2.2.1. The CE distribution functions

For a classical, collisional plasma – i.e. a plasma where the mean free path $\lambda _s$ of particles of species $s$ satisfies $\lambda _s/L \ll 1$ for all $s$, $L$ being the length scale over which the macroscopic properties of the plasma vary – a formal procedure exists for deriving a closed system of fluid equations from a kinetic description of the plasma. This procedure is the CE expansion, which gives distribution functions that are close to, but not exactly, Maxwellian. We call them CE distribution functions. The non-Maxwellian components of the CE distribution functions of particle species $s$ are proportional to $\lambda _s/L$, and must be present in order to support gradients of $n_s$, $\boldsymbol {V}_s$ and $T_s$ on ${O}(L)$ length scales, because (2.6b–e) are all zero for a Maxwellian plasma.

We consider a collisional electron–ion plasma (in which, by definition, $\mu _e \equiv m_e/m_i \ll 1$) with the property that all constituent particle species are strongly magnetised by the macroscopically varying magnetic field $\boldsymbol {B}$: that is, the Larmor radius $\rho _s \equiv m_s v_{\mathrm {th}s} c/|Z_s| e |\boldsymbol {B}|$ satisfies $\rho _s \ll \lambda _{s}$ both for the ions and for the electrons (here, $v_{\mathrm {th}s} \equiv \sqrt {2 T_s/m_s}$ is the thermal speed of species $s$). Equivalently, a strongly magnetised plasma is one in which the Larmor frequency $\varOmega _s \equiv e |Z_s|/m_s c$ satisfies $\varOmega _s \tau _s \gg 1$, where $\tau _s$ is the collision time of species $s$. In such a plasma, the macroscopic variation of the fluid moments is locally anisotropic with respect to $\boldsymbol {B}$; $L$ is the typical length scale of variation in the direction locally parallel to $\boldsymbol {B}$. It can then be shown that, to first order of the CE expansion in $\lambda _{s}/L \ll 1$, and to zeroth order in $\rho _s/\lambda _{s} \ll 1$, the CE distribution functions of the electrons and ions are

(2.8a)\begin{align} f_{e}(\tilde{v}_{e\|},\tilde{v}_{e\bot}) & = \frac{n_{e}}{v_{\mathrm{th}e}^3 {\rm \pi}^{3/2}} \exp \left(-\tilde{v}_{e}^2\right) \nonumber\\ &\quad \times \left\{1+\left[\eta_e^{T} A_e^T(\tilde{v}_e) + \eta_e^{R} A_e^R(\tilde{v}_e) + \eta_e^{u} A_e^u(\tilde{v}_e) \right] \tilde{v}_{e\|} + \epsilon_e C_e(\tilde{v}_e) \left(\tilde{v}_{e\|}^2- \frac{\tilde{v}_{e\bot}^2}{2}\right)\right\}, \end{align}

(2.8b)\begin{align} f_{i}(\tilde{v}_{i\|},\tilde{v}_{i\bot}) & = \frac{n_{i}}{v_{\mathrm{th}i}^3 {\rm \pi}^{3/2}} \exp \left(-\tilde{v}_{i}^2\right) \nonumber\\ & \quad \times \left\{1+\eta_i A_i(\tilde{v}_i) \tilde{v}_{i\|} + \epsilon_i C_i(\tilde{v}_i) \left(\tilde{v}_{i\|}^2- \frac{\tilde{v}_{i\bot}^2}{2} \right)\right\} . \end{align}

Let us define the various symbols employed in (2.8), before discussing the origin of these expressions and their significance for formulating fluid equations (see § 2.2.2).

The particle velocity $\boldsymbol {v}$ (with the corresponding speed $v = |\boldsymbol {v}|$) is split into components parallel and perpendicular to the macroscopic magnetic field $\boldsymbol {B} = B \hat {\boldsymbol {z}}$ as $\boldsymbol {v} = v_{\|} \hat {\boldsymbol {z}} + \boldsymbol {v}_{\bot }$, and the perpendicular plane is in turn characterised by two vectors $\hat {\boldsymbol {x}}$ and $\hat {\boldsymbol {y}}$ chosen so that $\{\hat {\boldsymbol {x}},\hat {\boldsymbol {y}},\hat {\boldsymbol {z}}\}$ is an orthonormal basis. The perpendicular velocity is related to these basis vectors by the gyrophase angle $\phi$:

(2.9)\begin{equation} \boldsymbol{v}_{\bot} = v_{\bot} \left(\cos\phi \, \hat{\boldsymbol{x}} - \sin\phi \, \hat{\boldsymbol{y}} \right) . \end{equation}

The non-dimensionalised peculiar velocity $\tilde {\boldsymbol {v}}_s$ in the rest frame of the ion fluid is defined by $\tilde {\boldsymbol {v}}_s \equiv (\boldsymbol {v}-\boldsymbol {V}_i)/v_{\mathrm {th}s}$, $\tilde {v}_s \equiv |\tilde {\boldsymbol {v}}_s|$, $\tilde {v}_{s\|} \equiv \hat {\boldsymbol {z}} \boldsymbol {\cdot } \tilde {\boldsymbol {v}}_s$ and $\tilde {v}_{s\bot } \equiv |\tilde {\boldsymbol {v}}_s-\tilde {v}_{s\|} \hat {\boldsymbol {z}}|$. The number densities satisfy the quasi-neutrality condition

(2.10)\begin{equation} Z_i n_i = n_e , \end{equation}

where we have utilised $Z_{e} = -1$. We emphasise that $n_{s}$, $\{\hat {\boldsymbol {x}},\hat {\boldsymbol {y}},\hat {\boldsymbol {z}}\}$ and $v_{\mathrm {th}s}$ all vary over length scales $L$ in the plasma, but not on shorter scales (at least not in the direction locally parallel to $\boldsymbol {B}$). The functions $A_e^T(\tilde {v}_e)$, $A_e^R(\tilde {v}_e)$, $A_e^u(\tilde {v}_e)$, $C_e(\tilde {v}_e)$, $A_i(\tilde {v}_i)$ and $C_i(\tilde {v}_i)$ are isotropic functions whose precise forms depend only on the collision operator assumed in the original kinetic equation. Their magnitude is ${O}(1)$ when $\tilde {v}_e \sim 1$ or $\tilde {v}_i \sim 1$, for electrons and ions, respectively. Finally, the parameters $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$ are defined as follows:

(2.11a)$$\begin{align} \eta_{e}^T &= \lambda_{e} \boldsymbol{\nabla}_{\|} \log T_e = \mathrm{sgn}(\boldsymbol{\nabla}_{\|} \log T_e) \frac{\lambda_{e}}{L_T} , \end{align}$$

(2.11b)$$\begin{align}\eta_{e}^R &= \lambda_{e} \frac{R_{e\|}}{p_e} , \end{align}$$

(2.11c)$$\begin{align}\eta_{e}^u &= \lambda_{e} \frac{m_e u_{ei\|}}{T_e \tau_{e}} , \end{align}$$

(2.11d)$$\begin{align}\eta_{i} &= \lambda_{i} \boldsymbol{\nabla}_{\|} \log T_i = \mathrm{sgn}(\boldsymbol{\nabla}_{\|} \log T_i) \frac{\lambda_{i}}{L_{T_i}}, \end{align}$$

(2.11e)$$\begin{align}\epsilon_{e} &= \frac{\lambda_{e}}{v_{\mathrm{th}e}} {\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) \!\boldsymbol{:}\! \boldsymbol{\mathsf{W}}_e} = \mathrm{sgn}\left[\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_e\right] \frac{V_e}{v_{\mathrm{th}e}} \frac{\lambda_e}{L_{V_e}} , \end{align}$$

(2.11f)$$\begin{align}\epsilon_{i} &= \frac{\lambda_{i}}{v_{\mathrm{th}i}} {\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_i} = \mathrm{sgn}\left[\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_i\right]\frac{V_i}{v_{\mathrm{th}i}} \frac{\lambda_i}{L_{V}} , \end{align}$$

where $\lambda _{e}$ is the electron mean free path, $\lambda _{i}$ the ion mean free path, $\tau _e$ the electron collision time, $R_{e\|} \equiv \hat {\boldsymbol {z}} \boldsymbol {\cdot } \boldsymbol {R}_{e}$ the parallel electron-friction force, $\boldsymbol {u}_{ei} \equiv \boldsymbol {V}_e - \boldsymbol {V}_i$ the relative electron–ion velocity, $u_{ei\|} \equiv \hat {\boldsymbol {z}} \boldsymbol {\cdot } \boldsymbol {u}_{ei}$,

(2.12)\begin{equation} \boldsymbol{\mathsf{W}}_s = \boldsymbol{\nabla} \boldsymbol{V}_s + \left(\boldsymbol{\nabla} \boldsymbol{V}_s\right)^{{\rm T}} -\tfrac{2}{3} \left(\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{V}_s\right) \boldsymbol{\mathsf{I}} \end{equation}

the traceless rate-of-strain tensor of species $s$, $V_e$ ($V_i$) the bulk electron-(ion-)fluid speed and

(2.13a)$$\begin{align} L_T &\equiv \left|\boldsymbol{\nabla}_{\|} \log T_e \right|^{{-}1} , \end{align}$$

(2.13b)$$\begin{align}L_{T_i} &\equiv \left|\boldsymbol{\nabla}_{\|} \log T_i \right|^{{-}1} , \end{align}$$

(2.13c)$$\begin{align}L_{V_e} &\equiv V_e \left|\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \tfrac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_e \right|^{{-}1} , \end{align}$$

(2.13d)$$\begin{align}L_V &\equiv V_i \left|\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \tfrac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_i \right|^{{-}1} , \end{align}$$

are, respectively, the electron and ion temperature and the electron- and ion-flow length scales parallel to the background magnetic field. The mean free paths are formally defined for a two-species plasma by

(2.14a)$$\begin{gather} \lambda_{e} \equiv {v_{\mathrm{th}e}}{\tau_{e}} , \end{gather}$$

(2.14b)$$\begin{gather}\lambda_{i} \equiv {v_{\mathrm{th}i}}{\tau_{i}} , \end{gather}$$

and the collision times $\tau _e$ and $\tau _i$ are given in terms of macroscopic plasma parameters by

(2.15a)$$\begin{gather} \tau_{e} \equiv \frac{3 m_e^{1/2} T_e^{3/2}}{4 \sqrt{2 {\rm \pi}} Z_i^2 e^4 n_i \log{\varLambda_{\mathrm{CL}}}} , \end{gather}$$

(2.15b)$$\begin{gather}\tau_{i} \equiv \frac{3 m_i^{1/2} T_i^{3/2}}{4 \sqrt{2 {\rm \pi}} Z_i^4 e^4 n_i \log{\varLambda_{\mathrm{CL}}}} , \end{gather}$$

where $\log {\varLambda _{\mathrm {CL}}}$ is the Coulomb logarithm (Braginskii Reference Braginskii1965).Footnote ¹ In a collisional plasma, $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$ are assumed small. We note that all these parameters can be either positive or negative, depending on the orientation of temperature and velocity gradients.

It is clear from their definitions that each of the non-Maxwellian terms associated with the parameters $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$ is linked to a different macroscopic physical quantity. Thus, $\eta _e^T$ and $\eta _i$ are proportional to the electron- and ion-temperature gradients, respectively; we will therefore refer to the associated non-Maxwellian terms as the CE electron-temperature-gradient term and the CE ion-temperature-gradient term. We refer to the non-Maxwellian term proportional to $\eta _e^R$ as the CE electron-friction term, to the non-Maxwellian term proportional to $\eta _e^u$ as the CE electron–ion-drift term, and the non-Maxwellian terms proportional to $\epsilon _e$ and $\epsilon _i$ as the CE electron-shear term and the CE ion-shear term. The friction and electron–ion-drift terms appear in the electron CE distribution function but not the ion CE distribution function because of our choice to define all velocities in the ion-fluid rest frame. We note that, while macroscopic variation of both the electron and ion densities is a priori allowed in our plasma, the derivation of the CE distribution functions reveals that such variation does not directly give rise to a non-Maxwellian perturbation to the CE distribution function (see Appendix B.1).

The derivation of the CE distribution functions (2.8) for a two-species strongly magnetised plasma undergoing sonic motions (that is, $V_i \sim v_{\mathrm {th}i}$) from the kinetic equation (2.1) was first completed by Braginskii (Reference Braginskii1965) for arbitrary values of $\rho _s/\lambda _{s}$. We do not reproduce the full derivation in the main text, but, for the reader's convenience, we provide a derivation of (2.8) in Appendix B.1. The gist of the full derivation is to assume that the distribution function is close to a Maxwellian, with parameters that only evolve on a slow time scale $t' \sim t L/\lambda _e \sim t L/\lambda _i \gg t$. The kinetic equation (2.1) is then expanded and solved order by order in $\lambda _e/L \sim \lambda _i/L \ll 1$, allowing for the calculation of the (small) non-Maxwellian components of the distribution function. The small parameters $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$, as well as the isotropic functions $A_e^T(\tilde {v}_e)$, $A_e^R(\tilde {v}_e)$, $A_e^u(\tilde {v}_e)$, $C_e(\tilde {v}_e)$, $A_i(\tilde {v}_i)$ and $C_i(\tilde {v}_i)$ emerge during this calculation. In Appendix B.2, we provide a simple illustration of how the isotropic functions are determined by the specific choice of collision operator; in Appendices B.2.1 and B.2.2, we calculate them explicitly for Krook and Lorentz collision operators, respectively. For the full Landau collision operator, the equivalent calculation is more complicated, but can be performed (for example) by expanding the isotropic functions in Sonine polynomials (see Helander & Sigmar Reference Helander and Sigmar2005).

2.2.2. Closure of fluid equations (2.4)

Once the CE distribution function has been calculated, the desired fluid closure can be obtained by evaluating the heat fluxes, the friction forces, and the momentum fluxes (2.6) associated with the non-Maxwellian components of the CE distribution functions. These calculations were carried out in full for arbitrary values of $\rho _s/\lambda _{s}$ by Braginskii (Reference Braginskii1965).

We do not reproduce the full fluid closure relations here; instead, we illustrate how the non-Maxwellian terms in the CE distribution functions (2.8) give rise to the friction force and heat fluxes parallel to the macroscopic magnetic field, as well as to the viscosity tensor. In a strongly magnetised two-species plasma (where $\rho _s \ll \lambda _s$), parallel friction forces and heat fluxes are typically much larger than their perpendicular or diamagnetic counterparts.

