2 The gyrofluid model

We consider as a starting point a simplified version of the gyrokinetic equation presented in Brizard (Reference Brizard1989, Reference Brizard1992), where background magnetic inhomogeneities and curvature have been neglected. It reads

(2.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}F_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}t}+\frac{c}{B}\left[\text{J}_{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}-\frac{v}{c}\text{J}_{0}\unicode[STIX]{x1D6FF}A_{\Vert }+\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}}{q_{\unicode[STIX]{x1D6FC}}}\frac{2}{k_{\bot }{\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}}}\text{J}_{1}\unicode[STIX]{x1D6FF}B_{z},F_{\unicode[STIX]{x1D6FC}}\right]+v\frac{\unicode[STIX]{x2202}F_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}z}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left(\frac{q_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}c}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\text{J}_{0}\unicode[STIX]{x1D6FF}A_{\Vert }+\frac{q_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\text{J}_{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}+\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}}\frac{2}{k_{\bot }{\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\text{J}_{1}\unicode[STIX]{x1D6FF}B_{z}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.-\,\frac{q_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}B}[\text{J}_{0}\unicode[STIX]{x1D6FF}A_{\Vert },\text{J}_{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}]-\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}}{m_{\unicode[STIX]{x1D6FC}}B}\left[\text{J}_{0}\unicode[STIX]{x1D6FF}A_{\Vert },\frac{2}{k_{\bot }{\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}}}\text{J}_{1}\unicode[STIX]{x1D6FF}B_{z}\right]\right)\frac{\unicode[STIX]{x2202}F_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}v}=0,\end{eqnarray}$$

where $F_{\unicode[STIX]{x1D6FC}}(x,y,z,v,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}},t)$ is the gyrocentre distribution function for the particle species $\unicode[STIX]{x1D6FC}$ and $1/k_{\bot }$ holds for $(-\unicode[STIX]{x1D6E5}_{\bot })^{-1/2}$ . For simplicity, we consider a plasma composed of electrons and one single ionized ion species, permeated by a constant magnetic guide field. The index $\unicode[STIX]{x1D6FC}$ will then take the values $i$ for ions and $e$ for electrons. In particular, one has $q_{i}=e$ and $q_{e}=-e$ to indicate the charge of the ions and of the electrons, respectively. On the other hand, we denote as $m_{i}=M$ and $m_{e}=m$ the ion and electron mass, respectively. The spatial Cartesian coordinates $x,y,z$ span a domain $[0,L_{x}]\times [0,L_{y}]\times [0,L_{z}]$ where we impose periodic boundary conditions, whereas $v\in \mathbb{R},\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}\in \mathbb{R},t\in \mathbb{R}$ with $t\geqslant 0$ and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}\geqslant 0$ . We denote as $\boldsymbol{k}=\boldsymbol{k}_{\bot }+\boldsymbol{k}_{\Vert }$ the wave vector of a generic field in Fourier space, with $\boldsymbol{k}_{\bot }=k_{x}\hat{\boldsymbol{x}}+k_{y}\hat{\boldsymbol{y}}$ and $\boldsymbol{k}_{\Vert }=k_{z}\hat{\boldsymbol{z}}$ indicating the components perpendicular and parallel to the strong background magnetic guide field directed along the $z$ axis. Based on the strong guide field ordering, we assume $k_{z}/k_{\bot }\sim \unicode[STIX]{x1D6FF}B_{z}/B_{0}\sim \unicode[STIX]{x1D716}\ll 1$ , with $k_{\bot }$ the modulus of the perpendicular wave vector, $B_{0}$ the (constant) amplitude of the strong guide field component and $\unicode[STIX]{x1D6FF}B_{z}$ the perturbation along the guide field direction. With such assumptions, the expression for the magnetic field (up to terms of order $\unicode[STIX]{x1D716}^{2}$ ) is given by $\boldsymbol{B}(x,y,z,t)=\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}A_{\Vert }(x,y,z,t)\times \hat{\boldsymbol{z}}+(B_{0}+\unicode[STIX]{x1D6FF}B_{z}(x,y,z,t))\hat{\boldsymbol{z}}$ . The field $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}(x,y,z,t)$ , on the other hand, denotes the electrostatic potential. The coordinate $v$ represents the velocity along the $z$ direction. The expression for the magnetic moment $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}$ is given, to lowest order, by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}=m_{\unicode[STIX]{x1D6FC}}v_{\bot }^{2}/2B_{0}$ , with $v_{\bot }$ indicating the modulus of the perpendicular velocity coordinate. The constant $c$ refers to the speed of light, whereas $\text{J}_{0}$ and $\text{J}_{1}$ are operators that, in Fourier space, correspond to the multiplicative factors $J_{0}(k_{\bot }{\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}})$ and $J_{1}(k_{\bot }{\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}})$ , with $J_{0}$ and $J_{1}$ denoting the zeroth- and first-order Bessel functions of the first kind. Here, ${\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}}=v_{\bot }/\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}}=|q_{\unicode[STIX]{x1D6FC}}|B_{0}/(m_{\unicode[STIX]{x1D6FC}}c)$ are the Larmor radius and the cyclotron frequency associated with the species $\unicode[STIX]{x1D6FC}$ , respectively. Finally, the canonical bracket of two functions $f$ and $g$ is defined as $[f,g]=\unicode[STIX]{x2202}_{x}f\unicode[STIX]{x2202}_{y}g-\unicode[STIX]{x2202}_{y}f\unicode[STIX]{x2202}_{x}g$ .

The distribution function of each gyrocentre species is assumed to be given by the sum

(2.2) $$\begin{eqnarray}\displaystyle F_{\unicode[STIX]{x1D6FC}}(x,y,z,v,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}},t)={{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}(v,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}})+\widetilde{f}_{\unicode[STIX]{x1D6FC}}(x,y,z,v,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}},t), & & \displaystyle\end{eqnarray}$$

of an equilibrium Maxwellian

(2.3) $$\begin{eqnarray}\displaystyle {{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}(v,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}})=n_{0}\left(\frac{m_{\unicode[STIX]{x1D6FC}}}{2\unicode[STIX]{x03C0}T_{\unicode[STIX]{x1D6FC}}}\right)^{3/2}\exp \left(-\frac{m_{\unicode[STIX]{x1D6FC}}v^{2}}{2T_{\unicode[STIX]{x1D6FC}}}-\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}B_{0}}{T_{\unicode[STIX]{x1D6FC}}}\right) & & \displaystyle\end{eqnarray}$$

where $n_{0}$ is the uniform equilibrium density, and a perturbation $\widetilde{f}_{\unicode[STIX]{x1D6FC}}$ so that $\widetilde{f}_{\unicode[STIX]{x1D6FC}}/{{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}=O(\unicode[STIX]{x1D716})$ .

The scalar electromagnetic fields $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}$ , $\unicode[STIX]{x1D6FF}A_{\Vert }$ and $\unicode[STIX]{x1D6FF}B_{z}$ are related to the perturbed distribution functions $\widetilde{f}_{\unicode[STIX]{x1D6FC}}$ by means of Poisson’s equation

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}=-4\unicode[STIX]{x03C0}\mathop{\sum }_{\unicode[STIX]{x1D6FC}=e,i}q_{\unicode[STIX]{x1D6FC}}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,\left(\text{J}_{0}\widetilde{f}_{\unicode[STIX]{x1D6FC}}+(\text{J}_{0}^{2}-1){{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}\frac{q_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}{T_{\unicode[STIX]{x1D6FC}}}+\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}B_{0}}{T_{\unicode[STIX]{x1D6FC}}}\text{J}_{0}\frac{2\text{J}_{1}}{k_{\bot }\unicode[STIX]{x1D70C}_{\bot \unicode[STIX]{x1D6FC}}}{{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x1D6FF}B_{z}}{B_{0}}\right), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and of the parallel and perpendicular components of Ampère’s law

(2.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D6FF}A_{\Vert }=-\frac{4\unicode[STIX]{x03C0}}{c}\mathop{\sum }_{\unicode[STIX]{x1D6FC}=e,i}q_{\unicode[STIX]{x1D6FC}}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,v\text{J}_{0}\widetilde{f}_{\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

and

(2.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D6FF}B_{z}\times \widehat{\boldsymbol{z}} & = & \displaystyle 4\unicode[STIX]{x03C0}\widehat{\boldsymbol{z}}\times \mathop{\sum }_{\unicode[STIX]{x1D6FC}=e,i}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,\left(\vphantom{\left(\frac{2\text{J}_{1}}{k_{\bot }\unicode[STIX]{x1D70C}_{\bot \unicode[STIX]{x1D6FC}}}\right)^{2}}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}\frac{2\text{J}_{1}}{k_{\bot }\unicode[STIX]{x1D70C}_{\bot \unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D735}_{\bot }\widetilde{f}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}\text{J}_{0}\frac{2\text{J}_{1}}{k_{\bot }\unicode[STIX]{x1D70C}_{\bot \unicode[STIX]{x1D6FC}}}{{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D735}_{\bot }\frac{q_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}{T_{\unicode[STIX]{x1D6FC}}}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.+\,\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}^{2}B}{T_{\unicode[STIX]{x1D6FC}}}\left(\frac{2\text{J}_{1}}{k_{\bot }\unicode[STIX]{x1D70C}_{\bot \unicode[STIX]{x1D6FC}}}\right)^{2}{{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D735}_{\bot }\frac{\unicode[STIX]{x1D6FF}B_{z}}{B_{0}}\right).\end{eqnarray}$$

We then introduce the following gyrofluid moments built from the perturbation of the distribution function:

(2.7a,b ) $$\begin{eqnarray}\displaystyle \widetilde{N}_{\unicode[STIX]{x1D6FC}}=\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,\widetilde{f}_{\unicode[STIX]{x1D6FC}},\quad n_{0}\widetilde{U}_{\unicode[STIX]{x1D6FC}}=\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,v\widetilde{f}_{\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

(2.8a,b ) $$\begin{eqnarray}\displaystyle \widetilde{P}_{\Vert _{\unicode[STIX]{x1D6FC}}}=m_{\unicode[STIX]{x1D6FC}}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,v^{2}\widetilde{f}_{\unicode[STIX]{x1D6FC}},\quad \widetilde{P}_{\bot _{\unicode[STIX]{x1D6FC}}}=\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}B_{0}\widetilde{f}_{\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

(2.9a,b ) $$\begin{eqnarray}\displaystyle \widetilde{Q}_{\Vert _{\unicode[STIX]{x1D6FC}}}=m_{\unicode[STIX]{x1D6FC}}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,v^{3}\widetilde{f}_{\unicode[STIX]{x1D6FC}}-3n_{0}T_{\unicode[STIX]{x1D6FC}}\widetilde{U}_{\unicode[STIX]{x1D6FC}},\quad \widetilde{Q}_{\bot _{\unicode[STIX]{x1D6FC}}}=\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}B_{0}v\widetilde{f}_{\unicode[STIX]{x1D6FC}}-n_{0}T_{\unicode[STIX]{x1D6FC}}\widetilde{U}_{\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}\widetilde{R}_{\Vert \bot \unicode[STIX]{x1D6FC}}=m_{\unicode[STIX]{x1D6FC}}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,v^{2}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}B_{0}\widetilde{f}_{\unicode[STIX]{x1D6FC}}-T_{\unicode[STIX]{x1D6FC}}\widetilde{P}_{\Vert _{\unicode[STIX]{x1D6FC}}}-T_{\unicode[STIX]{x1D6FC}}\widetilde{P}_{\bot _{\unicode[STIX]{x1D6FC}}}+T_{\unicode[STIX]{x1D6FC}}^{2}\widetilde{N}_{\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}\widetilde{R}_{\bot \bot \unicode[STIX]{x1D6FC}}=\frac{1}{2}\int \text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}^{2}B_{0}^{2}\widetilde{f}_{\unicode[STIX]{x1D6FC}}-2T_{\unicode[STIX]{x1D6FC}}\widetilde{P}_{\bot _{\unicode[STIX]{x1D6FC}}}+T_{\unicode[STIX]{x1D6FC}}^{2}\widetilde{N}_{\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

with the volume element $\text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}$ defined by $\text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}=(2\unicode[STIX]{x03C0}B_{0}/m_{\unicode[STIX]{x1D6FC}})\text{d}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}\text{d}v$ . In (2.7)–(2.11), $\widetilde{N}_{\unicode[STIX]{x1D6FC}}$ and $\widetilde{U}_{\unicode[STIX]{x1D6FC}}$ indicate the gyrofluid density and parallel velocity, $\widetilde{P}_{\Vert _{\unicode[STIX]{x1D6FC}}}$ and $\widetilde{P}_{\bot _{\unicode[STIX]{x1D6FC}}}$ the parallel and perpendicular pressures, $\widetilde{Q}_{\Vert _{\unicode[STIX]{x1D6FC}}}$ and $\widetilde{Q}_{\bot _{\unicode[STIX]{x1D6FC}}}$ the parallel and perpendicular components of the parallel heat fluxes and $\widetilde{R}_{\Vert \bot \unicode[STIX]{x1D6FC}}$ and $\widetilde{R}_{\bot \bot \unicode[STIX]{x1D6FC}}$ the perpendicular component of the parallel and perpendicular energy-weighted pressure tensors, respectively. Furthermore, $n_{0}$ and $T_{\unicode[STIX]{x1D6FC}}$ are the equilibrium density and temperature (assumed isotropic) of particles of species $\unicode[STIX]{x1D6FC}$ , respectively.

We then define the following non-dimensional quantities (denoted with overbars):

(2.12) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \bar{t}=\unicode[STIX]{x1D6FA}_{i}t,\quad \bar{x}=\frac{x}{\unicode[STIX]{x1D70C}_{s}},\quad \bar{y}=\frac{y}{\unicode[STIX]{x1D70C}_{s}},\quad \bar{z}=\frac{z}{\unicode[STIX]{x1D70C}_{s}},\quad \bar{k}_{\bot }=k_{\bot }\unicode[STIX]{x1D70C}_{s},\\ \displaystyle \bar{k}_{z}=k_{z}\unicode[STIX]{x1D70C}_{s},\quad \unicode[STIX]{x1D6E5}_{s}=\frac{\unicode[STIX]{x1D70C}_{s}^{2}}{2}\unicode[STIX]{x1D6FB}_{\bot }^{2},\\ \displaystyle \bar{N}_{\unicode[STIX]{x1D6FC}}=\frac{\widetilde{N}_{\unicode[STIX]{x1D6FC}}}{n_{0}},\quad \bar{U}_{\unicode[STIX]{x1D6FC}}=\frac{\widetilde{U}_{\unicode[STIX]{x1D6FC}}}{c_{s}},\quad \bar{P}_{\Vert _{\unicode[STIX]{x1D6FC}}}=\frac{\widetilde{P}_{\Vert _{\unicode[STIX]{x1D6FC}}}}{n_{0}T_{\unicode[STIX]{x1D6FC}}},\quad \bar{P}_{\bot _{\unicode[STIX]{x1D6FC}}}=\frac{\widetilde{P}_{\bot _{\unicode[STIX]{x1D6FC}}}}{n_{0}T_{\unicode[STIX]{x1D6FC}}},\\ \displaystyle \bar{Q}_{\Vert _{\unicode[STIX]{x1D6FC}}}=\frac{\widetilde{Q}_{\Vert _{\unicode[STIX]{x1D6FC}}}}{n_{0}T_{\unicode[STIX]{x1D6FC}}c_{s}},\quad \bar{Q}_{\bot _{\unicode[STIX]{x1D6FC}}}=\frac{\widetilde{Q}_{\bot _{\unicode[STIX]{x1D6FC}}}}{n_{0}T_{\unicode[STIX]{x1D6FC}}c_{s}},\quad \bar{R}_{x_{\unicode[STIX]{x1D6FC}}}=m_{\unicode[STIX]{x1D6FC}}\frac{\widetilde{R}_{\Vert \bot \unicode[STIX]{x1D6FC}}}{n_{0}T_{\unicode[STIX]{x1D6FC}}^{2}},\quad \bar{R}_{\bot _{\unicode[STIX]{x1D6FC}}}^{(\bot )}=m_{\unicode[STIX]{x1D6FC}}\frac{\widetilde{R}_{\bot \bot \unicode[STIX]{x1D6FC}}}{n_{0}T_{\unicode[STIX]{x1D6FC}}^{2}},\\ \displaystyle \unicode[STIX]{x1D711}=\frac{e\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}{T_{e}},\quad A_{\Vert }=\frac{\unicode[STIX]{x1D6FF}A_{\Vert }}{B_{0}\unicode[STIX]{x1D70C}_{s}},\quad B_{z}=\frac{\unicode[STIX]{x1D6FF}B_{z}}{B_{0}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

