a)}{a}F_{0}}=-[I_{0}(\unicode[STIX]{x1D6FC})\text{e}^{-\unicode[STIX]{x1D6FC}}]^{\prime }=1-\frac{3}{2}\unicode[STIX]{x1D6FC}+\cdots \,.\end{eqnarray}$$

The GK equation (3.7), rewritten in terms of $g$ , becomes

(3.29) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(g-\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}B}{B}F_{0}\right)+\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{t h }}}{2}\left(\left\{\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}},g-\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}B}{B}F_{0}\right\}+\left\{\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}B}{B},g\right\}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,v_{\Vert }\left\langle \unicode[STIX]{x1D735}_{\Vert }\left(g+\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}n}{n}F_{0}\right)+\unicode[STIX]{x1D70C}_{i}\{{\mathcal{A}}-\langle {\mathcal{A}}\rangle _{\boldsymbol{R}},\unicode[STIX]{x1D711}-\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}\}F_{0}\right\rangle _{\boldsymbol{R}}\nonumber\\ \displaystyle & & \displaystyle \qquad =C\left[g+\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}F_{0}\right]+\frac{2v_{\Vert }\langle a_{\text{ext}}\rangle _{\boldsymbol{R}}}{v_{\text{t h }}^{2}}F_{0}.\end{eqnarray}$$

This has been derived by using (3.10) to express $\langle \unicode[STIX]{x2202}{\mathcal{A}}/\unicode[STIX]{x2202}t\rangle _{\boldsymbol{R}}$ and after some manipulation of gyroaverages.Footnote ¹²

In terms of $g$ , the free energy (3.16) becomes

(3.30) $$\begin{eqnarray}\displaystyle W=\frac{v_{\text{t h }}^{2}}{4}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\frac{\overline{\langle g^{2}\rangle _{\boldsymbol{r}}}}{F_{0}}+\unicode[STIX]{x1D711}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}+\frac{Z}{\unicode[STIX]{x1D70F}}\left|(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}-\overline{{\hat{J}}_{0}g}\right|^{2}+\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\left(|\hat{\unicode[STIX]{x1D735}}_{\bot }{\mathcal{A}}|^{2}+\frac{\unicode[STIX]{x1D6FF}B^{2}}{B^{2}}\right)\right],\hspace{-10.00002pt} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where, by definition, $(1/V)\int \text{d}^{3}\boldsymbol{r}\unicode[STIX]{x1D711}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}=\sum _{\boldsymbol{k}}(1-\unicode[STIX]{x1D6E4}_{0})|\unicode[STIX]{x1D711}_{\boldsymbol{k}}|^{2}$ .

For some upcoming derivations, it will be useful to have the zeroth moment of (3.29). We integrate (3.29) over velocities at constant $\boldsymbol{r}$ , use (3.24) to express $\overline{{\hat{J}}_{0}g}$ and subtract (3.11) from the resulting equation, using (3.25) for $u_{\Vert e}$ and so far neglecting nothing. The outcome is

(3.31) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}t}\left[(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}+(1-\hat{\unicode[STIX]{x1D6E4}}_{1})\frac{\unicode[STIX]{x1D6FF}B}{B}\right]-v_{\text{th}}\unicode[STIX]{x1D735}_{\Vert }\left(\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+{\mathcal{J}}_{\text{ext}}\right)=\unicode[STIX]{x1D70C}_{i}\overline{\langle \{\langle {\mathcal{A}}\rangle _{\boldsymbol{R}}-{\mathcal{A}},v_{\Vert }g\}\rangle _{\boldsymbol{r}}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{t h }}}{2}\left[\overline{\left\langle \left\{\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}-\unicode[STIX]{x1D711},g-\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}B}{B}F_{0}\right\}+\left\{\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}B}{B},g\right\}\right\rangle _{\boldsymbol{r}}}-\left\{\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}n}{n},\frac{\unicode[STIX]{x1D6FF}B}{B}\right\}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\overline{\langle C[g+\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}F_{0}]\rangle _{\boldsymbol{r}}}.\end{eqnarray}$$

Note that (3.10) and (3.31) have the makings of the RMHD system (3.3): this emerges from any long-wavelength approximation where one can neglect the $\unicode[STIX]{x1D6FF}n$ term in (3.10), as well as $(1-\hat{\unicode[STIX]{x1D6E4}}_{1})\unicode[STIX]{x1D6FF}B/B$ and all of the right-hand side of (3.31). This is indeed how RMHD is derived from gyrokinetics in the limit of $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ (S09–§§ 5.2, 5.3). Below, we shall apply somewhat different orderings to work out the reduced dynamics at low beta.

4.1 Ordering

Working in the limit $\unicode[STIX]{x1D6FD}_{e}\sim \unicode[STIX]{x1D6FD}_{i}\ll 1$ , we let

(4.1) $$\begin{eqnarray}\frac{Z}{\unicode[STIX]{x1D70F}}=\frac{\unicode[STIX]{x1D6FD}_{e}}{\unicode[STIX]{x1D6FD}_{i}}\sim 1,\quad k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1,\end{eqnarray}$$

the latter assumption meaning that we are able to treat the Larmor-scale transition directly.

Since we wish to be able to handle Alfvénic perturbations, and since we wish their linear frequency ( $k_{\Vert }v_{\text{A}}$ ) and their nonlinear interaction rate ( $k_{\bot }u_{\bot }$ ) to be able to be comparable, we stipulate

(4.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}B_{\bot }}{B}\sim \frac{u_{\bot }}{v_{\text{A}}}\sim \frac{k_{\Vert }}{k_{\bot }}\sim \unicode[STIX]{x1D716},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}$ is the basic GK expansion parameter, with no further $\unicode[STIX]{x1D6FD}_{i}$ -related factors, of which we shall now keep a close watch. In view of (4.1), this assumption implies

(4.3) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\unicode[STIX]{x1D6FF}B_{\bot }}{B}\sim \frac{k_{\bot }A_{\Vert }}{B_{0}}\sim k_{\bot }\unicode[STIX]{x1D70C}_{i}{\mathcal{A}} & \quad \Rightarrow \quad & \displaystyle {\mathcal{A}}\sim \unicode[STIX]{x1D716},\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle \displaystyle \frac{u_{\bot }}{v_{\text{A}}}\sim \frac{ck_{\bot }\unicode[STIX]{x1D719}}{v_{\text{A}}B_{0}}\sim k_{\bot }\unicode[STIX]{x1D70C}_{i}\sqrt{\unicode[STIX]{x1D6FD}_{i}}\,\unicode[STIX]{x1D711} & \quad \Rightarrow \quad & \displaystyle \unicode[STIX]{x1D711}\sim \frac{\unicode[STIX]{x1D716}}{\sqrt{\unicode[STIX]{x1D6FD}_{i}}}.\end{eqnarray}$$

Examination of (3.24)–(3.26) then suggests that

(4.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}n}{n}\sim \frac{g}{F_{0}}\sim \unicode[STIX]{x1D711}\sim \frac{\unicode[STIX]{x1D716}}{\sqrt{\unicode[STIX]{x1D6FD}_{i}}},\quad \frac{\unicode[STIX]{x1D6FF}B}{B}\sim \unicode[STIX]{x1D716}\sqrt{\unicode[STIX]{x1D6FD}_{i}}.\end{eqnarray}$$

4.2 Equations

With this ordering, the kinetic equation (3.29) becomes, to lowest order in $\unicode[STIX]{x1D6FD}_{i}$ ,

(4.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{th}}}{2}\left\{\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}},g\right\}=C[g+\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}F_{0}]+\frac{2v_{\Vert }\langle a_{\text{ext}}\rangle _{\boldsymbol{R}}}{v_{\text{th}}^{2}}F_{0}.\end{eqnarray}$$

If we ignore collisions and assume no external forcing ( $a_{\text{ext}}=0$ ), then $g=0$ is a good solution of this equation (these assumptions will be relaxed in § 4.5). The field equations (3.24)–(3.26) turn into simple constitutive relations

(4.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}n}{n}=-(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711},\quad \frac{u_{\Vert e}}{v_{\text{th}}}=\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+{\mathcal{J}}_{\text{ext}},\quad \frac{\unicode[STIX]{x1D6FF}B}{B}=\frac{\unicode[STIX]{x1D6FD}_{i}}{2}\left[\frac{Z}{\unicode[STIX]{x1D70F}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})+(1-\hat{\unicode[STIX]{x1D6E4}}_{1})\right]\unicode[STIX]{x1D711}.\end{eqnarray}$$

Using the first two of these in (3.10)–(3.11), we find that the latter become, to lowest order,

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{A}}}{\unicode[STIX]{x2202}t}+\frac{v_{\text{t h }}}{2}\unicode[STIX]{x1D735}_{\Vert }\left[1+\frac{Z}{\unicode[STIX]{x1D70F}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\right]\unicode[STIX]{x1D711}=\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}{\mathcal{A}}, & \displaystyle\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}=\frac{v_{\text{t h }}}{\unicode[STIX]{x1D6FD}_{i}}\unicode[STIX]{x1D735}_{\Vert }\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+v_{\text{th}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{J}}_{\text{ext}}. & \displaystyle\end{eqnarray}$$

These equations are the same as those derived by Zocco & Schekochihin (Reference Zocco and Schekochihin2011) in the limit of ultra-low beta ( $\unicode[STIX]{x1D6FD}_{e}\sim m_{e}/m_{i}$ ), except the electron inertia and the coupling to non-isothermal electron kinetics have now been lost – the price (painless to pay, in the context of present study, because energetics are not affected) for considering somewhat higher  $\unicode[STIX]{x1D6FD}_{e}$ .

The system of (4.8)–(4.9) turns into RMHD (3.3) when $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ : this is shown by using $1-\hat{\unicode[STIX]{x1D6E4}}_{0}\approx -\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}/2$ (see (3.27)). In the opposite limit $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , using $1-\hat{\unicode[STIX]{x1D6E4}}_{0}\approx 1$ , one obtains the $\unicode[STIX]{x1D6FD}\ll 1$ limit of the ‘electron RMHD’ equations (ERMHD; see S09–§ 7.2 or Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013). In the more conventional notation involving stream and flux functions (defined in (3.1)), they are

(4.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}t}=v_{\text{A}}\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6F7}+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9},\quad \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}t}=-\frac{v_{\text{A}}}{2}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D70C}_{i}^{2}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}\end{eqnarray}$$

(we have dropped ${\mathcal{J}}_{\text{ext}}$ because it occurs at large scales). The relationship between the magnetic field and $\unicode[STIX]{x1D6F9}$ is still the same as in (3.2). While $\unicode[STIX]{x1D6F7}$ is still the stream function for the $\boldsymbol{E}\times \boldsymbol{B}$ velocity, this is now the velocity of the electron flow (ions are much slower because of gyroaveraging). These equations describe what is sometimes referred to as the turbulence of kinetic Alfvén waves (KAW) – although, like the Alfvénic (RMHD) turbulence in the inertial range, it is expected to be strong and critically balanced and so does not literally consist of waves (see S09–§ 7.5 and Cho & Lazarian Reference Cho and Lazarian2004, Reference Cho and Lazarian2009, Boldyrev & Perez Reference Boldyrev and Perez2012, TenBarge & Howes Reference TenBarge and Howes2012, TenBarge et al. Reference TenBarge, Howes and Dorland2013, Boldyrev & Loureiro Reference Boldyrev and Loureiro2019).

If we consider cold ions, $Z/\unicode[STIX]{x1D70F}\gg 1$ (but not so cold as to break $\unicode[STIX]{x1D6FD}_{e}\ll 1$ ), there is an intermediate regime with

(4.11) $$\begin{eqnarray}\frac{Z}{\unicode[STIX]{x1D70F}}(1-\unicode[STIX]{x1D6E4}_{0})\approx k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\text{s}}^{2}\sim 1,\quad \unicode[STIX]{x1D70C}_{\text{s}}=\sqrt{\frac{Z}{2\unicode[STIX]{x1D70F}}}\,\unicode[STIX]{x1D70C}_{i}=\frac{c_{\text{s}}}{\unicode[STIX]{x1D6FA}},\quad c_{\text{s}}=\sqrt{\frac{ZT_{e}}{m_{i}}},\end{eqnarray}$$

where $c_{\text{s}}$ is the sound speed and $\unicode[STIX]{x1D70C}_{\text{s}}\gg \unicode[STIX]{x1D70C}_{i}$ is the ‘sound radius’, setting a transition scale. In this regime, the electron-pressure-gradient term (the right-hand side of (3.11)) is non-negligible and so the AW dynamics become dispersive: using (3.1) and (4.11) in (4.8)–(4.9), we arrive at a simple modification of RMHD equations (3.3) (cf. Bian & Tsiklauri Reference Bian and Tsiklauri2009):

(4.12) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}t}=v_{\text{A}}\unicode[STIX]{x1D735}_{\Vert }(1-\unicode[STIX]{x1D70C}_{\text{s}}^{2}\unicode[STIX]{x1D6FB}_{\bot }^{2})\unicode[STIX]{x1D6F7},\quad \frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}=v_{\text{A}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}.\end{eqnarray}$$

There is then a second transition in (4.8)–(4.9) at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ , to ERMHD (4.10).