1. (i) Heat fluxes. Recalling (2.6c), the parallel heat flux $q_{s\|} \equiv \hat {\boldsymbol {z}} \boldsymbol {\cdot } \boldsymbol {q}_{s}$ associated with species $s$ is given by

   (2.16)\begin{equation} q_{s\|} = \frac{1}{2} \int \mathrm{d}^3\boldsymbol{v}_s' \, m_s \left|\boldsymbol{v}_s'\right|^2 v_{s\|}' \, f_{s} , \end{equation}
   where $\boldsymbol {v}_s' \equiv \boldsymbol {v}-\boldsymbol {V}_s$. Noting that the electron distribution function (2.8a) is specified in the rest frame of the ions, not electrons, it is necessary first to calculate the electron distribution function in the electron rest frame before calculating the parallel electron heat flux. An expression for this quantity is given by (B18) in Appendix B.1 as part of our derivation of (2.8a):
   (2.17)\begin{align} f_{e}(v_{e\|}',v_{e\bot}') & = \frac{n_{e}}{v_{\mathrm{th}e}^3 {\rm \pi}^{3/2}} \exp \left(-\frac{|\boldsymbol{v}_{e}'|^2}{v_{\mathrm{th}e}^2}\right) \nonumber\\ & \quad \times \left\{1 + \left[\eta_e^T A_e^{T}\!\left(\frac{|\boldsymbol{v}_e'|}{v_{\mathrm{th}e}}\right) +\eta_e^R A_e^{R}\!\left(\frac{|\boldsymbol{v}_e'|}{v_{\mathrm{th}e}}\right) + \eta_e^u \left(A_e^{u}\!\left(\frac{|\boldsymbol{v}_e'|}{v_{\mathrm{th}e}}\right) -1 \right) \right] \frac{v_{e\|}'}{v_{\mathrm{th}e}} \right. \nonumber\\ & \quad \left.+ \,\epsilon_e C_e\!\left(\frac{|\boldsymbol{v}_e'|}{v_{\mathrm{th}e}}\right) \left(\frac{v_{e\|}'^2}{v_{\mathrm{th}e}^2} - \frac{v_{e\bot}'^2}{2 v_{\mathrm{th}e}^2} \right) \right\}. \end{align}
   Now substituting (2.17) into (2.16) (with $s = e$), we find that the parallel electron heat flux is
   (2.18)\begin{equation} q_{e\|} ={-} n_{e} T_e v_{\mathrm{th}e} \left[\mathcal{A}_e^T \eta_e^T + \mathcal{A}_e^R \eta_e^R + \left(\mathcal{A}_e^u - \tfrac{1}{2}\right) \eta_e^u\right], \end{equation}
   where
   (2.19)\begin{equation} \mathcal{A}_e^{T,R,u} ={-}\frac{4}{3\sqrt{\rm \pi}} \int_0^{\infty} \mathrm{d}\tilde{v}_e \, \tilde{v}_e^6 A_e^{T,R,u}(\tilde{v}_e) \exp \left(-\tilde{v}_e^2\right) . \end{equation}
   The minus signs in the definitions of $\mathcal {A}_e^{T,R,u}$ have been introduced so that $\mathcal {A}_e^{T,R,u} \geqslant 0$ for a typical collision operator (determining that these constants are indeed positive for any given collision operator is non-trivial, but it is a simple exercise to show this for a Krook collision operator, using the expressions for $A_e^T(\tilde {v}_e)$, $A_e^R(\tilde {v}_e)$ and $A_e^u(\tilde {v}_e)$ given in Appendix B.2.1). Expression (2.18) for the electron heat flux can be rewritten as
   (2.20)\begin{equation} q_{e\|} ={-} \kappa_{e}^{\|} \boldsymbol{\nabla}_{\|} T_e - \left[ \mathcal{A}_e^u - \frac{1}{2} - \frac{ \mathcal{A}_e^R}{\tilde{\mathcal{A}}_e^R}\left(\tilde{\mathcal{A}}_e^u-\frac{1}{2}\right)\right] n_e T_e u_{ei\|} , \end{equation}
   where the parallel electron heat conductivity is defined by
   (2.21)\begin{equation} \kappa_{e}^{\|} = 2 \left(\mathcal{A}_e^T - \frac{\mathcal{A}_e^R}{\tilde{\mathcal{A}}_e^R} \tilde{\mathcal{A}}_e^T\right) \frac{n_e T_e \tau_e}{m_e} , \end{equation}
   and
   (2.22)\begin{equation} \tilde{\mathcal{A}}_e^{T,R,u} ={-}\frac{4}{3\sqrt{\rm \pi}} \int_0^{\infty} \mathrm{d}\tilde{v}_e \, \tilde{v}_e^4 A_e^{T,R,u}(\tilde{v}_e) \exp \left(-\tilde{v}_e^2\right) . \end{equation}
   Numerical evaluation of the coefficients $\mathcal {A}_e^{T,R,u}$ and $\tilde {\mathcal {A}}_e^{T,R,u}$ for the Landau collision operator gives (Braginskii Reference Braginskii1965)
   (2.23)\begin{equation} q_{e\|} \simeq{-} 3.16 \frac{n_e T_e \tau_e}{m_e} \boldsymbol{\nabla}_{\|} T_e + 0.71 n_e T_e u_{ei\|} . \end{equation}

   The ion heat flux can be calculated directly from (2.16) ($s = i$) using (2.8b):

   (2.24)\begin{equation} q_{i\|} ={-} n_{i} T_i v_{\mathrm{th}i} \mathcal{A}_i \eta_i , \end{equation}
   where
   (2.25)\begin{equation} \mathcal{A}_i ={-}\frac{4}{3\sqrt{\rm \pi}} \int_0^{\infty} \mathrm{d}\tilde{v}_i \, \tilde{v}_i^6 A_i(\tilde{v}_i) \exp \left(-\tilde{v}_i^2\right) . \end{equation}
   This becomes
   (2.26)\begin{equation} q_{i\|} ={-} \kappa_{i}^{\|} \boldsymbol{\nabla}_{\|} T_i , \end{equation}
   where the parallel ion heat conductivity is
   (2.27)\begin{equation} \kappa_{i}^{\|} ={-}2 \mathcal{A}_i \frac{n_i T_i \tau_i}{m_i} \simeq{-} 3.9 \frac{n_i T_i \tau_i}{m_i} . \end{equation}
   The last equality is for the Landau collision operator (Braginskii Reference Braginskii1965). Note that the absence of a term proportional to the electron–ion-drift in the ion heat flux (2.24) is physically due to the smallness of the ion–electron collision operator (Helander & Sigmar Reference Helander and Sigmar2005).

2. (ii) Friction force. We evaluate the friction force by considering the electron–ion-drift associated with electron CE distribution function. Namely, noting that

   (2.28)\begin{equation} u_{ei\|} = \frac{v_{\mathrm{th}e}^4}{n_e} \int \mathrm{d}^3 \tilde{\boldsymbol{v}}_e \, \tilde{v}_{e\|} f_{e} , \end{equation}
   it follows from (2.8a) that
   (2.29)\begin{equation} u_{ei\|} = v_{\mathrm{th}e} \left(\tilde{\mathcal{A}}_e^T \eta_e^T + \tilde{\mathcal{A}}_e^R \eta_e^R + \tilde{\mathcal{A}}_e^u \eta_e^u \right) . \end{equation}
   This expression can in turn be used to relate the parallel electron-friction force $R_{e\|}$, defined in (2.6d), to electron flows and temperature gradients
   (2.30)\begin{equation} R_{e\|} ={-}\left(\frac{2 \tilde{\mathcal{A}}_e^u + 1}{2 \tilde{\mathcal{A}}_e^R}\right) \frac{n_e m_e u_{ei\|}}{\tau_e} - \frac{\tilde{\mathcal{A}}_e^T}{\tilde{\mathcal{A}}_e^R} n_e \boldsymbol{\nabla}_{\|} T_e . \end{equation}
   Evaluating the coefficients $\tilde {\mathcal {A}}_e^T$, $\tilde {\mathcal {A}}_e^R$ and $\tilde {\mathcal {A}}_e^u$ for the full Landau collision operator, one finds (Braginskii Reference Braginskii1965)
   (2.31)\begin{equation} R_{e\|} \simeq{-}0.51 \frac{n_e m_e u_{ei\|}}{\tau_e} - 0.71 n_e \boldsymbol{\nabla}_{\|} T_e . \end{equation}

3. (iii) Viscosity tensor. For gyrotropic distributions such as the CE distribution functions (2.8), the viscosity tensor $\boldsymbol {{\rm \pi} }_s$ of species $s$ defined by (2.6b) – which is the momentum flux excluding the convective terms and isotropic pressure – is given by

   (2.32)\begin{equation} \boldsymbol{\rm \pi}_s = \left(p_{s\|} - p_{s\perp}\right) \left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}}-\tfrac{1}{3}\boldsymbol{\mathsf{I}}\right) , \end{equation}
   where the parallel pressure $p_{s\|}$ and the perpendicular pressure $p_{s\bot }$ are defined by
   (2.33a)$$\begin{align} p_{s\|} &\equiv \int \mathrm{d}^3 \boldsymbol{v}_s' \, m_s |v_{s\|}'|^2 f_{s} = n_{s} T_{s} \left(1-\frac{2}{3}\epsilon_s \mathcal{C}_s\right) , \end{align}$$
   (2.33b)$$\begin{align}p_{s\bot} &\equiv \frac{1}{2}\int \mathrm{d}^3 \boldsymbol{v}_{s}' \, m_s |\boldsymbol{v}_{s\bot}'|^2 f_{s} = n_{s} T_{s} \left(1+\frac{1}{3}\epsilon_s \mathcal{C}_s\right) , \end{align}$$
   with the last expressions having been obtained on substitution of the CE distribution function (2.8), and
   (2.34)\begin{equation} \mathcal{C}_s ={-}\frac{8}{5\sqrt{\rm \pi}} \int_0^{\infty} \mathrm{d}\tilde{v}_s \, \tilde{v}_s^6 C_s(\tilde{v}_s) \exp \left(-\tilde{v}_s^2\right) . \end{equation}
   The sign of the constant $\mathcal {C}_s$ is again chosen so that $\mathcal {C}_s > 0$ for typical collision operators; for the Landau collision operator, $\mathcal {C}_e \simeq 1.1$ and $\mathcal {C}_i \simeq 1.44$ (Braginskii Reference Braginskii1965). We note for reference that the parameter $\epsilon _s$ (see (2.11e–f)) has a simple relationship to the pressure anisotropy of species $s$: utilising (2.33), one finds
   (2.35)\begin{equation} \varDelta_s \equiv \frac{p_{s\bot}-p_{s\|}}{p_{s}} = \mathcal{C}_s \epsilon_s . \end{equation}
   Using (2.33), the viscosity tensor (2.32) can be written
   (2.36)\begin{equation} \boldsymbol{\rm \pi}_s ={-} \frac{\mu_{{v}s}}{2} \left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) {\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_s} , \end{equation}
   where the dynamic viscosity of species $s$ is
   (2.37)\begin{equation} \mu_{{v}s} \equiv 2 \mathcal{C}_s n_s T_s \tau_s . \end{equation}

4. (iv) Thermal energy transfer between species. It can be shown that for the CE distribution functions (2.8), the rate of thermal energy transfer from electrons to ions $\mathcal {Q}_e$ is simply

   (2.38)\begin{equation} \mathcal{Q}_e ={-} \boldsymbol{R}_e \boldsymbol{\cdot} \boldsymbol{u}_{ei} , \end{equation}
   while the rate of thermal energy transfer from ions to electrons vanishes: $\mathcal {Q}_i \approx 0$. This is because the ion–electron collision rate is assumed small (by a factor of the mass ratio) compared with the ion–ion collision rate when deriving (2.8b), and is thus neglected. Braginskii (Reference Braginskii1965) shows that, in fact, there is a non-zero (but small) rate of transfer
   (2.39)\begin{equation} \mathcal{Q}_i ={-}\mathcal{Q}_e - \boldsymbol{R}_e \boldsymbol{\cdot} \boldsymbol{u}_{ei} = \frac{3 n_e m_e}{m_i \tau_{e}} \left(T_e-T_i\right) . \end{equation}
   The time scale on which the ion and electron temperatures equilibrate is the ion–electron temperature equilibration time
   (2.40)\begin{equation} \tau_{ie}^{\rm eq} \equiv \tfrac{1}{2} \mu_e^{{-}1/2} \tau_{i} . \end{equation}

In summary, the non-Maxwellian components of the CE distribution function are essential for a collisional plasma to be able to support fluxes of heat and momentum. More specifically, (2.20) demonstrates that the electron heat fluxes in a CE plasma are proportional to both temperature gradients and electron–ion drifts, and are carried by the electron-temperature-gradient, friction and electron–ion-drift terms of the CE distribution function. In contrast, the ion heat fluxes (2.26) are proportional only to ion temperature gradients (and carried by the CE ion-temperature-gradient term). Momentum fluxes (2.36) for electrons and ions are carried by the CE electron- and ion-shear terms, respectively, and are proportional to components of the rate-of-strain tensor.

2.2.3. Relative size of non-Maxwellian terms in the CE distribution function

In the case of magnetised, two-species plasma satisfying $T_i \sim T_e$, (2.11) can be used to estimate the size of the small parameters $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$. Although these parameters are a priori proportional to $\lambda _s/L$ for both ions and electrons, their precise magnitudes are, in fact, subtly different. Namely, the terms associated with $\eta _e^T$, $\eta _e^R$, $\eta _e^u$ and $\eta _i$ are gradients of the electron and ion temperatures and electron–ion relative parallel drift velocities, whereas terms associated with $\epsilon _e$ and $\epsilon _i$ involve gradients of the bulk flows (cf. (2.11)) – and these gradients do not necessarily occur on the same length scale. Recalling that the (electron) temperature and the (ion) flow length scales parallel to the macroscopic magnetic field are defined by (cf. (2.13))

(2.41a)$$\begin{align} L_T &= \left|\boldsymbol{\nabla}_{\|} \log T_e \right|^{{-}1} , \end{align}$$

(2.41b)$$\begin{align}L_V &= \frac{1}{V_i}\left|\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\boldsymbol{\mathsf{I}} \right) \! \boldsymbol{:} \! \boldsymbol{\mathsf{W}}_i \right|^{{-}1} , \end{align}$$

where $\boldsymbol{\mathsf{W}}_i$ is the ion rate-of-strain tensor (2.12), and assuming that $L_{T_i} = |\boldsymbol {\nabla }_{\|} \log T_i |^{-1} \sim L_{T}$ (an assumption we will check a posteriori), it follows from (2.11) that

(2.42a)$$\begin{align} \eta_e^T &\sim \frac{\lambda_{e}}{L_T}, \end{align}$$

(2.42b)$$\begin{align}\eta_e^R &\sim \lambda_{e} \frac{R_{e\|}}{p_e} \sim \frac{\lambda_{e}}{L_T} \sim \eta_e^T, \end{align}$$

(2.42c)$$\begin{align}\eta_e^u &\sim \frac{u_{ei\|}}{v_{\mathrm{th}e}} \sim \frac{\lambda_{e}}{L_T} \sim \eta_e^T, \end{align}$$

(2.42d)$$\begin{align}\eta_i &\sim \frac{\lambda_{i}}{L_T} \sim \frac{1}{Z_i^2} \eta_e^T , \end{align}$$

(2.42e)$$\begin{align}\epsilon_e &\sim \frac{V_i}{v_{\mathrm{th}e}} \frac{\lambda_{e}}{L_V} \sim {Ma}\, \mu_e^{1/2} \frac{L_T}{L_V} \eta_e^T, \end{align}$$

(2.42f)$$\begin{align}\epsilon_i &\sim \frac{V_i}{v_{\mathrm{th}i}} \frac{\lambda_{i}}{L_V} \sim {Ma} \, \frac{L_T}{Z_i^2 L_V} \eta_e^T, \end{align}$$

where ${Ma} \equiv V_i/v_{\mathrm {th}i}$ is the Mach number. Note that, to arrive at (2.42b), we assumed that $R_{e\|} \sim p_e/L_{T}$ and $u_{ei\|} \sim v_{\mathrm {th}e} \lambda _e/L_T$, justified by (2.30) and (2.29), respectively. The relative magnitudes of $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$ therefore depend on the Mach number of the plasma, as well as on the length scales $L_T$ and $L_V$.

In the work of Braginskii (Reference Braginskii1965), who a priori presumes all ‘fluid’ quantities in the plasma to vary on just a single scale $L \sim L_T \sim L_V$, with sonic ordering ${Ma} \lesssim 1$, determining the relative size of these parameters for a hydrogen plasma ($Z_i = 1$) is simple:

(2.43)\begin{equation} \epsilon_e \sim \mu_e^{1/2} \epsilon_i \ll \epsilon_i \sim \eta_i \sim \eta_e^T \sim \eta_e^R \sim \eta_e^u . \end{equation}

However, in most interesting applications, this single-scale ordering is incorrect. In a plasma with $\lambda _s/L \ll 1$ under Braginskii's ordering, motions on many scales naturally arise. The fluid Reynolds number in such a plasma is given by

(2.44)\begin{equation} {Re} \equiv \frac{V_0 L_0}{\nu} , \end{equation}

where $V_0$ is the typical fluid velocity at the scale $L_0$ of driving motions and $\nu \equiv \mu _{{v}i}/m_i n_i \sim v_{ \mathrm {th}i} \lambda _i$ is the kinematic viscosity (see (2.37)). Typically, this number is large:

(2.45)\begin{equation} {Re} \sim \frac{V_0}{v_{\mathrm{th}i}} \frac{L_0}{\lambda_{i}} \gtrsim \frac{1}{\epsilon_i} \gg 1 , \end{equation}

where we have assumed ${Ma}_0 \equiv V_0/v_{\mathrm {th}i} \lesssim 1$, in line with Braginskii's sonic ordering. Therefore, such a plasma will naturally become turbulent and exhibit motions across a range of scales. As a consequence, velocity and temperature fluctuations on the smallest (fluid) scales must be considered, since the associated shears and temperature gradients are the largest. To estimate $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$ accurately, we must determine the magnitude of these gradients.