We introduce the non-dimensional parameters:

(2.13a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}=\frac{T_{i}}{T_{e}},\quad \unicode[STIX]{x1D6FF}=\sqrt{\frac{m}{M}},\quad \unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D6FC}}=8\unicode[STIX]{x03C0}\frac{n_{0}T_{\unicode[STIX]{x1D6FC}}}{B_{0}^{2}}, & & \displaystyle\end{eqnarray}$$

and assume the ordering

(2.14) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \bar{x}\sim \bar{y}\sim \bar{z}\sim \bar{k}_{\bot }\sim \unicode[STIX]{x1D6FD}_{e}=O(1),\\ \displaystyle \frac{\bar{k}_{\Vert }}{\bar{k}_{\bot }}\sim \frac{1}{\bar{t}}\sim \bar{N}_{\unicode[STIX]{x1D6FC}}\sim \bar{U}_{\unicode[STIX]{x1D6FC}}\sim \bar{P}_{\Vert _{\unicode[STIX]{x1D6FC}}}\sim \bar{P}_{\bot _{\unicode[STIX]{x1D6FC}}}\sim \bar{Q}_{\Vert _{\unicode[STIX]{x1D6FC}}}\sim \bar{Q}_{\bot _{\unicode[STIX]{x1D6FC}}}\sim \bar{R}_{x_{\unicode[STIX]{x1D6FC}}}\sim \bar{R}_{\bot _{\unicode[STIX]{x1D6FC}}}^{(\bot )}\\ \displaystyle \qquad \sim \unicode[STIX]{x1D711}\sim A_{\Vert }\sim B_{z}=O(\unicode[STIX]{x1D716})\ll 1,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $c_{s}=\sqrt{T_{e}/M}$ is the sound speed and $\unicode[STIX]{x1D70C}_{s}=c_{s}/\unicode[STIX]{x1D6FA}_{i}$ the sonic Larmor radius. Note that the normalization and the ordering differ from those adopted in Brizard (Reference Brizard1992).

Taking moments of (2.1) leads to the set of normalized evolution equations (where the overbars were removed from the non-dimensional quantities)

(2.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}N_{e}}{\unicode[STIX]{x2202}t}+[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},N_{e}]+\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{e}}-N_{e}]-[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },U_{e}]-[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{e}}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{e}}-N_{e}]+\frac{\unicode[STIX]{x2202}U_{e}}{\unicode[STIX]{x2202}z}=0,\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D6FF}^{2}U_{e}-\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert })+\unicode[STIX]{x1D6FF}^{2}[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},U_{e}]-[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },P_{\Vert _{e}}]-\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },P_{\bot _{e}}-N_{e}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\unicode[STIX]{x1D6FF}^{2}[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},U_{e}]-\unicode[STIX]{x1D6FF}^{2}[B_{z},Q_{\bot _{e}}]-\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{A})[\unicode[STIX]{x1D711},A_{\Vert }]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,(\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{B},\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{A})+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}\unicode[STIX]{x1D6E4}_{1}(\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{B},\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{A}))[B_{z},A_{\Vert }]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(P_{\Vert _{e}}-\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711}+\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z})=0,\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}P_{\Vert _{e}}}{\unicode[STIX]{x2202}t}+[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\Vert _{e}}]+\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{e}}-N_{e}]-[A_{\Vert },Q_{\Vert _{e}}]-3[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },U_{e}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\Vert _{e}}+P_{\bot _{e}}-N_{e}]-\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{e}}-N_{e}]-[B_{z},R_{\Vert \bot _{e}}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(Q_{\Vert _{e}}+3U_{e})=0,\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}P_{\bot _{e}}}{\unicode[STIX]{x2202}t}+[(1+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{e}}]+\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}(2+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{e}}-N_{e}]-[\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },U_{e}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,[A_{\Vert },Q_{\bot _{e}}]-[(2+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},2P_{\bot _{e}}-N_{e}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6E5}_{s}(3+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{e}}-N_{e}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,2[B_{z},R_{\bot \bot e}]+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(U_{e}+Q_{\bot _{e}})=0,\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}N_{i}}{\unicode[STIX]{x2202}t}+[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},N_{i}]+\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{i}}-N_{i}]-[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },U_{i}]+\unicode[STIX]{x1D70F}[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{i}}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{i}}-N_{i}]+\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}z}=0,\end{eqnarray}$$

(2.20) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(U_{i}+\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert }\right)+[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},U_{i}]-\unicode[STIX]{x1D70F}[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },P_{\Vert _{i}}]-\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },P_{\bot _{i}}-N_{i}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},U_{i}]+\unicode[STIX]{x1D70F}[B_{z},Q_{\bot _{i}}]+\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}^{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}^{A})[\unicode[STIX]{x1D711},A_{\Vert }]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D6E4}_{0}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}^{B},\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}^{A})+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}\unicode[STIX]{x1D6E4}_{1}(\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{B},\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}^{A}))[B_{z},A_{\Vert }]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}(\unicode[STIX]{x1D70F}P_{\Vert _{i}}+\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711}+\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z})=0,\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}P_{\Vert _{i}}}{\unicode[STIX]{x2202}t}+[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\Vert _{i}}]+\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{i}}-N_{i}]-[A_{\Vert },Q_{\Vert _{i}}]-3[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },U_{i}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\Vert _{i}}+P_{\bot _{i}}-N_{i}]+\unicode[STIX]{x1D70F}^{2}[\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{i}}-N_{i}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}[B_{z},R_{\Vert \bot i}]+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(Q_{\Vert _{i}}+3U_{i}\right)=0,\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}P_{\bot _{i}}}{\unicode[STIX]{x2202}t}+[(1+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{i}}]+\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E5}_{s}(2+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711},P_{\bot _{i}}-N_{i}]-[\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}A_{\Vert },U_{i}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,[A_{\Vert },Q_{\bot _{i}}]+\unicode[STIX]{x1D70F}[(2+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},2P_{\bot _{i}}-N_{i}]+\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E5}_{s}(3+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},P_{\bot _{i}}-N_{i}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2\unicode[STIX]{x1D70F}[B_{z},R_{\bot \bot i}]+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(U_{i}+Q_{\bot _{i}}\right)=0\end{eqnarray}$$

for the first four electron and ion gyrofluid moments, where terms of order up to $\unicode[STIX]{x1D716}^{2}$ have been retained, while FLR effects have been estimated only up to leading order in the terms involving heat fluxes and energy-weighted pressures. The operators $\unicode[STIX]{x1D6E4}_{0}$ and $\unicode[STIX]{x1D6E4}_{1}$ are defined as $\unicode[STIX]{x1D6E4}_{0}(z,z^{\prime })=I_{0}(zz^{\prime })\exp (z+z^{\prime })$ and $\unicode[STIX]{x1D6E4}_{1}(z,z^{\prime })=I_{1}(zz^{\prime })\exp (z+z^{\prime })$ , where $I_{n}$ is the modified Bessel function of the first kind of order $n$ . Furthermore, the symbols $\unicode[STIX]{x1D6E5}_{s}^{B},\unicode[STIX]{x1D6E5}_{s}^{A}$ and $\unicode[STIX]{x1D6E5}_{s}^{\unicode[STIX]{x1D711}}$ refer to the Laplacian operator $\unicode[STIX]{x1D6E5}_{s}$ acting on $B_{z}$ , $A_{\Vert }$ and $\unicode[STIX]{x1D711}$ , respectively. Equations (2.15)–(2.22) are supplemented by Poisson’s equation

(2.23) $$\begin{eqnarray}\displaystyle \frac{v_{A}^{2}}{c^{2}}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711} & = & \displaystyle \text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}N_{e}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}(P_{\bot _{e}}-N_{e})-(I_{0}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})-1)\text{e}^{2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & & \displaystyle +\,(I_{0}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})+I_{1}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}))\text{e}^{2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z}-\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}N_{i}-\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}(P_{\bot _{i}}-N_{i})\nonumber\\ \displaystyle & & \displaystyle -\,(I_{0}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})\text{e}^{2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}-1)\frac{\unicode[STIX]{x1D711}}{\unicode[STIX]{x1D70F}}-(I_{0}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})+I_{1}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}))\text{e}^{2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z},\end{eqnarray}$$

where $v_{A}=B_{0}/(4\unicode[STIX]{x03C0}Mn_{0})^{1/2}$ is the Alfvén speed at equilibrium, and by the parallel and perpendicular components of Ampère’s law which yield

(2.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{\unicode[STIX]{x1D6FD}_{e}}{2}(\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}U_{e}-\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}U_{i}), & & \displaystyle\end{eqnarray}$$

and

(2.25) $$\begin{eqnarray}\displaystyle B_{z} & = & \displaystyle -\frac{\unicode[STIX]{x1D6FD}_{e}}{2} (\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}P_{\bot _{e}}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}(P_{\bot _{e}}-N_{e})-(I_{0}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})+I_{1}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}))\text{e}^{2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & & \displaystyle +\,2(I_{0}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s})+I_{1}(2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}))\text{e}^{2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}}B_{z}+\unicode[STIX]{x1D70F}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}P_{\bot _{i}}+\unicode[STIX]{x1D70F}^{2}\unicode[STIX]{x1D6E5}_{s}\text{e}^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}(P_{\bot _{i}}-N_{i})\nonumber\\ \displaystyle & & \displaystyle +\,(I_{0}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})+I_{1}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}))\text{e}^{2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}\unicode[STIX]{x1D711}+2\unicode[STIX]{x1D70F}(I_{0}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s})+I_{1}(2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}))\text{e}^{2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}}B_{z} ).\end{eqnarray}$$

Equations (2.23), (2.24) and (2.25), obtained from (2.4), (2.5) and (2.6) respectively, provide relations that permit the fields $\unicode[STIX]{x1D711}$ , $A_{\Vert }$ and $B_{z}$ to be expressed in terms of the eight gyrofluid moments $N_{e}$ , $N_{i}$ , $U_{e}$ , $U_{i}$ , $P_{\Vert _{e}}$ , $P_{\Vert _{i}}$ , $P_{\bot _{e}}$ and $P_{\bot _{i}}$ .

3 Two asymptotic regimes

Because of the complexity of the gyrofluid model discussed above, we shall concentrate on two asymptotic regimes where drastic simplifications take place.

3.1 The cold-ion limit

We here consider the limit of negligible electron inertia ( $\unicode[STIX]{x1D6FF}\ll 1$ ), cold ions ( $\unicode[STIX]{x1D70F}\ll 1$ ) and non-relativistic Alfvén speed ( $v_{A}/c\ll 1$ ). Introducing the constant $b=(\unicode[STIX]{x1D6FD}_{e}/2)(1+\unicode[STIX]{x1D6FD}_{e})^{-1}$ and the parallel gradient operator

(3.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\Vert }=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}-[A_{\Vert },\,\cdot ], & & \displaystyle\end{eqnarray}$$

equations (2.15)–(2.22), (2.23), (2.24) and (2.25) become

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}N_{e}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},N_{e}]-[B_{z},P_{\bot _{e}}]+\unicode[STIX]{x1D735}_{\Vert }U_{e}=0, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D735}_{\Vert }\left(P_{\Vert _{e}}-\unicode[STIX]{x1D711}+B_{z}\right)=0, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}P_{\Vert _{e}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},P_{\Vert _{e}}]-[B_{z},P_{\Vert _{e}}]-[B_{z},P_{\bot _{e}}]+[B_{z},N_{e}]-[B_{z},R_{\Vert \bot _{e}}]+\unicode[STIX]{x1D735}_{\Vert }\left(Q_{\Vert _{e}}+3U_{e}\right)=0, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}P_{\bot _{e}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},P_{\bot _{e}}]-4[B_{z},P_{\bot _{e}}]+2[B_{z},N_{e}]-2[B_{z},R_{\bot \bot e}]+\unicode[STIX]{x1D735}_{\Vert }\left(U_{e}+Q_{\bot _{e}}\right)=0, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}N_{i}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},N_{i}]+\unicode[STIX]{x1D735}_{\Vert }U_{i}=0, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(U_{i}+A_{\Vert }\right)+[\unicode[STIX]{x1D711},U_{i}]+\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D711}=0, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}P_{\Vert _{i}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},P_{\Vert _{i}}]+\unicode[STIX]{x1D735}_{\Vert }\left(Q_{\Vert _{i}}+3U_{i}\right)=0, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}P_{\bot _{i}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},P_{\bot _{i}}]+\unicode[STIX]{x1D735}_{\Vert }\left(U_{i}+Q_{\bot _{i}}\right)=0, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711}=N_{e}-N_{i}, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{\unicode[STIX]{x1D6FD}_{e}}{2}(U_{e}-U_{i}), & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle B_{z}=-bP_{\bot _{e}}. & \displaystyle\end{eqnarray}$$

Provided that closures for heat fluxes and fourth-order moments are given, the system (3.2)–(3.12) yields a model for finite electron-beta plasmas, accounting for parallel magnetic perturbations and valid at scales larger than the electron thermal gyroradius, in the limit of low ion temperature.

3.2 Sub-ion-scale limit

We consider in this section a different situation, where we still have $v_{A}/c\ll 1$ and $2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}=\unicode[STIX]{x1D70C}_{e}^{2}\unicode[STIX]{x1D6FB}_{\bot }^{2}\ll 1$ , with $\unicode[STIX]{x1D70C}_{e}$ indicating the electron thermal gyroradius, but where the scales are now much smaller than the ion thermal radius $\unicode[STIX]{x1D70C}_{i}$ , i.e. $2\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E5}_{s}=\unicode[STIX]{x1D70C}_{i}^{2}\unicode[STIX]{x1D6FB}_{\bot }^{2}\gg 1$ . In order to derive a model in this sub-ion-scale limit, it is convenient to introduce an additional dimensionless parameter $\unicode[STIX]{x1D707}\ll 1$ . We then impose the following subsidiary ordering between the non-dimensional quantities:

(3.13) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\sim \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}=O(1/\unicode[STIX]{x1D707}),\\ \displaystyle \unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D6FD}_{e}=O(1),\\ \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}=O(\unicode[STIX]{x1D716}/\unicode[STIX]{x1D707}),\\ \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\sim U_{\unicode[STIX]{x1D6FC}}\sim Q_{\bot _{\unicode[STIX]{x1D6FC}}}\sim Q_{\Vert _{\unicode[STIX]{x1D6FC}}}=O(\unicode[STIX]{x1D716}),\\ \displaystyle N_{\unicode[STIX]{x1D6FC}}\sim P_{\Vert _{\unicode[STIX]{x1D6FC}}}\sim P_{\bot _{\unicode[STIX]{x1D6FC}}}\sim R_{x_{\unicode[STIX]{x1D6FC}}}\sim R_{\bot _{\unicode[STIX]{x1D6FC}}}^{(\bot )}\sim \unicode[STIX]{x1D711}\sim B_{z}=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}),\\ \displaystyle A_{\Vert }=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{2}).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

As a result of such an ordering and of the assumption $2\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}\ll 1$ , one still obtains from (2.15)–(2.18) the evolution equations (3.2)–(3.5) for the four electron moments, as in § 3.1. On the other hand, in the sub-ion-scale limit, Poisson’s equation (2.23) yields

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}=-\unicode[STIX]{x1D70F}N_{e}-\unicode[STIX]{x1D70F}B_{z}, & & \displaystyle\end{eqnarray}$$

whereas the parallel and perpendicular Ampère’s law (2.24) and (2.25) give

(3.15) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{\unicode[STIX]{x1D6FD}_{e}}{2}U_{e}, & & \displaystyle\end{eqnarray}$$

and

(3.16) $$\begin{eqnarray}\displaystyle B_{z}=b(\unicode[STIX]{x1D711}-P_{\bot _{e}}), & & \displaystyle\end{eqnarray}$$

respectively.