4.3 Linear theory

These transitions become particularly transparent if we consider the linear dispersion relation for the system (4.8)–(4.9):

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}=k_{\Vert }^{2}v_{\text{A}}^{2}\frac{k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}}{2}\left(\frac{1}{1-\unicode[STIX]{x1D6E4}_{0}}+\frac{Z}{\unicode[STIX]{x1D70F}}\right)\approx \left\{\begin{array}{@{}ll@{}}k_{\Vert }^{2}v_{\text{A}}^{2}(1+k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\text{s}}^{2}), & k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1,\\ \displaystyle \frac{1+Z/\unicode[STIX]{x1D70F}}{2}\,k_{\Vert }^{2}v_{\text{A}}^{2}k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}, & k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1.\end{array}\right.\end{eqnarray}$$

The $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ limit is the Alfvén wave with the dispersive correction due to the electron-pressure gradient. The $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ limit is the KAW dispersion relation with $\unicode[STIX]{x1D6FD}\ll 1$ (see S09–§ 7.3). When $Z/\unicode[STIX]{x1D70F}\gg 1$ , it becomes $\unicode[STIX]{x1D714}^{2}\approx k_{\Vert }^{2}v_{\text{A}}^{2}k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\text{s}}^{2}$ and so the transition between the long- and short-wavelength frequencies is seamless. Thus, in this limit, the transition between the AW and KAW cascades occurs at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{s}}\sim 1$ .

4.4 Free-energy budget

The nonlinear system (4.8)–(4.9) has a conserved energy

(4.14) $$\begin{eqnarray}W=\frac{v_{\text{th}}^{2}}{4}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\unicode[STIX]{x1D711}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}+\frac{Z}{\unicode[STIX]{x1D70F}}|(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}|^{2}+\frac{2}{\unicode[STIX]{x1D6FD}_{i}}|\hat{\unicode[STIX]{x1D735}}_{\bot }{\mathcal{A}}|^{2}\right],\end{eqnarray}$$

which is the appropriate low-beta, $g=0$ limit of (3.30). Note that, whereas $\unicode[STIX]{x1D6FF}n/n$ does appear in (4.14) (the second term), $\unicode[STIX]{x1D6FF}B$ is energetically (and dynamically) insignificant (see (4.7)).

At $k_{\bot }\unicode[STIX]{x1D70C}_{\text{s}}\ll 1$ , equation (4.14) reduces to the energy of Alfvén waves

(4.15) $$\begin{eqnarray}W_{\text{AW}}=\frac{1}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}(|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F7}|^{2}+|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}|^{2})=\frac{1}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}(|\boldsymbol{u}_{\bot }|^{2}+|\boldsymbol{b}_{\bot }|^{2}),\end{eqnarray}$$

conserved by RMHD (3.3). When $k_{\bot }\unicode[STIX]{x1D70C}_{\text{s}}\sim 1$ but $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ ,

(4.16) $$\begin{eqnarray}W=\frac{1}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}(|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F7}|^{2}+\unicode[STIX]{x1D70C}_{\text{s}}^{2}|\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}|^{2}+|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}|^{2}),\end{eqnarray}$$

the energy of the system (4.12). At $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , $W$ becomes the energy of low-beta KAW perturbations described by (4.10):Footnote ¹³

(4.17) $$\begin{eqnarray}W_{\text{KAW}}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)\frac{\unicode[STIX]{x1D6F7}^{2}}{\unicode[STIX]{x1D70C}_{i}^{2}}+\frac{1}{2}|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}|^{2}\right].\end{eqnarray}$$

The existence of the invariant (4.14), valid uniformly at small, order-unity and large $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , means that no damping of anything and, therefore, no ion heating occurs at any wavenumber, until resistivity kicks in and causes electron heating: it is easy to ascertain that

(4.18) $$\begin{eqnarray}\frac{\text{d}W}{\text{d}t}=\unicode[STIX]{x1D700}_{\text{AW}}-Q_{e},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{\text{AW}}$ is given by (3.18) and $Q_{e}$ by (3.21). In steady state, $Q_{e}=\unicode[STIX]{x1D700}_{\text{AW}}$ .

4.5 Energy partition in the presence of compressive cascade

In the above, we assumed the $g=0$ solution for the kinetic equation (4.6). This corresponds to a situation in which only Alfvénic perturbations are stirred up at the largest scales: indeed, the relations (4.7) imply that the compressive fields $\unicode[STIX]{x1D6FF}n$ and $\unicode[STIX]{x1D6FF}B$ peter out at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ . Let us now relax this assumption. Mathematically, this would correspond, e.g. to restoring the external parallel acceleration term in (4.6). The variance of the forced kinetic scalar described by (4.6) with $a_{\text{ext}}\neq 0$ is conserved by the nonlinearity:

(4.19) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\frac{v_{\text{th}}^{2}}{4}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\overline{\frac{\langle g^{2}\rangle _{\boldsymbol{ r}}}{F_{0}}}-\frac{v_{\text{th}}^{2}}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\overline{\left\langle \frac{gC[g+\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}F_{0}]}{F_{0}}\right\rangle _{\boldsymbol{r}}}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\overline{a_{\text{ext}}\langle v_{\Vert }g\rangle _{\boldsymbol{r}}}=\unicode[STIX]{x1D700}_{\text{compr}},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{\text{compr}}$ is the energy flux in the compressive cascade (cf. (3.19)). In steady state ( $\text{d}/\text{d}t=0$ ), we have a balance between this compressive input power and the collisional terms (cf. (3.17)):

(4.20) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{\text{compr}}=Q_{i}+Q_{\text{x}},\quad Q_{\text{x}}=\frac{v_{\text{th}}^{2}}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\overline{\langle \langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}C[h]\rangle _{\boldsymbol{r}}}=\frac{v_{\text{th}}^{2}}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\unicode[STIX]{x1D711}\overline{\langle C[h]\rangle _{\boldsymbol{r}}},\end{eqnarray}$$

where $Q_{i}$ is given by (3.20). Thus, all the compressive energy becomes ion heat, with the exception of the collisional energy exchange $Q_{\text{x}}$ with Alfvénic perturbations, which is, as we are about to argue, small when collisions are weak. The implication is that all the Alfvénic energy is destined, via the AW cascade smoothly transitioning into the KAW cascade, to be dissipated into electron heat,

(4.21) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{\text{AW}}=\unicode[STIX]{x1D700}_{\text{KAW}}=Q_{e}.\end{eqnarray}$$

We will confirm this directly in § 4.6.

If the collision frequency is small compared to the forcing or nonlinear-advection time scales in (4.6), the only way for the collision terms to balance the finite energy flux is for $g$ to develop small scales in phase space, thus activating large derivatives in $C[h]$ . This is indeed what happens, as the nonlinear term successfully pushes $g$ towards small scales in both $\boldsymbol{R}$ and $v_{\bot }$ , viz., towards $\unicode[STIX]{x1D6FF}v_{\bot }/v_{\text{th}}\sim (k_{\bot }\unicode[STIX]{x1D70C}_{i})^{-1}\ll 1$ , via a process known as the entropy cascade (see S09–§ 7.9). This is a route to ion heating that requires no parallel streaming and is, therefore, the only feasible one for the effectively 2-D kinetic equation (4.6), where the parallel streaming has been ordered out due to low $\unicode[STIX]{x1D6FD}_{i}$ (cf. Tatsuno et al. Reference Tatsuno, Dorland, Schekochihin, Plunk, Barnes, Cowley and Howes2009; Plunk et al. Reference Plunk, Cowley, Schekochihin and Tatsuno2010). Recent numerical results by Kawazura et al. (Reference Kawazura, Barnes and Schekochihin2019) appear to confirm the presence of such an ion-heating route in low-beta GK turbulence.

The ion-heating rate $Q_{i}$ (see (3.20)) is positive definite and by this process it will be rendered finite, i.e. independent of the ion collision rate, however small the latter is. Let us estimate the size of $Q_{\text{x}}$ in comparison to $Q_{i}$ . Clearly, only the parts of $\unicode[STIX]{x1D711}$ and $h$ that vary on fine scales in position and velocity space matter in $Q_{\text{x}}$ and $Q_{i}$ , the contribution from large scales being small because the collision frequency is small. The GK collision operator is a diffusion operator both in velocity and position (see, e.g. Abel et al. Reference Abel, Barnes, Cowley, Dorland and Schekochihin2008), with the size of the position and velocity gradients comparable in the entropy cascade. At $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , when collisions become important,

(4.22) $$\begin{eqnarray}\displaystyle \displaystyle Q_{i} & {\sim} & \displaystyle v_{\text{th}}^{2}\unicode[STIX]{x1D708}_{ii}(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{2}\frac{h^{2}}{F_{0}^{2}},\end{eqnarray}$$

(4.23) $$\begin{eqnarray}\displaystyle \displaystyle Q_{\text{x}} & {\sim} & \displaystyle v_{\text{th}}^{2}\unicode[STIX]{x1D708}_{ii}(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{3/2}\frac{h}{F_{0}}\unicode[STIX]{x1D711}\sim v_{\text{th}}^{2}\unicode[STIX]{x1D708}_{ii}k_{\bot }\unicode[STIX]{x1D70C}_{i}\frac{h^{2}}{F_{0}^{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}_{ii}$ is the ion collision frequency. Thus, $Q_{\text{x}}\ll Q_{i}$ . Here $Q_{\text{x}}$ loses out compared to $Q_{i}$ by one factor of $(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{1/2}$ because of the gyroaveraging under the velocity integral of $C[h]$ and by another factor of $(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{1/2}$ because, as will be evident from (4.24)–(4.25), we must order $\unicode[STIX]{x1D711}\sim \overline{{\hat{J}}_{0}g}\sim (k_{\bot }\unicode[STIX]{x1D70C}_{i})^{-1/2}h/F_{0}$ in order for the compressive perturbations to have any relevance. In fact, (4.23) is probably an overestimate because $Q_{\text{x}}$ is not sign definite and so there will also be a tendency for the small-scale variation within it to average out under integration. In any event, it is clear that when collisions are weak, the collisional energy exchange can be neglected.

4.6 Effect of compressive cascade on Alfvénic cascade

For completeness, let us ascertain that the notion that non-zero $g$ has no energetic effect on the AW and KAW cascades is consistent with the dynamical equations for the latter. We allow $g/F_{0}\sim \unicode[STIX]{x1D711}$ as per (4.5). In this case, $\overline{v_{\Vert }{\hat{J}}_{0}g}$ is still one-order subdominant in (3.25) and $\unicode[STIX]{x1D6FF}B/B$ is still small compared to $\unicode[STIX]{x1D6FF}n/n$ , but there is now a contribution from $g$ to $\unicode[STIX]{x1D6FF}n/n$ in (3.24). The resulting pair of equations, replacing (4.8)–(4.9), is

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{A}}}{\unicode[STIX]{x2202}t}+\frac{v_{\text{t h }}}{2}\unicode[STIX]{x1D735}_{\Vert }\left\{\unicode[STIX]{x1D711}+\frac{Z}{\unicode[STIX]{x1D70F}}\left[(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}-\overline{{\hat{J}}_{0}g}\right]\right\}=\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}{\mathcal{A}}, & \displaystyle\end{eqnarray}$$

(4.25) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left[(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}-\overline{{\hat{J}}_{0}g}\right]-\frac{v_{\text{t h }}}{\unicode[STIX]{x1D6FD}_{i}}\unicode[STIX]{x1D735}_{\Vert }\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}=v_{\text{th}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{J}}_{\text{ext}}, & \displaystyle\end{eqnarray}$$

coupled to (4.6).

The quantity in the square brackets in (4.24) and (4.25) is $-\unicode[STIX]{x1D6FF}n/n$ , so these equations can be thought of as evolution equations of ${\mathcal{A}}$ and $\unicode[STIX]{x1D6FF}n/n$ , the latter’s relationship to $\unicode[STIX]{x1D711}$ now involving  $g$ . Alternatively, (4.25) can be recast as

(4.26) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}-\frac{v_{\text{th}}}{\unicode[STIX]{x1D6FD}_{i}}\unicode[STIX]{x1D735}_{\Vert }\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}=v_{\text{t h}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{J}}_{\text{ext}}-\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{th}}}{2}\overline{\langle \left\{\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}-\unicode[STIX]{x1D711},g\right\}\rangle _{\boldsymbol{r}}}+\overline{\langle C[g+\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}F_{0}]\rangle _{\boldsymbol{r}}}\end{eqnarray}$$

if one uses the evolution equation for $\overline{{\hat{J}}_{0}g}$ derived by integrating (4.6) over the velocity space ((4.26) can also be obtained by applying the ordering (4.5) to (3.31)).Footnote ¹⁴ This emphasises the nonlinear FLR coupling of $\unicode[STIX]{x1D711}$ to $g$ .