First, let $\ell _{\nu }$ be the smallest scale on which the velocity varies due to turbulent motions (the Kolmogorov scale), with velocity fluctuations on scales $\ell \ll \ell _{\nu }$ being suppressed by viscous diffusion. Then it follows that ${Re}_{\ell _{\nu }} \sim 1$, where ${Re}_\ell \equiv V(\ell ) \ell /\nu$ is the scale-dependent Reynolds number and $V(\ell )$ is the typical fluid velocity on scale $\ell$. For Kolmogorov turbulence,

(2.46)\begin{equation} \frac{V\!\left(\ell\right)}{V_0} \sim \left(\frac{\ell}{L_0}\right)^{1/3} \sim \left(\frac{{Re}_\ell}{Re} \right)^{1/4}, \end{equation}

and $\ell /L_0 \sim ({{Re}_\ell }/{Re} )^{3/4}$, which gives $V(\ell )/\ell \sim (V_0/L_0) ({{Re}_\ell }/{Re} )^{-1/2}$, and thus, from (2.45),

(2.47)\begin{equation} \frac{V\!\left(\ell_\nu\right)}{\ell_\nu} \sim \frac{V_0}{L_0} \left(\frac{{Re}_{\ell_\nu}}{Re} \right)^{{-}1/2} \sim {Ma}_0^{1/2} \left(\frac{\lambda_{i}}{L_0}\right)^{{-}1/2} \frac{V_0}{L_0} . \end{equation}

We therefore conclude that

(2.48)\begin{equation} L_V \sim \ell_{\nu} \frac{V_0}{V\!\left(\ell_\nu\right)} \sim L_0 {Ma}_0^{{-}1/2} \left(\frac{\lambda_{i}}{L_0}\right)^{1/2} . \end{equation}

Next, the smallest scale on which the electron temperature varies, $\ell _{\chi }$, is the scale below which temperature fluctuations are suppressed by thermal diffusion; it satisfies ${Pe}_{\ell _{\chi }} \sim 1$, where ${Pe}_\ell \equiv V(\ell ) \ell /\chi$ is the scale-dependent Péclet number and $\chi \equiv 2 \kappa _{e}^{\|}/3 n_e \sim v_{\mathrm {th}e} \lambda _{e}$ is the (parallel) thermal diffusivity (see (2.21)). Because temperature is passively advected by the flow, the temperature fluctuation $T(\ell )$ at any scale $\ell > \ell _\chi$ obeys the same scaling as the bulk velocity:

(2.49)\begin{equation} \frac{T\!\left(\ell\right)}{T(L_0)} \sim \frac{V\!\left(\ell\right)}{V_0} \sim \left(\frac{{Pe}_\ell}{Pe} \right)^{1/4} . \end{equation}

In addition, the magnitude of temperature fluctuations at the driving scale is related to the mean temperature by the Mach number of the driving-scale motions, $T(L_0) \sim T_0 {Ma}_0$, which then gives

(2.50)\begin{equation} \frac{T\!\left(\ell\right)}{T_0} \sim {Ma}_0 \left(\frac{{Pe}_\ell}{Pe} \right)^{1/4} , \end{equation}

where ${Pe} \equiv {Pe}_{L_0}$. It follows from an analogous argument to that just given for the velocity fluctuations that

(2.51)\begin{equation} \frac{T\!\left(\ell_\chi\right)}{\ell_\chi} \sim \frac{T_0}{L_0} {Ma}_0 {Pe}^{1/2} . \end{equation}

Under Braginskii's ordering, the Prandtl number of CE plasma is

(2.52)\begin{equation} {Pr} \equiv \frac{\nu}{\chi} = \frac{Pe}{Re} \sim \frac{v_{\mathrm{th}i} \lambda_{i}}{v_{\mathrm{th}e} \lambda_{e} } \sim \mu_e^{1/2} \ll 1 , \end{equation}

and, therefore,

(2.53)\begin{equation} L_T \sim \ell_{\chi} \frac{T_0}{T\!\left(\ell_\chi\right)} \sim L_0 \mu_e^{{-}1/4} {Ma}_0^{{-}3/2} \left(\frac{\lambda_{i}}{L_0}\right)^{1/2} . \end{equation}

Thus, $L_V \sim {Ma}_0 \mu _e^{1/4} L_T \ll L_T$ under the assumed ordering.

Finally, we consider whether our a priori assumption that $L_{T_i} \sim L_T$ is, in fact, justified. A sufficient condition for ion-temperature gradients to be the same as electron-temperature gradients is for the evolution time $\tau _{L}$ of all macroscopic motions to be much longer than the ion–electron equilibration-time $\tau _{ie}^{\rm eq}$ defined by (2.40). Since $\tau _{L} \gtrsim {\ell _\nu }/{V(\ell _\nu )}$, it follows that

(2.54)\begin{equation} \frac{\tau_{L}}{\tau_{ie}^{\rm eq}} \sim \left(\frac{m_i}{m_e}\right)^{1/2}{Ma}_0^{3/2} \left(\frac{\lambda_i}{L_0}\right)^{1/2} \sim \epsilon_i \left(\frac{m_i}{m_e}\right)^{1/2} . \end{equation}

Thus, if $\epsilon _i \gg \mu _e^{1/2}$, we conclude that collisional equilibration of ion and electron temperatures might be too inefficient to regulate small-scale ion-temperature fluctuations, in which case it would follow that $L_{T_i} < L_{T}$. However, it has been previously demonstrated via numerical solution of the Vlasov–Fokker–Planck equation that the CE expansion procedure breaks down due to non-local transport effects if $\lambda _e/L$ is only moderately small (Bell, Evans & Nicholas Reference Bell, Evans and Nicholas1981); thus, the only regime in which there is not ion–electron equilibration over all scales is one where the CE expansion is not valid anyway. In short, we conclude that assuming $L_{T_i} \sim L_T$ is reasonable.

Bringing these considerations together with (2.42), we find that

(2.55a)$$\begin{align} &\eta_e^T \sim \mu_e^{1/4} {Ma}_0 \frac{\lambda_i}{L_V} \sim {Ma}_0^{3/2} \mu_e^{1/4} \left(\frac{\lambda_i}{L_0}\right)^{1/2} \sim \eta_e^{R} \sim \eta_e^{u} \sim \eta_i , \end{align}$$

(2.55b)$$\begin{align} &\epsilon_e \sim \mu_e^{1/2} {Ma}_0 \frac{\lambda_i}{L_V}\sim \mu_e^{1/2} {Ma}_0^{3/2} \left(\frac{\lambda_i}{L_0}\right)^{1/2} , \end{align}$$

(2.55c)$$\begin{align} &\epsilon_i \sim {Ma}_0 \frac{\lambda_i}{L_V} \sim {Ma}_0^{3/2} \left(\frac{\lambda_i}{L_0}\right)^{1/2} . \end{align}$$

Thus, we conclude that the largest distortions of the ion CE distribution are due to flow gradients, while temperature gradients cause the greatest distortions of the electron CE distribution function.

In deriving the estimates (2.55) of the small parameters $\eta _e^T$, $\eta _e^R$, $\eta _e^u$, $\eta _i$, $\epsilon _e$ and $\epsilon _i$, we assumed that the motions of the plasma (including those at the Kolmogorov scale) satisfied Kolmogorov's statistical theory of turbulence. There are several reasonable grounds on which this assumption could be questioned. First of these is intermittency: it is a long-standing result that the distribution of turbulent velocities at a given scale in hydrodynamic turbulence becomes increasingly non-Gaussian as that scale decreases (see, e.g. Davidson Reference Davidson2015). This has the consequence that (2.55) may overestimate the size of the non-Maxwellian terms in the CE distribution function for the majority of the plasma's volume, but underestimate it significantly in localised regions. Another objection is the now generally accepted finding that turbulent fluids with sufficiently strong magnetic fields do not behave like hydrodynamic fluids: instead, turbulent motions tend to become anisotropic with respect to the background magnetic field if the latter's energy becomes greater than the former (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). When applied to turbulent motions at the Kolmogorov scale where motions are assumed to behave as we have proposed, this condition becomes

(2.56)\begin{equation} \frac{m_i n_i V(\ell_{\nu})^2/2}{B^2/8{\rm \pi}} \sim \frac{m_i n_i V_0^2/2}{B^2/8{\rm \pi}} \frac{V(\ell_{\nu})^2}{V_0^2} \sim {Ma}_0^{3/2} \left(\frac{\lambda_i}{L_0}\right)^{1/2} \beta_i \sim \epsilon_i \beta_i , \end{equation}

where

(2.57)\begin{equation} \beta_s \equiv \frac{8 {\rm \pi}n_s T_s}{B^2}\end{equation}

is the plasma beta of species $s$. So, whenever $\beta _i \ll \epsilon _i^{-1}$, the assumption that the turbulence is unmagnetised is unlikely to be a good one, whereas if $\beta _i$ is sufficiently large ($\beta _i \gtrsim \epsilon _i^{-1}$), it is more reasonable. As we have already outlined in the introduction (cf. figure 1), the case in which $\beta _i$ is large will tend to be our focus in this paper; so we conclude that the scalings (2.55) will generally be the pertinent ones.

2.3.1. Overview

We have seen that the CE expansion provides a procedure for the calculation of the distribution functions arising in a classical, collisional plasma in terms of gradients of temperature, electron–ion drifts and bulk fluid velocities; these calculations in turn allow for the closure of the system (2.4) of fluid equations. However, these same gradients are sources of free energy in the plasma, so they can lead to instabilities. Some of these instabilities will be ‘fluid’, i.e. they are captured within the CE description and are features of the fluid dynamics of plasmas; others are kinetic (‘microinstabilities’), and their existence implies that the CE expansion is, in fact, illegitimate. Our primary purpose in this paper is to determine when such microinstabilities do not occur in a strongly magnetised two-species plasma. If, however, they do occur, we wish to determine their growth rates. We begin by making a few general qualitative comments concerning the existence and nature of these microinstabilities, before presenting the technical details of their derivation.

2.3.2. Existence of microinstabilities in classical, collisional plasma

It might naively be assumed that a classical, collisional plasma is kinetically stable, on two grounds. The first of these is that the distribution function of such a plasma is ‘almost’ Maxwellian, and thus stable. While it is certainly the case that a plasma whose constituent particles have Maxwellian distribution functions is kinetically stable (Bernstein Reference Bernstein1958; Krall & Trivelpiece Reference Krall and Trivelpiece1973), it is also known that a plasma with anisotropic particle distribution functions is typically not (Furth Reference Furth1963; Kalman et al. Reference Kalman, Montes and Quemada1968; Davidson Reference Davidson1983; Gary Reference Gary1993). The (small) non-Maxwellian component of the CE distribution function is anisotropic (as, e.g. was explicitly demonstrated by the calculation of pressure anisotropy in § 2.2.2), and thus we cannot a priori rule out microinstabilities associated with this anisotropy.

The second naive reason for dismissing the possibility of microinstabilities in classical, collisional plasma is the potentially stabilising effect of collisional damping on microinstability growth rates. If collisional processes are sufficiently dominant to be responsible for the mediation of macroscopic momentum and heat fluxes in the plasma, it might be naively inferred that they would also suppress microinstabilities. This is, in fact, far from guaranteed, for the following reason. The characteristic scales of the microinstabilities are not fluid scales, but are rather intrinsic plasma length scales related to quantities such as the Larmor radius $\rho _s$ or the inertial scale $d_s$ of species $s$, or the Debye length $\lambda _{{\rm D}}$ – quantities given in terms of macroscopic physical properties of plasma by

(2.58a)$$\begin{align} &\rho_s = \frac{m_s v_{\mathrm{th}s} c}{|Z_s| e |\boldsymbol{B}|} , \end{align}$$

(2.58b)$$\begin{align} &d_s \equiv \left(\frac{4 {\rm \pi}Z_s^2 e^2 n_s}{m_s c^2}\right)^{{-}1/2} = \rho_s \beta_s^{{-}1/2}, \end{align}$$

(2.58c)$$\begin{align} &\lambda_{{\rm D}} \equiv \left(\sum_s \frac{4 {\rm \pi}Z_s^2 e^2 n_s}{T_s}\right)^{{-}1/2} = \left(\sum_s \frac{2 c}{d_s^2 v_{\mathrm{th}s}} \right)^{{-}1/2} . \end{align}$$

The crucial observation is then that the dynamics on characteristic microinstability scales may be collisionless. For a classical, collisional hydrogen plasma (where $\lambda \equiv \lambda _e \sim \lambda _i$ for $T_e \sim T_i$), the mean free path is much larger than the Debye length: $\lambda /\lambda _{{\rm D}} \sim n_e \lambda _{{\rm D}}^3 \gg 1$; so there exists a range of wavenumbers $k$ on which microinstabilities are both possible ($k \lambda _{{\rm D}} \lesssim 1$) and collisionless ($k \lambda \gg 1$). For a strongly magnetised collisional plasma, $\lambda _s \gg \rho _s$ for all species by definition; thus, any microinstability with a characteristic scale comparable to the Larmor radius of any constituent particle will be effectively collisionless. We note that such a range of collisionless wavenumbers only exists in classical (viz. weakly coupled) plasmas; in strongly coupled plasmas, for which $\lambda \lesssim \lambda _{{\rm D}}$, all hypothetically possible microinstability wavenumber scales are collisional. Thus the phenomenon of microinstabilities in collisional plasmas is solely a concern for the classical regime.

2.3.3. A simple example: the firehose instability in CE plasmas

Perhaps the simplest example of a microinstability that can occur in CE plasma is the firehose instability. This example was previously discussed by Schekochihin et al. (Reference Schekochihin, Cowley, Kulsrud, Hammett and Sharma2005), but we nonetheless outline it here to illustrate the central concept of our paper.

Consider bulk fluid motions of the plasma on length scales $L_V$ that are much smaller than the mean free path $\lambda _i$, but much larger than the ion-Larmor radius $\rho _i$; the characteristic frequencies associated with these motions are assumed to be much smaller that the ion Larmor frequency $\varOmega _i$, but much larger than the inverse of the ion collision time $\tau _{i}^{-1}$. Under these assumptions, the following four statements can be shown to be true (Schekochihin et al. Reference Schekochihin, Cowley, Kulsrud, Hammett and Sharma2005):

1. (i) The bulk velocities of the electron and ion species are approximately equal: $\boldsymbol {V}_e \approx \boldsymbol {V}_i$.

2. (ii) The electric field in a frame co-moving with the ion fluid vanishes; transforming to the stationary frame of the system, this gives

   (2.59)\begin{equation} \boldsymbol{E} ={-}\frac{\boldsymbol{V}_i \times \boldsymbol{B}}{c} . \end{equation}

3. (iii) The contribution of the displacement current to the Maxwell–Ampère law (2.2d) is negligible, and so

   (2.60)\begin{equation} e n_e \left(\boldsymbol{V}_i - \boldsymbol{V}_e \right) \approx \frac{c}{4 {\rm \pi}} \boldsymbol{\nabla} \times \boldsymbol{B} . \end{equation}

4. (iv) The electron and ion viscosity tensors both take the form (2.32), and the electron-pressure anisotropy, defined by (2.35), is small compared with the ion pressure anisotropy: $\varDelta _e \ll \varDelta _i$.

It then follows directly from (2.4b), summed over both ion and electron species, that

(2.61)\begin{equation} m_i n_i \left.\frac{\mathrm{D} \boldsymbol{V}_i }{\mathrm{D} t} \right|_i ={-}\boldsymbol{\nabla} \left(\frac{B^2}{8 {\rm \pi}} + p_{e\perp} + p_{i\perp}\right) - \boldsymbol{\nabla} \boldsymbol{\cdot} \left[ \hat{\boldsymbol{z}} \hat{\boldsymbol{z}} \left(p_{i\perp}-p_{i\|}\right) \right] + \frac{\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{B}}{4 {\rm \pi}} . \end{equation}

We remind the reader that $\hat {\boldsymbol {z}} = \boldsymbol {B}/B$, and emphasise that we have neglected the electron inertial term on the grounds that it is small compared with the ion inertial term:

(2.62)\begin{equation} \left.m_e n_e \frac{\mathrm{D} \boldsymbol{V}_e }{\mathrm{D} t} \right|_e \ll m_i n_i \left.\frac{\mathrm{D} \boldsymbol{V}_i }{\mathrm{D} t} \right|_i . \end{equation}

The evolution of the magnetic field is described by the induction equation

(2.63)\begin{equation} \left.\frac{\mathrm{D} \boldsymbol{B} }{\mathrm{D} t} \right|_i = \boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{V}_{i} - \boldsymbol{B} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{V}_{i} , \end{equation}

which is derived by substituting (2.59) into Faraday's law (2.2c).