From (3.14) and (3.16) one obtains the relations

(3.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}=-\frac{\unicode[STIX]{x1D70F}}{1+\unicode[STIX]{x1D70F}b}N_{e}+\frac{\unicode[STIX]{x1D70F}b}{1+\unicode[STIX]{x1D70F}b}P_{\bot _{e}},\quad B_{z}=-\frac{\unicode[STIX]{x1D70F}b}{1+\unicode[STIX]{x1D70F}b}N_{e}-\frac{b}{1+\unicode[STIX]{x1D70F}b}P_{\bot _{e}}, & & \displaystyle\end{eqnarray}$$

which, together with (3.15), permit, in this regime, electromagnetic fluctuations to be expressed in terms of gyrofluid moments. Note that evolution equations for the ion moments do not enter this model.

Thus, equations (3.2)–(3.5), together with equations (3.15) and (3.17), yield a set of equations applicable to plasmas with finite beta for both ions and electrons, accounting for parallel magnetic perturbations and valid on scales larger than the electron thermal gyroradius but smaller than the ion thermal gyroradius.

3.3 Energy conservation

The two asymptotic models derived in the previous subsections are not closed because expressions for the electron and ion heat fluxes $Q_{\Vert _{e}}$ , $Q_{\bot _{e}}$ , $Q_{\Vert _{i}}$ , $Q_{\bot _{i}}$ as well as for the electron energy-weighted pressure tensor components $R_{\Vert \bot _{e}}$ and $R_{\bot \bot e}$ , in terms of lower-order moments, are not provided. Because no dissipative effect has been included in the derivation of the models, it is important to verify that, unless dissipation is introduced through closure relations on the heat fluxes and energy-weighted pressure tensors, both systems possess conserved energy functionals. For this purpose, it is convenient to rewrite the two models replacing the pressures in favour of the parallel and perpendicular temperature fluctuations, defined as $T_{\Vert _{\unicode[STIX]{x1D6FC}}}=P_{\Vert _{\unicode[STIX]{x1D6FC}}}-N_{\unicode[STIX]{x1D6FC}}$ and $T_{\bot _{\unicode[STIX]{x1D6FC}}}=P_{\bot _{\unicode[STIX]{x1D6FC}}}-N_{\unicode[STIX]{x1D6FC}}$ , respectively, for $\unicode[STIX]{x1D6FC}=e,i$ .

The cold-ion model then reads

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}N_{e}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},N_{e}]+\unicode[STIX]{x1D735}_{\Vert }U_{e}=0, & \displaystyle\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},A_{\Vert }]-\unicode[STIX]{x1D735}_{\Vert }\left(T_{\Vert _{e}}-bT_{\bot _{e}}+(1-b)N_{e}-\unicode[STIX]{x1D711}\right)=0, & \displaystyle\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}T_{\Vert _{e}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},T_{\Vert _{e}}]+~b[T_{\bot _{e}},T_{\Vert _{e}}]+b[N_{e},T_{\Vert _{e}}] & \displaystyle \nonumber\\ \displaystyle & +b[T_{\bot _{e}},R_{\Vert \bot _{e}}]+b[N_{e},R_{\Vert \bot _{e}}]+\unicode[STIX]{x1D735}_{\Vert }\left(Q_{\Vert _{e}}+2U_{e}\right)=0, & \displaystyle\end{eqnarray}$$

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}T_{\bot _{e}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},T_{\bot _{e}}]-2b[T_{\bot _{e}},N_{e}]+2b[T_{\bot _{e}},R_{\bot \bot e}]+2b[N_{e},R_{\bot \bot e}]+\unicode[STIX]{x1D735}_{\Vert }Q_{\bot _{e}}=0, & \displaystyle\end{eqnarray}$$

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}N_{i}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},N_{i}]+\unicode[STIX]{x1D735}_{\Vert }U_{i}=0, & \displaystyle\end{eqnarray}$$

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(U_{i}+A_{\Vert }\right)+[\unicode[STIX]{x1D711},U_{i}]+\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D711}=0, & \displaystyle\end{eqnarray}$$

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}T_{\Vert _{i}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},T_{\Vert _{i}}]+\unicode[STIX]{x1D735}_{\Vert }\left(Q_{\Vert _{i}}+2U_{i}\right)=0, & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}T_{\bot _{i}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},T_{\bot _{i}}]+\unicode[STIX]{x1D735}_{\Vert }Q_{\bot _{i}}=0, & \displaystyle\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711}=N_{e}-N_{i}, & \displaystyle\end{eqnarray}$$

(3.27) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{b}{1-2b}(U_{e}-U_{i}), & \displaystyle\end{eqnarray}$$

where we also made use of relation (3.12) to replace $B_{z}$ in favour of $P_{\bot _{e}}$ .

If we denote with $D=A_{\Vert }+U_{i}$ the ion parallel canonical momentum, and consider the energy functional

(3.28) $$\begin{eqnarray}\displaystyle H & = & \displaystyle \frac{1}{2}\int \text{d}^{3}x\,\left((1-b)N_{e}^{2}+\frac{T_{\Vert _{e}}^{2}}{2}+(1-b)T_{\bot _{e}}^{2}-2bN_{e}T_{\bot _{e}}-\unicode[STIX]{x1D711}(N_{e}-N_{i})\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,A_{\Vert }(U_{e}-U_{i})+A_{\Vert }^{2}+D^{2}-2A_{\Vert }D\vphantom{\frac{T_{\Vert _{e}}^{2}}{2}}\right)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\int \text{d}^{3}x\,\left(N_{e}^{2}+\frac{T_{\Vert _{e}}^{2}}{2}+T_{\bot _{e}}^{2}-bP_{\bot _{e}}^{2}+|\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}|^{2}+\frac{1-2b}{b}|\unicode[STIX]{x1D735}A_{\Vert }|^{2}+U_{i}^{2}\right),\end{eqnarray}$$

we obtain, from (3.18)–(3.27), the following relation

(3.29) $$\begin{eqnarray}\displaystyle {\dot{H}} & = & \displaystyle \int \text{d}^{3}x\,\left(-T_{\Vert _{e}}\unicode[STIX]{x1D735}_{\Vert }Q_{\Vert _{e}}+bP_{\bot _{e}}\unicode[STIX]{x1D735}_{\Vert }Q_{\bot _{e}}-T_{\bot _{e}}\unicode[STIX]{x1D735}_{\Vert }Q_{\bot _{e}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,(b/2)T_{\Vert _{e}}[P_{\bot _{e}},R_{\Vert \bot _{e}}]-2bT_{\bot _{e}}[P_{\bot _{e}},R_{\bot \bot e}]\right),\end{eqnarray}$$

where a dot indicates a time derivative. It follows in particular that, if the closures on the heat flux and energy-weighted pressure tensor fluctuations are such that the integral on the right-hand side of (3.29) vanishes, the energy functional $H$ is conserved.

We remark that the functional $H$ consists in the sum of terms involving free energy fluctuations (first three terms on the third line of (3.28)), energy due to parallel and perpendicular magnetic perturbations (fourth and sixth term, respectively), energy due to electrostatic fluctuations (fifth term) and parallel ion kinetic energy (seventh term).

The expression for the functional $H$ can be constructed starting from the expression of the conserved energy functional for the gyrokinetic equations

(3.30) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}g_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}t}+\frac{c}{B}\left[\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}-\frac{v}{c}\unicode[STIX]{x1D6FF}A_{\Vert }+\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}}{q_{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6FF}B_{z},g_{\unicode[STIX]{x1D6FC}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,v\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(\widetilde{g}_{\unicode[STIX]{x1D6FC}}+\frac{q_{\unicode[STIX]{x1D6FC}}}{T_{\unicode[STIX]{x1D6FC}}}\left(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}-\frac{v}{c}\unicode[STIX]{x1D6FF}A_{\Vert }+\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}}{q_{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6FF}B_{z}\right){{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}\right)=0,\end{eqnarray}$$

where $\widetilde{g}_{\unicode[STIX]{x1D6FC}}=\widetilde{f}_{\unicode[STIX]{x1D6FC}}+(q_{\unicode[STIX]{x1D6FC}}/T_{\unicode[STIX]{x1D6FC}})(v/c){{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}A_{\Vert }$ , for $\unicode[STIX]{x1D6FC}=e,i$ and where (3.10)–(3.12) have to be used to determine the electromagnetic perturbations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}$ , $\unicode[STIX]{x1D6FF}A_{\Vert }$ and $\unicode[STIX]{x1D6FF}B_{z}$ .

Equation (3.30) is obtained from the original gyrokinetic equation (2.1) by assuming $\widetilde{f}_{\unicode[STIX]{x1D6FC}}/{{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}=O(\unicode[STIX]{x1D716})$ , applying the ordering (2.14) and retaining only terms of order $\unicode[STIX]{x1D716}^{2}$ . The gyroaveraging operators have also been approximated by $\text{J}_{0}\sim 1$ and $\text{J}_{1}\sim k_{\bot }{\unicode[STIX]{x1D70C}_{\bot }}_{\unicode[STIX]{x1D6FC}}/2$ , which can be seen a posteriori to be consistent with the asymptotic limits $\unicode[STIX]{x1D70F}\ll 1$ and $\unicode[STIX]{x1D6FF}\ll 1$ (the model (3.18)–(3.27) could indeed be alternatively derived directly by taking moments of (3.30) and then applying the limit $\unicode[STIX]{x1D70F}\ll 1$ , $\unicode[STIX]{x1D6FF}\ll 1$ to the resulting equations). The system composed by (3.30), (3.10), (3.11) and (3.12) admits the conserved energy functional

(3.31) $$\begin{eqnarray}\displaystyle {\mathcal{H}}=\frac{1}{2}\mathop{\sum }_{\unicode[STIX]{x1D6FC}=e,i}\int \text{d}^{3}x\,\text{d}{\mathcal{W}}_{\unicode[STIX]{x1D6FC}}\left(T_{\unicode[STIX]{x1D6FC}}\frac{\widetilde{g}_{\unicode[STIX]{x1D6FC}}^{2}}{{{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}}+q_{\unicode[STIX]{x1D6FC}}\left(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}-\frac{v}{c}\unicode[STIX]{x1D6FF}A_{\Vert }+\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}}{q_{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6FF}B_{z}\right)\widetilde{g}_{\unicode[STIX]{x1D6FC}}\right). & & \displaystyle\end{eqnarray}$$

One can then suppose that the perturbed functions $\widetilde{g}_{\unicode[STIX]{x1D6FC}}$ are amenable to the following expansion in terms of Hermite–Laguerre polynomials, truncated to account for the dynamical variables present in the fluid model, that is, density, canonical momentum, parallel and perpendicular temperatures:

(3.32) $$\begin{eqnarray}\displaystyle \widetilde{g}_{\unicode[STIX]{x1D6FC}}={{\mathcal{F}}_{eq}}_{\unicode[STIX]{x1D6FC}}\left[\frac{\widetilde{N}_{\unicode[STIX]{x1D6FC}}}{n_{0}}+\left(\frac{\widetilde{U}_{\unicode[STIX]{x1D6FC}}}{v_{t\unicode[STIX]{x1D6FC}}}+\frac{q_{\unicode[STIX]{x1D6FC}}A_{\Vert }}{m_{\unicode[STIX]{x1D6FC}}v_{t\unicode[STIX]{x1D6FC}}c}\right)\frac{v}{v_{t\unicode[STIX]{x1D6FC}}}+\frac{1}{2}\frac{\widetilde{T}_{\Vert _{\unicode[STIX]{x1D6FC}}}}{T_{\unicode[STIX]{x1D6FC}}}\left(\frac{v^{2}}{v_{t\unicode[STIX]{x1D6FC}}^{2}}-1\right)-\frac{\widetilde{T}_{\bot _{\unicode[STIX]{x1D6FC}}}}{T_{\unicode[STIX]{x1D6FC}}}\left(1-\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}B}{T_{\unicode[STIX]{x1D6FC}}}\right)\right], & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $v_{t\unicode[STIX]{x1D6FC}}=\sqrt{T_{\unicode[STIX]{x1D6FC}}/m_{\unicode[STIX]{x1D6FC}}}$ . One can insert expression (3.32) into the functional (3.31) and then take the limit $\unicode[STIX]{x1D70F}\ll 1$ and $\unicode[STIX]{x1D6FF}\ll 1$ . The resulting functional is (3.28). Therefore the origin of the expression for the functional (3.28) is explained by the consistency with the conserved energy functional of the underlying gyrokinetic model. A similar procedure was used to establish energy conservation theorems for gyrofluid models in Scott (Reference Scott2010).

With regard to the second model, valid for the sub-ionic scales, its equations can be rewritten as

(3.33) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}N_{e}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},N_{e}]+\frac{\unicode[STIX]{x1D70F}}{1+\unicode[STIX]{x1D70F}b}[N_{e},T_{\bot _{e}}]+\unicode[STIX]{x1D735}_{\Vert }U_{e}=0,\end{eqnarray}$$

(3.34) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D735}_{\Vert }\left(T_{\Vert _{e}}-\frac{b}{1+\unicode[STIX]{x1D70F}b}T_{\bot _{e}}+\frac{1-b}{1+\unicode[STIX]{x1D70F}b}N_{e}-\unicode[STIX]{x1D711}\right)=0,\end{eqnarray}$$

(3.35) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}T_{\Vert _{e}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},T_{\Vert _{e}}]+\frac{b}{1+\unicode[STIX]{x1D70F}b}[T_{\bot _{e}},T_{\Vert _{e}}]+b\frac{1+\unicode[STIX]{x1D70F}}{1+\unicode[STIX]{x1D70F}b}[N_{e},T_{\Vert _{e}}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{b}{1+\unicode[STIX]{x1D70F}b}[T_{\bot _{e}},R_{\Vert \bot _{e}}]+b\frac{1+\unicode[STIX]{x1D70F}}{1+\unicode[STIX]{x1D70F}b}[N_{e},R_{\Vert \bot _{e}}]+\unicode[STIX]{x1D735}_{\Vert }\left(Q_{\Vert _{e}}+2U_{e}\right)=0,\end{eqnarray}$$

(3.36) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}T_{\bot _{e}}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},T_{\bot _{e}}]-\frac{3\unicode[STIX]{x1D70F}b+2b}{1+\unicode[STIX]{x1D70F}b}[T_{\bot _{e}},N_{e}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{2b}{1+\unicode[STIX]{x1D70F}b}[T_{\bot _{e}},R_{\bot \bot e}]+\frac{2b(1+\unicode[STIX]{x1D70F})}{1+\unicode[STIX]{x1D70F}b}[N_{e},R_{\bot \bot e}]+\unicode[STIX]{x1D735}_{\Vert }Q_{\bot _{e}}=0,\end{eqnarray}$$

(3.37) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}=\unicode[STIX]{x1D70F}\frac{b-1}{1+\unicode[STIX]{x1D70F}b}N_{e}+\frac{\unicode[STIX]{x1D70F}b}{1+\unicode[STIX]{x1D70F}b}T_{\bot _{e}},\end{eqnarray}$$

(3.38) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{b}{1-2b}U_{e},\end{eqnarray}$$

where we made use of (3.17) and of the relation $P_{\bot _{e}}=T_{\bot _{e}}+N_{e}$ to express $B_{z}$ in terms of $N_{e}$ and $T_{\bot _{e}}$ .