These equations support a generalised version of the (collisionless) invariant (4.14):

(4.27) $$\begin{eqnarray}\widetilde{W}=\frac{v_{\text{th}}^{2}}{4}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\unicode[STIX]{x1D711}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}+\frac{Z}{\unicode[STIX]{x1D70F}}\left|(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\unicode[STIX]{x1D711}-\overline{{\hat{J}}_{0}g}\right|^{2}+\frac{2}{\unicode[STIX]{x1D6FD}_{i}}|\hat{\unicode[STIX]{x1D735}}_{\bot }{\mathcal{A}}|^{2}\right],\end{eqnarray}$$

which is the low-beta limit of (3.30), excluding the variance of $g$ , which is still conserved independently (see (4.19)). Indeed, using (4.26) to work out the time derivative of the first term and (4.24) and (4.25) for the other two terms, we get

(4.28) $$\begin{eqnarray}\frac{\text{d}\widetilde{W}}{\text{d}t}=\unicode[STIX]{x1D700}_{\text{AW}}-Q_{e}+Q_{\text{x}},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{\text{AW}}$ is given by (3.18), $Q_{e}$ by (3.21) and $Q_{\text{x}}$ in (4.20). The nonlinear terms have vanished by cancellation and because

(4.29) $$\begin{eqnarray}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\unicode[STIX]{x1D711}\overline{\langle \left\{\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}-\unicode[STIX]{x1D711},g\right\}\rangle _{\boldsymbol{r}}}=\overline{\int \frac{\text{d}^{3}\boldsymbol{R}}{V}\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}}\left\{\langle \unicode[STIX]{x1D711}\rangle _{\boldsymbol{R}},g\right\}}=0\end{eqnarray}$$

(after swapping the order of the $\boldsymbol{v}$ and $\boldsymbol{r}$ integration and changing the integration variable from $\boldsymbol{r}$ to $\boldsymbol{R}$ ).

Combining (4.28) and (4.19), we recover the overall conservation law (3.17), as indeed we must, because the free energy is

(4.30) $$\begin{eqnarray}W=\widetilde{W}+\frac{v_{\text{th}}^{2}}{4}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\overline{\frac{\langle g^{2}\rangle _{\boldsymbol{ r}}}{F_{0}}}.\end{eqnarray}$$

However, we now have more restrictive and, therefore, more informative energy balances (4.20) and (4.28) (with $\text{d}\widetilde{W}/\text{d}t=0$ in steady state). Since, as we argued in § 4.5, $Q_{\text{x}}$ is small, we conclude that

(4.31) $$\begin{eqnarray}Q_{i}=\unicode[STIX]{x1D700}_{\text{compr}},\quad Q_{e}=\unicode[STIX]{x1D700}_{\text{AW}},\end{eqnarray}$$

so compressive energy goes into ions, Alfvénic into electrons. Thus, while non-zero $g$ does insinuate itself into the dynamics of Alfvénic perturbations, there is no energy exchange between the two cascades.

4.7 Ultra-low beta

Formally, there is an interesting very-low-beta limit that is outside the validity of our theory so far. Namely, if $\unicode[STIX]{x1D6FD}_{e}\sim m_{e}/m_{i}$ , we can no longer use the isothermal-electron-fluid approximation introduced in § 3.3. The equations in this case are quite similar to (4.6) and (4.24)–(4.25), except in (4.24) there is now an electron-inertia term and a piece of parallel pressure gradient that contains a non-zero parallel electron temperature perturbation. The latter has to be calculated from the electron drift-kinetic equation, thus opening up an electron-heating route via parallel heat transport and Landau damping. With $g=0$ , the appropriate equations were worked out by Zocco & Schekochihin (Reference Zocco and Schekochihin2011) and proved to be a useful model for numerical experimentation (Loureiro et al. Reference Loureiro, Schekochihin and Zocco2013, Reference Loureiro, Dorland, Fazendeiro, Kanekar, Mallet, Vilelas and Zocco2016; Grošelj et al. Reference Grošelj, Cerri, Bañón Navarro, Willmott, Told, Loureiro, Califano and Jenko2017); they can be generalised to $g\neq 0$ in exactly the same way as the system (4.8)–(4.9) was generalised in §§ 4.5 and 4.6. There is no change in the energy partition: by the same arguments as above, the energy of compressive perturbations goes into ions, the energy of Alfvénic ones into electrons.

5.1 Ordering

In this limit, since $\unicode[STIX]{x1D6FD}_{e}=Z\unicode[STIX]{x1D6FD}_{i}/\unicode[STIX]{x1D70F}$ , the ions are cold and, as we anticipate based on § 4, the AW physics will become dispersive at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{s}}\sim 1$ ,

(5.1) $$\begin{eqnarray}\frac{Z}{\unicode[STIX]{x1D70F}}\sim \frac{1}{\unicode[STIX]{x1D6FD}_{i}}\gg 1,\quad k_{\bot }\unicode[STIX]{x1D70C}_{\text{s}}\sim 1\quad \Rightarrow \quad k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim \sqrt{\frac{\unicode[STIX]{x1D70F}}{Z}}\sim \sqrt{\unicode[STIX]{x1D6FD}_{i}}\ll 1\quad \Rightarrow \quad k_{\bot }d_{i}\sim 1,\end{eqnarray}$$

where $d_{i}=\unicode[STIX]{x1D70C}_{i}/\sqrt{\unicode[STIX]{x1D6FD}_{i}}=\unicode[STIX]{x1D70C}_{\text{s}}\sqrt{2/\unicode[STIX]{x1D6FD}_{e}}$ is the ion inertial scale, which is of the same order as $\unicode[STIX]{x1D70C}_{\text{s}}$ in this limit.

We must adjust all expansions and equations accordingly. Instead of (4.3) and (4.4), we have

(5.2) $$\begin{eqnarray}{\mathcal{A}}\sim \frac{\unicode[STIX]{x1D716}}{k_{\bot }\unicode[STIX]{x1D70C}_{i}}\sim \frac{\unicode[STIX]{x1D716}}{\sqrt{\unicode[STIX]{x1D6FD}_{i}}},\quad \unicode[STIX]{x1D711}\sim \frac{\unicode[STIX]{x1D716}}{k_{\bot }\unicode[STIX]{x1D70C}_{i}\sqrt{\unicode[STIX]{x1D6FD}_{i}}}\sim \frac{\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6FD}_{i}}.\end{eqnarray}$$

Since the sound speed and the Alfvén speed are of the same order in this limit, viz.,

(5.3) $$\begin{eqnarray}c_{\text{s}}=\sqrt{\frac{ZT_{e}}{m_{i}}}=v_{\text{th}}\sqrt{\frac{Z}{2\unicode[STIX]{x1D70F}}}=v_{\text{A}}\sqrt{\frac{\unicode[STIX]{x1D6FD}_{e}}{2}}\sim v_{\text{A}},\end{eqnarray}$$

the AW and the compressive modes (slow waves) have similar frequencies. This allows us to handle both cascades simultaneously. To avoid prejudice, we order the compressive perturbations to have similar amplitudes to the Alfvénic ones:

(5.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}n}{n}\sim \frac{u_{\Vert i}}{v_{\text{A}}}\sim \frac{\unicode[STIX]{x1D6FF}B}{B}\sim \frac{\unicode[STIX]{x1D6FF}B_{\bot }}{B}\sim \unicode[STIX]{x1D716},\end{eqnarray}$$

where $u_{\Vert i}=\overline{v_{\Vert }{\hat{J}}_{0}g}$ is the parallel ion flow velocity. The requirement that (3.24)–(3.26) be consistent with (5.4) implies that we ought to order

(5.5) $$\begin{eqnarray}\frac{g}{F_{0}}\sim \frac{\unicode[STIX]{x1D716}}{\sqrt{\unicode[STIX]{x1D6FD}_{i}}},\quad \overline{{\hat{J}}_{0}g}\sim \unicode[STIX]{x1D716},\end{eqnarray}$$

i.e. to lowest order, the distribution function should have no density moment.

5.2 Equations

With these orderings, (3.24)–(3.26) become, to lowest order in $\unicode[STIX]{x1D6FD}_{i}$ (and $\unicode[STIX]{x1D70F}$ ),

(5.6) $$\begin{eqnarray}\overline{g}=\frac{\unicode[STIX]{x1D6FF}n}{n}-\frac{1}{2}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}\unicode[STIX]{x1D711},\quad u_{\Vert e}=u_{\Vert i}+v_{\text{th}}\left(\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+{\mathcal{J}}_{\text{ext}}\right),\quad \frac{\unicode[STIX]{x1D6FF}n}{n}=-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\frac{\unicode[STIX]{x1D6FF}B}{B}.\end{eqnarray}$$

The last of these equations is the balance between the magnetic and electron pressure, ions being too cold to matter. Using this relationship in (3.10)–(3.11), we getFootnote ¹⁵

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{A}}}{\unicode[STIX]{x2202}t}+\frac{v_{\text{t h }}}{2}\unicode[STIX]{x1D735}_{\Vert }\left(\unicode[STIX]{x1D711}+\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}B}{B}\right)=\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}{\mathcal{A}}, & \displaystyle\end{eqnarray}$$

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle \left(1+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\right)\frac{\text{d}}{\text{d}t}\frac{\unicode[STIX]{x1D6FF}B}{B}=\unicode[STIX]{x1D735}_{\Vert }\left[u_{\Vert i}+v_{\text{th}}\left(\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}+{\mathcal{J}}_{\text{ext}}\right)\right]. & \displaystyle\end{eqnarray}$$

So all four fields ${\mathcal{A}}$ , $\unicode[STIX]{x1D711}$ , $\unicode[STIX]{x1D6FF}B$ and $u_{\Vert i}$ (the latter representing $g$ ) are coupled and we need two more equations to close the system.

One of these is (3.31), where applying the ordering of § 5.1 leads to the disappearance of the entire right-hand side, as well as of the $\unicode[STIX]{x1D6FF}B$ term under the time derivative. To lowest order, therefore, we are left with a rather familiar equation (cf. the second RMHD equation in (3.3)):

(5.9) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\frac{1}{2}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}\unicode[STIX]{x1D711}+\frac{v_{\text{th}}}{\unicode[STIX]{x1D6FD}_{i}}\unicode[STIX]{x1D735}_{\Vert }\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}{\mathcal{A}}=-v_{\text{t h}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{J}}_{\text{ext}}.\end{eqnarray}$$

The last required equation is the lowest-order version of the kinetic equation (3.29):

(5.10) $$\begin{eqnarray}\frac{\text{d}g}{\text{d}t}=v_{\Vert }\unicode[STIX]{x1D735}_{\Vert }\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}B}{B}F_{0}+C[g]+\frac{2v_{\Vert }a_{\text{ext}}}{v_{\text{th}}^{2}}F_{0},\end{eqnarray}$$

where we again used the last equation in (5.6). This is consistent with $\overline{g}=0$ to lowest order, as anticipated in (5.5). If we split off the velocity moment from $g$ , viz.,

(5.11) $$\begin{eqnarray}g=\frac{2u_{\Vert i}v_{\Vert }}{v_{\text{th}}^{2}}F_{0}+{\mathcal{G}},\quad \overline{{\mathcal{G}}}=0,\quad \overline{v_{\Vert }{\mathcal{G}}}=0,\end{eqnarray}$$

then (5.10) becomes

(5.12) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}u_{\Vert i}}{\text{d}t} & = & \displaystyle v_{\text{A}}^{2}\unicode[STIX]{x1D735}_{\Vert }\frac{\unicode[STIX]{x1D6FF}B}{B}+a_{\text{ext}},\end{eqnarray}$$

(5.13) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}{\mathcal{G}}}{\text{d}t} & = & \displaystyle C[{\mathcal{G}}].\end{eqnarray}$$

The first of these is the final equation that we needed to close the system comprising already (5.7)–(5.9). The second equation, (5.13), describes a passively advected kinetic field, which, however, is not coupled to anything and so can be safely put to zeroFootnote ¹⁶ – it is the kinetic version of the MHD entropy mode, whereas the rest of our equations describe linearly and nonlinearly coupled AW and slow waves (SW). Note finally that (5.9) is needed because it is not possible to calculate $\unicode[STIX]{x1D711}$ from the first and last of the field equations (5.6) and the kinetic equation (5.10). This is because, as assumed in (5.5), the density moment $\overline{g}$ comes from the next-order part of $g$ not captured in (5.10).