Now consider small-amplitude perturbations with respect to a particular macroscale state of the plasma

(2.64a)$$\begin{gather} \delta \boldsymbol{V}_{i} = \widehat{\delta \boldsymbol{V}}_{i\perp} \exp\left\{\mathrm{i}\left(\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{r} - \omega t\right)\right\} , \end{gather}$$

(2.64b)$$\begin{gather}\delta \boldsymbol{B} = \widehat{\delta \boldsymbol{B}}_{{\perp}} \exp\left\{\mathrm{i}\left(\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{r} - \omega t\right)\right\}, \end{gather}$$

whose characteristic frequency $\omega$ is much greater than that of the plasma's bulk fluid motions (but is still much smaller than $\varOmega _i$), whose wavevector $\boldsymbol {k} = k_{\|} \hat {\boldsymbol {z}}$ is parallel to $\boldsymbol {B}$, and assume also that the velocity and magnetic-field perturbations are perpendicular to $\boldsymbol {B}$. It is then easy to show that (2.61) and (2.63) become

(2.65a)$$\begin{align} - \mathrm{i} m_i n_i \omega \widehat{\delta \boldsymbol{V}}_{i\perp} &= \mathrm{i} \left(\frac{B^2}{4 {\rm \pi}} + p_{i\perp} - p_{i\|}\right) k_{\|} \frac{ \widehat{\delta \boldsymbol{B}}_{{\perp}} }{B} , \end{align}$$

(2.65b)$$\begin{align}- \mathrm{i} \omega \widehat{\delta \boldsymbol{B}}_{{\perp}} &= \mathrm{i} B k_{\|} \widehat{\delta \boldsymbol{V}}_{i\perp} , \end{align}$$

where $p_{i\perp }$ and $p_{i\|}$ are the perpendicular and parallel ion pressures associated with the macroscale state (which, on account of its comparatively slow evolution compared with the perturbation, can be regarded a quasi-equilibrium). Physically, the macroscale flow gives rise to different values of $p_{i\perp }$ and $p_{i\|}$, and thereby an ion-pressure anisotropy $\varDelta _i$, because it changes the strength $B$ of the macroscale magnetic field; thanks to the effective conservation of the first and second adiabatic moments of the ions on the evolution time scale of the macroscale flow (Chew et al. Reference Chew, Goldberger, Low and Chandrasekhar1956), an increase (decrease) in $B$ results in an increase (decrease) in $p_{i\perp }$, and a decrease (increase) in $p_{i\|}$. The dispersion relation for the perturbation is then

(2.66)\begin{equation} \omega^2 = k_{\|}^2 v_{\mathrm{th}i}^2 \left(\frac{1}{\beta_i} + \frac{\varDelta_i}{2}\right) , \end{equation}

where $\beta _i$, defined by (2.57), is the ion plasma beta. For a sufficiently negative ion-pressure anisotropy, viz. $\varDelta _i < -2/\beta _i$, the perturbation is unstable. This instability is known as the (parallel) firehose instability.

The underlying physics of the parallel firehose instability has been discussed extensively elsewhere (see Rosin et al. (Reference Rosin, Schekochihin, Rincon and Cowley2011), and references therein; also see § 4.4.1). Here, we simply note that the firehose instability arises in a magnetised plasma with sufficiently negative pressure anisotropy as compared with the inverse of the ion plasma beta; because the ion CE distribution function has a small, non-zero pressure anisotropy, this statement applies to CE plasma at large $\beta _i$. We also observe that the product of the growth rate (2.66) of the firehose instability with the ion–ion collision time satisfies

(2.67)\begin{equation} \omega \tau_i \sim k_{\|} \lambda_i \left|\frac{1}{\beta_i} + \varDelta_i \right|^{1/2} \sim \frac{1}{\beta_i} \frac{\lambda_i}{\rho_i} , \end{equation}

where we have assumed that $\varDelta _i \lesssim -2 \beta _i^{-1}$, and employed the (non-trivial) result that the peak growth of the parallel firehose instability occurs at wavenumbers satisfying $k_{\|} \rho _i \sim \beta _i^{-1/2}$ (see §§ 4.4.1 and 4.4.2). Thus, if $\beta _i \ll \lambda _i/\rho _i$ – a condition easily satisfied in weakly collisional astrophysical environments such as the ICM (see table 4) – it follows that $\omega \tau _i \gg 1$, and so collisional damping is unable to inhibit the parallel firehose in a CE plasma.Footnote ² This failure is directly attributable to its characteristic wavelength being at collisionless scales: the parallel wavenumber satisfies $k_{\|} \lambda _i \sim \beta _i^{-1/2} \lambda _i/\rho _i \gg 1$.

This simple example clearly illustrates that microinstabilities are indeed possible in a classical, collisional plasma, for precisely the reasons given in § 2.3.2.

2.3.4. Which microinstabilities are relevant

Although the naive arguments described in § 2.3.2 do not imply kinetic stability of CE plasma, these same arguments do lead to significant restrictions on the type of microinstabilities that can arise. Namely, for some plasma modes, the small anisotropy of CE distribution functions is an insufficient free-energy source for overcoming the competing collisionless damping mechanisms that ensure stability for pure Maxwellian distribution functions – e.g. Landau damping or cyclotron damping. For other plasma modes, the characteristic length scales are so large that collisional damping does suppress growth. In magnetised plasmas, there also exist cyclotron harmonic oscillations that, despite minimal damping, can only become unstable for sufficiently large anisotropy of the particle distribution function: e.g. the electrostatic Harris instability (Harris Reference Harris1959; Hall, Heckrotte & Kammash Reference Hall, Heckrotte and Kammash1964). Since the anisotropy threshold for such microinstabilities is typically $\varDelta _s \gtrsim 1$ (Shima & Hall Reference Shima and Hall1965), they cannot operate in a CE plasma.

We claim that there are only two classes of microinstabilities that can be triggered in a CE plasma. The first are quasi-cold-plasma modes: these are modes whose frequency is so large that resonant wave–particle interactions (Landau or cyclotron resonances) only occur with electrons whose speed greatly exceeds the electron thermal speed $v_{\mathrm {th}e}$. Collisionless damping of such modes is typically very weak, and thus small anisotropies of particle distribution functions can be sufficient to drive an instability. Well-known examples of a small non-Maxwellian part of the distribution function giving rise to microinstabilities include the bump-on-tail instability associated with a fast beam of electrons (see § 3.3.3 of Davidson Reference Davidson1983), or the whistler instability for small temperature anisotropies (see § 3.3.5 of Davidson Reference Davidson1983). The existence of such instabilities for the CE distribution can be demonstrated explicitly: e.g. the peak growth rate of the bump-on-tail instability associated with the CE distribution function (‘the CE bump-on-tail instability’) is calculated in Appendix D.3. However, the growth rates $\gamma$ of such instabilities are exponentially small in $\lambda _e/L \ll 1$. This claim, which is explicitly proven for the CE bump-on-tail instability in Appendix D.3, applies to all electrostatic instabilities (see Appendix D.4), and it can be argued that it also applies to all quasi-cold-plasma modes (see Appendix E). When combined with the constraint that the resonant wave–particle interactions required for such instabilities cannot occur if $\gamma \tau _{r} \lesssim 1$, where $\tau _{r}$ is the collision time of the resonant particles, the exponential smallness of the growth rate suggests that such microinstabilities will not be significant provided $\lambda _e/L$ really is small. As discussed in § 2.2.3, plasmas in which $\lambda _e/L$ is only moderately small are not well modelled as CE plasmas anyway, and thus, for the rest of this paper, we will not study quasi-cold-plasma-mode instabilities.

The second class of allowed microinstabilities comprises modes that are electromagnetic and low frequency in the sense that the complex frequency $\omega$ of the microinstability satisfies, for at least one particle species $s$,

(2.68)\begin{equation} \frac{\omega}{k v_{\mathrm{th}s}} \sim \left(\frac{\lambda_s}{L}\right)^{\iota} \ll 1 , \end{equation}

where $\iota$ is some order-unity number. Low-frequency electromagnetic modes are in general only subject to weak Landau and cyclotron damping (of order $\omega /k v_{\mathrm {th}s} \ll 1$ or less), and thus can become unstable for small distribution-function anisotropies. By contrast, electromagnetic modes satisfying $\omega \sim k v_{\mathrm {th}s}$ would typically generate strong inductive electric fields, which would in turn be subject to significant Landau or cyclotron damping, overwhelming any unstable tendency. The firehose instability introduced in § 2.3.3 is one example of this type of microinstability: it satisfies (2.68) with $\iota = 1/2$, provided its $\beta$-stabilisation threshold is surpassed.

In this paper, we will focus on microinstabilities in this second class. Whilst small compared with the streaming rate $k v_{\mathrm {th}s}$ of species $s$, the growth rates satisfying (2.68) can still be significantly larger than the rate at which the plasma evolves on macroscopic scales, and thus invalidate the CE expansion. We do not in this paper present a rigorous proof that there are no microinstabilities of the CE distribution function which do not fall into either of the two classes considered above. However, there do exist more precise arguments supporting the latter claim than those based on physical intuition just presented; these are discussed further in §§ 2.4.2 and 2.5.8.

The microinstabilities satisfying (2.68) fall into two sub-classes. The first sub-class consists of microinstabilities driven by the CE temperature-gradient, CE electron-friction and CE electron–ion-drift terms in the CE distribution functions (2.8); we refer to these collectively as CE temperature-gradient-driven microinstabilities, or CET microinstabilities, on account of the parameters $\eta _e^R$ and $\eta _e^u$ scaling with temperature gradients (see § 2.2.2). The second sub-class is microinstabilities driven by the CE shear terms, or CE shear-driven microinstabilities (CES microinstabilities). This sub-classification is necessary for two reasons. First, the velocity-space anisotropy associated with the CE shear terms is different from other non-Maxwellian terms, and thus different types of microinstabilities can emerge for the two sub-classes. Secondly, as was discussed in § 2.2.3 for the case of CE plasma, the typical size of small parameters $\eta _e^T$, $\eta _e^R$, $\eta _e^u$ and $\eta _i$ is different from that of $\epsilon _e$ and $\epsilon _i$. In our initial overview of our calculations (§ 2.4) and in the more detailed discussion of our method (§ 2.5), we will consider all microinstabilities driven by the non-Maxwellian terms of the CE distribution together; however, when it comes to presenting detailed results, we will consider CET and CES microinstabilities separately (§§ 3 and 4, respectively).

2.4.1. General dispersion relation

Our linear kinetic stability calculation proceeds as follows: we consider an electromagnetic perturbation with wavevector $\boldsymbol {k}$ and (complex) frequency $\omega$ of the form

(2.69a)$$\begin{gather} \delta \boldsymbol{E} = \widehat{\delta \boldsymbol{E}} \exp\left\{\mathrm{i}\left(\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{r} - \omega t\right)\right\} , \end{gather}$$

(2.69b)$$\begin{gather}\delta \boldsymbol{B} = \widehat{\delta \boldsymbol{B}} \exp\left\{\mathrm{i}\left(\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{r} - \omega t\right)\right\} , \end{gather}$$

in a plasma with the equilibrium electron and ion distribution functions given by (2.8a) and (2.8b), respectively. We assume that all macroscopic parameters in the CE distribution function are effectively constant on the time scales and length scales associated with microinstabilities: this is equivalent to assuming that $k \lambda _e, k \lambda _i \gg 1$ (where $k \equiv |\boldsymbol {k}|$ is the wavenumber of the perturbation), and $|\omega | \tau _L \gg 1$. To minimise confusion between quantities evolving on short, collisionless time scales, and those on long, fluid time scales, we relabel the equilibrium number density of species $s$ as $n_{s0}$, and the macroscopic magnetic field as $\boldsymbol {B}_0$ in subsequent calculations. For notational convenience, we define

(2.70)\begin{equation} \eta_e \equiv \eta_e^T , \end{equation}

and

(2.71)\begin{equation} A_e(\tilde{v}_e) \equiv A_e^T(\tilde{v}_e) + \frac{\eta_e^R}{\eta_e^T} A_e^R(\tilde{v}_e) + \frac{\eta_e^u}{\eta_e^T} A_e^u(\tilde{v}_e) , \end{equation}

which in turn allows for the equilibrium distribution function of species $s$ to be written as

(2.72)\begin{equation} f_{s0}(\tilde{v}_{s\|},\tilde{v}_{s\bot}) = \frac{n_{s0}}{v_{\mathrm{th}s}^3 {\rm \pi}^{3/2}} \exp \left(-\tilde{v}_{s}^2\right) \left[1+\eta_s A_s(\tilde{v}_s) \tilde{v}_{s\|} + \epsilon_s C_s(\tilde{v}_s) \left(\tilde{v}_{s\|}^2- \frac{\tilde{v}_{s\bot}^2}{2} \right)\right] . \end{equation}

Finally, without loss of generality, we can set $\boldsymbol {V}_{i} = 0$ by choosing to perform the kinetic calculation in the frame of the ions; thus, $\tilde {\boldsymbol {v}}_s = \boldsymbol {v}/v_{\mathrm {th}s}$.

It is well known (Stix Reference Stix1962; Parra Reference Parra2017) that the electric field of all linear electromagnetic perturbations in a collisionless, magnetised plasma with equilibrium distribution function $f_{s0}$ must satisfy

(2.73)\begin{equation} \left[\frac{c^2 k^2}{\omega^2} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right)+ \boldsymbol{\mathfrak{E}} \right] \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} = 0 , \end{equation}

where $\hat {\boldsymbol {k}} \equiv \boldsymbol {k}/k$ is the direction of the perturbation,

(2.74)\begin{equation} \boldsymbol{\mathfrak{E}} \equiv \boldsymbol{\mathsf{I}} + \frac{4 {\rm \pi}\mathrm{i}}{\omega} \boldsymbol{\sigma} ,\end{equation}

the plasma dielectric tensor and $\boldsymbol {\sigma }$ the plasma conductivity tensor. The hot-plasma dispersion relation is then given by

(2.75)\begin{equation} \mbox{det}\left[\frac{c^2 k^2}{\omega^2} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right)+ \boldsymbol{\mathfrak{E}} \right]=0 . \end{equation}

The conductivity tensor in a hot, magnetised plasma is best displayed in an orthogonal coordinate system with basis vectors $\{\hat {\boldsymbol {x}},\hat {\boldsymbol {y}},\hat {\boldsymbol {z}}\}$ defined in terms of $\boldsymbol {B}_0$ and $\boldsymbol {k}$:

(2.76a–c)\begin{equation} \hat{\boldsymbol{z}} \equiv \frac{\boldsymbol{B}_0}{B_0}, \quad \hat{\boldsymbol{x}} \equiv \frac{\boldsymbol{k}_{{\perp}}}{k_{{\perp}}} \equiv \frac{\boldsymbol{k}- k_{\|} \hat{\boldsymbol{z}}}{k_{\bot}} , \quad \hat{\boldsymbol{y}} \equiv \hat{\boldsymbol{z}} \times \hat{\boldsymbol{x}} , \end{equation}

where $B_0 \equiv |\boldsymbol {B}_0|$, $k_{\|} \equiv \boldsymbol {k}\boldsymbol {\cdot } \hat {\boldsymbol {z}}$, and $k_{\bot } \equiv |\boldsymbol {k}_{\perp }|$. In this notation, $\boldsymbol {k} = k_{\|} \hat {\boldsymbol {z}} + k_{\bot } \hat {\boldsymbol {x}}$. The conductivity tensor is then given by

(2.77)\begin{align} \boldsymbol{\sigma} = \sum_s \boldsymbol{\sigma} _s & ={-} \frac{\mathrm{i}}{4 {\rm \pi}\omega} \sum_s \omega_{{\rm p}s}^2 \left[ \frac{2}{\sqrt{\rm \pi}} \frac{k_{\|}}{|k_{\|}|} \int_{-\infty}^{\infty} \mathrm{d} \tilde{w}_{s\|} \, \tilde{w}_{s\|} \int_0^{\infty} \mathrm{d} \tilde{v}_{s\bot}\, \varLambda_s(\tilde{w}_{s\|},\tilde{v}_{s\bot}) \hat{\boldsymbol{z}} \hat{\boldsymbol{z}}\right. \nonumber\\ &\quad \left.+ \,\tilde{\omega}_{s\|} \frac{2}{\sqrt{\rm \pi}} \int_{C_L} \mathrm{d} \tilde{w}_{s\|} \int_0^{\infty} \mathrm{d} \tilde{v}_{s\bot}\, \tilde{v}_{s\bot}^2 \varXi_s(\tilde{w}_{s\|},\tilde{v}_{s\bot}) \sum_{n ={-}\infty}^{\infty} \frac{\boldsymbol{\mathsf{R}}_{sn}}{\zeta_{sn} -\tilde{w}_{s\|}} \right] , \end{align}

where

(2.78)$$\begin{gather} \omega_{{\rm p}s} \equiv \sqrt{\frac{4 {\rm \pi}Z_s^2 e^2 n_{s0}}{m_s}} , \end{gather}$$