By means of a procedure analogous to that adopted for the cold-ion model, one obtains that the energy functional for the system (3.33)–(3.38) reads

(3.39) $$\begin{eqnarray}\displaystyle H=\frac{1}{2}\int \text{d}^{3}x\!\left(\!N_{e}^{2}+\frac{1-2b}{2b}|\unicode[STIX]{x1D735}A_{\Vert }|^{2}+\frac{T_{\Vert _{e}}^{2}}{2}+T_{\bot _{e}}^{2}\vphantom{\frac{b}{1+\unicode[STIX]{x1D70F}b}P_{\bot _{e}}^{2}}-\unicode[STIX]{x1D711}N_{e}-\frac{\unicode[STIX]{x1D70F}b}{1+\unicode[STIX]{x1D70F}b}N_{e}P_{\bot _{e}}-\frac{b}{1+\unicode[STIX]{x1D70F}b}P_{\bot _{e}}^{2}\!\right)\!, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and that its time evolution is governed by

(3.40) $$\begin{eqnarray}\displaystyle {\dot{H}} & = & \displaystyle \int \text{d}^{3}x\,\left(T_{\Vert _{e}}\left(\frac{1}{2}[B_{z},R_{\Vert \bot _{e}}]-\unicode[STIX]{x1D735}_{\Vert }Q_{\Vert _{e}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\left(T_{\bot _{e}}-\frac{\unicode[STIX]{x1D70F}b}{1+\unicode[STIX]{x1D70F}b}N_{e}-\frac{b}{1+\unicode[STIX]{x1D70F}b}P_{\bot _{e}}\right)\left(2[B_{z},R_{\bot \bot e}]-\unicode[STIX]{x1D735}_{\Vert }Q_{\bot _{e}}\right)\right).\end{eqnarray}$$

Also in this case, depending on the closure relation, the functional $H$ can be conserved.

A natural question that arises in connection with the energy conservation properties of the systems concerns the existence of a Hamiltonian structure for such models. Typically, in the absence of dissipative terms, fluid models for plasmas expressed in terms of Eulerian variables possess a non-canonical Hamiltonian structure (see, e.g. Morrison (Reference Morrison1998)). In particular, reduced drift or gyrofluids models in their two-dimensional limit possess non-canonical Poisson brackets of Lie–Poisson type (see, e.g. Hazeltine, Hsu & Morrison (Reference Hazeltine, Hsu and Morrison1987), Thiffeault & Morrison (Reference Thiffeault and Morrison2000), Waelbroeck & Tassi (Reference Waelbroeck and Tassi2012)) and the extension of the Hamiltonian structure to three dimensions, in the limit $k_{z}/k_{\bot }\ll 1$ , can be obtained following the prescription of Tassi et al. (Reference Tassi, Morrison, Grasso and Pegoraro2010). Clearly, the possibility for the existence of a Hamiltonian structure depends on the adopted closures. A systematic identification of all possible closures leading to Hamiltonian models with a Lie–Poisson bracket lies outside the scope of this paper, but we note that, for example, by choosing the closure

(3.41) $$\begin{eqnarray}\displaystyle Q_{\Vert _{e}}=Q_{\bot _{e}}=Q_{\Vert _{i}}=Q_{\bot _{i}}=0, & & \displaystyle\end{eqnarray}$$

(3.42a,b ) $$\begin{eqnarray}\displaystyle R_{\bot \bot e}=0,\quad R_{\Vert \bot _{e}}=-\frac{1}{1+2b}T_{\Vert _{e}}. & & \displaystyle\end{eqnarray}$$

the cold-ion model acquires a Hamiltonian structure consisting of the functional (3.28) as Hamiltonian with the Poisson bracket

(3.43) $$\begin{eqnarray}\displaystyle \{F,G\}=\{F,G\}_{e}-\{F,G\}_{i}, & & \displaystyle\end{eqnarray}$$

where

(3.44) $$\begin{eqnarray}\displaystyle \{F,G\}_{e} & = & \displaystyle \int \text{d}^{3}x\,\left(N_{e}\left([F_{N_{e}},G_{N_{e}}]+2\frac{1+b}{1+2b}[F_{T_{\Vert _{e}}},G_{T_{\Vert _{e}}}]\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\,A_{\Vert }\left([F_{N_{e}},G_{A_{\Vert }}]+[F_{A_{\Vert }},G_{N_{e}}]+2([F_{A_{\Vert }},G_{T_{\Vert _{e}}}]+[F_{T_{\Vert _{e}}},G_{A_{\Vert }}])\right)\nonumber\\ \displaystyle & & \displaystyle +\,T_{\Vert _{e}}\left([F_{N_{e}},G_{T_{\Vert _{e}}}]+[F_{T_{\Vert _{e}}},G_{N_{e}}]+\frac{1+3b}{1+2b}[F_{T_{\Vert _{e}}},G_{T_{\Vert _{e}}}]\right)\nonumber\\ \displaystyle & & \displaystyle +\,T_{\bot _{e}}\left([F_{N_{e}},G_{T_{\bot _{e}}}]+[F_{T_{\bot _{e}}},G_{N_{e}}]-\frac{2b}{1+2b}[F_{T_{\Vert _{e}}},G_{T_{\Vert _{e}}}]+\frac{1+b}{b}[F_{T_{\bot _{e}}},G_{T_{\bot _{e}}}]\right)\nonumber\\ \displaystyle & & \displaystyle \left.+\,F_{N_{e}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{A_{\Vert }}+F_{A_{\Vert }}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{N_{e}}+2F_{A_{\Vert }}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{T_{\Vert _{e}}}+2F_{T_{\Vert _{e}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{A_{\Vert }}\right),\end{eqnarray}$$

and

(3.45) $$\begin{eqnarray}\displaystyle \{F,G\}_{i} & = & \displaystyle \int \text{d}^{3}x\,\left(N_{i}[F_{N_{i}},G_{N_{i}}]\vphantom{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,D([F_{N_{i}},G_{D}]+[F_{D},G_{N_{i}}]+2([F_{D},G_{T_{\Vert _{i}}}]+[F_{T_{\Vert _{i}}},G_{D}]))\nonumber\\ \displaystyle & & \displaystyle +\,T_{\Vert _{i}}([F_{N_{i}},G_{T_{\Vert _{i}}}]+[F_{T_{\Vert _{i}}},G_{N_{i}}]+2[F_{T_{\Vert _{i}}},G_{T_{\Vert _{i}}}])\nonumber\\ \displaystyle & & \displaystyle +\,T_{\bot _{i}}([F_{N_{i}},G_{T_{\bot _{i}}}]+[F_{T_{\bot _{i}}},G_{N_{i}}])\nonumber\\ \displaystyle & & \displaystyle \left.+\,F_{N_{i}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{D}+F_{D}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{N_{i}}+2F_{T_{\Vert _{i}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{D}+2F_{D}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}G_{T_{\Vert _{i}}}\right).\end{eqnarray}$$

Here, the subscripts on functionals indicate functional derivatives. The bracket (3.43) is the direct sum of two Poisson brackets, acting on electron and ion moments, respectively. The coupling in the dynamics, on the other hand, comes from the Hamiltonian (3.28). It can be verified that the two operations (3.44) and (3.45) satisfy bilinearity, antisymmetry as well as Leibniz and Jacobi identities.

3.4 Particle moment formulation

The cold-ion model (3.2)–(3.12) and the sub-ion-scale model consisting of (3.2)–(3.5), together with (3.15) and (3.17), are written in terms of gyrofluid moments. For a more complete physical interpretation, it is however important to determine their expression also in terms of particle moments. This is accomplished in the following §§ 3.4.1 and 3.4.2.

3.4.1 Cold-ion limit

In the limit of cold ions and for scales larger than the electron thermal gyroradius, the transformation between the gyrofluid and the particle moments is given by Brizard (Reference Brizard1992) as

(3.46a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle N_{e}=n_{e}-\widehat{B_{z}},\quad U_{e}=u_{e}, & \displaystyle\end{eqnarray}$$

(3.47a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\Vert _{e}}=p_{\Vert _{e}}-\widehat{B_{z}},\quad P_{\bot _{e}}=p_{\bot _{e}}-2\widehat{B_{z}}, & \displaystyle\end{eqnarray}$$

(3.48a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle N_{i}=n_{i}-\widehat{B_{z}}-\unicode[STIX]{x1D6FB}_{\bot }^{2}\widehat{\unicode[STIX]{x1D711}},\quad U_{i}=u_{i}, & \displaystyle\end{eqnarray}$$

(3.49a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\Vert _{i}}=p_{\Vert _{i}}-\widehat{B_{z}}-\unicode[STIX]{x1D6FB}_{\bot }^{2}\widehat{\unicode[STIX]{x1D711}},\quad P_{\bot _{i}}=p_{\bot _{i}}-2\widehat{B_{z}}-2\unicode[STIX]{x1D6FB}_{\bot }^{2}\widehat{\unicode[STIX]{x1D711}}, & \displaystyle\end{eqnarray}$$

(3.50a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{\Vert _{\unicode[STIX]{x1D6FC}}}=q_{\Vert _{\unicode[STIX]{x1D6FC}}},\quad Q_{\bot _{\unicode[STIX]{x1D6FC}}}=q_{\bot _{\unicode[STIX]{x1D6FC}}}, & \displaystyle\end{eqnarray}$$

(3.51a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle R_{x_{\unicode[STIX]{x1D6FC}}}=r_{x_{\unicode[STIX]{x1D6FC}}}-2\widehat{B_{z}},\quad R_{\bot _{\unicode[STIX]{x1D6FC}}}^{(\bot )}=r_{\bot _{\unicode[STIX]{x1D6FC}}}^{(\bot )}-6\widehat{B_{z}}, & \displaystyle\end{eqnarray}$$

for $\unicode[STIX]{x1D6FC}=e,i$ . In (3.46)–(3.51), the quantities $n_{\unicode[STIX]{x1D6FC}}$ , $u_{\unicode[STIX]{x1D6FC}}$ , $p_{\Vert _{\unicode[STIX]{x1D6FC}}}$ , $p_{\bot _{\unicode[STIX]{x1D6FC}}}$ , $q_{\Vert _{\unicode[STIX]{x1D6FC}}}$ , $q_{\bot _{\unicode[STIX]{x1D6FC}}}$ , $r_{x_{\unicode[STIX]{x1D6FC}}}$ and $r_{\bot _{\unicode[STIX]{x1D6FC}}}^{(\bot )}$ , for $\unicode[STIX]{x1D6FC}=e,i$ , indicate the moments with respect to the perturbed particle distribution function. We indicated with a hat the electromagnetic fluctuations expressed in terms of the particle moments, so that, for instance, $B_{z}(P_{\bot _{e}})=\widehat{B_{z}}(p_{\bot _{e}})$ . Transformations (3.46)–(3.51) are consistent with the asymptotic limits and the assumptions made to derive (3.2)–(3.12).

In terms of the particle moments, the system (3.2)-(3.12) can be written (after suppression of the hats) as

(3.52) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(n_{e}-B_{z})+\unicode[STIX]{x1D735}_{\Vert }u_{e}=0, & \displaystyle\end{eqnarray}$$

(3.53) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}A_{\Vert }}{\text{d}t}-\unicode[STIX]{x1D735}_{\Vert }p_{\Vert _{e}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}z}=0, & \displaystyle\end{eqnarray}$$

(3.54) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\Vert _{e}}-B_{z})-[B_{z},p_{\Vert _{e}}]+[B_{z},n_{e}]-[B_{z},r_{\Vert \bot e}]+\unicode[STIX]{x1D735}_{\Vert }\left(q_{\Vert _{e}}+3u_{e}\right)=0, & \displaystyle\end{eqnarray}$$

(3.55) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\bot _{e}}-2B_{z})+2[B_{z},n_{e}]-2[B_{z},r_{\bot \bot e}]+\unicode[STIX]{x1D735}_{\Vert }\left(u_{e}+q_{\bot _{e}}\right)=0, & \displaystyle\end{eqnarray}$$

(3.56) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(n_{i}-B_{z}-\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711})+\unicode[STIX]{x1D735}_{\Vert }u_{i}=0, & \displaystyle\end{eqnarray}$$

(3.57) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(u_{i}+A_{\Vert }\right)+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}z}=0, & \displaystyle\end{eqnarray}$$

(3.58) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\Vert _{i}}-B_{z}-\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711})+\unicode[STIX]{x1D735}_{\Vert }\left(q_{\Vert _{i}}+3u_{i}\right)=0, & \displaystyle\end{eqnarray}$$

(3.59) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\bot _{i}}-2B_{z}-2\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711})+\unicode[STIX]{x1D735}_{\Vert }\left(u_{i}+q_{\bot _{i}}\right)=0, & \displaystyle\end{eqnarray}$$

(3.60) $$\begin{eqnarray}\displaystyle & \displaystyle n_{e}=n_{i}, & \displaystyle\end{eqnarray}$$

(3.61) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{\unicode[STIX]{x1D6FD}_{e}}{2}(u_{e}-u_{i}), & \displaystyle\end{eqnarray}$$

(3.62) $$\begin{eqnarray}\displaystyle & \displaystyle B_{z}=-\frac{\unicode[STIX]{x1D6FD}_{e}}{2}p_{\bot _{e}}, & \displaystyle\end{eqnarray}$$

where we introduced the operator

(3.63) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+[\unicode[STIX]{x1D711},]. & & \displaystyle\end{eqnarray}$$

We remark that, in terms of the particle perturbed moments, (3.60) and (3.62) take the familiar form of the quasi-neutrality condition and of the balance between magnetic and pressure perturbations (in this case, the latter are restricted to the electron pressure because of the cold-ion limit). In particular, from (3.60), (3.61), (3.52) and (3.56), one gets

(3.64) $$\begin{eqnarray}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711}}{\text{d}t}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=0, & & \displaystyle\end{eqnarray}$$

which corresponds to the usual vorticity equation of reduced magnetohydrodynamics.

By imposing isotropic and isothermal electrons, i.e. $p_{\Vert _{e}}=p_{\bot _{e}}=n_{e}=-(2/\unicode[STIX]{x1D6FD}_{e})B_{z}$ , one obtains, from (3.52), (3.53) and (3.57), the set of equations

(3.65) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}B_{z}}{\text{d}t}-\frac{\unicode[STIX]{x1D6FD}_{e}}{2+\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }\left(\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }+u_{i}\right)=0, & \displaystyle\end{eqnarray}$$

(3.66) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}A_{\Vert }}{\text{d}t}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }B_{z}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}z}=0, & \displaystyle\end{eqnarray}$$

(3.67) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}u_{i}}{\text{d}t}-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }B_{z}=0, & \displaystyle\end{eqnarray}$$

which, together with the vorticity equation (3.64), forms the closed system of Hall reduced MHD (Schekochihin et al. Reference Schekochihin, Cowley, W. Dorland, Howes, Quataert and Tatsuno2009). From such a system, if the evolution equation (3.67) is replaced by $u_{i}=0$ , one recovers the three-field low-ion beta model of Boldyrev et al. (Reference Boldyrev, Horaites, Xia and Perez2013). On the other hand, (3.52) and (3.53) are compatible with (107) and (106) of Boldyrev et al. (Reference Boldyrev, Horaites, Xia and Perez2013), and the model (3.52)–(3.62) generalizes the electron dynamics to account for anisotropic pressure evolution when compared to the models of Boldyrev et al. (Reference Boldyrev, Horaites, Xia and Perez2013). However, the ion dynamics is treated in a different way, for in our case the cold-ion, low ion-beta limit is taken.