It is instructive to rewrite the Hall-RMHD equations (5.7)–(5.9) and (5.12) in ‘fluid’ notation, dropping the forcing terms and resistivity (cf. Gómez, Mahajan & Dmitruk Reference Gómez, Mahajan and Dmitruk2008):

(5.14) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}t}=v_{\text{A}}\unicode[STIX]{x1D735}_{\Vert }(\unicode[STIX]{x1D6F7}+v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}{\mathcal{B}}), & \displaystyle\end{eqnarray}$$

(5.15) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}{\mathcal{B}}}{\text{d}t}=\unicode[STIX]{x1D735}_{\Vert }(v_{\text{s}}\,{\mathcal{U}}-\unicode[STIX]{x1D70C}_{\text{H}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}), & \displaystyle\end{eqnarray}$$

(5.16) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}=v_{\text{A}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}, & \displaystyle\end{eqnarray}$$

(5.17) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\,{\mathcal{U}}}{\text{d}t}=v_{\text{s}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{B}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D6F9}$ are defined by (3.1), we have denoted

(5.18) $$\begin{eqnarray}{\mathcal{B}}=\frac{\unicode[STIX]{x1D6FF}B}{B}\sqrt{1+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}},\quad {\mathcal{U}}=\frac{u_{\Vert i}}{v_{\text{A}}},\end{eqnarray}$$

and introduced the Hall transition scale

(5.19) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\text{H}}=\frac{d_{i}}{\sqrt{1+2/\unicode[STIX]{x1D6FD}_{e}}}=\frac{\unicode[STIX]{x1D70C}_{\text{s}}}{\sqrt{1+\unicode[STIX]{x1D6FD}_{e}/2}}=\unicode[STIX]{x1D70C}_{i}\sqrt{\frac{Z/\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D6FD}_{e}}}\end{eqnarray}$$

and the SW phase speed

(5.20) $$\begin{eqnarray}v_{\text{s}}=\frac{v_{\text{A}}}{\sqrt{1+2/\unicode[STIX]{x1D6FD}_{e}}}=\frac{c_{\text{s}}}{\sqrt{1+\unicode[STIX]{x1D6FD}_{e}/2}}.\end{eqnarray}$$

At $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\ll 1$ , the Alfvénic and the SW-like perturbations decouple from each other and revert to standard RMHD equations (see S09–§ 2.4): the AW equations (5.14) and (5.16) become (3.3) and the SW equations (5.15) and (5.17) become

(5.21) $$\begin{eqnarray}\frac{\text{d}{\mathcal{B}}}{\text{d}t}=v_{\text{s}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{U}},\quad \frac{\text{d}\,{\mathcal{U}}}{\text{d}t}=v_{\text{s}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{B}}\end{eqnarray}$$

(passively advected by the AW via $\text{d}/\text{d}t$ and $\unicode[STIX]{x1D735}_{\Vert }$ , without energy exchange). Thus, our new system of equations (5.14)–(5.17) captures the RMHD regime and describes its transformation, at the Hall transition scale $\unicode[STIX]{x1D70C}_{\text{H}}$ , into one in which all four fields $\unicode[STIX]{x1D6F7}$ , $\unicode[STIX]{x1D6F9}$ , ${\mathcal{B}}$ and ${\mathcal{U}}$ are coupled.

The system (5.14)–(5.17) also contains the low- $\unicode[STIX]{x1D6FD}_{e}$ limit (4.12). This corresponds to taking the limit $v_{\text{s}}\rightarrow 0$ . Combining (5.15) and (5.16), we get

(5.22) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\left({\mathcal{B}}+\frac{\unicode[STIX]{x1D70C}_{\text{H}}}{v_{\text{A}}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}\right)=v_{\text{s}}\unicode[STIX]{x1D735}_{\Vert }{\mathcal{U}}\rightarrow 0\quad \Rightarrow \quad {\mathcal{B}}=-\frac{\unicode[STIX]{x1D70C}_{\text{H}}}{v_{\text{A}}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}.\end{eqnarray}$$

Using this in (5.14) and setting $\unicode[STIX]{x1D70C}_{\text{H}}=\unicode[STIX]{x1D70C}_{\text{s}}$ , we get the first equation in (4.12). The second is the same as (5.16). The parallel velocity in this limit decouples and cascades independently:

(5.23) $$\begin{eqnarray}\frac{\text{d}\,{\mathcal{U}}}{\text{d}t}=0,\end{eqnarray}$$

just like ${\mathcal{G}}$ does in (5.13) and like $g$ did in (4.6).

5.3 Free energy and heating

The conserved free energy for (5.14)–(5.17) (equivalently, for (5.7)–(5.9) and (5.12)) is

(5.24) $$\begin{eqnarray}\displaystyle \widetilde{W} & = & \displaystyle \frac{1}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}[|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F7}|^{2}+|\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9}|^{2}+v_{\text{A}}^{2}({\mathcal{U}}^{2}+{\mathcal{B}}^{2})]\nonumber\\ \displaystyle & = & \displaystyle \int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\frac{v_{\text{t h }}^{2}}{4}\left(\frac{1}{2}|\hat{\unicode[STIX]{x1D735}}_{\bot }\unicode[STIX]{x1D711}|^{2}+\frac{2}{\unicode[STIX]{x1D6FD}_{i}}|\hat{\unicode[STIX]{x1D735}}_{\bot }{\mathcal{A}}|^{2}\right)+\frac{u_{\Vert i}^{2}}{2}+\frac{\unicode[STIX]{x1D6FF}B^{2}}{8\unicode[STIX]{x03C0}m_{i}n_{i}}\left(1+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\right)\right].\qquad\end{eqnarray}$$

The free energy has no access to  ${\mathcal{G}}$ , whose variance is individually conserved, as is obvious from (5.13). If we forced ${\mathcal{G}}$ (without breaking the ordering of § 5.1), the free energy injected in this way would remain decoupled and travel all through the Hall range of scales unconcerned with the wave dynamics, eventually arriving at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ and transiting into the sub-Larmor-scale ion-entropy cascade and eventually into ion heat (see § 5.7.4). As we already mentioned in footnote 5.13, a natural physical way in which ${\mathcal{G}}$ might be forced is by the presence of an ion temperature gradient. However, there cannot be net heating of the plasma by turbulence produced by temperature gradients: any energy thus ‘borrowed’ from the ion thermal bath may only be redistributed between species (Abel et al. Reference Abel, Plunk, Wang, Barnes, Cowley, Dorland and Schekochihin2013). In the present case, all of it is destined for ions.

The first two terms in (5.24) are the Alfvénic energy (4.15), conserved by the RMHD (3.3); the last two terms are the slow-wave energy

(5.25) $$\begin{eqnarray}W_{\text{SW}}=\frac{v_{\text{A}}^{2}}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\left({\mathcal{U}}^{2}+{\mathcal{B}}^{2}\right),\end{eqnarray}$$

conserved by (5.21). When $\unicode[STIX]{x1D6FD}_{e}\ll 1$ , the substitution of (5.22) turns (5.24) into (4.16), with the ${\mathcal{U}}^{2}$ part of the free energy splitting off, destined for ion heating. In contrast, at $\unicode[STIX]{x1D6FD}_{e}\sim 1$ , the decoupling between the Alfvénic and compressive cascades is broken at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\sim 1$ , so we can no longer conclude that the former must heat electrons and the latter ions. In order to work out what happens (see § 5.5.4 for a preview of the answer), we must shift our focus to $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ (§ 5.7), but for that, we must first investigate into what kind of turbulence the Hall turbulence turns at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ (§ 5.5). In working this out, we will find linear theory to be a valuable guide.

5.4 Linear theory

The dispersion relation is

(5.26) $$\begin{eqnarray}(\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{AW}}^{2})(\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{SW}}^{2})=\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D714}_{\text{KAW}}^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\text{AW}}=k_{\Vert }v_{\text{A}}$ is the AW frequency, $\unicode[STIX]{x1D714}_{\text{SW}}=k_{\Vert }v_{\text{s}}$ the SW frequency and $\unicode[STIX]{x1D714}_{\text{KAW}}=k_{\Vert }v_{\text{A}}k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}$ the KAW frequency in the $Z/\unicode[STIX]{x1D70F}\gg 1$ limit (cf. S09–§ 7.3). There is no damping of anything here because ions cannot stream along the field lines as fast as waves propagate (see (5.10)).

At $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\ll 1$ , $\unicode[STIX]{x1D714}_{\text{KAW}}\ll \unicode[STIX]{x1D714}_{\text{AW}},\unicode[STIX]{x1D714}_{\text{SW}}$ and we recover from (5.26) four low-frequency MHD waves

(5.27) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{\text{AW}},\quad \unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{\text{SW}}.\end{eqnarray}$$

At $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ , if $\unicode[STIX]{x1D714}\gg \unicode[STIX]{x1D714}_{\text{AW}},\unicode[STIX]{x1D714}_{\text{SW}}$ , the linear response assumes its KAW form:Footnote ¹⁷

(5.28) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{\text{KAW}}=\pm k_{\Vert }v_{\text{A}}k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}.\end{eqnarray}$$

This is not particularly surprising: the KAW response is the Alfvénic response with (nearly) immobile ions – and the ion-flow terms in the two magnetic-field equations (5.14) and (5.15) do indeed become subdominant at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ . Linearly, the KAW are then described by

(5.29) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}t}=v_{\text{A}}^{2}\unicode[STIX]{x1D70C}_{\text{H}}\frac{\unicode[STIX]{x2202}{\mathcal{B}}}{\unicode[STIX]{x2202}z},\quad \frac{\unicode[STIX]{x2202}{\mathcal{B}}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D70C}_{\text{H}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}.\end{eqnarray}$$

In the Hall limit, there is nothing particularly kinetic about kinetic Alfvén waves, so they should probably be called Hall Alfvén waves (but are sometimes called whistlers); we shall keep the KAW moniker to avoid multiplying entities beyond necessity.

There is more to the story at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ . In this limit, besides the two KAW, (5.26) has two other, low-frequency, solutions:

(5.30) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm \frac{\unicode[STIX]{x1D714}_{\text{AW}}\unicode[STIX]{x1D714}_{\text{SW}}}{\unicode[STIX]{x1D714}_{\text{KAW}}}=\pm \frac{k_{\Vert }v_{\text{s}}}{k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}}=\pm \unicode[STIX]{x1D6FA}\,\frac{k_{\Vert }}{k_{\bot }}\equiv \pm \unicode[STIX]{x1D714}_{\text{ICW}}.\end{eqnarray}$$

These are oblique ion-cyclotron waves (ICW; cf. Sahraoui, Galtier & Belmont Reference Sahraoui, Galtier and Belmont2007). For these perturbations, (5.14) and (5.15) become quasistatic, viz.,

(5.31) $$\begin{eqnarray}{\mathcal{B}}=-\frac{\unicode[STIX]{x1D6F7}}{v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}},\quad \unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F9}=\frac{v_{\text{s}}}{\unicode[STIX]{x1D70C}_{\text{H}}}\,{\mathcal{U}}=\unicode[STIX]{x1D6FA}\,{\mathcal{U}},\end{eqnarray}$$

and, consequently, the linearised versions of (5.16) and (5.17) turn into

(5.32) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6FA}\,\frac{\unicode[STIX]{x2202}u_{\Vert i}}{\unicode[STIX]{x2202}z},\quad \frac{\unicode[STIX]{x2202}u_{\Vert i}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D6FA}\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}z}.\end{eqnarray}$$

It is more transparent here to go back from ${\mathcal{U}}$ (defined in (5.18)) to $u_{\Vert i}$ as the Alfvénic normalisation is no longer physically relevant. These equations, and the corresponding dispersion relation (5.30), are mathematically the same as the equations and the dispersion relation for inertial waves in rigidly rotating (with angular velocity $\unicode[STIX]{x1D6FA}/2$ ) neutral fluids (see, e.g. Nazarenko & Schekochihin Reference Nazarenko and Schekochihin2011; Davidson Reference Davidson2013). We shall see momentarily that the analogy survives also nonlinearly and that, therefore, ICW turbulence displays some familiar features.

Figure 1. Solutions (5.33) of the Hall dispersion relation (5.26) with $\unicode[STIX]{x1D6FD}_{e}=1$ .