(2.79)$$\begin{gather}\tilde{w}_{s\|} \equiv \frac{k_{\|} \tilde{v}_{s\|}}{|k_{\|}|}, \end{gather}$$

(2.80)$$\begin{gather}\tilde{\rho}_s \equiv \frac{m_s c v_{\mathrm{th}s}}{Z_{s} e B_0} = \frac{|Z_{s}|}{Z_{s}} \rho_s, \end{gather}$$

(2.81)$$\begin{gather}\tilde{\omega}_{s\|} \equiv \frac{\omega}{|k_{\|}| v_{\mathrm{th}s}} , \end{gather}$$

(2.82)$$\begin{gather}\zeta_{sn} \equiv \tilde{\omega}_{s\|} - \frac{n}{|k_{\|}| \tilde{\rho}_s} , \end{gather}$$

(2.83)$$\begin{gather}\tilde{f}_{s0}(\tilde{v}_{s\|},\tilde{v}_{s\bot}) \equiv \frac{{\rm \pi}^{3/2} v_{\mathrm{th}s}^3}{n_{s0}} f_{s0}\left(\frac{k_{\|}}{|k_{\|}|} v_{\mathrm{th}s} \tilde{w}_{s\|},v_{\mathrm{th}s} \tilde{v}_{s\bot}\right) , \end{gather}$$

(2.84)$$\begin{gather}\varLambda_s(\tilde{w}_{s\|},\tilde{v}_{s\bot}) \equiv \tilde{v}_{s\bot} \frac{\partial \tilde{f}_{s0}}{\partial \tilde{w}_{s\|}}-\tilde{w}_{s\|} \frac{\partial \tilde{f}_{s0}}{\partial \tilde{v}_{s\bot}} , \end{gather}$$

(2.85)$$\begin{gather}\varXi_s(\tilde{w}_{s\|},\tilde{v}_{s\bot}) \equiv \frac{\partial \tilde{f}_{s0}}{\partial \tilde{v}_{s\bot}} + \frac{\varLambda_s(\tilde{w}_{s\|},\tilde{v}_{s\bot})}{\tilde{\omega}_{s\|}} , \end{gather}$$

(2.86a)$$\begin{align} (\boldsymbol{\mathsf{R}}_{sn} )_{xx} &\equiv \frac{n^2 {J}_n(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot})^2}{k_{\bot}^2 \tilde{\rho}_s^2 \tilde{v}_{s\bot}^2} , \end{align}$$

(2.86b)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{xy} &\equiv \frac{\mathrm{i} n {J}_n(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot}) {J}_n'(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot})}{k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot}} , \end{align}$$

(2.86c)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{xz} &\equiv \frac{n {J}_n(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot})^2}{k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot}} \frac{k_{\|} \tilde{w}_{s\|}}{|k_{\|}| \tilde{v}_{s\bot}} , \end{align}$$

(2.86d)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{yx} &\equiv{-} (\boldsymbol{\mathsf{R}}_{sn} )_{xy} , \end{align}$$

(2.86e)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{yy} &\equiv {J}_n'(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot})^2 , \end{align}$$

(2.86f)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{yz} &\equiv{-}\mathrm{i} n {J}_n(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot}) {J}_n'(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot}) \frac{k_{\|} \tilde{w}_{s\|}}{|k_{\|}| \tilde{v}_{s\bot}} , \end{align}$$

(2.86g)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{zx} &\equiv (\boldsymbol{\mathsf{R}}_{sn} )_{xz} , \end{align}$$

(2.86h)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{zy} &\equiv{-}(\boldsymbol{\mathsf{R}}_{sn} )_{yz} , \end{align}$$

(2.86i)$$\begin{align}(\boldsymbol{\mathsf{R}}_{sn} )_{zz} &\equiv \frac{\tilde{w}_{s\|}^2}{\tilde{v}_{s\bot}^2} {J}_n(k_{\bot} \tilde{\rho}_s \tilde{v}_{s\bot})^2 . \end{align}$$

Here, $(\boldsymbol{\mathsf{R}}_{sn})_{xy} = \hat {\boldsymbol {x}} \boldsymbol {\cdot } \boldsymbol{\mathsf{R}}_{sn} \boldsymbol {\cdot } \hat {\boldsymbol {y}}$, and similarly for other components of $\boldsymbol{\mathsf{R}}_{sn}$. The integration in (2.77) is carried out over the Landau contour $C_L$; this is simply the real axis for growing perturbations, but has a more complicated form for non-growing ones (Landau Reference Landau1946; Schekochihin Reference Schekochihin2024). For the reader's convenience, a summary of the derivation of the hot-plasma dispersion relation is given in Appendix C.

We note that the dielectric and conductivity tensors have the following symmetries:

(2.87a–c)$$\begin{gather} \mathfrak{E}_{yx} ={-} \mathfrak{E}_{xy} , \quad \mathfrak{E}_{zx} = \mathfrak{E}_{xz} , \quad \mathfrak{E}_{zy} ={-} \mathfrak{E}_{yz} , \end{gather}$$

(2.88a–c)$$\begin{gather}\sigma_{yx} ={-} \sigma_{xy} , \quad \sigma_{zx} = \sigma_{xz} , \quad \sigma_{zy} ={-} \sigma_{yz} , \end{gather}$$

where, for tensors with no species subscript, we use the notation $\mathfrak {E}_{xy} \equiv \hat {\boldsymbol {x}} \boldsymbol {\cdot } \boldsymbol {\mathfrak {E}} \boldsymbol {\cdot } \hat {\boldsymbol {y}}$. We also observe that if $f_{s0}(v_{\|},v_{\bot })$ is an even function with respect to $v_{\|}$, then, for $k_{\|} > 0$,

(2.89a)$$\begin{align} \sigma_{xx}({-}k_{\|}) &= \sigma_{xx}(k_{\|}) , \end{align}$$

(2.89b)$$\begin{align}\sigma_{xy}({-}k_{\|}) &= \sigma_{xy}(k_{\|}) , \end{align}$$

(2.89c)$$\begin{align}\sigma_{xz}({-}k_{\|}) &={-}\sigma_{xz}(k_{\|}) , \end{align}$$

(2.89d)$$\begin{align}\sigma_{yy}({-}k_{\|}) &= \sigma_{yy}(k_{\|}) , \end{align}$$

(2.89e)$$\begin{align}\sigma_{yz}({-}k_{\|}) &={-}\sigma_{yz}(k_{\|}) , \end{align}$$

(2.89f)$$\begin{align}\sigma_{zz}({-}k_{\|}) &= \sigma_{zz}(k_{\|}) , \end{align}$$

with the remaining components of the conductivity tensor given by (2.87a–c). If $f_{s0}(v_{\|},v_{\bot })$ is an odd function with respect to $v_{\|}$, then

(2.90a)$$\begin{align} \sigma_{xx}({-}k_{\|}) &={-}\sigma_{xx}(k_{\|}) , \end{align}$$

(2.90b)$$\begin{align}\sigma_{xy}({-}k_{\|}) &={-}\sigma_{xy}(k_{\|}) , \end{align}$$

(2.90c)$$\begin{align}\sigma_{xz}({-}k_{\|}) &= \sigma_{xz}(k_{\|}) , \end{align}$$

(2.90d)$$\begin{align}\sigma_{yy}({-}k_{\|}) &={-}\sigma_{yy}(k_{\|}) , \end{align}$$

(2.90e)$$\begin{align}\sigma_{yz}({-}k_{\|}) &= \sigma_{yz}(k_{\|}) , \end{align}$$

(2.90f)$$\begin{align}\sigma_{zz}({-}k_{\|}) &={-}\sigma_{zz}(k_{\|}) . \end{align}$$

These symmetries can be used to determine completely the behaviour of perturbations with $k_{\|} < 0$ directly from perturbations with $k_{\|} > 0$, without any additional calculations. Thus, unless stated otherwise, from this point on, we assume $k_{\|} > 0$, and thus $\tilde {w}_{s\|} = \tilde {v}_{s\|}$ (see (2.79)).

2.4.2. Simplifications of dispersion relation: overview of our approach

The full hot-plasma dispersion relation (2.75) is a transcendental equation, and thus, for general distribution functions, the growth rates of perturbations can only be determined numerically; this hinders the systematic investigation of stability over wide-ranging parameter regimes. However, adopting a few simplifications both to the form of the CE distribution functions (2.72) and to the type of microinstabilities being considered (see § 2.3.4) turns out to be advantageous when attempting a systematic study. It enables us to obtain simple analytical results for microinstability growth rates and characteristic wavenumbers, as well as greatly reducing the numerical cost of evaluating these quantities. The former allows us to make straightforward comparisons between microinstabilities, while the latter facilitates the calculation of stability plots over a wide range of parameters without requiring intensive computational resources.

First, we choose a Krook collision operator, with constant collision time $\tau _s$ for each species $s$ (Bhatnagar et al. Reference Bhatnagar, Gross and Krook1954), when evaluating the isotropic functions $A_e^T(\tilde {v}_e)$, $A_e^R(\tilde {v}_e)$, $A_e^u(\tilde {v}_e)$, $A_i(\tilde {v}_i)$, $C_e(\tilde {v}_e)$ and $C_i(\tilde {v}_i)$ in (2.72). As was explained in § 2.2.1, these functions are determined by the collision operator. While the full Landau collision operator might seem to be the most appropriate choice, the conductivity tensor $\boldsymbol {\sigma }$ defined by (2.77) cannot be written in terms of standard mathematical functions if this choice is made. Instead, the relevant integrals must be done numerically. If a simplified collision operator is assumed, $\boldsymbol {\sigma }$ can be evaluated analytically with only a moderate amount of algebra. In Appendix B.2.1, we show that for the Krook collision operator

(2.91a)$$\begin{align} A_e^T(\tilde{v}_e) &={-}\left(\tilde{v}_e^2-\tfrac{5}{2}\right) , \end{align}$$

(2.91b)$$\begin{align}A_e^R(\tilde{v}_e) &={-}1 , \end{align}$$

(2.91c)$$\begin{align}A_e^u(\tilde{v}_e) &= 0 , \end{align}$$

(2.91d)$$\begin{align}A_i(\tilde{v}_e) &={-}\left(\tilde{v}_i^2-\tfrac{5}{2}\right) , \end{align}$$

(2.91e)$$\begin{align}C_e(\tilde{v}_e) &={-}1 , \end{align}$$

(2.91f)$$\begin{align}C_i(\tilde{v}_i) &={-}1 , \end{align}$$

where it is assumed that $\tilde {v}_e, \tilde {v}_i \ll \eta _e^{-1/3}, \epsilon _i^{-1/2}$ in order that the CE distribution functions retain positive signs (the vanishing of the CE electron–ion-drift term is discussed in Appendix B.2.1). Adopting the Krook collision operator has the additional advantage of allowing a simple prescription for collisional damping of microinstabilities to be introduced self-consistently into our stability calculation (see § 2.5.7 for further discussion of this).

Secondly, as discussed in § 2.3.4, the most important microinstabilities associated with the CE distribution function are low frequency, i.e. they satisfy (2.68). Therefore, instead of solving the full hot-plasma dispersion relation, we can obtain a less complicated algebraic dispersion relation. We also always consider electromagnetic rather than electrostatic perturbations. This is because it can be shown for a CE plasma that purely electrostatic microinstabilities are limited to the quasi-cold-plasma modes (see Appendix D). Describing how the simplified dispersion relation for low-frequency, electromagnetic perturbations is obtained from the full hot-plasma dispersion relation requires a rather lengthy exposition, and necessitates the introduction of a substantial amount of additional mathematical notation. In addition to this, certain shortcomings of this approach warrant an extended discussion. Readers who are interested these details will find them in the next section (§ 2.5). Readers who are instead keen to see the results of the stability calculations as soon as possible are encouraged to jump to §§ 3 and 4.

2.5.1. Low-frequency condition in a magnetised plasma

Before applying to the hot-plasma dispersion relation (2.75) the simplifications discussed in § 2.4.2, we refine the low-frequency condition (2.68) based on the specific form (2.77) of the conductivity tensor for a magnetised plasma. It is clear that the equilibrium distribution function only affects the conductivity tensor via the functions $\varLambda _s(\tilde {v}_{s\|},\tilde {v}_{s\bot })$ and $\varXi _s(\tilde {v}_{s\|},\tilde {v}_{s\bot })$ (see (2.84) and (2.85)). For a distribution function of the form (2.72), it can be shown that

(2.92)\begin{equation} \varLambda_s(\tilde{v}_{s\|},\tilde{v}_{s\bot})={-} \tilde{v}_{s\bot} \exp \left(-\tilde{v}_{s}^2\right) \left[\eta_s A_s(\tilde{v}_s) - 3 \epsilon_s C_s(\tilde{v}_s) \tilde{v}_{s\|} \right] , \end{equation}

and

(2.93)\begin{align} \varXi_s(\tilde{v}_{s\|},\tilde{v}_{s\bot}) & ={-} \tilde{v}_{s\bot} \exp \left(-\tilde{v}_{s}^2\right) \left[ 2 + 2 \tilde{v}_{s\|} \eta_s A_s(\tilde{v}_s) - \frac{\tilde{v}_{s\|}}{\tilde{v}_s} \eta_s A_s'(\tilde{v}_s)\right. \nonumber\\ & \quad + 2 \epsilon_s C_s(\tilde{v}_s) \left(\tilde{v}_{s\|}^2-\frac{\tilde{v}_{s\bot}^2}{2}+\frac{1}{2}\right) - \frac{1}{\tilde{v}_s} \left(\tilde{v}_{s\|}^2-\frac{\tilde{v}_{s\bot}^2}{2}\right) \epsilon_s C_s'(\tilde{v}_s) \nonumber\\ &\left.\quad +\, \frac{\eta_s}{ \tilde{\omega}_{s\|}} A_s(\tilde{v}_s) - 3 \frac{\epsilon_s}{ \tilde{\omega}_{s\|}} C_s(\tilde{v}_s) \tilde{v}_{s\|} \right] , \end{align}

where the first term in the square brackets in (2.93) originates from the Maxwellian part of the distribution function. A comparison of the size of the second, third, fourth and fifth terms with the first indicates that for $\tilde {v}_s \sim 1$ – for which $\varXi _s$ attains its largest characteristic values – the non-Maxwellian terms of the CE distribution function only provide a small, ${O}(\eta _e, \epsilon _e)$ contribution, and thus the conductivity is only altered slightly. However, considering the sixth and seventh terms in the square brackets in (2.93) (which are only present thanks to the anisotropy of the CE distribution function), it is clear that the non-Maxwellian contribution to the conductivity tensor can be significant for $\tilde {v}_s \sim 1$ provided the frequency (2.81) satisfies one of

(2.94)\begin{equation} \tilde{\omega}_{s\|} \sim \eta_s \ll 1 \quad \mathrm{or} \quad \tilde{\omega}_{s\|} \sim \epsilon_s \ll 1 . \end{equation}

Thus, the relevant low-frequency condition in a magnetised plasma involves the parallel particle streaming rate $k_{\|} v_{\mathrm {th}s}$.

There do exist certain caveats to the claim that it is necessary for microinstabilities of CE plasma to satisfy (2.94); we defer detailed statement and discussion of these caveats – as well as of other potential shortcomings of our approach – to §§ 2.5.6, 2.5.7 and 2.5.8.