3.4.2 Sub-ion limit

In this regime, for the electron moments, the mapping between gyrofluid and particle moments is the same as in § 3.4.1 and is thus given by

(3.68a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle N_{e}=n_{e}-\widehat{B_{z}},\quad U_{e}=u_{e}, & \displaystyle\end{eqnarray}$$

(3.69a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\Vert _{e}}=p_{\Vert _{e}}-\widehat{B_{z}},\quad P_{\bot _{e}}=p_{\bot _{e}}-2\widehat{B_{z}}, & \displaystyle\end{eqnarray}$$

(3.70a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{\Vert _{e}}=q_{\Vert _{e}},\quad Q_{\bot _{e}}=q_{\bot _{e}}, & \displaystyle\end{eqnarray}$$

(3.71a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle R_{x_{e}}=r_{x_{e}}-2\widehat{B_{z}},\quad R_{\bot _{e}}^{(\bot )}=r_{\bot _{e}}^{(\bot )}-6\widehat{B_{z}}, & \displaystyle\end{eqnarray}$$

When transformed in terms of particle moments, the sub-ion model becomes

(3.72) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(n_{e}-B_{z})-[B_{z},p_{\bot _{e}}]+\unicode[STIX]{x1D735}_{\Vert }u_{e}=0, & \displaystyle\end{eqnarray}$$

(3.73) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}A_{\Vert }}{\text{d}t}-\unicode[STIX]{x1D735}_{\Vert }p_{\Vert _{e}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}z}=0, & \displaystyle\end{eqnarray}$$

(3.74) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\Vert _{e}}-B_{z})-[B_{z},p_{\Vert _{e}}]-[B_{z},p_{\bot _{e}}]+[B_{z},n_{e}]-[B_{z},r_{\Vert \bot e}]+\unicode[STIX]{x1D735}_{\Vert }(q_{\Vert _{e}}+3u_{e})=0,\quad & \displaystyle\end{eqnarray}$$

(3.75) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\bot _{e}}-2B_{z})-4[B_{z},p_{\bot _{e}}]+2[B_{z},n_{e}]-2[B_{z},r_{\bot \bot e}]+\unicode[STIX]{x1D735}_{\Vert }(u_{e}+q_{\bot _{e}})=0, & \displaystyle\end{eqnarray}$$

(3.76) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D711}=-\unicode[STIX]{x1D70F}n_{e}, & \displaystyle\end{eqnarray}$$

(3.77) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=\frac{\unicode[STIX]{x1D6FD}_{e}}{2}u_{e}, & \displaystyle\end{eqnarray}$$

(3.78) $$\begin{eqnarray}\displaystyle & \displaystyle B_{z}=-\unicode[STIX]{x1D70F}\frac{\unicode[STIX]{x1D6FD}_{e}}{2}n_{e}-\frac{\unicode[STIX]{x1D6FD}_{e}}{2}p_{\bot _{e}}. & \displaystyle\end{eqnarray}$$

Note that the electromagnetic fluctuations $\unicode[STIX]{x1D711}$ , $A_{\Vert }$ and $B_{z}$ are entirely expressed in terms of electron particle moments and agree with those of Schekochihin et al. (Reference Schekochihin, Cowley, W. Dorland, Howes, Quataert and Tatsuno2009) and Boldyrev et al. (Reference Boldyrev, Horaites, Xia and Perez2013). The evolution equations for the electron moments, on the other hand, remain unchanged with respect to those of the previous section because in both cases we assumed $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{s}\ll 1$ . Furthermore, if instead of (3.74) and (3.75), an isothermal, isotropic electron pressure ( $p_{\Vert _{e}}=p_{\bot _{e}}=n_{e}$ ) is prescribed, the system reduces to

(3.79) $$\begin{eqnarray}\displaystyle & \displaystyle \left(1+(1+\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x1D6FD}_{e}}{2}\right)\frac{\unicode[STIX]{x2202}n_{e}}{\unicode[STIX]{x2202}t}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=0, & \displaystyle\end{eqnarray}$$

(3.80) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}-(1+\unicode[STIX]{x1D70F})\unicode[STIX]{x1D735}_{\Vert }n_{e}=0, & \displaystyle\end{eqnarray}$$

which corresponds, although written with a different normalization, to the model of Boldyrev et al. (Reference Boldyrev, Horaites, Xia and Perez2013) for the sub-proton scales.

We remark that, for the ions, the transformation from gyrofluid to particle moments in the limit $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$ is not invertible and yields, to leading order,

(3.81) $$\begin{eqnarray}\displaystyle & \displaystyle n_{i}=p_{\Vert _{i}}=p_{\bot _{i}}=-\frac{\unicode[STIX]{x1D711}}{\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(3.82) $$\begin{eqnarray}\displaystyle & \displaystyle u_{i}=0 & \displaystyle\end{eqnarray}$$

which corresponds to an isothermal isotropic fluid with a Boltzmann response to the potential fluctuations and with negligible parallel ion velocity. This corresponds to the hypotheses adopted in Boldyrev et al. (Reference Boldyrev, Horaites, Xia and Perez2013).

4 A reduced FLR–Landau fluid model

The starting point is provided by the FLR–Landau fluid model discussed in § 5 of Sulem & Passot (Reference Sulem and Passot2015), considered at the lowest relevant order relative to the ordering defined in (2.14), and written in a non-dimensional form, using the units defined by (2.12). In contrast with the previous section, it is not convenient here to separate everywhere the equilibrium profiles from the fluctuations. We thus keep the original notations of the various variables and indicate, when needed, fluctuations by means of the symbol $\unicode[STIX]{x1D6FF}$ . Furthermore, we use the notation $u_{\Vert \unicode[STIX]{x1D6FC}}$ for the parallel (hydrodynamic) velocity of particles of species $\unicode[STIX]{x1D6FC}$ , which was referred to as $u_{\unicode[STIX]{x1D6FC}}$ in the previous sections.

When neglecting electron inertia, the electron velocity equation leads to

(4.1) $$\begin{eqnarray}\boldsymbol{u}_{\bot e}=\boldsymbol{E}\times \widehat{\boldsymbol{b}}-\frac{1}{n}\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D665}_{e},\end{eqnarray}$$

where $\widehat{\boldsymbol{b}}$ denotes the unit vector in the direction of the ambient field, and $n$ is the density of both the ions and the electrons, due to electric quasi-neutrality.

Within the ordering defined by (2.14), and assuming isotropic equilibrium pressures and scales large compared with the electron gyroradius (which makes electron finite Larmor radius effect negligible) one has

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D665}_{e}=\unicode[STIX]{x1D735}p_{\bot e}+\unicode[STIX]{x1D735}_{\Vert }(p_{\Vert e}-p_{\bot e})\widehat{\boldsymbol{b}},\end{eqnarray}$$

It follows that, to leading order, $(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D665}_{e})_{\bot }=\unicode[STIX]{x1D735}_{\bot }p_{\bot e}$ and $\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D665}_{e})=\unicode[STIX]{x1D735}_{\Vert }p_{\Vert e}$ , where, in the present asymptotics, transverse differential operators are defined with respect to the guide field. Similarly, one can approximate $\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}$ by $\widehat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}$ , which gives

(4.3) $$\begin{eqnarray}\boldsymbol{u}_{\bot e}=\widehat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}(\unicode[STIX]{x1D711}-p_{\bot e}).\end{eqnarray}$$

From the Faraday equation and Ohm’s law ( $\boldsymbol{E}=-\boldsymbol{u}_{e}\times \boldsymbol{B}-(1/n)\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D665}_{e}$ ), one has to leading order

(4.4) $$\begin{eqnarray}\frac{\text{d}B_{z}}{\text{d}t}-[p_{\bot e},B_{z}]-\unicode[STIX]{x1D735}_{\Vert }u_{\Vert e}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}-[p_{\bot e},n]=0,\end{eqnarray}$$

where, when not differently specified, the parallel component of the various vector fields and their $z$ -components can be identified. In order to eliminate $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}$ , it is convenient to use the electron continuity equation

(4.5) $$\begin{eqnarray}\frac{\text{d}n}{\text{d}t}-[p_{\bot e},n]+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}=0.\end{eqnarray}$$

One gets

(4.6) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(n-B_{z})+\unicode[STIX]{x1D735}_{\Vert }u_{\Vert e}-[B_{z},p_{\bot e}]=0.\end{eqnarray}$$

The equation for the electron pressures read

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}p_{\Vert e}+\boldsymbol{u}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}p_{\Vert e}+p_{\Vert e}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}+2p_{\Vert e}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{e}\boldsymbol{\cdot }\widehat{\boldsymbol{b}}+(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D666}_{e})\boldsymbol{ : }\unicode[STIX]{x1D749}=0 & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}p_{\bot e}+\boldsymbol{u}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}p_{\bot e}+2p_{\bot e}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}-p_{\bot e}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{e}\boldsymbol{\cdot }\widehat{\boldsymbol{b}}+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D666}_{e})\boldsymbol{ : }\unicode[STIX]{x1D663}=0, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D749}=\widehat{\boldsymbol{b}}\otimes \widehat{\boldsymbol{b}}$ and $\unicode[STIX]{x1D663}=\unicode[STIX]{x1D644}-\unicode[STIX]{x1D749}$ . To leading order, one can write

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D666}_{e})\,\boldsymbol{ : }\,\unicode[STIX]{x1D749}=\unicode[STIX]{x1D735}_{\Vert }q_{\Vert e}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{S}_{\bot e}^{\Vert } & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{1}{2}}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D666}_{e})\,\boldsymbol{ : }\,\unicode[STIX]{x1D663}=\unicode[STIX]{x1D735}_{\Vert }q_{\bot e}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{S}_{\bot r}^{\bot }. & \displaystyle\end{eqnarray}$$

Here, the non-gyrotropic heat flux contributions, described by the vectors $\boldsymbol{S}_{\bot e}^{\Vert }$ and $\boldsymbol{S}_{\bot r}^{\bot }$ , are estimated for scales much larger than the electron gyroscales by (3.8) and (3.9) of Sulem & Passot (Reference Sulem and Passot2015), in the form

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{S}_{\bot e}^{\Vert }=-\frac{1}{B}\widehat{\boldsymbol{b}}\times (p_{\bot e}\unicode[STIX]{x1D735}T_{\Vert e}+\unicode[STIX]{x1D735}\widetilde{r}_{\Vert \bot e}) & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{S}_{\bot e}^{\bot }=-\frac{2}{B}\widehat{\boldsymbol{b}}\times (p_{\bot _{e}}\unicode[STIX]{x1D735}T_{\bot e}+\unicode[STIX]{x1D735}\widetilde{r}_{\bot \bot e}), & \displaystyle\end{eqnarray}$$

where, at the order of the present analysis, the fluctuations of the magnetic field originating from the local Larmor radius are to be retained. Furthermore, the contribution of the gyrotropic fourth-order cumulant $\widetilde{r}_{\Vert \bot e}$ , not retained in (3.7) of Sulem & Passot (Reference Sulem and Passot2015), turns out to be potentially relevant, for example within the closure model defined by (3.24)–(3.27) and (3.17)–(3.19) of the above reference, where $\widetilde{r}_{\Vert \bot e}$ scales like $M/m$ in the units of that paper. Differently, within this framework, $\widetilde{r}_{\bot \bot e}$ is negligible.

Combining with the equation for $B_{z}$ , one gets

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\Vert e}-B_{z})-[B_{z},p_{\bot e}]-[B_{z},p_{\Vert e}-n+\widetilde{r}_{\Vert \bot e}]+\unicode[STIX]{x1D735}_{\Vert }(3u_{\Vert e}+q_{\Vert e})=0 & \displaystyle\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(p_{\bot e}-2B_{z})-4[B_{z},p_{\bot e}]+2[B_{z},n-\widetilde{r}_{\bot \bot e}]+\unicode[STIX]{x1D735}_{\Vert }(u_{\Vert e}+q_{\bot e})=0. & \displaystyle\end{eqnarray}$$

Ampère law directly gives that, to leading order,

(4.15) $$\begin{eqnarray}u_{\Vert e}=u_{\Vert i}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }.\end{eqnarray}$$

Equating the expressions of the electric field given by Ohm’s law and the Faraday equation (taken in the form $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}-\unicode[STIX]{x2202}_{t}\boldsymbol{A}$ ), leads, for the parallel component, to

(4.16) $$\begin{eqnarray}\frac{\text{d}A_{\Vert }}{\text{d}t}-\unicode[STIX]{x1D735}_{\Vert }p_{\Vert e}+\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D711}=0,\end{eqnarray}$$

and for the perpendicular one to

(4.17) $$\begin{eqnarray}-\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D711}=\widehat{\boldsymbol{z}}\times \left(\boldsymbol{u}_{i}-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}\times \boldsymbol{B}\right)-\unicode[STIX]{x1D735}_{\bot }p_{\bot e},\end{eqnarray}$$

or when writing $\boldsymbol{u}_{\bot i}=-\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D712}_{c}+\unicode[STIX]{x1D735}_{\bot }\times (\unicode[STIX]{x1D712}_{s}\widehat{\boldsymbol{z}})$ and taking the transverse divergence,

(4.18) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\bot }^{2}\left(\unicode[STIX]{x1D711}+\unicode[STIX]{x1D712}_{s}-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}B_{z}-p_{\bot e}\right)=0.\end{eqnarray}$$

Note that the quantities $\unicode[STIX]{x1D711}$ , $B_{z}$ and $A_{\Vert }$ determine the full electromagnetic field, as the condition $\boldsymbol{B}=\unicode[STIX]{x1D735}\times \boldsymbol{A}$ , together with the gauge condition $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{A}=0$ gives $\unicode[STIX]{x1D6FB}_{\bot }^{2}\boldsymbol{A}_{\bot }=\widehat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}B_{z}-\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D735}A_{\Vert }$ .

One now turns to the equation for the ion velocity where the electric field has been expressed using Ohm’s law. It reads

(4.19) $$\begin{eqnarray}\frac{D^{(i)}}{Dt}\boldsymbol{u}_{i}+\frac{1}{n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D665}^{(G)}+\unicode[STIX]{x1D72B})-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\frac{1}{n}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}=0,\end{eqnarray}$$

where $D^{(i)}/Dt=\unicode[STIX]{x2202}_{t}+\boldsymbol{u}_{i}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ , while $\unicode[STIX]{x1D665}^{(G)}$ denotes the total gyrotropic pressure tensor and $\unicode[STIX]{x1D72B}$ the non-gyrotropic contribution to the ion pressure. One writes $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(-{\mathcal{A}}\unicode[STIX]{x1D663}+{\mathcal{B}}\boldsymbol{\unicode[STIX]{x1D716}}\boldsymbol{\cdot }\boldsymbol{b}+\boldsymbol{b}\otimes \unicode[STIX]{x1D72B}_{\Vert }+\unicode[STIX]{x1D72B}_{\Vert }\otimes \boldsymbol{b})$ , where $\unicode[STIX]{x1D663}=\unicode[STIX]{x1D644}-\boldsymbol{b}\otimes \boldsymbol{b}$ . The scalar quantities ${\mathcal{A}}$ and ${\mathcal{B}}$ , together with the vector $\unicode[STIX]{x1D72B}_{\Vert }$ , are defined by (5.7)–(5.9) of Sulem & Passot (Reference Sulem and Passot2015).