Finally, figure 1 shows the full solutions of (5.26),

(5.33) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}=\frac{k_{\Vert }^{2}v_{\text{A}}^{2}}{2}\left\{1+\unicode[STIX]{x1D70E}^{2}+k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\text{H}}^{2}\pm \sqrt{[k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\text{H}}^{2}+(1+\unicode[STIX]{x1D70E})^{2}][k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\text{H}}^{2}+(1-\unicode[STIX]{x1D70E})^{2}]}\right\},\end{eqnarray}$$

where

(5.34) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=\frac{v_{\text{s}}}{v_{\text{A}}}=\frac{1}{\sqrt{1+2/\unicode[STIX]{x1D6FD}_{e}}}\end{eqnarray}$$

(the only parameter in the problem). Note that there is no mode conversion, the AW continuously turn into KAW and SW into ICW. The two curves separate ever further at smaller  $\unicode[STIX]{x1D6FD}_{e}$ , tending towards the limit described by (4.12) and (5.22).

5.5 Hall turbulence at short wavelengths

The nature of Hall turbulence at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ is determined by the the way in which fast (KAW) and slow (ICW) perturbations interact with themselves and (potentially) with each other.

We are dealing with a two-time-scale problem, so let us split all our fields into slow and fast components, with ‘slow’ defined as the average of the relevant field over the KAW period and ‘fast’ as the difference between that and the exact field:

(5.35) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}=\overline{\unicode[STIX]{x1D6F9}}+\widetilde{\unicode[STIX]{x1D6F9}},\quad \unicode[STIX]{x1D6F7}=\overline{\unicode[STIX]{x1D6F7}}+\widetilde{\unicode[STIX]{x1D6F7}},\quad {\mathcal{B}}=\overline{{\mathcal{B}}}+\widetilde{{\mathcal{B}}},\quad {\mathcal{U}}=\overline{{\mathcal{U}}}+\widetilde{{\mathcal{U}}}.\end{eqnarray}$$

In everything that follows, overbar will mean KAW-time-scale averaging and overtilde will designate KAW-time-scale quantities, which average to zero (with apologies to the reader, who should now forget what overbars and overtildes have been used for previously). The slow quantities will represent the ICW turbulence and the fast ones the KAW turbulence.

We shall venture an a priori guess that the two cascades will decouple completely at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ , work out the scalings of all the fields on that basis and then confirm a posteriori that those are consistent with such a decoupling. Namely, we anticipate that the nonlinear version of the KAW equations (5.29) will be

(5.36) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D6F9}}}{\unicode[STIX]{x2202}t}=v_{\text{A}}^{2}\unicode[STIX]{x1D70C}_{\text{H}}\widetilde{\unicode[STIX]{x1D735}_{\Vert }\widetilde{{\mathcal{B}}}},\quad \frac{\unicode[STIX]{x2202}\widetilde{{\mathcal{B}}}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D70C}_{\text{H}}\widetilde{\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}},\quad \unicode[STIX]{x1D735}_{\Vert }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}+\frac{1}{v_{\text{A}}}\{\widetilde{\unicode[STIX]{x1D6F9}},\ldots \},\end{eqnarray}$$

and the nonlinear version of the ICW equations (5.32)

(5.37) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F7}}=\unicode[STIX]{x1D6FA}v_{\text{A}}\frac{\unicode[STIX]{x2202}\,\overline{{\mathcal{U}}}}{\unicode[STIX]{x2202}z},\quad \frac{\text{d}\,\overline{{\mathcal{U}}}}{\text{d}t}=-\frac{\unicode[STIX]{x1D6FA}}{v_{\text{A}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}z},\quad \frac{\text{d}}{\text{d}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\{\overline{\unicode[STIX]{x1D6F7}},\ldots \}.\end{eqnarray}$$

In each case, the other two fields play a subordinate role: for KAW turbulence, from (5.16) and (5.17),

(5.38) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F7}}=v_{\text{A}}\widetilde{\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}},\quad \frac{\unicode[STIX]{x2202}\,\widetilde{{\mathcal{U}}}}{\unicode[STIX]{x2202}t}=v_{\text{s}}\widetilde{\unicode[STIX]{x1D735}_{\Vert }\widetilde{{\mathcal{B}}}},\end{eqnarray}$$

for ICW turbulence, (5.31) hold nonlinearly, viz.,

(5.39) $$\begin{eqnarray}\overline{{\mathcal{B}}}=-\frac{\overline{\unicode[STIX]{x1D6F7}}}{v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}},\quad \unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F9}}=\unicode[STIX]{x1D6FA}\,\overline{{\mathcal{U}}}.\end{eqnarray}$$

The physics of these ‘constitutive relations’ will be made transparent in (5.86). Note that the first equation in (5.38) combined with the second equation in (5.36) also turns into a ‘constitutive relation’ between $\widetilde{{\mathcal{B}}}$ and  $\widetilde{\unicode[STIX]{x1D6F7}}$ (cf. (5.22)):

(5.40) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F7}}=-\frac{v_{\text{A}}}{\unicode[STIX]{x1D70C}_{\text{H}}}\,\widetilde{{\mathcal{B}}}.\end{eqnarray}$$

The pieces of the free energy (5.24) individually conserved by the systems (5.36) and (5.37) are, respectively,

(5.41) $$\begin{eqnarray}\displaystyle & \displaystyle W_{\text{KAW}}=\frac{1}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}[|\unicode[STIX]{x1D735}_{\bot }\widetilde{\unicode[STIX]{x1D6F9}}|^{2}+v_{\text{A}}^{2}\,\widetilde{{\mathcal{B}}}^{2}], & \displaystyle\end{eqnarray}$$

(5.42) $$\begin{eqnarray}\displaystyle & \displaystyle W_{\text{ICW}}=\frac{1}{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}[|\unicode[STIX]{x1D735}_{\bot }\overline{\unicode[STIX]{x1D6F7}}|^{2}+v_{\text{A}}^{2}\,\overline{{\mathcal{U}}}^{2}]. & \displaystyle\end{eqnarray}$$

Here $W_{\text{ICW}}$ is just the kinetic energy of the ion motion, perpendicular plus parallel, whereas $W_{\text{KAW}}$ is the total magnetic energy plus the free energy of the electron distribution – the latter is the $\unicode[STIX]{x1D6FF}n^{2}/n^{2}$ term in (3.16), now absorbed into $\widetilde{{\mathcal{B}}}^{2}$ by way of the last equation in (5.6).

We are now going to work out all the relevant scalings for KAW (§ 5.5.1) and ICW (§ 5.5.2) turbulence, then use these scalings to confirm that (5.36)–(5.39) are correct (§ 5.5.3), and finally propose what the energy partition in these circumstances should be (§ 5.5.4).

5.5.1 KAW scalings

The scalings for a critically balanced cascade of KAW-like fluctuations are a standard proposition (see S09–§ 7.5 and Cho & Lazarian Reference Cho and Lazarian2004).Footnote ¹⁸ The magnetic energy has a constant flux $\unicode[STIX]{x1D700}_{\text{KAW}}$ , with the cascade time scale set by the magnetic nonlinearity inside $\unicode[STIX]{x1D735}_{\Vert }$ in, e.g. the first equation in (5.36):

(5.43) $$\begin{eqnarray}(k_{\bot }\widetilde{\unicode[STIX]{x1D6F9}})^{2}\unicode[STIX]{x1D70F}_{\text{nl}}^{-1}\sim \unicode[STIX]{x1D700}_{\text{KAW}},\quad \unicode[STIX]{x1D70F}_{\text{nl}}^{-1}\sim v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}k_{\bot }^{2}\widetilde{{\mathcal{B}}}.\end{eqnarray}$$

The relationship between $\widetilde{{\mathcal{B}}}$ and $\widetilde{\unicode[STIX]{x1D6F9}}$ , and hence the scaling of field amplitudes, is then fixed by the second equation in (5.36):

(5.44) $$\begin{eqnarray}\widetilde{{\mathcal{B}}}\sim \unicode[STIX]{x1D714}_{\text{KAW}}^{-1}\unicode[STIX]{x1D70C}_{\text{H}}k_{\Vert \text{KAW}}k_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}\sim \frac{k_{\bot }\widetilde{\unicode[STIX]{x1D6F9}}}{v_{\text{A}}}\sim \left(\frac{\unicode[STIX]{x1D700}_{\text{KAW}}}{\unicode[STIX]{x1D70C}_{\text{H}}v_{\text{A}}^{3}}\right)^{1/3}k_{\bot }^{-2/3},\end{eqnarray}$$

the last relation following from (5.43). Finally, the relationship between the wave frequency $\unicode[STIX]{x1D714}_{\text{KAW}}$ (and, therefore, $k_{\Vert \text{KAW}}$ ) and the nonlinear decorrelation rate $\unicode[STIX]{x1D70F}_{\text{nl}}^{-1}$ (and, therefore, $k_{\bot }$ ) is set by the critical-balance conjecture:

(5.45) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{KAW}}=k_{\Vert \text{KAW}}v_{\text{A}}k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\sim \unicode[STIX]{x1D70F}_{\text{nl}}^{-1}\quad \Rightarrow \quad k_{\Vert \text{KAW}}\sim \left(\frac{\unicode[STIX]{x1D700}_{\text{KAW}}}{\unicode[STIX]{x1D70C}_{\text{H}}v_{\text{A}}^{3}}\right)^{1/3}k_{\bot }^{1/3}.\end{eqnarray}$$

The two subordinate fields are found from (5.38):

(5.46) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6F7}}\sim \unicode[STIX]{x1D714}_{\text{KAW}}^{-1}k_{\Vert \text{KAW}}v_{\text{A}}\widetilde{\unicode[STIX]{x1D6F9}}=\frac{\widetilde{\unicode[STIX]{x1D6F9}}}{k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}},\quad \widetilde{{\mathcal{U}}}\sim \unicode[STIX]{x1D714}_{\text{KAW}}^{-1}k_{\Vert }v_{\text{s}}\widetilde{{\mathcal{B}}}=\frac{v_{\text{s}}}{v_{\text{A}}}\frac{\widetilde{{\mathcal{B}}}}{k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}}.\end{eqnarray}$$

It follows from all this that the magnetic and velocity spectra are

(5.47) $$\begin{eqnarray}E_{\widetilde{B}}\propto k_{\bot }^{-7/3},\quad E_{\widetilde{u}}\propto k_{\bot }^{-13/3}\end{eqnarray}$$

(cf. Galtier & Buchlin Reference Galtier and Buchlin2007; Meyrand & Galtier Reference Meyrand and Galtier2012).

5.5.2 ICW scalings

The scalings for a critically balanced ICW cascade are perhaps less well established, but also known, in the guise of the scalings for rotating hydrodynamic turbulence (Nazarenko & Schekochihin Reference Nazarenko and Schekochihin2011). Assuming constant energy flux $\unicode[STIX]{x1D700}_{\text{ICW}}$ and using the first equation in (5.37), we find the Kolmogorov scaling (which is no surprise, the nonlinear coupling being hydrodynamic):

(5.48) $$\begin{eqnarray}(k_{\bot }\overline{\unicode[STIX]{x1D6F7}})^{2}\unicode[STIX]{x1D70F}_{\text{nl}}^{-1}\sim \unicode[STIX]{x1D700}_{\text{ICW}},\quad \unicode[STIX]{x1D70F}_{\text{nl}}^{-1}\sim k_{\bot }^{2}\overline{\unicode[STIX]{x1D6F7}}\quad \Rightarrow \quad k_{\bot }\overline{\unicode[STIX]{x1D6F7}}\sim \unicode[STIX]{x1D700}_{\text{ICW}}^{1/3}k_{\bot }^{-1/3}.\end{eqnarray}$$

From either equation in (5.37),

(5.49) $$\begin{eqnarray}\overline{{\mathcal{U}}}\sim \frac{k_{\bot }\overline{\unicode[STIX]{x1D6F7}}}{v_{\text{A}}}.\end{eqnarray}$$

The critical-balance conjecture implies

(5.50) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{ICW}}=\unicode[STIX]{x1D6FA}\,\frac{k_{\Vert \text{ICW}}}{k_{\bot }}\sim \unicode[STIX]{x1D70F}_{\text{nl}}^{-1}\quad \Rightarrow \quad k_{\Vert \text{ICW}}\sim \frac{\unicode[STIX]{x1D700}_{\text{ICW}}^{1/3}}{\unicode[STIX]{x1D6FA}}\,k_{\bot }^{5/3}.\end{eqnarray}$$

Interestingly, it follows from (5.50) that ICW turbulence becomes less anisotropic at smaller scales.Footnote ¹⁹ Finally, the subordinate fields (5.39) are

(5.51) $$\begin{eqnarray}\overline{{\mathcal{B}}}\sim \frac{\overline{\unicode[STIX]{x1D6F7}}}{v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}}\sim \frac{\overline{{\mathcal{U}}}}{k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}},\quad \frac{k_{\bot }\overline{\unicode[STIX]{x1D6F9}}}{v_{\text{A}}}\sim \frac{v_{\text{s}}}{v_{\text{A}}}\frac{\overline{{\mathcal{U}}}}{k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}}.\end{eqnarray}$$

The velocity and magnetic energy spectra are, therefore,

(5.52) $$\begin{eqnarray}E_{\overline{u}}\propto k_{\bot }^{-5/3},\quad E_{\overline{B}}\propto k_{\bot }^{-11/3}\end{eqnarray}$$

(cf. Krishan & Mahajan Reference Krishan and Mahajan2004; Galtier & Buchlin Reference Galtier and Buchlin2007; Meyrand & Galtier Reference Meyrand and Galtier2012).