2.5.2. Simplification I: non-relativistic electromagnetic fluctuations

The requirement that the mode be electromagnetic, combined with the fact we are interested in non-relativistic fluctuations ($\omega \ll k c$) enables our first simplification. We see from (2.75) that for any perturbation of interest, the dielectric tensor must satisfy $\|\boldsymbol {\mathfrak {E}}\| \gtrsim k^2 c^2/\omega ^2 \gg 1$ (where $\| \cdot \|$ is the Euclidean tensor norm); therefore, it simplifies to

(2.95)\begin{equation} \boldsymbol{\mathfrak{E}} \approx \frac{4 {\rm \pi}\mathrm{i}}{\omega} \boldsymbol{\sigma} . \end{equation}

This amounts to ignoring the displacement current in the Ampère–Maxwell law, leaving Ampère's original equation. For convenience of exposition, we denote the contribution of each species $s$ to (2.95) by

(2.96)\begin{equation} \boldsymbol{\mathfrak{E}}_s \equiv \frac{4 {\rm \pi}\mathrm{i}}{\omega} \boldsymbol{\sigma}_s . \end{equation}

2.5.3. Simplification II: expansion of dielectric tensor in $\omega \ll k_{\|} v_{\mathrm {th}s}$

The next simplification involves an expansion of the matrices $\boldsymbol {\mathfrak {E}}_s$ in the small parameters $\tilde {\omega }_{s\|} \sim \eta _s \sim \epsilon _s \ll 1$. The general principle of the expansion is as follows. We first divide the matrix $\boldsymbol {\mathfrak {E}}_s$ (see (2.74), (2.77) and (2.96)) into the Maxwellian contribution $\boldsymbol{\mathsf{M}}_s$ and the non-Maxwellian one $\boldsymbol{\mathsf{P}}_s$:

(2.97)\begin{equation} \boldsymbol{\mathfrak{E}}_s = \frac{\omega_{{\rm p}s}^2}{\omega^2} \left(\boldsymbol{\mathsf{M}}_s + \boldsymbol{\mathsf{P}}_s \right) , \end{equation}

where the $\omega _{{\rm p}s}^2/\omega ^2$ factor is introduced for later convenience. Next, we note that, for a Maxwellian distribution, $\varLambda _s(\tilde {v}_{s\|},\tilde {v}_{s\bot }) = 0$ (see (2.84)), whereas $\varLambda _s \sim \epsilon _{s}, \eta _s$ for the non-Maxwellian component of the CE distribution function. Thus, from (2.77) considered under the ordering $k \rho _s \sim 1$, $\boldsymbol{\mathsf{M}}_s = {O}(\tilde {\omega }_{s\|})$ as $\tilde {\omega }_{s\|} \rightarrow 0$, while $\boldsymbol{\mathsf{P}}_s = {O}(\eta _s, \epsilon _s)$. The expansion of $\boldsymbol{\mathsf{M}}_s$ and $\boldsymbol{\mathsf{P}}_s$ in $\tilde {\omega }_{s\|}$ is, therefore,

(2.98a)$$\begin{align} \boldsymbol{\mathsf{M}}_s\!\left(\tilde{\omega}_{s\|},\boldsymbol{k}\right) &\equiv \tilde{\omega}_{s\|} \boldsymbol{\mathsf{M}}_s^{(0)} \!\left(\boldsymbol{k} \right) + \tilde{\omega}_{s\|}^2 \boldsymbol{\mathsf{M}}_s^{(1)}\!\left(\boldsymbol{k} \right) + \dots , \end{align}$$

(2.98b)$$\begin{align}\boldsymbol{\mathsf{P}}_s\!\left(\tilde{\omega}_{s\|},\boldsymbol{k}\right) &\equiv \boldsymbol{\mathsf{P}}_s^{(0)}\!\left(\boldsymbol{k} \right) + \tilde{\omega}_{s\|} \boldsymbol{\mathsf{P}}_s^{(1)}\!\left(\boldsymbol{k}\right) + \dots, \end{align}$$

where the matrices $\boldsymbol{\mathsf{M}}_s^{(0)}$ and $\boldsymbol{\mathsf{M}}_s^{(1)}$ are ${O}(1)$ functions of $\boldsymbol {k}$ only, and $\boldsymbol{\mathsf{P}}_s^{(0)}$ and $\boldsymbol{\mathsf{P}}_s^{(1)}$ are ${O}(\eta _s, \epsilon _s)$. We then expand $\boldsymbol {\mathfrak {E}}_s$ as follows:

(2.99)\begin{equation} \boldsymbol{\mathfrak{E}}_s = \tilde{\omega}_{s\|} \boldsymbol{\mathfrak{E}}_s^{(0)} + \tilde{\omega}_{s\|}^2 \boldsymbol{\mathfrak{E}}_s^{(1)} + \dots , \end{equation}

where

(2.100a)$$\begin{gather} \boldsymbol{\mathfrak{E}}_s^{(0)} \equiv \frac{\omega_{{\rm p}s}^2}{\omega^2} \left[ \boldsymbol{\mathsf{M}}_s^{(0)}\!\left(\boldsymbol{k} \right) + \frac{1}{\tilde{\omega}_{s\|}} \boldsymbol{\mathsf{P}}_s^{(0)}\!\left(\boldsymbol{k}\right) \right] , \end{gather}$$

(2.100b)$$\begin{gather}\boldsymbol{\mathfrak{E}}_s^{(1)} \equiv \frac{\omega_{{\rm p}s}^2}{\omega^2} \left[ \boldsymbol{\mathsf{M}}_s^{(1)}\!\left(\boldsymbol{k} \right) + \frac{1}{\tilde{\omega}_{s\|}} \boldsymbol{\mathsf{P}}_s^{(1)}\!\left(\boldsymbol{k}\right) \right] . \end{gather}$$

2.5.4. Additional symmetries of low-frequency dielectric tensor ${\mathfrak{E}}$$_s^{(0)}$

The tensor $\boldsymbol {\mathfrak {E}}_s^{(0)}$ defined by (2.100a) has some rather convenient additional symmetries, which lead to significant simplification of the dispersion relation. In Appendix F we show that in combination with the general symmetries (2.87a–c), which apply to $\boldsymbol {\mathfrak {E}}_s^{(0)}$ in addition to $\boldsymbol {\mathfrak {E}}$, for any distribution function of particle species $s$ with a small anisotropy:

(2.101a)$$\begin{gather} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{xz} ={-} \frac{k_{\bot}}{k_{\|}} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{xx} , \end{gather}$$

(2.101b)$$\begin{gather}(\boldsymbol{\mathfrak{E}}_s^{(0)})_{yz} = \frac{k_{\bot}}{k_{\|}} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{xy} , \end{gather}$$

(2.101c)$$\begin{gather}(\boldsymbol{\mathfrak{E}}_s^{(0)})_{zz} = \frac{k_{\bot}^2}{k_{\|}^2} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{xx} . \end{gather}$$

These symmetries have the consequence that

(2.102)\begin{equation} \hat{\boldsymbol{k}} \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}}_s^{(0)} = \boldsymbol{\mathfrak{E}}_s^{(0)} \boldsymbol{\cdot} \hat{\boldsymbol{k}} = 0 . \end{equation}

As a result of this identity, it is convenient to calculate the components of $\boldsymbol {\mathfrak {E}}_s^{(0)}$ (and $\boldsymbol {\mathfrak {E}}_s$) in the coordinate basis $\{\boldsymbol {e}_1,\boldsymbol {e}_2,\boldsymbol {e}_3\}$ defined by

(2.103a–c)\begin{equation} \boldsymbol{e}_1 \equiv \hat{\boldsymbol{y}} \times \hat{\boldsymbol{k}} , \quad \boldsymbol{e}_2 \equiv \hat{\boldsymbol{y}} , \quad \boldsymbol{e}_3 \equiv \hat{\boldsymbol{k}} . \end{equation}

Carrying out this calculation (see Appendix F), we find

(2.104a)$$\begin{align} &(\boldsymbol{\mathfrak{E}}_s^{(0)})_{11} = \frac{k^2}{k_{\|}^2} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{xx} , \end{align}$$

(2.104b)$$\begin{align} &(\boldsymbol{\mathfrak{E}}_s^{(0)})_{12} ={-} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{21} = \frac{k}{k_{\|}} (\boldsymbol{\mathfrak{E}}_s^{(0)})_{xy} , \end{align}$$

(2.104c)$$\begin{align} &(\boldsymbol{\mathfrak{E}}_s^{(0)})_{22} = (\boldsymbol{\mathfrak{E}}_s^{(0)})_{yy} , \end{align}$$

(2.104d)$$\begin{align} &(\boldsymbol{\mathfrak{E}}_s^{(0)})_{13} = (\boldsymbol{\mathfrak{E}}_s^{(0)})_{31} = (\boldsymbol{\mathfrak{E}}_s^{(0)})_{23} = (\boldsymbol{\mathfrak{E}}_s^{(0)})_{32} = (\boldsymbol{\mathfrak{E}}_s^{(0)})_{33} = 0 , \end{align}$$

where $(\boldsymbol {\mathfrak {E}}_s^{(0)})_{ij}$ is the $(i,j)$th component of $\boldsymbol {\mathfrak {E}}_s^{(0)}$ in the basis $\{\boldsymbol {e}_1,\boldsymbol {e}_2,\boldsymbol {e}_3 \}$. We conclude that, if $k \rho _s \sim 1$ and $\tilde {\omega }_{s\|} \ll 1$, the components of $\boldsymbol {\mathfrak {E}}_s$ satisfy

(2.105)\begin{equation} (\boldsymbol{\mathfrak{E}}_s)_{13} \sim (\boldsymbol{\mathfrak{E}}_s)_{23} \sim (\boldsymbol{\mathfrak{E}}_s)_{33} \sim \tilde{\omega}_{s\|} (\boldsymbol{\mathfrak{E}}_s)_{11} \sim \tilde{\omega}_{s\|} (\boldsymbol{\mathfrak{E}}_s)_{12} \sim \tilde{\omega}_{s\|} (\boldsymbol{\mathfrak{E}}_s)_{22} .\end{equation}

These components can be written in terms of the components of $\boldsymbol {\mathfrak {E}}_s$ in the $\{\hat {\boldsymbol {x}},\hat {\boldsymbol {y}},\hat {\boldsymbol {z}}\}$ coordinate frame (see (2.76a–c)) via a coordinate transformation; the resulting expressions are rather bulky, so we do not reproduce them here – they are detailed in Appendix G.

2.5.5. Consequences for dispersion relation

On account of the additional symmetries described in the previous section, a simplified dispersion relation for low-frequency modes can be derived in place of the full hot-plasma dispersion relation (2.75). However, depending on the frequency and characteristic wavelengths of modes, this derivation has a subtlety because of the large discrepancy between ion and electron masses. In, e.g. a two-species plasma with $\mu _e = m_e/m_i \ll 1$ (and ion charge $Z_i$), we have

(2.106)\begin{equation} \frac{\tilde{\omega}_{e\|}}{\tilde{\omega}_{i\|}} = \sqrt{\mu_e \tau} , \end{equation}

where $\tau = T_i/T_e$. If $\tau \sim 1$ (as would be expected in a collisional plasma on macroscopic evolution time scales $\tau _L$ greater than the ion–electron temperature-equilibration time $\tau _{ie}^{\rm eq}$ – cf. (2.54)), then $\tilde {\omega }_{i\|} \sim {\mu _e}^{-1/2} \tilde {\omega }_{e\|} \gg \tilde {\omega }_{e\|}$. Thus, in general, $\tilde {\omega }_{i\|} \not \sim \tilde {\omega }_{e\|}$, and any dispersion relation will, in principle, depend on an additional (small) dimensionless parameter $\mu _e$. This introduces various complications to the simplified dispersion relation's derivation, most significant of which being that, since $\rho _e = Z_i \mu _e^{1/2} \tau ^{-1/2} \rho _i \ll \rho _i$ (for $Z_i \gtrsim 1$), to assume the ordering $k \rho _s \sim 1$ for both ions and electrons is inconsistent (see § 2.5.6).

To avoid the description of our approach being obscured by these complications, we consider a special case at first: we adopt the ordering $k \rho _e \sim 1$ in a two-species plasma and assume that $\tilde {\omega }_{i\|} \sim \mu _e^{-1/2} \tilde {\omega }_{e\|} \ll 1$. In this case, $\tilde {\omega }_{i\|} \|\boldsymbol {\mathfrak {E}}_i^{(0)}\| \sim \mu _e^{1/2} Z_i \tau ^{-1/2} \tilde {\omega }_{e\|} \|\boldsymbol {\mathfrak {E}}_e^{(0)}\| \ll \tilde {\omega }_{e\|} \|\boldsymbol {\mathfrak {E}}_e^{(0)}\|$, and so the dielectric tensor $\boldsymbol {\mathfrak {E}}$ is given by

(2.107)\begin{equation} \boldsymbol{\mathfrak{E}} = \tilde{\omega}_{e\|} \boldsymbol{\mathfrak{E}}^{(0)} + \tilde{\omega}_{e\|}^2 \boldsymbol{\mathfrak{E}}^{(1)} + \dots , \end{equation}

where

(2.108a)$$\begin{align} \boldsymbol{\mathfrak{E}}^{(0)} &\equiv \boldsymbol{\mathfrak{E}}_e^{(0)} + \frac{\tilde{\omega}_{i\|}}{\tilde{\omega}_{e\|}} \boldsymbol{\mathfrak{E}}_i^{(0)} \approx \boldsymbol{\mathfrak{E}}_e^{(0)} , \end{align}$$

(2.108b)$$\begin{align}\boldsymbol{\mathfrak{E}}^{(1)} &\equiv \boldsymbol{\mathfrak{E}}_e^{(1)} + \frac{\tilde{\omega}_{i\|}^2}{\tilde{\omega}_{e\|}^2} \boldsymbol{\mathfrak{E}}_i^{(1)} . \end{align}$$

Thus, to leading order in the $\tilde {\omega }_{e\|} \ll 1$ expansion, only the electron species contributes to the dielectric tensor for electron-Larmor-scale modes. We revisit the derivation of simplified dispersion relations for CE microinstabilities more generally in § 2.5.6.

To derive the simplified dispersion relation for electron-Larmor-scale modes, we start by considering the component of (2.73) for the electric field that is parallel to the wavevector $\hat {\boldsymbol {k}}$,

(2.109)\begin{equation} \hat{\boldsymbol{k}} \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}} \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} = 0 , \end{equation}

and then substitute the expanded form (2.107) of the dielectric tensor (with $s = e$). The orthogonality of $\boldsymbol {\mathfrak {E}}_e^{(0)}$ to $\hat {\boldsymbol {k}}$ – viz. (2.102) – implies that (2.109) becomes

(2.110)\begin{equation} \hat{\boldsymbol{k}} \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}}^{(1)} \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} =\mathfrak{E}_{33}^{(1)} \hat{\boldsymbol{k}} \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} + \hat{\boldsymbol{k}} \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}}^{(1)} \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}}_T ={O}(\tilde{\omega}_{e\|} |\widehat{\delta \boldsymbol{E}}|) , \end{equation}

where the transverse electric field is defined by $\widehat {\delta \boldsymbol {E}}_{T} \equiv \widehat {\delta \boldsymbol {E}} \boldsymbol {\cdot } (\boldsymbol{\mathsf{I}} - \hat {\boldsymbol {k}} \hat {\boldsymbol {k}} )$. In Appendix D.2, we show that for $\tilde {\omega }_{e\|}, \tilde {\omega }_{i\|} \ll 1$,

(2.111)\begin{equation} \mathfrak{E}_{33}^{(1)} \approx \frac{\omega_{{\rm p}e}^2}{\omega^2} \frac{2 k_{\|}^2}{k^2} (1+Z_i\tau^{{-}1}) \left[1+ {O}(\eta_e, \epsilon_e)\right] . \end{equation}

Since this is strictly positive, we can rewrite (2.110) to give the electrostatic field in terms of the transverse electric field:

(2.112)\begin{equation} \hat{\boldsymbol{k}} \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} ={-}\left(\mathfrak{E}_{33}^{(1)} \right)^{{-}1} \left(\hat{\boldsymbol{k}} \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}}^{(1)} \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}}_T \right) . \end{equation}

We conclude that $|\hat {\boldsymbol {k}} \boldsymbol {\cdot } \widehat {\delta \boldsymbol {E}}| \sim |\widehat {\delta \boldsymbol {E}}_T|$ for all low-frequency perturbations with $k_{\|} \sim k$; a corollary of this result is that there can be no low-frequency purely electrostatic perturbations (see Appendix D.4.1 for an alternative demonstration of this).