The equation for $u_{\Vert i}$ is obtained by estimating $\unicode[STIX]{x1D72B}_{\Vert }$ from (5.9) of Sulem & Passot (Reference Sulem and Passot2015), which, to leading order, gives

(4.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}u_{\Vert i}+[u_{\Vert i},\unicode[STIX]{x1D712}_{s}]+\unicode[STIX]{x1D735}_{\Vert }(\unicode[STIX]{x1D70F}p_{\Vert i}+p_{\Vert e})+\unicode[STIX]{x1D70F}\widehat{\mathfrak{C}}_{2}\left(\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D735}_{\Vert }p_{\Vert e}+\unicode[STIX]{x1D735}_{\Vert }(\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{E})\right)=0, & & \displaystyle\end{eqnarray}$$

where it is sufficient to write $(\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{E})=-\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711}$ . Furthermore, $\widehat{\mathfrak{C}}_{2}$ is an operator which in Fourier space reduces to the multiplication by the factor

(4.21) $$\begin{eqnarray}\mathfrak{C}_{2}=\frac{1}{b_{i}}(1-\unicode[STIX]{x1D6E4}_{0}(b_{i})).\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6E4}_{n}(b_{i})=\exp (-b_{i})\text{I}_{n}(b_{i})$ where $\text{I}_{n}$ denotes the modified Bessel function of order $n$ . The variable $b_{i}$ , which is defined in the physical variables by $b_{i}=(k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2})/2$ (where $\unicode[STIX]{x1D70C}_{i}$ holds for the ion thermal gyroradius) reduces in the present units to $b_{i}=\unicode[STIX]{x1D70F}k_{\bot }^{2}$ . Furthermore, in the limit $b_{i}\rightarrow 0$ , $\mathfrak{C}_{2}\rightarrow 1$ while for $b\rightarrow \infty$ , $\mathfrak{C}_{2}=1/b_{i}+O(1/b_{i}^{3/2})$ .

Considering the transverse component of (4.19) and taking the transverse divergence, one gets to leading order the transverse pressure balance equation

(4.22) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\bot }^{2}\left(\unicode[STIX]{x1D70F}p_{\bot i}+p_{\bot e}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}B_{z}-{\mathcal{A}}\right)=0,\end{eqnarray}$$

where, from (5.7) of Sulem & Passot (Reference Sulem and Passot2015), ${\mathcal{A}}=\unicode[STIX]{x1D70F}(\widehat{\mathfrak{A}}_{1}\unicode[STIX]{x1D714}_{\Vert i}+\widehat{\mathfrak{A}}_{2}T_{\bot i})$ , with $\unicode[STIX]{x1D714}_{\Vert i}=-\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D712}_{s}$ denoting the parallel ion vorticity. The operators $\widehat{\mathfrak{A}}_{1}$ and $\widehat{\mathfrak{A}}_{2}$ reduce in Fourier space to the multiplicative factors

(4.23) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{A}_{1}=1-\frac{\unicode[STIX]{x1D6E4}_{1}(b_{i})}{b_{i}[\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})]}+\frac{\unicode[STIX]{x1D6E4}_{1}(b_{i})}{\unicode[STIX]{x1D6E4}_{0}(b_{i})} & \displaystyle\end{eqnarray}$$

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{A}_{2}=-\frac{\unicode[STIX]{x1D6E4}_{1}(b_{i})}{\unicode[STIX]{x1D6E4}_{0}(b_{i})}. & \displaystyle\end{eqnarray}$$

For $b_{i}\rightarrow 0$ , $\mathfrak{A}_{1}\rightarrow 1/2$ and $\mathfrak{A}_{2}\rightarrow 0$ , while for $b_{i}\rightarrow \infty$ , $\mathfrak{A}_{1}\sim 1/b_{i}$ and $\mathfrak{A}_{2}\rightarrow -1$ .

Equation (4.22) rewrites

(4.25) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\bot }^{2}\left[\frac{2}{\unicode[STIX]{x1D6FD}_{e}}B_{z}+\unicode[STIX]{x1D70F}n+p_{\bot e}-\unicode[STIX]{x1D70F}\widehat{\mathfrak{A}}_{1}\unicode[STIX]{x1D714}_{\Vert i}+\unicode[STIX]{x1D70F}(1-\widehat{\mathfrak{A}}_{2})T_{\bot i}\right]=0.\end{eqnarray}$$

Combining this equation with (4.18) also gives

(4.26) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\bot }^{2}[\unicode[STIX]{x1D70F}p_{\bot i}+\unicode[STIX]{x1D711}+\unicode[STIX]{x1D712}_{s}-\unicode[STIX]{x1D70F}\widehat{\mathfrak{A}}_{1}\unicode[STIX]{x1D714}_{\Vert i}-\unicode[STIX]{x1D70F}\widehat{\mathfrak{A}}_{2}T_{\bot i}]=0.\end{eqnarray}$$

The equation for the parallel ion vorticity reads

(4.27) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{D^{(i)}\unicode[STIX]{x1D714}_{\Vert i}}{Dt}-(\unicode[STIX]{x1D714}\boldsymbol{\cdot }\unicode[STIX]{x1D735})u_{\Vert i}+(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i})\unicode[STIX]{x1D714}_{\Vert i}+\widehat{\boldsymbol{z}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \left[\frac{1}{n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D665}_{i}+\unicode[STIX]{x1D665}_{e})\right]-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\widehat{\boldsymbol{z}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \left[\frac{1}{n}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}\right]=0.\end{eqnarray}$$

The ion continuity equation shows that $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}$ is second-order in magnitude, making $(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i})\unicode[STIX]{x1D714}_{\Vert i}$ negligible and indicating that, to leading order, $\boldsymbol{u}_{\bot i}=-\widehat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D712}_{s}$ , and thus $D^{(i)}/\text{d}t=\unicode[STIX]{x2202}_{t}+[\cdot ,\unicode[STIX]{x1D712}_{s}]$ . The ion vorticity equation rewrites

(4.28) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D714}_{\Vert i}+[\unicode[STIX]{x1D714}_{\Vert i},\unicode[STIX]{x1D712}_{s}]-\left[n,\unicode[STIX]{x1D70F}p_{\bot i}+p_{\bot e}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}B_{z}\right]+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }-\unicode[STIX]{x1D6FB}_{\bot }^{2}{\mathcal{B}}=0,\end{eqnarray}$$

where, from (5.8) of Sulem & Passot (Reference Sulem and Passot2015),

(4.29) $$\begin{eqnarray}{\mathcal{B}}=\unicode[STIX]{x1D70F}\left[\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}_{i}-\widehat{\mathfrak{B}}_{1}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{E}_{H})\right]\end{eqnarray}$$

with

(4.30) $$\begin{eqnarray}\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}_{i}=\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}_{\bot i}+[A_{\Vert },u_{\Vert i}].\end{eqnarray}$$

and

(4.31) $$\begin{eqnarray}\boldsymbol{E}_{H}=\frac{1}{n}\left(\frac{2}{\unicode[STIX]{x1D6FD}_{e}}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D665}_{e}\right)\end{eqnarray}$$

or

(4.32) $$\begin{eqnarray}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{E}_{H})=-\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert }+\left[n,p_{\bot e}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}B_{z}\right].\end{eqnarray}$$

Furthermore,

(4.33) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{B}_{1}=[\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})]\left(\frac{\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})}{1-\unicode[STIX]{x1D6E4}_{0}(b_{i})}+2\right)-\frac{1}{b_{i}} & \displaystyle\end{eqnarray}$$

(4.34) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{B}_{3}=-\mathfrak{B}_{1}-\frac{1}{b_{i}}+\frac{\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})}{1-\unicode[STIX]{x1D6E4}_{0}(b_{i})}. & \displaystyle\end{eqnarray}$$

For $b_{i}\rightarrow 0$ , $\mathfrak{B}_{1}\rightarrow -1/4$ and $\mathfrak{B}_{3}\rightarrow -1/2$ , while for $b_{i}\rightarrow \infty$ , $\mathfrak{B}_{1}\sim -1/b_{i}$ and $\mathfrak{B}_{3}\sim -1/(2\sqrt{2\unicode[STIX]{x03C0}}b_{i}^{3/2})$ .

At this stage, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}$ is eliminated by using the continuity equation for the ions

(4.35) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}_{\bot i}=-\unicode[STIX]{x2202}_{t}n-[n,\unicode[STIX]{x1D712}_{s}]-\unicode[STIX]{x2202}_{z}u_{\Vert i}.\end{eqnarray}$$

Equation (4.28) becomes

(4.36) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D714}_{\Vert i}+\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}n)+[\unicode[STIX]{x1D714}_{\Vert i},\unicode[STIX]{x1D712}_{s}]+\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}([n,\unicode[STIX]{x1D712}_{s}]+\unicode[STIX]{x2202}_{z}u_{\Vert i})\nonumber\\ \displaystyle & & \displaystyle \qquad -~\unicode[STIX]{x1D70F}[n,p_{\bot i}]-(1-\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{1}\unicode[STIX]{x1D6FB}_{\bot }^{2})\left[n,p_{\bot e}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}B_{z}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +~\frac{2}{\unicode[STIX]{x1D6FD}_{e}}(1-\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{1}\unicode[STIX]{x1D6FB}_{\bot }^{2})\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }-\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}[A_{\Vert },u_{\Vert i}]=0.\end{eqnarray}$$

It is also of interest to write the equation for the ion pressures. Neglecting terms that are subdominant in the present ordering in the case of an isotropic equilibrium pressure, the equations for the parallel and perpendicular ion pressure write

(4.37) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{D^{(i)}}{Dt}p_{\Vert i}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}+2\widehat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i}\boldsymbol{\cdot }\widehat{\boldsymbol{b}}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{\unicode[STIX]{x1D64E}}\boldsymbol{ : }\,\unicode[STIX]{x1D749}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\Vert i}\widehat{\boldsymbol{b}})=0, & \displaystyle\end{eqnarray}$$

(4.38) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{D^{(i)}}{Dt}p_{\bot i}+2\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}-\widehat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i}\boldsymbol{\cdot }\widehat{\boldsymbol{b}}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{\unicode[STIX]{x1D64E}}\boldsymbol{ : }\,\unicode[STIX]{x1D663}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\bot i}\widehat{\boldsymbol{b}}+\boldsymbol{S}_{\bot i}^{\bot })=0, & \displaystyle\end{eqnarray}$$

where the superscript $\unicode[STIX]{x1D64E}$ indicates that the tensor is symmetrized. In (4.37) we omitted the $\unicode[STIX]{x1D735}\boldsymbol{S}_{\bot i}^{\Vert }$ term which leads to purely nonlinear contributions that turn out to be negligible in both asymptotic regimes of cold ions and sub-ion scales. The last term in (4.38) reduces to

(4.39) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{S}_{\bot i}^{\bot }=\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FB}_{\bot }^{2}\left(\widehat{\mathfrak{E}_{4}}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{E}_{H})+\widehat{\mathfrak{E}_{6}}\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}_{i}\right),\end{eqnarray}$$

where

(4.40) $$\begin{eqnarray}\displaystyle \mathfrak{E}_{4} & = & \displaystyle [\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})]\left(\frac{1}{b_{i}}-\frac{\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})}{1-\unicode[STIX]{x1D6E4}_{0}(b_{i})}\right)\nonumber\\ \displaystyle & & \displaystyle +\,2(\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i}))-\frac{1}{b_{i}}\unicode[STIX]{x1D6E4}_{1}(b_{i})\end{eqnarray}$$

(4.41) $$\begin{eqnarray}\displaystyle \mathfrak{E}_{6} & = & \displaystyle \mathfrak{E}_{4}+\frac{1}{b_{i}}-\frac{\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})}{1-\unicode[STIX]{x1D6E4}_{0}(b_{i})}.\end{eqnarray}$$

For $b_{i}\rightarrow 0$ , $\mathfrak{E}_{4}\rightarrow 9/4$ and $\mathfrak{E}_{6}\rightarrow 3$ , while for $b_{i}\rightarrow \infty$ , $\mathfrak{E}_{4}\sim 160/(128b_{i}^{2}\sqrt{2\unicode[STIX]{x03C0}b_{i}})$ and $\mathfrak{E}_{6}\sim 1/b_{i}$ . Furthermore, in the present asymptotics, one has for the quantities associated with the work of the non-gyrotropic pressure forces

(4.42) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D663}=2\unicode[STIX]{x1D72B}_{\bot }\boldsymbol{ : }\unicode[STIX]{x1D735}_{\bot }\boldsymbol{u}_{\bot i} & \displaystyle\end{eqnarray}$$

(4.43) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D749}=2\unicode[STIX]{x1D72B}_{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }u_{\Vert i}. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D72B}_{\bot }=(\unicode[STIX]{x1D735}_{\bot }\otimes \unicode[STIX]{x1D735}_{\bot }-(\widehat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot })\otimes (\widehat{\boldsymbol{z}}\times \unicode[STIX]{x1D735}_{\bot }))(-\unicode[STIX]{x1D6FB}_{\bot }^{2})^{-1}{\mathcal{A}}$ and $\unicode[STIX]{x1D6F1}_{\Vert }=-\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}{\mathcal{D}}_{2}$ , with ${\mathcal{D}}_{2}$ given, up to normalization, by (5.11) of Sulem & Passot (Reference Sulem and Passot2015), where the last term is negligible.

The ion temperatures thus obey

(4.44) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}T_{\Vert i}+[T_{\Vert i},\unicode[STIX]{x1D712}_{s}]+2\unicode[STIX]{x1D735}_{\Vert }u_{\Vert i}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D749}+\unicode[STIX]{x1D735}_{\Vert }q_{\Vert i}=0,\end{eqnarray}$$

(4.45) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}T_{\bot i}+[T_{\bot i},\unicode[STIX]{x1D712}_{s}]+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D663}\nonumber\\ \displaystyle & & \displaystyle \qquad +~\unicode[STIX]{x1D735}_{\Vert }q_{\bot i}-\widehat{b_{i}\mathfrak{E}_{4}}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{E}_{H})+(1-\widehat{{b_{i}\mathfrak{E}}_{6}})\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}_{i}=0,\end{eqnarray}$$

where, in the latter equation, one can use (4.30) and (4.32). In this equation, $\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}_{\bot i}$ is again eliminated using the ion continuity equation, which leads to

(4.46) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}\left(T_{\bot i}-(1-\widehat{{b_{i}\mathfrak{E}}_{6}})n\right)-(1-\widehat{{b_{i}\mathfrak{E}}_{6}})\left([n,\unicode[STIX]{x1D712}_{s}]+\unicode[STIX]{x2202}_{z}u_{\Vert i}\right)+[T_{\bot i},\unicode[STIX]{x1D712}_{s}]+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D663}\nonumber\\ \displaystyle & & \displaystyle \qquad +~\unicode[STIX]{x1D735}_{\Vert }q_{\bot i}-\widehat{b_{i}\mathfrak{E}_{4}}\widehat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{E}_{H})+(1-\widehat{{b_{i}\mathfrak{E}}_{6}})[A_{\Vert },u_{\Vert i}]=0.\end{eqnarray}$$

A closed system (up to the closure of the fluid hierarchy) is thus provided by the dynamical equations (4.6) for $(n-B_{z})$ , (4.13) for $(p_{\Vert e}-B_{z})$ , (4.14) for $(p_{\bot e}-2B_{z})$ , (4.16) for $A_{\Vert }$ , (4.20) for $u_{\Vert i}$ , (4.44) for $T_{\Vert i}$ , (4.46) for $(T_{\bot i}-(1-\widehat{{b_{i}\mathfrak{E}}_{6}})n)$ and (4.36) for $(\unicode[STIX]{x1D714}_{\Vert i}+\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}n)$ (with $\unicode[STIX]{x1D714}_{\Vert i}=-\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D712}_{s}$ ), together with the explicit expressions (4.15) for $u_{\Vert e}$ and (4.18) for $\unicode[STIX]{x1D711}$ , provided $B_{z}$ given by (4.25) can be expressed in terms of the above quantities. For this purpose, this equation is rewritten

(4.47) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(\unicode[STIX]{x1D70F}+2\left(1+\frac{1}{\unicode[STIX]{x1D6FD}_{e}}\right)+\unicode[STIX]{x1D70F}^{2}\widehat{\mathfrak{A}}_{1}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}+\unicode[STIX]{x1D70F}(1-\widehat{\mathfrak{A}}_{2})(1-\widehat{{b_{i}\mathfrak{E}}_{6}})\right)\unicode[STIX]{x1D6FF}B_{z}\nonumber\\ \displaystyle & & \displaystyle \quad =\left(\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D70F}^{2}\widehat{\mathfrak{A}}_{1}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}+\unicode[STIX]{x1D70F}(1-\widehat{\mathfrak{A}}_{2})(1-\widehat{{b_{i}\mathfrak{E}}_{6}})\right)(\unicode[STIX]{x1D6FF}n-\unicode[STIX]{x1D6FF}B_{z})-(\unicode[STIX]{x1D6FF}p_{\bot e}-2\unicode[STIX]{x1D6FF}B_{z})\nonumber\\ \displaystyle & & \displaystyle \qquad -~\unicode[STIX]{x1D70F}\widehat{\mathfrak{A}}_{1}\left(\unicode[STIX]{x1D714}_{\Vert i}+\unicode[STIX]{x1D70F}\widehat{\mathfrak{B}}_{3}\unicode[STIX]{x1D6FB}_{\bot }^{2}n\right)+\unicode[STIX]{x1D70F}(1-\widehat{\mathfrak{A}}_{2})\left(\unicode[STIX]{x1D6FF}T_{\bot i}-(1-\widehat{{b_{i}\mathfrak{E}}_{6}})\unicode[STIX]{x1D6FF}n\right).\end{eqnarray}$$

It follows that $B_{z}$ is well defined, provided

(4.48) $$\begin{eqnarray}\unicode[STIX]{x1D70F}+2\left(1+\frac{1}{\unicode[STIX]{x1D6FD}_{e}}\right)-\unicode[STIX]{x1D70F}b_{i}\mathfrak{A}_{1}\mathfrak{B}_{3}+\unicode[STIX]{x1D70F}(1-\mathfrak{A}_{2})(1-b\mathfrak{E}_{6})\neq 0,\end{eqnarray}$$

be satisfied, a condition which was numerically checked to hold for any transverse wavenumber.