5.5.3 Decoupling of cascades

The above scalings appear to be consistent with the numerical evidence recently reported by Meyrand et al. (Reference Meyrand, Kiyani, Gürcan and Galtier2018), who solved the traditional Hall-MHD equations that effectively describe the $\unicode[STIX]{x1D6FD}_{e}\gg 1$ limit of our system ( $v_{\text{s}}=v_{\text{A}}$ and $\unicode[STIX]{x1D70C}_{\text{H}}=d_{i}$ ). They did see $E_{\widetilde{u}}\ll E_{\overline{B}}\ll E_{\widetilde{B}}\ll E_{\overline{u}}\propto k_{\bot }^{-5/3}$ ; the $k_{\bot }^{-7/3}$ and $k_{\bot }^{-11/3}$ spectra of the magnetic perturbations associated with the two different wave modes (see (5.47) and (5.52)) had previously been extracted numerically from Hall MHD by Meyrand & Galtier (Reference Meyrand and Galtier2012) (and from a shell model by Galtier & Buchlin Reference Galtier and Buchlin2007). Unlike us, Meyrand et al. (Reference Meyrand, Kiyani, Gürcan and Galtier2018) think that the KAW turbulence is weak, rather than critically balanced, but we consider the evidence that they present in fact consistent with the possibility of a critically balanced KAW cascade: in particular, both their KAW fluctuations and their ICW fluctuations have broad frequency spectra and are spatially anisotropic in a scale-dependent way, the former becoming more anisotropic and the latter less, as $k_{\bot }$ increases – in agreement with (5.45) and (5.50). They also see striking evidence that $k_{\Vert \text{KAW}}\ll k_{\Vert \text{ICW}}$ , which is indeed what (5.45) and (5.50) imply. Finally, and crucially, they show quite unambiguously that energy exchange between velocity and magnetic fields (and, therefore, between ICW and KAW fluctuations) peters out at $k_{\bot }d_{i}\gg 1$ , i.e. the two cascades are energetically decoupled.

As promised above, we now confirm that the scalings of §§ 5.5.1 and 5.5.2, if adopted as orderings, do indeed allow the two cascades to decouple (an impatient reader willing to trust us may skip to § 5.5.4). Let

(5.53) $$\begin{eqnarray}\unicode[STIX]{x1D716}=\frac{(\unicode[STIX]{x1D700}_{\text{ICW}}\unicode[STIX]{x1D70C}_{\text{H}})^{1/3}}{v_{\text{A}}}\sim \frac{(\unicode[STIX]{x1D700}_{\text{KAW}}\unicode[STIX]{x1D70C}_{\text{H}})^{1/3}}{v_{\text{A}}},\quad \unicode[STIX]{x1D6FF}=\frac{1}{(k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}})^{1/3}}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D716}$ is just the GK expansion parameter that must enter all field amplitudes. The only non-trivial choice about $\unicode[STIX]{x1D716}$ is that $\unicode[STIX]{x1D700}_{\text{KAW}}\sim \unicode[STIX]{x1D700}_{\text{ICW}}$ , i.e. that the KAW and ICW fluctuations receive a priori comparable amounts of energy – equivalently, we assume that the KAW and ICW amplitudes are similar at the Hall transition scale (at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\sim 1$ ). We now use $\unicode[STIX]{x1D6FF}$ as a subsidiary ordering parameter for the Hall-MHD equations (5.14)–(5.17):

(5.54) $$\begin{eqnarray}\frac{k_{\bot }\overline{\unicode[STIX]{x1D6F7}}}{v_{\text{A}}}\sim \overline{{\mathcal{U}}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF},\quad \frac{k_{\bot }\overline{\unicode[STIX]{x1D6F9}}}{v_{\text{A}}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}^{4},\quad \overline{{\mathcal{B}}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}^{4},\end{eqnarray}$$

(5.55) $$\begin{eqnarray}k_{\Vert \text{ICW}}\unicode[STIX]{x1D70C}_{\text{H}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D70E}^{-1}\unicode[STIX]{x1D6FF}^{-5},\quad \frac{\unicode[STIX]{x1D714}_{\text{ICW}}}{\unicode[STIX]{x1D6FA}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D70E}^{-1}\unicode[STIX]{x1D6FF}^{-2},\end{eqnarray}$$

(5.56) $$\begin{eqnarray}\frac{k_{\bot }\widetilde{\unicode[STIX]{x1D6F9}}}{v_{\text{A}}}\sim \widetilde{{\mathcal{B}}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}^{2},\quad \frac{k_{\bot }\widetilde{\unicode[STIX]{x1D6F7}}}{v_{\text{A}}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}^{5},\quad \widetilde{{\mathcal{U}}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}^{5},\end{eqnarray}$$

(5.57) $$\begin{eqnarray}k_{\Vert \text{KAW}}\unicode[STIX]{x1D70C}_{\text{H}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}^{-1},\quad \frac{\unicode[STIX]{x1D714}_{\text{KAW}}}{\unicode[STIX]{x1D6FA}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D70E}^{-1}\unicode[STIX]{x1D6FF}^{-4},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}$ is defined in (5.34).

Applying the decomposition (5.35) and the above ordering to (5.14), we get, keeping two lowest orders,

(5.58) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D6F9}}}{\unicode[STIX]{x2202}t}=v_{\text{A}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(\overline{\unicode[STIX]{x1D6F7}}+v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}\overline{{\mathcal{B}}})+v_{\text{A}}^{2}\unicode[STIX]{x1D70C}_{\text{H}}\unicode[STIX]{x1D735}_{\Vert }\widetilde{{\mathcal{B}}},\quad \unicode[STIX]{x1D735}_{\Vert }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}+\frac{1}{v_{\text{A}}}\{\widetilde{\unicode[STIX]{x1D6F9}},\ldots \}.\end{eqnarray}$$

Averaging this equation over the KAW time scale gives us

(5.59) $$\begin{eqnarray}v_{\text{A}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(\overline{\unicode[STIX]{x1D6F7}}+v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}\overline{{\mathcal{B}}})+v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}\overline{\{\widetilde{\unicode[STIX]{x1D6F9}},\widetilde{{\mathcal{B}}}\}}=0.\end{eqnarray}$$

Subtracting this from (5.58), we end up with the first KAW equation in (5.36). Retaining the lowest order only in (5.59) results in the first ICW constitutive relation in (5.39), assuming that we can ignore any additive corrections to this that are constant along the magnetic field.

From (5.17), again keeping only two lowest orders, we get

(5.60) $$\begin{eqnarray}\frac{\text{d}\,\overline{{\mathcal{U}}}}{\text{d}t}+\frac{\unicode[STIX]{x2202}\,\widetilde{{\mathcal{U}}}}{\unicode[STIX]{x2202}t}=v_{\text{s}}\left(\frac{\unicode[STIX]{x2202}\overline{{\mathcal{B}}}}{\unicode[STIX]{x2202}z}+\unicode[STIX]{x1D735}_{\Vert }\widetilde{{\mathcal{B}}}\right),\quad \frac{\text{d}}{\text{d}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\{\overline{\unicode[STIX]{x1D6F7}},\ldots \}.\end{eqnarray}$$

Averaging and using (5.59) gives us

(5.61) $$\begin{eqnarray}\frac{\text{d}\,\overline{{\mathcal{U}}}}{\text{d}t}=v_{\text{s}}\left(\frac{\unicode[STIX]{x2202}\overline{{\mathcal{B}}}}{\unicode[STIX]{x2202}z}+\frac{1}{v_{\text{A}}}\overline{\{\widetilde{\unicode[STIX]{x1D6F9}},\widetilde{{\mathcal{B}}}\}}\right)=-\frac{v_{\text{s}}}{v_{\text{A}}\unicode[STIX]{x1D70C}_{\text{H}}}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}z},\end{eqnarray}$$

which is the second ICW equation in (5.37). Subtracting (5.61) from (5.60) leaves us with the second equation in (5.38), describing small parallel ion flows associated with KAW.

Continuing in the same vein, we find that (5.15) becomes, to two lowest orders,

(5.62) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{{\mathcal{B}}}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(v_{\text{s}}\overline{{\mathcal{U}}}-\unicode[STIX]{x1D70C}_{\text{H}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F9}})-\unicode[STIX]{x1D70C}_{\text{H}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}.\end{eqnarray}$$

The average of this is

(5.63) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(v_{\text{s}}\overline{{\mathcal{U}}}-\unicode[STIX]{x1D70C}_{\text{H}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F9}})-\frac{\unicode[STIX]{x1D70C}_{\text{H}}}{v_{\text{A}}}\overline{\{\widetilde{\unicode[STIX]{x1D6F9}},\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}\}}=0,\end{eqnarray}$$

which, to lowest order, becomes the second ICW constitutive relation in (5.39) (again ignoring any contributions that do not vary along the magnetic field). Subtracting (5.63) from (5.62) gets us the second KAW equation in (5.36).

Finally, (5.16) to two lowest orders is

(5.64) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F7}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F7}}=v_{\text{A}}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F9}}+\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}\right).\end{eqnarray}$$

Its average is, via (5.63),

(5.65) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F7}}=v_{\text{A}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F9}}+\overline{\{\widetilde{\unicode[STIX]{x1D6F9}},\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{\unicode[STIX]{x1D6F9}}\}}=\frac{v_{\text{s}}v_{\text{A}}}{\unicode[STIX]{x1D70C}_{\text{H}}}\frac{\unicode[STIX]{x2202}\,\overline{{\mathcal{U}}}}{\unicode[STIX]{x2202}z},\end{eqnarray}$$

which is the first ICW equation in (5.37). Subtracting (5.65) from (5.64), we get the first equation in (5.38) for the small perpendicular flows present in KAW.

Thus, the equations for decoupled KAW and ICW cascades, (5.36)–(5.39), which were our basis for developing the scalings in §§ 5.5.1 and 5.5.2, can indeed be extracted from Hall equations (5.14)–(5.17) if those scalings are assumed.Footnote ²⁰ Consistency is the least – and the most – that we can ask for in this approach.

5.5.4 Energy partition

We anticipate, and will prove in § 5.7, that the sub-Hall-scale KAW cascade all goes into the sub-Larmor-scale KAW cascade and thence to electron heating, whereas the ICW cascade is destined for ion-entropy cascade and thence to ion heating. Thus, in the Hall regime, the energy partition is decided at the Hall scale  $\unicode[STIX]{x1D70C}_{\text{H}}$ . While we do not know how to determine this energy partition rigorously, a plausible conjecture can be made.

The only parameter in the problem is the ratio $\unicode[STIX]{x1D70E}=v_{\text{s}}/v_{\text{A}}$ (equivalently, $\unicode[STIX]{x1D6FD}_{e}$ : see (5.34)). As explained at the end of § 5.2, (5.14)–(5.17) reduce to (4.12) in the limit of $\unicode[STIX]{x1D70E}\ll 1$ (low $\unicode[STIX]{x1D6FD}_{e}$ ). This happens because, sufficiently far into that limit, the finite- $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}$ contribution to ${\mathcal{B}}$ from the Alfvénic fluctuations overwhelms the SW part of ${\mathcal{B}}$ (see (5.22)), while what remains of the SW cascades independently according to (5.23), unbothered by the Hall-scale transition. The result is again (4.31): the Alfvénic energy goes into electrons, the compressive one into ions.