We can now derive the dispersion relation from the other two components of (2.73),

(2.113)\begin{equation} \left[\frac{c^2 k^2}{\omega^2} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right)+ \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right) \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}} \right] \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} = 0 , \end{equation}

by (again) substituting the expanded dielectric tensor (2.107) into (2.113):

(2.114)\begin{equation} \left[\tilde{\omega}_{e\|} \boldsymbol{\mathfrak{E}}^{(0)} + \frac{c^2 k^2}{\omega^2} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right) \right] \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}}_T ={-} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right) \boldsymbol{\cdot} \left( \boldsymbol{\mathfrak{E}} - \tilde{\omega}_{e\|} \boldsymbol{\mathfrak{E}}^{(0)} \right) \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}} , \end{equation}

where we have used the identity

(2.115)\begin{equation} \boldsymbol{\mathfrak{E}}^{(0)} = \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right) \boldsymbol{\cdot} \boldsymbol{\mathfrak{E}}^{(0)} \boldsymbol{\cdot} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right) , \end{equation}

and ordered $k^2 c^2/\omega ^2 \sim \tilde {\omega }_{e\|} \|\boldsymbol {\mathfrak {E}}^{(0)}\|$. The ratio of the right-hand side of (2.114) to the left-hand side is ${O}(\tilde {\omega }_{e\|})$; we thus conclude that, to leading order in the $\tilde {\omega }_{e\|} \ll 1$ expansion,

(2.116)\begin{equation} \left[\tilde{\omega}_{e\|} \boldsymbol{\mathfrak{E}}_e^{(0)} + \frac{c^2 k^2}{\omega^2} \left(\hat{\boldsymbol{k}}\hat{\boldsymbol{k}}-\boldsymbol{\mathsf{I}}\right) \right] \boldsymbol{\cdot} \widehat{\delta \boldsymbol{E}}_T = 0 , \end{equation}

and the dispersion relation is approximately

(2.117)\begin{equation} \left[\tilde{\omega}_{e\|} (\boldsymbol{\mathfrak{E}}_e^{(0)})_{11}-\frac{k^2 c^2}{\omega^2}\right]\left[\tilde{\omega}_{e\|} (\boldsymbol{\mathfrak{E}}_e^{(0)})_{22}-\frac{k^2 c^2}{\omega^2}\right]+\left[\tilde{\omega}_{e\|} (\boldsymbol{\mathfrak{E}}_e^{(0)})_{12}\right]^2 = 0 . \end{equation}

Finally, writing the dielectric tensor in terms of $\boldsymbol{\mathsf{M}}_{e}$ and $\boldsymbol{\mathsf{P}}_{e}$ as defined by (2.97), we find

(2.118) \begin{align} &\left[\tilde{\omega}_{e\|} (\boldsymbol{\mathsf{M}}_{e}^{(0)})_{11} + (\boldsymbol{\mathsf{P}}_{e}^{(0)})_{11}- k^2 d_e^2\right]\left[\tilde{\omega}_{e\|} (\boldsymbol{\mathsf{M}}_{e}^{(0)})_{22} + (\boldsymbol{\mathsf{P}}_{e}^{(0)})_{22} - k^2 d_e^2\right] \nonumber\\ & + \left[\tilde{\omega}_{e\|} (\boldsymbol{\mathsf{M}}_{e}^{(0)})_{12} + (\boldsymbol{\mathsf{P}}_{e}^{(0)})_{12}\right]^2 = 0 , \end{align}

where $d_e = c/\omega _{{\rm p}e}$ is the electron inertial scale (see (2.58b)). This can be re-written as a quadratic equation in $\omega$ – and thus, expressions for the complex frequency of any low-frequency perturbation can be found for any given positive wavenumber. We note that the electron inertial scale is related to the electron-Larmor radius by $d_e = \rho _e \beta _e^{-1/2}$; therefore, our expansion scheme is only consistent with the low-frequency assumption (2.94) under our assumed ordering, $\tilde {\omega }_{e\|} \sim \beta _e^{-1}$, when $\beta _e \gg 1$.

We note that one only needs to know $\boldsymbol {\mathfrak {E}}_e^{(0)}$ in order to obtain the dispersion relation of low-frequency perturbations and the transverse component of the electric field, whereas to determine the electrostatic component of the electric field (and other quantities, such as the density perturbation – see Appendix H), one must go to higher order in the $\tilde {\omega }_{e\|} \ll 1$ expansion. Since we are primarily interested in microinstability growth rates and wavenumber scales, we will not explicitly calculate the electrostatic fields associated with perturbations using (2.112), and thus can avoid the rather laborious calculation of $\boldsymbol {\mathfrak {E}}^{(1)}$ for CE distribution functions. We do, however, in Appendix G.1.3 derive an explicit expression for $\boldsymbol {\mathfrak {E}}^{(1)}$ for a plasma with Maxwellian distribution functions for all particle species; this in turn allows us to relate the electrostatic electric field to the transverse field for such a plasma (see Appendix I).

For the sake of completeness, we also observe that if the non-Maxwellian part of the CE distribution function is even with respect to $v_{\|}$, the transformation rules (2.89) combined with (2.104) imply that a perturbation with a negative parallel wavenumber $k_{\|}$ will obey exactly the same dispersion relation as a perturbation for a positive parallel wavenumber, viz. for $k_{\|} > 0$,

(2.119)\begin{equation} \boldsymbol{\mathsf{P}}_e^{(0)}\!\left({-}k_{\|}, k_{\bot}\right) = \boldsymbol{\mathsf{P}}_e^{(0)}\!\left(k_{\|}, k_{\bot} \right) . \end{equation}

If instead the non-Maxwellian part is odd, then, for $k_{\|} > 0$,

(2.120)\begin{equation} \boldsymbol{\mathsf{P}}_e^{(0)}\!\left({-}k_{\|}, k_{\bot}\right) ={-}\boldsymbol{\mathsf{P}}_e^{(0)}\!\left(k_{\|}, k_{\bot} \right) . \end{equation}

The dispersion relation for perturbations with $k_{\|} < 0$ can, therefore, be recovered by considering perturbations with $k_{\|} > 0$, but under the substitution $\boldsymbol{\mathsf{P}}_{e}^{(0)} \rightarrow -\boldsymbol{\mathsf{P}}_{e}^{(0)}$. Thus, we can characterise all unstable perturbations under the assumption that $k_{\|} > 0$.

In all subsequent calculations, we require the Maxwellian part $\boldsymbol{\mathsf{M}}_e^{(0)}$ of the dielectric tensor. The elements of the matrix $\boldsymbol{\mathsf{M}}_s^{(0)}$ of species $s$ are as follows:

(2.121a)$$\begin{align} (\boldsymbol{\mathsf{M}}_{s}^{(0)})_{11} &= \mathrm{i} \frac{k^2}{k_{\|}^2} F\!\left(k_{\|} \tilde{\rho}_s,k_{\bot} \tilde{\rho}_s\right) , \end{align}$$

(2.121b)$$\begin{align}(\boldsymbol{\mathsf{M}}_{s}^{(0)})_{12} &={-}\mathrm{i} \frac{k}{k_{\|}} G\!\left(k_{\|} \tilde{\rho}_s,k_{\bot} \tilde{\rho}_s\right) , \end{align}$$

(2.121c)$$\begin{align}(\boldsymbol{\mathsf{M}}_{s}^{(0)})_{21} &= \mathrm{i} \frac{k}{k_{\|}} G\!\left(k_{\|} \tilde{\rho}_s,k_{\bot} \tilde{\rho}_s\right) , \end{align}$$

(2.121d)$$\begin{align}(\boldsymbol{\mathsf{M}}_{s}^{(0)})_{22} &= \mathrm{i} H\!\left(k_{\|} \tilde{\rho}_s,k_{\bot} \tilde{\rho}_s\right) , \end{align}$$

where the functions $F(x,y)$, $G(x,y)$ and $H(x,y)$ are

(2.122a)$$\begin{align} F\!\left(x,y\right) &\equiv \frac{4 \sqrt{\rm \pi}}{y^2} \exp{\left(-\frac{y^2}{2}\right)} \sum_{m=1}^{\infty} m^2 {I}_{m}\!\left(\frac{y^2}{2}\right)\exp{\left(-\frac{m^2}{x^2}\right)} , \end{align}$$

(2.122b)$$\begin{align}G\!\left(x,y\right) &\equiv \exp{\left(-\frac{y^2}{2}\right)} \sum_{m ={-}\infty}^{\infty} m \, {\rm Re}\,{Z\!\left(\frac{m}{x}\right)} \left[ {I}_{m}'\!\left(\frac{y^2}{2}\right)- {I}_{m}\! \left(\frac{y^2}{2}\right)\right] , \end{align}$$

(2.122c)$$\begin{align}H\!\left(x,y\right) &\equiv F\!\left(x,y\right) + \sqrt{\rm \pi} y^2 \exp{\left(-\frac{y^2}{2}\right)} \sum_{m ={-}\infty}^{\infty} \left[ {I}_{m}\!\left(\frac{y^2}{2}\right)- {I}_{m}'\!\left(\frac{y^2}{2}\right)\right] \exp{\left(-\frac{m^2}{x^2}\right)} , \end{align}$$

where ${I}_{m}(\alpha )$ is the $m$th modified Bessel function, and

(2.123)\begin{equation} Z\!\left(z\right) = \frac{1}{\sqrt{\rm \pi}} \int_{C_L} \frac{\mathrm{d}u \exp{\left({-}u^2\right)}}{u-z} ,\end{equation}

is the plasma dispersion function (Fried & Conte Reference Fried and Conte1961). The derivation of these results from the full dielectric tensor (which is calculated in Appendix G.1.1) for a plasma whose constituent particles all have Maxwellian distributions is presented in Appendices G.1.2 (expansion in the $\{\hat {\boldsymbol {x}},\hat {\boldsymbol {y}},\hat {\boldsymbol {z}}\}$ basis) and G.1.3 (expansion in the $\{\boldsymbol {e}_1,\boldsymbol {e}_2,\boldsymbol {e}_3 \}$ basis).

2.5.6. Effect of multiple species on dispersion-relation derivations

We now relax the assumptions adopted in § 2.5.5 that the low-frequency modes of interest are on electron-Larmor scales, and discuss how we derive simplified dispersion relations for (low-frequency) CE microinstabilities more generally.

First, it is unnecessarily restrictive to assume that, for all CE microinstabilities, $\tilde {\omega }_{s\|} \ll 1$ for all particle species. There are some instabilities for which $\tilde {\omega }_{e\|} \sim \eta _e \sim \epsilon _e \ll 1$ while $\tilde {\omega }_{i\|} \gtrsim 1$. Recalling the orderings $\tilde {\omega }_{e\|} \sim \beta _e^{-1}$ and $k \rho _e \sim 1$ that were adopted for the electron-Larmor-scale instabilities described in § 2.5.5, it follows that $\tilde {\omega }_{i\|} \gtrsim 1$ whenever $\beta _e \lesssim \tau ^{-1/2} \mu _e^{-1/2}$; in other words, electron-Larmor-scale CE microinstabilities in plasmas with $\beta _e$ that is not too large will satisfy $\tilde {\omega }_{i\|} \gtrsim 1$. Therefore, we cannot naively apply our low-frequency approximation to both $\boldsymbol {\mathfrak {E}}_e$ and $\boldsymbol {\mathfrak {E}}_i$ in all cases of interest. We will remain cognisant of this in the calculations that follow – a concrete example of $\tilde {\omega }_{i\|} \gtrsim 1$ will be considered in § 3.3.1.

Secondly, because of the large separation between electron- and ion-Larmor scales, it is necessary to consider whether the approximation $\boldsymbol{\mathsf{M}}_s(\tilde {\omega }_{s\|},\boldsymbol {k}) \approx \tilde {\omega }_{s\|} \boldsymbol{\mathsf{M}}_s^{(0)}(\boldsymbol {k})$ remains valid for parallel or perpendicular wavenumbers much larger or smaller than the inverse Larmor radii of each species. We show in Appendix G.1.6 that the leading-order term in the $\tilde {\omega }_{s\|} \ll 1$ expansion remains larger than higher-order terms for all $k_{\|} \rho _s \gtrsim 1$ (as, indeed, was implicitly assumed in § 2.5.5). However, for $k_{\|} \rho _s$ sufficiently small, the same statement does not hold for all components of $\boldsymbol{\mathsf{M}}_s$. More specifically, it is shown in the same appendix that the dominant contribution to $\boldsymbol{\mathsf{M}}_s(\boldsymbol {k})$ when $k_{\|} \rho _s \ll 1$ instead comes from the quadratic term $\tilde {\omega }_{s\|}^2 \boldsymbol{\mathsf{M}}_s^{(1)}(\boldsymbol {k})$ (rather than any higher-order term). Thus, in general, our simplified dispersion relation for low-frequency modes in a two-species plasma has the form of a quartic in $\omega$, rather than a quadratic, if $k_{\|} \rho _s \ll 1$ for at least the electron species. Physically, the reason why a quadratic dispersion relation is no longer a reasonable approximation is the existence of more than two low-frequency modes in a two-species Maxwellian plasma in certain wavenumber regimes. For example, for quasi-parallel modes with characteristic parallel wavenumbers satisfying $k_{\|} \rho _i \ll 1$, there are four low-frequency modes (see § 4.4.1). Nevertheless, in other situations, the components of $\boldsymbol{\mathsf{M}}_s(\boldsymbol {k})$ for which the $\boldsymbol{\mathsf{M}}_s(\tilde {\omega }_{s\|},\boldsymbol {k}) \approx \tilde {\omega }_{s\|} \boldsymbol{\mathsf{M}}_s^{(0)}(\boldsymbol {k})$ approximation breaks down are not important, on account of their small size compared with terms in the dispersion relation associated with other Maxwellian components. In this case, the original quadratic dispersion relation is sufficient. An explicit wavenumber regime in which this is realised is $k_{\|} \rho _e \sim k_{\perp } \rho _e \ll 1$ but $k \rho _i \gg 1$ – see §§ 4.3.4 and 4.4.7.

Taking these multiple-species effects into account, the reasons behind the decision made in § 2.3.4 to consider the CES microinstabilities separately from the CET microinstabilities come into plain focus. First, the characteristic sizes of the CE electron-temperature-gradient and ion-temperature-gradient terms are comparable ($\eta _i \sim \eta _e$), while the CE ion-shear term is much larger than the CE electron-shear term: $\epsilon _i \sim \mu _e^{-1/2} \epsilon _e$. This has the consequence that the natural orderings of $\tilde {\omega }_{e\|}$ and $\tilde {\omega }_{i\|}$ with respect to other parameters are different for CES and CET microinstabilities. Secondly, the fact that the velocity-space anisotropy associated with the CE temperature-gradient terms differs from the CE shear terms – which excite microinstabilities with different characteristic wavevectors – means that the form of the dispersion relations of CET and CES microinstabilities are distinct. More specifically, the dispersion relation for CET microinstabilities at both electron and ion scales can always be simplified to a quadratic equation in $\omega$; in contrast, for CES microinstabilities, the dispersion relation cannot in general be reduced to anything simpler than a quartic.

2.5.7. Modelling collisional effects on CE microinstabilities

As proposed thus far, our method for characterising microinstabilities in a CE plasma does not include explicitly the effect of collisions on the microinstabilities themselves. In principle, this can be worked out by introducing a collision operator into the linearised Maxwell–Vlasov–Landau equation from which the hot-plasma dispersion relation (2.75) is derived. Indeed, if a Krook collision operator is assumed (as was done in § 2.4.2 when determining the precise form of the CE distribution functions of ions and electrons), the resulting modification of the hot-plasma dispersion relation is quite simple: the conductivity tensor (2.77) remains the same, but with the substitution

(2.124)\begin{equation} \tilde{\omega}_{s\|} \rightarrow {\hat{\omega}}_{s\|} \equiv \tilde{\omega}_{s\|} + \frac{\mathrm{i}}{k_{\|} \lambda_s} , \end{equation}

in the resonant denominators (see Appendix C). As for how this affects the simplifications to the dispersion relation outlined in § 2.5.3, the expansion parameter in the dielectric tensor's expansion (2.99) is altered, becoming ${\hat {\omega }}_{s\|} \ll 1$ (as opposed to $\tilde {\omega }_{s\|} \ll 1$); in other words, $\|\boldsymbol {\mathfrak {E}}_s^{(1)}\|/\|\boldsymbol {\mathfrak {E}}_s^{(0)}\| \sim {\hat {\omega }}_{s\|}$.

The latter result leads to an seemingly counterintuitive conclusion: collisions typically fail to stabilise low-frequency instabilities in CE plasma if $\omega \tau _s \lesssim 1$ (where $\tau _s$ is the collision time of species $s$) but $k_{\|} v_{\mathrm {th}i} \tau _s = k_{\|} \lambda _s \gg 1$. This is because the simplified dispersion relation (2.118) only involves leading-order terms in the expanded dielectric tensor. These terms are independent of ${\hat {\omega }}_{s\|}$, and thus the growth rate of any microinstability that is adequately described by (2.118) does not depend on the size of $\omega \tau _s$. For these microinstabilities, the effect of collisions only becomes relevant if

(2.125)\begin{equation} k_{\|} \lambda_s \lesssim 1 . \end{equation}

This is inconsistent with the assumptions $k \lambda _e \gg 1$, $k \lambda _i \gg 1$ made when setting up our calculation in § 2.4.1. Thus, the only regime where collisions can reasonably be included in our calculation is one where they are typically not important. An exception to this rule arises when two-species plasma effects mean that the first-order terms in the ${\hat {\omega }}_{s\|} \ll 1$ expansion are needed for a correct characterisation of the growth rate of certain microinstabilities (see § 2.5.6); for these instabilities, we include the effect of collisions using (2.124).