Appropriate fluid closures can then be chosen in order to express heat fluxes in terms of lower-order moments. The simplest assumption is the adiabatic approximation which prescribes zero heat fluxes. A more refined approach is provided by Landau fluid closures (Hammett & Perkins Reference Hammett and Perkins1990; Snyder & Hammett Reference Snyder and Hammett2001; Sulem & Passot Reference Sulem and Passot2015) which are aimed at reproducing the linear kinetic theory, and thus include Landau damping. Note that in § 6, the heat fluxes are directly computed from the linearized kinetic theory in the low-frequency limit. Because it involves operators that are non-local in time, this approach is limited to the linearized problem and thus cannot be viewed as a genuine closure.

5 Asymptotic regimes

The above model strongly simplifies in two asymptotic regimes which are discussed below. In particular, it will be shown that in such regimes, one can retrieve the reduced models derived in §§ 3.4.1 and 3.4.2 from the gyrofluid approach.

5.1 The limit of cold ions

Equations (4.13) and (4.14) are unchanged, while in the limit $\unicode[STIX]{x1D70F}\rightarrow 0$ , (4.25) reduces to $p_{\bot e}=-(2/\unicode[STIX]{x1D6FD}_{e})B_{z}$ . Consequently, (4.6) becomes

(5.1) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(n-B_{z})+\unicode[STIX]{x1D735}_{\Vert }u_{\Vert e}=0,\end{eqnarray}$$

and (4.18) gives $\unicode[STIX]{x1D712}_{s}=-\unicode[STIX]{x1D711}$ . Furthermore, combining (4.20) with (4.16) gives

(5.2) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(u_{\Vert i}+A_{\Vert })+\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D711}=0.\end{eqnarray}$$

Using that $\unicode[STIX]{x1D714}_{\Vert i}=\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711}$ , (4.36) reduces to

(5.3) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711}+\frac{2}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FB}_{\bot }^{2}A_{\Vert }=0.\end{eqnarray}$$

Although subdominant, the ion pressures can be evaluated. We first note that, in the limit $\unicode[STIX]{x1D70F}\rightarrow 0$ , the considered scales are much larger than the ion thermal gyroradius, so one can consider the expression of the non-gyrotropic pressure tensor given by (3.1)–(3.3) of Sulem & Passot (Reference Sulem and Passot2015) and derived for example in Schekochihin et al. (Reference Schekochihin, Cowley, Rincon and Rosin2010). In evaluating the contributions of the work of the non-gyrotropic pressure forces in the parallel and perpendicular pressure equations, it is enough to evaluate $\unicode[STIX]{x1D72B}$ to first order, in the form

(5.4) $$\begin{eqnarray}\unicode[STIX]{x1D72B}=\unicode[STIX]{x1D70F}\left(\widehat{\boldsymbol{b}}\times (\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{\cdot }(\unicode[STIX]{x1D644}+3\unicode[STIX]{x1D749})-(\unicode[STIX]{x1D644}+3\unicode[STIX]{x1D749})\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\times \boldsymbol{b}\right).\end{eqnarray}$$

One easily checks that in this case, $(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D663}$ and $(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\,\unicode[STIX]{x1D749}$ are subdominant and can be neglected.

Combining (4.44) with (5.1), and using (4.15) together with (5.3), one gets

(5.5) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(p_{\Vert i}-B_{z}-\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711})+\unicode[STIX]{x1D735}_{\Vert }(3u_{\Vert i}+q_{\Vert i})=0.\end{eqnarray}$$

Similarly, from (4.46) and (4.6), it follows that

(5.6) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(p_{\bot i}-2B_{z}-2\unicode[STIX]{x1D6FB}_{\bot }^{2}\unicode[STIX]{x1D711})+\unicode[STIX]{x1D735}_{\Vert }(u_{\Vert i}+q_{\bot i})=0.\end{eqnarray}$$

To summarize, in the cold-ion asymptotic regime, the reduced model derived from the FLR–Landau fluid yields the model of § 3.4.1. Indeed, the electron equations (5.1), (4.16), (4.13), (4.14) correspond to (3.52), (3.53), (3.54) and (3.55), respectively. Similarly, the ion equations (5.2), (5.5) and (5.6), correspond to (3.57), (3.58) and (3.59), respectively. The vorticity equation (5.3), on the other hand, together with the quasi-neutrality assumption $n_{e}=n_{i}$ , which is intrinsic to the FLR reduced fluid model, and with the electron continuity equation (5.1), yields (3.56). Finally, (4.15) corresponds to (3.61) and, as stated above, (4.25) reduces to (3.62).

Note that when assuming a polytropic scalar electron pressure, the present cold-ion model reduces to the Hamiltonian four-field model of Tassi et al. (Reference Tassi, Morrison, Grasso and Pegoraro2010) (when neglecting electron inertia), which extends to three dimensions the two-dimensional model of Fitzpatrick & Porcelli (Reference Fitzpatrick and Porcelli2004, Reference Fitzpatrick and Porcelli2007).

5.2 The sub-ion-scale limit

The equations for the electrons remain valid in this regime, and the main issue concerns the ion dynamics. In particular, it will be shown that ion velocities are subdominant and, as a consequence, the perpendicular ion pressure satisfies an isothermal equation of state. In order to address this issue, the additional small parameter $\unicode[STIX]{x1D707}\propto k_{\bot }^{-1}$ , measuring the magnitude of the inverse transverse wavenumber, must be introduced. While the smallness of the parameter $\unicode[STIX]{x1D707}$ is physically limited by the ratio of the electron to ion gyroradii, and thus by the square root of the electron to ion mass ratio, it is technically relevant to take the limit $\unicode[STIX]{x1D707}\rightarrow 0$ where important simplifications take place. In particular, the operators $(1-\widehat{b_{i}\mathfrak{A}}_{1})$ , $(1+\widehat{b_{i}\mathfrak{B}}_{1})$ and $\widehat{b_{i}\mathfrak{E}}_{4}$ vanish asymptotically, while $\widehat{b_{i}\mathfrak{E}}_{6}=1$ and $\widehat{b_{i}\mathfrak{B}}_{3}=-1/(2\sqrt{2\unicode[STIX]{x03C0}}b_{i}^{3/2})$ .

The amplitude of the fields $\unicode[STIX]{x1D6FF}n$ , $\unicode[STIX]{x1D711}$ , $\unicode[STIX]{x1D6FF}B_{z}$ , $\unicode[STIX]{x1D6FF}p_{\bot e}$ and $\unicode[STIX]{x1D6FF}p_{\Vert e}$ are now scaled by the small parameter $\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}$ . The parallel space variable is stretched by a factor $\unicode[STIX]{x1D716}$ , while the time variable is scaled by the factor $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D707}$ , that results from a dominant balance between the linear terms of the electronic equations. This regime is indeed dominated by the dynamics of KAWs whose dispersion relation is $\unicode[STIX]{x1D714}\propto k_{\Vert }k_{\bot }$ .

A first remark is that in the material derivative, the transport operator $[\cdot ,\unicode[STIX]{x1D712}_{s}]=(1/\unicode[STIX]{x1D707}^{2})O(\unicode[STIX]{x1D712}_{s})$ , is negligible compared to $\unicode[STIX]{x2202}_{t}=O(\unicode[STIX]{x1D716}/\unicode[STIX]{x1D707})$ . Indeed, otherwise, the order of magnitude of $\unicode[STIX]{x1D712}_{s}$ would be larger than $\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}$ and thus that of $\boldsymbol{u}_{\bot i}$ larger than $\unicode[STIX]{x1D716}$ , which is unrealistic in the present ordering. This point will be checked a posteriori.

From (4.16), it follows that $A_{\Vert }=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{2})$ , and from (4.15) that $u_{\Vert e}=O(\unicode[STIX]{x1D716})$ , provided the magnitude of $u_{\Vert i}$ does not exceed $\unicode[STIX]{x1D716}$ . From the density equation, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}=O(\unicode[STIX]{x1D716}^{2})$ , and so is the transverse part $\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}_{\bot i}$ .

Equation (4.28) then reduces to

(5.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D714}_{\Vert i}-\unicode[STIX]{x1D70F}[n,p_{\bot i}]-\widehat{1/(2\sqrt{2\unicode[STIX]{x03C0}b_{i}})}(\unicode[STIX]{x1D663}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})=0.\end{eqnarray}$$

The last term is $O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D707})$ . It follows that, provided $\unicode[STIX]{x1D6FF}T_{\bot i}\lesssim {\mathcal{O}}(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{2})$ , one has $\unicode[STIX]{x1D712}_{s}=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{4})$ and $|\boldsymbol{u}_{\bot i}|=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{3})$ , consistent with the previous assumptions.

Let us now turn to the equation for the ion perpendicular pressure. As in this limit, $\unicode[STIX]{x1D72B}_{\bot }$ involves only the temperature contribution, $(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{S}\boldsymbol{ : }\unicode[STIX]{x1D663}=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707})O(\unicode[STIX]{x1D6FF}T_{\bot i})$ and is thus negligible in (4.45) which reduces to

(5.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}T_{\bot i}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}_{\Vert }q_{\bot i}=O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D707}).\end{eqnarray}$$

The latter equation describes the relaxation of the parallel temperature fluctuations under the effect of the heat flux, leading indeed to $\unicode[STIX]{x1D6FF}T_{\bot i}=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{2})$ .

Returning to (4.26), we find that the fluctuations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}$ of $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D6FF}n$ of $n$ satisfy

(5.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}=-\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FF}n,\end{eqnarray}$$

which is (3.76). The term ${\mathcal{A}}$ in the pressure balance equation (4.22) is also subdominant and condition (3.78) is recovered.

The last point concerns (3.77), which requires that the ion parallel velocity be smaller than $\unicode[STIX]{x1D716}$ . In the small-scale limit, the electron pressure gradient terms cancel out in (4.20). Using (5.9), this equation reduces to

(5.10) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}u_{\Vert i}+\unicode[STIX]{x1D735}_{\Vert }T_{\Vert i}=O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D707}^{2}),\end{eqnarray}$$

where the driving term on the right-hand side originates from the subdominant contribution to $\mathfrak{C}_{2}$ .

Furthermore, in the equation for the parallel pressure, one has to evaluate the work of the non-gyrotropic pressure given by $2\unicode[STIX]{x1D72B}_{\Vert }\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }u_{\Vert i}$ where in the small-scale limit $\unicode[STIX]{x1D72B}_{\Vert }=-\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}_{\bot }{\mathcal{D}}_{2}$ . The evaluation of this term requires considering the asymptotic behaviour for $b_{i}\rightarrow \infty$ of the various coefficients constructed with the functions $\unicode[STIX]{x1D6E4}_{0}$ and $\unicode[STIX]{x1D6E4}_{1}$ in the expression of ${\mathcal{D}}_{2}$ . It turns out that ${\mathcal{D}}_{2}$ scales like $\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{2}$ , so $\unicode[STIX]{x1D72B}_{\Vert }=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707})$ and $2\unicode[STIX]{x1D72B}_{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }u_{\Vert i}=O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D707}^{2})O(|u_{\Vert }|)$ . This term is asymptotically small compared to $\unicode[STIX]{x1D735}_{\Vert }u_{\Vert i}$ . The equation for the ion parallel temperature thus reduces to

(5.11) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}T_{\Vert i}+2\unicode[STIX]{x1D735}_{\Vert }u_{\Vert i}+\unicode[STIX]{x2202}_{\Vert }q_{\Vert i}=0.\end{eqnarray}$$

Without specifying the heat flux $q_{\Vert }$ , it is possible to derive magnitude estimates from (5.10) and (5.11). The latter equation can be viewed as describing the relaxation of the parallel temperature fluctuations under the effect of the heat flux, in the presence of a driving originating from the parallel velocity fluctuations. This suggests that the first term in (5.11) cannot exceed the second one, indicating that the magnitude of the ratio $T_{\Vert i}/u_{\Vert i}$ cannot exceed $\unicode[STIX]{x1D707}$ . This makes the temperature contribution negligible in (5.10), which ensures that $u_{\Vert i}=O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{3})$ . Consequently, $T_{\Vert i}\leqslant O(\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}^{4})$ . It follows that both the ion temperature fluctuations and the ion parallel velocity are subdominant and can be neglected in this regime.

Interestingly, the above estimates are consistent with the linear kinetic theory. Indeed, inspection of (B5) and (B7) of Passot & Sulem (Reference Passot and Sulem2007) indicates that, at the level of the linear kinetic theory,

(5.12) $$\begin{eqnarray}\frac{T_{\Vert i}}{u_{\Vert i}}=\frac{1}{\sqrt{2\unicode[STIX]{x1D70F}}}\frac{1-R(\unicode[STIX]{x1D701}_{i})+2\unicode[STIX]{x1D701}_{i}^{2}R(\unicode[STIX]{x1D701}_{i})}{\unicode[STIX]{x1D701}_{i}R(\unicode[STIX]{x1D701}_{i})},\end{eqnarray}$$

where $R$ is the plasma response function related to the plasma dispersion function $Z$ by $R(\unicode[STIX]{x1D701})=1+\unicode[STIX]{x1D701}Z(\unicode[STIX]{x1D701})$ , and $\unicode[STIX]{x1D701}_{i}=(1/\sqrt{2\unicode[STIX]{x1D70F}})(\unicode[STIX]{x1D714}/|k_{z}|)$ . It follows that for large $\unicode[STIX]{x1D701}_{i}$ , $T_{\Vert i}/u_{\Vert i}=O(1/\unicode[STIX]{x1D701}_{i})=O(\unicode[STIX]{x1D707})$ .