In contrast, when $\unicode[STIX]{x1D70E}\sim 1$ , there are no small parameters left in the problem and all time scales and all parts of the free energy (5.24) are of the same order at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\sim 1$ . At $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\ll 1$ , $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D6F9}$ fluctuations carry $\unicode[STIX]{x1D700}_{\text{AW}}$ , while ${\mathcal{U}}$ and ${\mathcal{B}}$ fluctuations carry $\unicode[STIX]{x1D700}_{\text{compr}}$ . On the other side of the transition, at $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ , the $\unicode[STIX]{x1D6F9}$ and ${\mathcal{B}}$ fluctuations (the magnetic energy) are picked up by the KAW cascade ( $\unicode[STIX]{x1D700}_{\text{KAW}}$ ) and the $\unicode[STIX]{x1D6F7}$ and ${\mathcal{U}}$ fluctuations (the kinetic energy) by the ICW cascade ( $\unicode[STIX]{x1D700}_{\text{ICW}}$ ). It is then natural to conjecture an equal split of the power of the original MHD cascade between $\unicode[STIX]{x1D700}_{\text{KAW}}$ and $\unicode[STIX]{x1D700}_{\text{ICW}}$ , and, therefore, between electron and ion heating – independently of the relative size of $\unicode[STIX]{x1D700}_{\text{AW}}$ and $\unicode[STIX]{x1D700}_{\text{compr}}$ :

(5.66) $$\begin{eqnarray}Q_{e}\sim \unicode[STIX]{x1D700}_{\text{KAW}}\sim \frac{\unicode[STIX]{x1D700}_{\text{AW}}+\unicode[STIX]{x1D700}_{\text{compr}}}{2}\sim \unicode[STIX]{x1D700}_{\text{ICW}}\sim Q_{i}.\end{eqnarray}$$

Numerical simulations of the full system (5.14)–(5.17) with external driving are needed (and will be done) to test this reasoning. A parameter scan in $\unicode[STIX]{x1D70E}$ should reveal a gradual transition from $\left[Q_{i}/Q_{e}\right]_{\unicode[STIX]{x1D70E}\rightarrow 0}\rightarrow \unicode[STIX]{x1D700}_{\text{compr}}/\unicode[STIX]{x1D700}_{\text{AW}}$ to $\left[Q_{i}/Q_{e}\right]_{\unicode[STIX]{x1D70E}\rightarrow 1}\rightarrow 1$ .

5.6 Helicities

Before, as promised, moving on to the Larmor-scale dynamics, let us, for the sake of completeness and for the benefit of those readers who might be interested in Hall-RMHD turbulence per se, offer some discussion of other invariants of the system (5.14)–(5.17). Famously, Hall-MHD equations conserve two helicities, magnetic and ‘hybrid’ (Turner Reference Turner1986; Mahajan & Yoshida Reference Mahajan and Yoshida1998). However, in Hall RMHD, owing to the presence of a strong background magnetic field, magnetic helicity is not conserved, except in two dimensions:

(5.67) $$\begin{eqnarray}H=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,\unicode[STIX]{x1D6F9}{\mathcal{B}},\quad \frac{\text{d}H}{\text{d}t}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,\left(v_{\text{s}}\unicode[STIX]{x1D6F9}\,\frac{\unicode[STIX]{x2202}\,{\mathcal{U}}}{\unicode[STIX]{x2202}z}+v_{\text{A}}{\mathcal{B}}\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}z}\right)\end{eqnarray}$$

(see S09–§  F.4 and references therein for a discussion of helicity non-conservation in a system with a mean field). What is conserved, however, is the sum of three other ‘helicities’ present in the system, viz., the Alfvénic cross-helicity, the compressive cross-helicity and the kinetic helicity (note that $\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}$ is the $z$ component of the vorticity of the plasma motions):

(5.68) $$\begin{eqnarray}X=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,\left[(\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F7})\boldsymbol{\cdot }(\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6F9})+\frac{v_{\text{A}}^{3}}{v_{\text{s}}}\,{\mathcal{U}}\left({\mathcal{B}}+\frac{\unicode[STIX]{x1D70C}_{\text{H}}}{v_{\text{A}}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}\right)\right].\end{eqnarray}$$

Note that the Hall-MHD ‘hybrid’ helicity referred to above is then just $H-(\unicode[STIX]{x1D70C}_{\text{H}}/v_{\text{A}})X$ (not conserved because $H$ is not conserved).

In the RMHD limit ( $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\ll 1$ ), $X$ loses its last term (the kinetic helicity) and turns into the standard RMHD cross-helicity, whose conservation reflects the energetic decoupling of the cascades of the four Elsasser fields $\unicode[STIX]{x1D6F7}\pm \unicode[STIX]{x1D6F9}$ and ${\mathcal{U}}\pm {\mathcal{B}}$ (see S09–§ 2.7). In the opposite limit, $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ , when Hall-RMHD splits into the KAW equations (5.36) and the ICW equations (5.37), each of these systems conserves its own piece of  $X$ :

(5.69) $$\begin{eqnarray}X_{\text{KAW}}=\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,\widetilde{\unicode[STIX]{x1D6F9}}\widetilde{{\mathcal{B}}},\quad X_{\text{ICW}}=\frac{v_{\text{A}}^{2}}{\unicode[STIX]{x1D6FA}}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\,\overline{{\mathcal{U}}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\overline{\unicode[STIX]{x1D6F7}}.\end{eqnarray}$$

The first of these is the helicity of the KAW turbulence, which is in fact the cross-helicity (5.68) by way of (5.40) and integration by parts (cf. S09–§  F.3); the second is the kinetic helicity of the ICW turbulence – the last term in (5.68), dominant when $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ .

While $X_{\text{KAW}}$ or $X_{\text{ICW}}$ being non-zero would indicate an imbalance between counterpropagating KAW-like or ICW-like perturbations, respectively, there is no corollary that such counterpropagating perturbations have energetically decoupled cascades in the way that Elsasser fields do in RMHD. This is because while the conserved quantity $X$ is the difference between the ‘energies’ of the generalised Elsasser fields $\unicode[STIX]{x1D6F7}\pm \unicode[STIX]{x1D6F9}$ and ${\mathcal{U}}\pm ({\mathcal{B}}+\unicode[STIX]{x1D70C}_{\text{H}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6F7}/v_{\text{A}})$ , the sum of these ‘energies’ is not the free energy (5.24) and is not conserved, and neither, therefore, are these ‘energies’ conserved individually. Note also that these fields are not the eigenfunctions associated with the counterpropagating modes: in the $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ limit, those are $v_{\text{A}}\widetilde{{\mathcal{B}}}\mp k_{\bot }\widetilde{\unicode[STIX]{x1D6F9}}$ for $\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{\text{KAW}}$ and $v_{\text{A}}\overline{{\mathcal{U}}}\pm k_{\bot }\overline{\unicode[STIX]{x1D6F7}}$ for $\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}_{\text{ICW}}$ . The energies of these fields are not individually conserved either.

The presence of extra invariants does open the possibility of dual or even triple cascades in Hall-RMHD turbulence: in particular, $X_{\text{KAW}}$ will cascade to larger scales if it is injected at small scales (see S09–§  F.6; Cho & Kim Reference Cho and Kim2016 and references therein), whereas $X_{\text{ICW}}$ is expected to cascade forward, together with the free energy (Chen, Chen & Eyink Reference Chen, Chen and Eyink2003; Banerjee & Galtier Reference Banerjee and Galtier2016). Thus, the Hall-RMHD system can offer some considerable rewards to a devoted turbulence theorist.

5.7 Larmor-scale transition

We are now going to prove that, once the KAW and ICW cascades hit the Larmor scale, the former will be channelled into electron heating (via a sub-Larmor KAW cascade) and the latter into ion heating (via an electrostatic ion-entropy cascade). What follows is formally necessary, as due diligence, but a reader who is not a particular GK aficionada need not read it if she trusts our algebra.Footnote ²¹ Qualitative physics discussion resumes in § 6.

5.7.1 Ordering

We continue to assume the Hall ordering of the temperature ratio versus plasma beta (see (5.1)), but focus on scales that are of the order of the Larmor radius, a regime that does not appear to have been studied before:

(5.70) $$\begin{eqnarray}\frac{Z}{\unicode[STIX]{x1D70F}}\sim \frac{1}{\unicode[STIX]{x1D6FD}_{i}}\gg 1,\quad k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1\quad \Rightarrow \quad k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\sim k_{\bot }\unicode[STIX]{x1D70C}_{i}\sqrt{\frac{Z}{\unicode[STIX]{x1D70F}}}\sim \frac{1}{\sqrt{\unicode[STIX]{x1D6FD}_{i}}}\gg 1.\end{eqnarray}$$

The ordering of the time scales and amplitudes must now be adjusted. How to do this can be deduced a priori from the $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\gg 1$ orderings (5.53)–(5.57) by taking them to the illegitimate extreme $k_{\bot }\unicode[STIX]{x1D70C}_{\text{H}}\sim 1/\sqrt{\unicode[STIX]{x1D6FD}_{i}}$ , or $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D6FD}_{i}^{1/6}$ . Having obtained the orderings, we will then backtrack to the hybrid ion–electron equations of §§ 3.3 and 3.5 and derive a new set of equations valid under our new ordering.

Reverting to our old notation, we convert the $\unicode[STIX]{x1D6FF}$ orderings (5.54)–(5.57) into $\unicode[STIX]{x1D6FD}_{i}$ orderings using $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D6FD}_{i}^{1/6}$ , $\unicode[STIX]{x1D70E}\sim 1$ , $k_{\Vert }\unicode[STIX]{x1D70C}_{\text{H}}\sim k_{\Vert }\unicode[STIX]{x1D70C}_{i}\sqrt{\unicode[STIX]{x1D6FD}_{i}}$ , and

(5.71) $$\begin{eqnarray}\frac{k_{\bot }\unicode[STIX]{x1D6F7}}{v_{\text{A}}}\sim \frac{k_{\bot }\unicode[STIX]{x1D70C}_{i}v_{\text{th}}}{v_{\text{A}}}\,\unicode[STIX]{x1D711}\sim \unicode[STIX]{x1D711}\sqrt{\unicode[STIX]{x1D6FD}_{i}},\quad {\mathcal{U}}\sim \frac{u_{\Vert i}}{v_{\text{th}}}\sqrt{\unicode[STIX]{x1D6FD}_{i}},\quad \frac{k_{\bot }\unicode[STIX]{x1D6F9}}{v_{\text{A}}}\sim k_{\bot }\unicode[STIX]{x1D70C}_{i}{\mathcal{A}}\sim {\mathcal{A}},\quad {\mathcal{B}}\sim \frac{\unicode[STIX]{x1D6FF}B}{B}\end{eqnarray}$$

(recall (3.1) and (5.18)). The resulting ordering is

(5.72) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D711}}\sim \frac{\overline{u}_{\Vert i}}{v_{\text{th}}}\sim \frac{\overline{g}}{F_{0}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FD}_{i}^{-1/3},\quad \overline{{\mathcal{A}}}\sim \frac{\unicode[STIX]{x1D6FF}\overline{B}}{B}\sim \frac{\unicode[STIX]{x1D6FF}\overline{n}}{n}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FD}_{i}^{2/3},\quad \frac{\unicode[STIX]{x1D714}_{\text{ICW}}}{\unicode[STIX]{x1D6FA}}\sim \frac{k_{\Vert \text{ICW}}v_{\text{th}}}{\unicode[STIX]{x1D6FA}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FD}_{i}^{-1/3},\end{eqnarray}$$

(5.73) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D711}}\sim \frac{\widetilde{u}_{\Vert i}}{v_{\text{th}}}\sim \frac{\widetilde{g}}{F_{0}}\sim \widetilde{{\mathcal{A}}}\sim \frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}\sim \frac{\unicode[STIX]{x1D6FF}\widetilde{n}}{n}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FD}_{i}^{1/3},\quad \frac{\unicode[STIX]{x1D714}_{\text{KAW}}}{\unicode[STIX]{x1D6FA}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FD}_{i}^{-2/3},\quad \frac{k_{\Vert \text{KAW}}v_{\text{th}}}{\unicode[STIX]{x1D6FA}}\sim \unicode[STIX]{x1D716}\unicode[STIX]{x1D6FD}_{i}^{1/3},\end{eqnarray}$$

where bars and tildes continue to mean averaged and fluctuating quantities over the KAW time scale. Note that for ICW, the wave frequency and the ion streaming rate have turned out to be the same size, whereas for KAW, the former is much larger than the latter. This is the basic physical reason why ICW will couple into ion kinetics and, eventually, ion heating, while KAW will not.

5.7.2 Field equations

The field equations (3.24)–(3.26) are linear, so can be split cleanly into slow- and fast-varying parts. To lowest order (in all cases, ${\sim}\unicode[STIX]{x1D6FD}_{i}^{-1/3}$ for the slow fields and ${\sim}\unicode[STIX]{x1D6FD}_{i}^{-2/3}$ for the fast ones), they areFootnote ²²

(5.74) $$\begin{eqnarray}\displaystyle & \displaystyle (1-\hat{\unicode[STIX]{x1D6E4}}_{0})\widetilde{\unicode[STIX]{x1D711}}=-\frac{\unicode[STIX]{x1D6FF}\widetilde{n}}{n}+\frac{1}{n_{i}}\int \text{d}^{3}\boldsymbol{v}{\hat{J}}_{0}\widetilde{g}, & \displaystyle\end{eqnarray}$$

(5.75) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\widetilde{u}_{\Vert e}}{v_{\text{t h }}}=\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}\widetilde{{\mathcal{A}}}+\widetilde{{\mathcal{J}}}_{\text{ext}}, & \displaystyle\end{eqnarray}$$

(5.76) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\widetilde{n}}{n}=-\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}, & \displaystyle\end{eqnarray}$$

(5.77) $$\begin{eqnarray}\displaystyle & \displaystyle (1-\hat{\unicode[STIX]{x1D6E4}}_{0})\overline{\unicode[STIX]{x1D711}}=\frac{1}{n_{i}}\int \text{d}^{3}\boldsymbol{v}{\hat{J}}_{0}\overline{g}, & \displaystyle\end{eqnarray}$$

(5.78) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\overline{u}_{\Vert e}}{v_{\text{t h }}}=\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}\overline{{\mathcal{A}}}+\frac{\overline{u}_{\Vert i}}{v_{\text{t h }}}+\overline{{\mathcal{J}}}_{\text{ext}}, & \displaystyle\end{eqnarray}$$

(5.79) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\overline{n}}{n}=-\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}\overline{B}}{B}+(1-\hat{\unicode[STIX]{x1D6E4}}_{1})\overline{\unicode[STIX]{x1D711}}-\frac{1}{n_{i}}\int \text{d}^{3}\boldsymbol{v}\hat{v}_{\bot }^{2}{\hat{J}}_{1}\overline{g}. & \displaystyle\end{eqnarray}$$

The external energy-injecting currents are, in fact, supposed to represent energy arriving from much larger scales. It is then a logical choice to set $\overline{{\mathcal{J}}}_{\text{ext}}=0$ and treat $\widetilde{{\mathcal{J}}}_{\text{ext}}$ as representing the incoming KAW energy (see (5.84)). In a similar vein, we shall, in (5.91), let $\widetilde{a}_{\text{ext}}=0$ and treat $\overline{a}_{\text{ext}}$ as representing the incoming ICW energy.

5.7.3 Electron equations

The treatment of the electron equations (3.10) and (3.11) is completely analogous to the treatment of their counterparts (5.14) and (5.15) in § 5.5.3. We retain terms to two leading orders, $\unicode[STIX]{x1D6FD}_{i}^{-2/3}$ and $\unicode[STIX]{x1D6FD}_{i}^{-1/3}$ :

(5.80) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{{\mathcal{A}}}}{\unicode[STIX]{x2202}t}+\frac{v_{\text{t h }}}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(\overline{\unicode[STIX]{x1D711}}-\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\overline{n}}{n}\right)=\frac{v_{\text{t h }}}{2}\unicode[STIX]{x1D735}_{\Vert }\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\widetilde{n}}{n}+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{{\mathcal{A}}},\end{eqnarray}$$

(5.81) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{\unicode[STIX]{x1D6FF}\widetilde{n}}{n}-\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}\right)+\frac{\unicode[STIX]{x2202}\overline{u}_{\Vert e}}{\unicode[STIX]{x2202}z}=-\unicode[STIX]{x1D735}_{\Vert }\widetilde{u}_{\Vert e},\quad \text{where}\;\unicode[STIX]{x1D735}_{\Vert }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}-\unicode[STIX]{x1D70C}_{i}\{\widetilde{{\mathcal{A}}},\ldots \}.\end{eqnarray}$$

If these are averaged over the KAW time scale and then its average is subtracted from each equation, we obtain, using also (5.75) and (5.76),

(5.82) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{{\mathcal{A}}}}{\unicode[STIX]{x2202}t}=-\frac{v_{\text{t h }}}{\unicode[STIX]{x1D6FD}_{i}}\widetilde{\unicode[STIX]{x1D735}_{\Vert }\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}}+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}_{\bot }^{2}\widetilde{{\mathcal{A}}}, & \displaystyle\end{eqnarray}$$

(5.83) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(1+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\right)\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}=v_{\text{th}}\widetilde{\unicode[STIX]{x1D735}_{\Vert }\left(\frac{1}{\unicode[STIX]{x1D6FD}_{i}}\hat{\unicode[STIX]{x1D6FB}}_{\bot }^{2}\widetilde{{\mathcal{A}}}+\widetilde{{\mathcal{J}}}_{\text{ext}}\right)}. & \displaystyle\end{eqnarray}$$

These are just the KAW equations (5.36) in different notation, but now they are valid at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ , i.e. both above and below the Larmor scale. They are entirely decoupled from ion dynamics and so the KAW energy will cascade right through the Larmor scale and eventually dissipate into electron heat.

To restate the last point in terms of a free-energy budget, the system (5.82)–(5.83) obeys

(5.84) $$\begin{eqnarray}\frac{\text{d}W_{\text{KAW}}}{\text{d}t}+Q_{e}=v_{\text{th}}^{2}\int \frac{\text{d}^{3}\boldsymbol{r}}{V}\frac{\unicode[STIX]{x2202}\widetilde{{\mathcal{A}}}}{\unicode[STIX]{x2202}t}\widetilde{{\mathcal{J}}}_{\text{ext}}=\unicode[STIX]{x1D700}_{\text{KAW}},\end{eqnarray}$$

where $W_{\text{KAW}}$ is given by (5.41) (it is the same as the last two terms of (3.30), after using (5.76)), $Q_{e}$ is given by (3.21) (with ${\mathcal{A}}\rightarrow \widetilde{{\mathcal{A}}}$ ) and $\widetilde{{\mathcal{J}}}_{\text{ext}}$ now represents the KAW cascade from $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ . In steady state,

(5.85) $$\begin{eqnarray}Q_{e}=\unicode[STIX]{x1D700}_{\text{KAW}}.\end{eqnarray}$$

Returning to the averaged versions of (5.80) and (5.81) and retaining only the lowest order, we find

(5.86) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(\overline{\unicode[STIX]{x1D711}}-\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\overline{n}}{n}\right)=0,\quad \frac{\unicode[STIX]{x2202}\overline{u}_{\Vert e}}{\unicode[STIX]{x2202}z}=0.\end{eqnarray}$$

With the aid of (5.78) and (5.79), these are readily seen to be the $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ counterparts of the ICW ‘constitutive relations’ (5.39). When they are written in the form (5.86), their physical meaning becomes particularly transparent: these are statements of Boltzmann (‘adiabatic’) electrons and zero electron current, usually associated with the electrostatic approximation. We shall see in § 5.7.4 that the ion dynamics on ICW time scales are indeed electrostatic.

5.7.4 Ion equations

Finally, we treat the ion GK equation (3.29) in the same manner as we did the electron equations in § 5.7.3. To two lowest orders, it is

(5.87) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\overline{g}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{t h }}}{2}\{\langle \overline{\unicode[STIX]{x1D711}}\rangle _{\boldsymbol{R}},\overline{g}\}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\!\left(\widetilde{g}-\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}F_{0}\right)+v_{\Vert }\!\left\langle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\!\left(\overline{g}+\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\overline{n}}{n}F_{0}\right)+\unicode[STIX]{x1D735}_{\Vert }\frac{Z}{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D6FF}\widetilde{n}}{n}F_{0}\right\rangle _{\boldsymbol{R}}\nonumber\\ \displaystyle & & \displaystyle \quad =C[\overline{g}+\widetilde{g}+\langle \overline{\unicode[STIX]{x1D711}}+\widetilde{\unicode[STIX]{x1D711}}\rangle _{\boldsymbol{R}}F_{0}]+\frac{2v_{\Vert }\langle \overline{a}_{\text{ext}}+\widetilde{a}_{\text{ext}}\rangle _{\boldsymbol{R}}}{v_{\text{t h }}^{2}}F_{0}.\end{eqnarray}$$

Taking the KAW-time-scale average of (5.87) and then subtracting it from the equation, we get

(5.88) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\widetilde{g}-\hat{v}_{\bot }^{2}{\hat{J}}_{1}\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}F_{0}\right)=v_{\Vert }\left\langle \widetilde{\unicode[STIX]{x1D735}_{\Vert }\frac{2}{\unicode[STIX]{x1D6FD}_{i}}\frac{\unicode[STIX]{x1D6FF}\widetilde{B}}{B}}\right\rangle _{\boldsymbol{R}}F_{0}+C[\widetilde{g}+\langle \widetilde{\unicode[STIX]{x1D711}}\rangle _{\boldsymbol{R}}F_{0}],\end{eqnarray}$$

where have used also (5.76) and set $\widetilde{a}_{\text{ext}}=0$ as promised at the end of § 5.7.2. This is an (irrelevant) imprint of the KAW turbulence on the ion distribution function – the FLR version of (5.38).Footnote ²³ Note that there is no phase mixing here, either parallel or perpendicular, so, in a weakly collisional plasma, these small perturbations of the ion distribution function have no means of accessing the collision operator and thermalising.

Returning to the KAW-time-scale average of (5.87), retaining the lowest-order terms and using the first equation in (5.86), we get

(5.89) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{g}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x1D70C}_{i}v_{\text{th}}}{2}\{\langle \overline{\unicode[STIX]{x1D711}}\rangle _{\boldsymbol{R}},\overline{g}\}+v_{\Vert }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(\overline{g}+\langle \overline{\unicode[STIX]{x1D711}}\rangle _{\boldsymbol{R}}F_{0}\right)=C[\overline{g}+\langle \overline{\unicode[STIX]{x1D711}}\rangle _{\boldsymbol{R}}F_{0}]+\frac{2v_{\Vert }\langle \overline{a}_{\text{ext}}\rangle _{\boldsymbol{R}}}{v_{\text{th}}^{2}}F_{0}.\end{eqnarray}$$

Together with (5.77), this is a closed system – the standard electrostatic GK equation supporting ion hydrodynamics (5.37) at long scales ( $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ )Footnote ²⁴ and the ion-entropy cascade at sub-Larmor scales (see S09–§ 7.10 and Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Plunk, Quataert and Tatsuno2008). There is no coupling to any other dynamics and so the ICW energy arriving from $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ flows into the ion entropy cascade at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ , to become, upon reaching collisional phase-space scales, ion heat.

For the reference of a meticulous reader, the dispersion relation that follows from (5.89) and (5.77) is

(5.90) $$\begin{eqnarray}1+\unicode[STIX]{x1D701}{\mathcal{Z}}(\unicode[STIX]{x1D701})=1-\frac{1}{\unicode[STIX]{x1D6E4}_{0}},\quad \unicode[STIX]{x1D701}=\frac{\unicode[STIX]{x1D714}}{|k_{\Vert }|v_{\text{th}}},\end{eqnarray}$$

where ${\mathcal{Z}}(\unicode[STIX]{x1D701})$ is the plasma dispersion function (Fried & Conte Reference Fried and Conte1961). When $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ and (consequently) $\unicode[STIX]{x1D701}\gg 1$ , the right-hand side of (5.90) is ${\approx}-k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}/2$ and the left-hand side is ${\approx}-1/2\unicode[STIX]{x1D701}^{2}$ (the ‘fluid’ limit). The result is the ICW dispersion relation (5.30). When $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ , we must have $\unicode[STIX]{x1D701}\sim 1$ and the solutions of (5.90) contain heavy Landau damping on the ions – the linear signature of ion heating.

Finally, the free-energy budget of the system (5.89) and (5.77) is

(5.91) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\frac{v_{\text{th}}^{2}}{4}\!\int \!\frac{\text{d}^{3}\boldsymbol{r}}{V}\left[\frac{1}{n_{i}}\!\int \!\text{d}^{3}\boldsymbol{v}\frac{\langle \overline{g}^{2}\rangle _{\boldsymbol{ r}}}{F_{0}}+\overline{\unicode[STIX]{x1D711}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\overline{\unicode[STIX]{x1D711}}\right]+Q_{i}=\int \!\frac{\text{d}^{3}\boldsymbol{r}}{V}\frac{1}{n_{i}}\!\int \!\text{d}^{3}\boldsymbol{v}\,\overline{a}_{\text{ext}}\langle v_{\Vert }g\rangle _{\boldsymbol{r}}=\unicode[STIX]{x1D700}_{\text{ICW}},\end{eqnarray}$$

where $Q_{i}$ is given by (3.20) (with $h\rightarrow \overline{h}$ ) and, as promised in § 5.7.2, $\overline{a}_{\text{ext}}$ now represents the energy flux into the ICW cascade. In the long-wavelength limit $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ , the individually conserved piece of free energy appearing on the left-hand side turns into the ICW free energy (5.42) plus the variance of the passive kinetic field ${\mathcal{G}}$ (see (5.13)). The difference between (5.91) and the analogous low- $\unicode[STIX]{x1D6FD}_{e}$ equation (4.19) is that the ‘kinetic-energy’ term $\overline{\unicode[STIX]{x1D711}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\overline{\unicode[STIX]{x1D711}}$ has now migrated into the ion-heating balance (cf. (4.27) and (3.30)) (removing also the technical complications associated with $Q_{\text{x}}$ ). In steady state, (5.91) tells us that

(5.92) $$\begin{eqnarray}Q_{i}=\unicode[STIX]{x1D700}_{\text{ICW}},\end{eqnarray}$$

restating again that all the ICW energy goes into ion heating.