Similarly, at the level of the linear kinetic theory, one also has,

(5.13) $$\begin{eqnarray}\frac{q_{\Vert i}}{T_{\Vert i}}=\sqrt{2\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D701}_{i}[1-3R(\unicode[STIX]{x1D701}_{i})+2\unicode[STIX]{x1D701}_{i}^{2}R(\unicode[STIX]{x1D701}_{i})]}{1-R(\unicode[STIX]{x1D701}_{i})+2\unicode[STIX]{x1D701}_{i}^{2}R(\unicode[STIX]{x1D701}_{i})},\end{eqnarray}$$

which behaves like $3/(2\unicode[STIX]{x1D701}_{i})$ when $\unicode[STIX]{x1D701}_{i}\rightarrow \infty$ , indicating that $q_{\Vert i}$ is subdominant in (5.11). The two other terms balance each other, consistent with $T_{\Vert i}/u_{\Vert i}=O(\unicode[STIX]{x1D707})$ .

The complete system of equations of the small-scale gyrofluid model can thus be recovered by the FLR–Landau fluid approach in the limit of small amplitude and large transverse wavenumbers. Indeed, as previously mentioned, the electron equations (4.6), (4.16), (4.13), (4.14) yield equations (3.72), (3.73), (3.74) and (3.75), respectively. Equation (5.9) corresponds to (3.76). Finally, the above arguments leading to an isotropic, isothermal ion fluid with negligible parallel velocity, yield (3.77) and (3.78) from (4.15) and (4.18), respectively.

Note that the order of magnitude estimates for the various fields (and in particular for the ion vorticity and divergence) from the fluid system are consistent with the kinetic low-frequency linear formulae of appendix B in Passot & Sulem (Reference Passot and Sulem2007). The non-trivial cancellations with FLR corrections mentioned above are crucial to obtain these estimates.

6 Validity range of the small-scale gyrofluid model

It is of interest to study the validity of the small-scale model by analysing, in the linear regime, its prediction for the dispersion and damping of kinetic Afvén waves. For this purpose, and in order not to introduce additional uncertainty related to the hierarchy closure, electron heat fluxes are evaluated from the low-frequency kinetic theory, as done in Passot & Sulem (Reference Passot and Sulem2007). Indeed, in the present framework where the electron FLR corrections are neglected, (B 12) and (B 7) of this reference give

(6.1) $$\begin{eqnarray}q_{\Vert e}=\sqrt{\frac{2M}{m}}\frac{\unicode[STIX]{x1D701}_{e}[1-3R(\unicode[STIX]{x1D701}_{e})+2\unicode[STIX]{x1D701}_{e}^{2}R(\unicode[STIX]{x1D701}_{e})]}{1-R(\unicode[STIX]{x1D701}_{e})+2\unicode[STIX]{x1D701}_{e}^{2}R(\unicode[STIX]{x1D701}_{e})}\unicode[STIX]{x1D6FF}T_{\Vert e}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D701}_{e}=(1/\sqrt{2M/m})(\unicode[STIX]{x1D714}/|k_{z}|)$ , where $\unicode[STIX]{x1D714}$ is the complex frequency. Similarly, (B 13) and (B 6) lead to

(6.2) $$\begin{eqnarray}q_{\bot e}=-\sqrt{\frac{2M}{m}}\frac{\unicode[STIX]{x1D701}_{e}R(\unicode[STIX]{x1D701}_{e})}{1-R(\unicode[STIX]{x1D701}_{e})}\unicode[STIX]{x1D6FF}T_{\bot e}.\end{eqnarray}$$

In order to simplify the writing, it is convenient to use the notations $q_{\Vert e}={\mathcal{Q}}_{\Vert }(\unicode[STIX]{x1D701}_{e})\unicode[STIX]{x1D6FF}T_{\Vert e}$ and $q_{\bot e}={\mathcal{Q}}_{\bot }(\unicode[STIX]{x1D701}_{e})\unicode[STIX]{x1D6FF}T_{\bot e}$ . The linear system reads (keeping the same notation for the fields and their Fourier modes)

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}\left(\left(1+\frac{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{e}}{2}\right)\unicode[STIX]{x1D6FF}n+\frac{\unicode[STIX]{x1D6FD}_{e}}{2}\unicode[STIX]{x1D6FF}p_{\bot e}\right)+\frac{2k_{z}k_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{e}}A_{\Vert }=0 & \displaystyle\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}A_{\Vert }+k_{z}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FF}n+\unicode[STIX]{x1D6FF}p_{\Vert e})=0 & \displaystyle\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}\left(\frac{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{e}}{2}\unicode[STIX]{x1D6FF}n+\unicode[STIX]{x1D6FF}p_{\Vert e}+\frac{\unicode[STIX]{x1D6FD}_{e}}{2}\unicode[STIX]{x1D6FF}p_{\bot e}\right)+k_{z}\left(-{\mathcal{Q}}_{\Vert }(\unicode[STIX]{x1D701}_{e})(\unicode[STIX]{x1D6FF}p_{\Vert e}-\unicode[STIX]{x1D6FF}n)+\frac{6k_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{e}}A_{\Vert }\right)=0 & \displaystyle\end{eqnarray}$$

(6.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}\left(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{e}\unicode[STIX]{x1D6FF}n+(1+\unicode[STIX]{x1D6FD}_{e})\unicode[STIX]{x1D6FF}p_{\bot e}\right)+k_{z}\left(-{\mathcal{Q}}(\unicode[STIX]{x1D701}_{e})(\unicode[STIX]{x1D6FF}p_{\bot e}-\unicode[STIX]{x1D6FF}n)+\frac{2k_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{e}}A_{\Vert }\right)=0. & \displaystyle\end{eqnarray}$$

It is easily seen that the parallel wavenumber only enters in the combination $\unicode[STIX]{x1D714}/k_{z}$ .

Equations (6.3)–(6.6) are conveniently solved numerically. In order to test the validity of this gyrofluid model, we also solve for the linear fully kinetic solution using the WHAMP software (Rönnmark Reference Rönnmark1982), and for a low-frequency limit equivalent to the gyrokinetic approach. It consists in substituting the kinetic expressions of the ion and electron density and velocity, as given in Appendix B of Passot & Sulem (Reference Passot and Sulem2007), within the quasi-neutrality condition, the parallel component of the Ampère equation and of its curl (which isolates the contribution of the solenoidal velocities). This leads to a linear system of three equations. The dispersion relation is then obtained by solving for the determinant of a $3\times 3$ matrix $\mathfrak{M}$ whose elements (in the case of isotropic pressures) are given by (Kuznetsov, Passot & Sulem Reference Kuznetsov, Passot and Sulem2012)

(6.7) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{11}=-(\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i}))(R(\unicode[STIX]{x1D701}_{i})-1)+(\unicode[STIX]{x1D6E4}_{0}(b_{e})-\unicode[STIX]{x1D6E4}_{1}(b_{e}))(R(\unicode[STIX]{x1D701}_{e})-1) & \displaystyle\end{eqnarray}$$

(6.8) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{12}=\unicode[STIX]{x1D6E4}_{0}(b_{i})-1+\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D6E4}_{0}(b_{e})-1) & \displaystyle\end{eqnarray}$$

(6.9) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{13}=-[\unicode[STIX]{x1D6E4}_{0}(b_{i})R(\unicode[STIX]{x1D701}_{i})+1-\unicode[STIX]{x1D6E4}_{0}(b_{i})]-\unicode[STIX]{x1D70F}[\unicode[STIX]{x1D6E4}_{0}(b_{e})R(\unicode[STIX]{x1D701}_{e})+1-\unicode[STIX]{x1D6E4}_{0}(b_{e})] & \displaystyle\end{eqnarray}$$

(6.10) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{21}=(\unicode[STIX]{x1D6E4}_{1}(b_{i})-\unicode[STIX]{x1D6E4}_{0}(b_{i}))R(\unicode[STIX]{x1D701}_{i})-(\unicode[STIX]{x1D6E4}_{1}(b_{e})-\unicode[STIX]{x1D6E4}_{0}(b_{e}))R(\unicode[STIX]{x1D701}_{e}) & \displaystyle\end{eqnarray}$$

(6.11) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{22}=-\frac{k_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{e}\unicode[STIX]{x1D701}_{i}^{2}} & \displaystyle\end{eqnarray}$$

(6.12) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{23}=-[\unicode[STIX]{x1D6E4}_{0}(b_{i})R(\unicode[STIX]{x1D701}_{i})+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6E4}_{0}(b_{e})R(\unicode[STIX]{x1D701}_{e})] & \displaystyle\end{eqnarray}$$

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{31}=\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{e}(\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i}))(R(\unicode[STIX]{x1D701}_{i})-1)+\unicode[STIX]{x1D6FD}_{e}(\unicode[STIX]{x1D6E4}_{0}(b_{e})-\unicode[STIX]{x1D6E4}_{1}(b_{e}))(R(\unicode[STIX]{x1D701}_{e})-1)-1-\frac{k_{z}^{2}}{k_{\bot }^{2}} & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(6.14) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{32}=-\frac{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{e}}{2}(\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i})-\unicode[STIX]{x1D6E4}_{0}(b_{e})+\unicode[STIX]{x1D6E4}_{1}(b_{e})) & \displaystyle\end{eqnarray}$$

(6.15) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{M}_{33}=\frac{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{e}}{2}[(\unicode[STIX]{x1D6E4}_{0}(b_{i})-\unicode[STIX]{x1D6E4}_{1}(b_{i}))(R(\unicode[STIX]{x1D701}_{i})-1)-(\unicode[STIX]{x1D6E4}_{0}(b_{e})-\unicode[STIX]{x1D6E4}_{1}(b_{e}))(R(\unicode[STIX]{x1D701}_{e})-1)], & \displaystyle\end{eqnarray}$$

where $b_{\unicode[STIX]{x1D6FC}}=(k_{\bot }^{2}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}^{2})/2$ , with $\unicode[STIX]{x1D6FC}=i$ for the ions and $\unicode[STIX]{x1D6FC}=e$ for the electrons. In the latter approach, two cases can be distinguished, depending on whether the electron FLR effects are retained or neglected (by taking $b_{e}=0$ ). In all the three considered kinetic approaches, electron inertia is implicitly built in, while it is neglected in the small-scale gyrofluid model and thus in (6.3)–(6.6).

In the linear regime, the angle of propagation $\unicode[STIX]{x1D703}$ between the ambient field and the wave vector has to be specified. While it has almost no influence in the low-frequency asymptotic description, it has to be chosen close to $90^{\circ }$ in the Vlasov description in order to avoid resonances. We have taken $\unicode[STIX]{x1D703}=89.9^{\circ }$ in order to prevent any finite angle dependence in the considered range of wavenumbers). We further assume $\unicode[STIX]{x1D6FD}_{e}=\unicode[STIX]{x1D70F}=1$ .

For comparison, we also consider the adiabatic and isothermal closures. In the adiabatic case ( $q_{\Vert e}=q_{\bot e}=0$ ), an analytic solution is found for the frequency in the form

(6.16) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k_{z}^{2}k_{\bot }^{2}}=\frac{4\unicode[STIX]{x1D70F}+12+(6\unicode[STIX]{x1D70F}+10)\unicode[STIX]{x1D6FD}_{e}}{\unicode[STIX]{x1D6FD}_{e}[2+(\unicode[STIX]{x1D70F}+2)\unicode[STIX]{x1D6FD}_{e}]}.\end{eqnarray}$$

This solution contrasts from the isothermal one

(6.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k_{z}^{2}k_{\bot }^{2}}=\frac{2(1+\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D6FD}_{e}[1+(1+\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FD}_{e}/2]},\end{eqnarray}$$

obtained by solving the subsystem of equations for $\unicode[STIX]{x1D6FF}n_{e}$ and $A_{\Vert }$ , when imposing $\unicode[STIX]{x1D6FF}p_{\Vert e}=\unicode[STIX]{x1D6FF}p_{\bot e}=\unicode[STIX]{x1D6FF}n$ . In this regime, following Howes et al. (Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006), an estimate of the Landau damping can be obtained in the form

(6.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FE}}{k_{z}k_{\bot }^{2}}=-\frac{2\unicode[STIX]{x03C0}m}{\unicode[STIX]{x1D6FD}_{e}M}\left(1-\frac{1+(1+\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FD}_{e}}{2[1+(1+\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FD}_{e}/2]^{2}}\right).\end{eqnarray}$$

Figure 1 displays the real and imaginary parts $\text{Re}(\unicode[STIX]{x1D714})/k_{z}$ (a) and $\text{Im}(\unicode[STIX]{x1D714})/k_{z}$ (b) of the complex frequency as a function of $k_{\bot }$ , resulting from the various approximations described above. The solid lines correspond to the fully kinetic system. The low-frequency kinetic description, for which the dispersion relation is given by the condition $\text{det}(\mathfrak{M})=0$ , is considered in two different cases. In the first one (referred to by $+$ symbols), no assumptions are made on $b_{i}$ and $b_{e}$ but for numerical convenience we take $R(\unicode[STIX]{x1D701}_{i})=0$ which is valid for $k_{\bot }$ larger than a few units, as expected for small-scale kinetic Alfvén waves for which $\unicode[STIX]{x1D714}/k_{z}\sim k_{\bot }\gg 1$ . Furthermore, $R(\unicode[STIX]{x1D701}_{e})$ is approximated by a four-pole Padé approximant with the parameters $(N,p)$ defined in table 1 of Hedrick & Leboeuf (Reference Hedrick and Leboeuf1992), are taken equal to $(4,1)$ , for which the plasma response function is globally approximated with a good accuracy. In the second case (referred to by $\ast$ symbols), we concentrate on the small-scale asymptotic regime corresponding in the present notations to $b_{e}=0$ and $b_{i}=\infty$ (here taken equal to $10^{9}$ ). In this setting, the functions $R(\unicode[STIX]{x1D701}_{i})$ and $R(\unicode[STIX]{x1D701}_{e})$ are both approximated by the above four-pole Padé approximant. The reduced model given by (6.1)–(6.6) is indicated by $\diamond$ symbols. Finally, the upper (lower) dashed lines correspond to the adiabatic (isothermal) solutions, the dashed line of figure 1(b) corresponding to the isothermal analytic damping rate, as given by (6.18).

Figure 1.

Real part (a) and imaginary part (b) of $\unicode[STIX]{x1D714}/k_{z}$ versus $k_{\bot }$ (in units of $\unicode[STIX]{x1D70C}_{s}^{-1}$ ) for KAWs, in the conditions of the small-scale model, within various descriptions: adiabatic and isothermal regimes (upper and lower dashed lines respectively, in a) together with the corresponding analytic damping rate in the isothermal case (dashed in the b), gyrofluid model (diamonds), full kinetic theory assuming $\unicode[STIX]{x1D703}=89.9^{\circ }$ (solid line), low-frequency kinetic theory including ions and electrons FLR corrections (crosses) and without electron FLR contributions (stars).

It is to be noticed that for the considered large angle of propagation, the low-frequency limit with ions and electrons FLR corrections ( $+$ symbols) displays almost no deviation with respect to the fully kinetic solution up to $k_{\bot }=30$ (in $\unicode[STIX]{x1D70C}_{s}^{-1}$ units), confirming the validity of the small-frequency asymptotics. When neglecting electron FLR correction in the low-frequency kinetic theory ( $\ast$ symbols), some inaccuracy starts to be visible on the real frequency (but not on the damping rate) by $k_{\bot }=15$ , but remains moderate for all the considered scales. Differently, the gyrofluid ( $\diamond$ symbols) precisely predicts the real frequency and the damping rate at moderately small scales but significantly departs from the kinetic theory when $k_{\bot }>15$ , suggesting the importance of electron inertia (neglected in the present theory) for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gtrapprox 20$ . When comparing with the real frequency predicted by the conservative closures, we note that, while the adiabatic approximation is inaccurate at all the scales, assuming isothermal electron provides a rather good approximation of the real frequency up to the electron gyroscale.