Although our calculation is not formally valid when (2.125) holds, so we cannot show explicitly that growth ceases, this condition nonetheless represents a sensible criterion for suppression of microinstabilities by collisional damping. Physically, it signifies that collisions are strong enough to scatter a particle before it has streamed across a typical wavelength of fluctuation excited by a microinstability. This collisional scattering prevents particles from being resonant, which in turn would suppress the growth of many different microinstabilities. However, we acknowledge that there exist microinstabilities that do not involve resonant-particle populations (e.g., the firehose instability – see §§ 2.3.3 and 4.4.1), and thus it cannot be rigorously concluded from our work that all microinstabilities are suppressed when (2.125) applies.

Yet even without an actual proof of collisional stabilisation, there is another reason implying that (2.125) is a reasonable threshold for microinstabilities: the characteristic growth time of microinstabilities at wavenumbers satisfying (2.125) is comparable to the evolution time $\tau _L$ of macroscopic motions in the plasma. To illustrate this idea, we consider the ordering (2.94) relating the complex frequency of microinstabilities to the small parameter $\epsilon _s$ for CES (CE shear-driven) microinstabilities, and use it to estimate

(2.126)\begin{equation} \omega \tau_L \sim \epsilon_s k_{\|} v_{\mathrm{th}s} \tau_L \lesssim \epsilon_s \frac{L_V}{\lambda_s} \frac{v_{\mathrm{th}s}}{V} , \end{equation}

where $V \sim L_V/\tau _L$ is the characteristic ion bulk-flow velocity. Considering orderings (2.55), it follows that $\epsilon _e \sim \mu _e^{1/2} \epsilon _i$, and so

(2.127)\begin{equation} \epsilon_i \frac{v_{\mathrm{th}i}}{V} \sim \epsilon_e \frac{v_{\mathrm{th}e}}{V} \sim \frac{\lambda_e}{L_V} \sim \frac{\lambda_i}{L_V} . \end{equation}

Then (2.126) becomes

(2.128)\begin{equation} \omega \tau_L \lesssim 1 , \end{equation}

implying (as claimed) that the CES microinstability growth rate is smaller than the fluid turnover rate $\tau _L^{-1}$. Spelled out clearly, this means that the underlying quasiequilibrium state changes before going unstable. Similar arguments can be applied to CET (CE temperature-gradient-driven) microinstabilities.

Thus, (2.125) represents a lower bound on the characteristic wavenumbers at which microinstabilities can operate. We shall therefore assume throughout the rest of this paper that microinstabilities are suppressed (or rendered irrelevant) if they satisfy (2.125).

2.5.8. Caveats: microinstabilities in CE plasma where $\omega /k_{\|} v_{\mathrm {th}s} \not \sim \eta _s, \epsilon _s$

As mentioned in § 2.4.2, there are a number of important caveats to the claim that the ordering (2.94) must be satisfied by microinstabilities in a CE plasma.

The first of these is that our comparison of non-Maxwellian with the Maxwellian terms in expression (2.93) for $\varXi _s$ is in essence a pointwise comparison at characteristic values of $\tilde {v}_s$ for which $\varXi _s$ attains its largest typical magnitude. However, $\varXi _s$ affects the components of the conductivity tensor via the velocity integral of its product with a complicated function of frequency and wavenumber (see (2.77)). Thus, it does not necessarily follow that the ratio of the integrated responses of the Maxwellian and non-Maxwellian contributions to the conductivity tensor is the same as the pointwise ratio of the respective contributions to $\varXi _s$. In some circumstances, this can result in the Maxwellian part being smaller than anticipated, leading to faster microinstabilities. An example of this phenomenon was given in § 2.5.6: for $k_{\|} \rho _s \ll 1$, the characteristic magnitude of the Maxwellian contribution to some components of the dielectric tensor is ${O}(\tilde {\omega }_{s\|}^2)$, as compared with the naive estimate ${O}(\tilde {\omega }_{s\|})$. This leads to certain CES microinstabilities (for example, the CE ion-shear-driven firehose instability – § 4.4.1) satisfying a modified low-frequency condition

(2.129)\begin{equation} \tilde{\omega}_{s\|} \sim \epsilon_s^{1/2} \ll 1 . \end{equation}

A similar phenomenon affects the limit $k_{\|} \rightarrow 0$ for fixed $k_{\bot }$, in which case it can be shown that the Maxwellian contribution to $\sigma _{zz}$ is ${O}(k_{\|}/k_{\bot })$; this leads to a CES microinstability (the CE electron-shear-driven ordinary-mode instability – see § 4.4.11) satisfying a modified ordering

(2.130)\begin{equation} \frac{\omega}{k_{\bot} v_{\mathrm{th}s}}\sim \epsilon_s \ll 1 . \end{equation}

The second caveat is that for some plasma modes, the particles predominantly responsible for collisionless damping or growth are suprathermal, i.e. $\tilde {v}_s \gg 1$. Then the previous comparison of terms in (2.93) is not applicable. Modes of this sort are the quasi-cold plasma modes discussed in § 2.3.4 and Appendix D. They can be unstable, but always with a growth rate that is exponentially small in $\eta _s$ and $\epsilon _s$.

In spite of these two caveats, we proceed by considering the full hot-plasma dispersion relation (2.75) in the low-frequency limit $\omega \ll k_{\|} v_{\mathrm {th}s}$. This approach enables the treatment of all microinstabilities satisfying condition

(2.131a,b)\begin{equation} \tilde{\omega}_{s\|} \sim \eta_s^{\iota_\eta} , \quad\epsilon_s^{\iota_\epsilon} \ll 1 , \end{equation}

where ${\iota _\eta }$ and ${\iota _\epsilon }$ are fractional powers not much smaller than unity. Similarly to the discussion in § 2.3.4, we claim that the microinstabilities satisfying the low-frequency condition (2.131a,b) are likely to be the most rapid of all possible microinstabilities in CE plasma. A formal justification of this claim relies on the argument – presented in Appendix E – that for all plasma modes satisfying $\omega \gtrsim k_{\|} v_{\mathrm {th}s}$ and $|{\rm Re}\,{\omega }| \gg |{\rm Im}\,{\omega }|$, the growth rate is exponentially small in $\eta _s$ and $\epsilon _s$. By definition, this class of modes includes the quasi-cold modes. In a plasma where $\epsilon _s, \eta _s \ll 1$, the growth rates of such microinstabilities will be exponentially small, and thus of little significance. The only situation that we are aware of in which the low-frequency condition (2.131a,b) is not appropriate is the aforementioned CES ordinary-mode instability; a separate treatment of it involving the full hot-plasma dispersion relation is provided in Appendix K.3.13.

3.3.1. Parallel whistler (heat-flux) instability

The CET whistler instability, which has been studied previously by a number of authors (Levinson & Eichler Reference Levinson and Eichler1992; Pistinner & Eichler Reference Pistinner and Eichler1998; Gary & Li Reference Gary and Li2000; Roberg-Clark et al. Reference Roberg-Clark, Drake, Reynolds and Swisdak2016; Komarov et al. Reference Komarov, Schekochihin, Churazov and Spitkovsky2018; Roberg-Clark et al. Reference Roberg-Clark, Drake, Reynolds and Swisdak2018a,Reference Roberg-Clark, Drake, Swisdak and Reynoldsb; Kuzichev et al. Reference Kuzichev, Vasko, Soto-Chavez, Tong, Artemyev, Bale and Spitkovsky2019; Shaaban et al. Reference Shaaban, Lazar, Yoon, Poedts and López2019; Drake et al. Reference Drake, Pfrommer, Reynolds, Ruszkowski, Swisdak, Einarsson, Thomas, Hassam and Roberg-Clark2021), is driven by parallel electron heat fluxes. These heat fluxes introduce the asymmetry to the CE electron distribution function (i.e. the electron-temperature-gradient term), which, if it is sufficiently large, can overcome electron-cyclotron damping of (electromagnetic) whistler waves and render them unstable. The instability is mediated by gyroresonant wave–particle interactions that allow whistlers to drain free energy from electrons with parallel velocities $v_{\|} = \pm \varOmega _e/k_{\|}$. For a positive, parallel electron heat flux, which is driven by an anti-parallel temperature gradient ($\boldsymbol {\nabla }_{\|} T_e < 0$, so $\eta _e^T < 0$), it is only whistlers with a positive parallel wavenumber that are unstable. Whistler waves with both parallel and oblique wavevectors with respect to the magnetic field can be destabilised, although the parallel modes are the fastest-growing ones.

The CET whistler instability is most simply characterised analytically for parallel wavenumbers (i.e. $k = k_{\|}$). Then, it can be shown (see Appendix J.3.1, and also Levinson & Eichler Reference Levinson and Eichler1992; Roberg-Clark et al. Reference Roberg-Clark, Drake, Reynolds and Swisdak2016) that the real frequency $\varpi$ and growth rate $\gamma$ at arbitrary $k_{\|} > 0$ are given by

(3.5a)$$\begin{align} \frac{\varpi}{\varOmega_e} &= \eta_e^T \left(\frac{k_\| \rho_e}{4} - \frac{1}{2 k_\| \rho_e}\right) - \frac{\left(\eta_e^T/2 + k_\|^3 \rho_e^3/\beta_e\right) {\rm Re}{\,Z\!\left({1}/{k_\| \rho_e}\right)} }{ \left[{\rm Re}{\,Z\!\left({1}/{k_\| \rho_e}\right)}\right]^2 + {\rm \pi}\exp{\left(-{2}/{k_\|^2 \rho_e^2}\right)}} , \end{align}$$

(3.5b)$$\begin{align}\frac{\gamma}{\varOmega_e} &={-}\frac{\sqrt{\rm \pi}\left(\eta_e^T/2 + k_\|^3 \rho_e^3/\beta_e\right) }{\left[{\rm Re}{\,Z\!\left({1}/{k_\| \rho_e}\right)}\right]^2 \exp{\left({1}/{k_\|^2 \rho_e^2}\right)}+ {\rm \pi}\exp{\left(-{1}/{k_\|^2 \rho_e^2}\right)}} . \end{align}$$

For $\eta _e^T > 0$, $\gamma < 0$, but if $\eta _e^T < 0$, then $\gamma$ is non-negative for $k_{\|} \rho _e \leqslant (\eta _e^T \beta _e/2)^{1/3}$. The dispersion curves $\varpi = \varpi (k_{\|})$ and $\gamma = \gamma (k_\|)$ of unstable whistler waves with parallel wavevectors for three different values of $|\eta _e^T| \beta _e$ are plotted in figure 3 using the above formulae. For $|\eta _e^T| \beta _e \gtrsim 1$, the range of unstable parallel wavenumbers, $\Delta k_{\|}$, is comparable to the characteristic wavenumber of the instability: $\Delta k_{\|} \sim k_{\|} \sim \rho _e^{-1}$.

Figure 3.

Parallel CET whistler instability. Dispersion curves of unstable whistler modes, whose instability is driven by the electron-temperature-gradient term in the CE distribution function (3.1a), for wavevectors that are co-parallel with the background magnetic field (viz. $\boldsymbol {k} = k_{\|} \hat {\boldsymbol {z}}$). The frequency (solid blue) and growth rates (solid red) of the modes are calculated using (3.5a) and (3.5b), respectively. The resulting frequencies and growth rates, when normalised as $\gamma \beta _e/\varOmega _e$, are functions of the dimensionless quantity $\eta _e^{T} \beta _e$; we show the dispersion curves for three different values of $\eta _e^{T} \beta _e$. The approximations (3.6a) and (3.6b) for the frequency (dotted blue) and growth rate (dotted red) in the limit $k_{\|} \rho _e \ll 1$ are also plotted, as are the approximations (3.9a) and (3.9b) for the frequency (dashed blue) and growth rate (dashed red) in the limit $k_{\|} \rho _e \gg 1$.

The expressions (3.5a) and (3.5b) can be simplified in two subsidiary limits, which in turn allows for the derivation of analytic expressions for the maximum growth rate of the instability and the (parallel) wavenumber at which that growth rate is realised.

First, adopting the ordering $k_{\|} \rho _e \sim (|\eta _e^T| \beta _e)^{1/3} \ll 1$ under which the destabilising $\eta _e^T$ terms and the stabilising electron finite-Larmor-radius (FLR) terms are the same order, we find

(3.6a)$$\begin{align} \varpi&\approx \frac{k_\|^2 \rho_e^2}{\beta_e} \varOmega_e , \end{align}$$

(3.6b)$$\begin{align}\gamma &\approx{-}\frac{\sqrt{\rm \pi}}{k_\|^2 \rho_e^2} \left( \frac{\eta_e^T}{2} + \frac{k_\|^3 \rho_e^3}{\beta_e}\right) \exp{\left(-\frac{1}{k_\|^2 \rho_e^2}\right)} \varOmega_e . \end{align}$$

The frequency corresponds to that of a whistler wave in the $k_{\|} \rho _e \ll 1$ limit (Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013). The fastest growth, which occurs at the wavenumber

(3.7)\begin{equation} k_{\|} \rho_e \approx \left(\frac{|\eta_e^T| \beta_e}{2}\right)^{1/3} -\frac{|\eta_e^T| \beta_e}{4} , \end{equation}

is exponentially slow in $|\eta _e^T| \beta _e \ll 1$:

(3.8)\begin{equation} \gamma_{\rm max} \approx \frac{3 \sqrt{\rm \pi}}{4} |\eta_e^T| \exp{\left[-\frac{2^{2/3}}{\left(|\eta_e^T| \beta_e\right)^{2/3}}-1\right]} \varOmega_e . \end{equation}

Next, considering the opposite limit $k_\| \rho _e \gg 1$, we obtain

(3.9a)$$\begin{align} \varpi &\approx \left[\eta_e^T \beta_e\left(\frac{1}{4} k_\| \rho_e-\frac{{\rm \pi}-2}{2 {\rm \pi}k_\| \rho_e}\right)+ \frac{2}{\rm \pi} k_\|^2 \rho_e^2 \right] \frac{\varOmega_e}{\beta_e} , \end{align}$$

(3.9b)$$\begin{align}\gamma &\approx{-}\frac{1}{\sqrt{\rm \pi}} \left[\eta_e^T \beta_e\left(\frac{1}{2}-\frac{4-{\rm \pi}}{2 {\rm \pi}k_{\|}^2 \rho_e^2}\right) + k_{\|}^3 \rho_e^3 \right] \frac{\varOmega_e}{\beta_e} . \end{align}$$

We then find that the maximum growth rate of the parallel mode is given by

(3.10)\begin{align} \gamma_{\rm max} & \approx \frac{|\eta_e^{T}|}{\sqrt{\rm \pi}} \left\{1- \left[\frac{1}{\sqrt{\rm \pi}}\left(\frac{4}{\rm \pi}-1\right)\right]^{3/5} \left[\left(\frac{3}{2}\right)^{2/5}-\left(\frac{2}{3}\right)^{3/5}\right]\left(|\eta_e^{T}|\beta_e \right)^{{-}2/5} \right\}\varOmega_e \nonumber\\ & \approx 0.56 |\eta_e^{T}| \left[1-0.13 \left(|\eta_e^{T}|\beta_e \right)^{{-}2/5} \right] \varOmega_e , \end{align}

at the parallel wavenumber

(3.11)\begin{equation} k_{\|} \rho_e = \left[\frac{2}{3\sqrt{\rm \pi}}\left(\frac{4}{\rm \pi}-1\right)\right]^{1/5} \left(|\eta_e^{T}|\beta_e \right)^{1/5} \approx 0. 63 \left(|\eta_e^{T}|\beta_e \right)^{1/5} . \end{equation}

In addition, we see that the real frequency of modes with $k_\| \rho _e \lesssim (|\eta _e^T| \beta _e/{2})^{1/3}$ is larger than the growth rate of the mode: $\varpi \sim k_\| \rho _e \gamma \gg \gamma$. Thus, these modes oscillate more rapidly than they grow.

The approximate expressions for (3.6) and (3.9) are valid in the limits $|\eta _e^T| \beta _e \ll 1$ and $|\eta _e^T| \beta _e \gg 1$, respectively, and are plotted in figure 3 alongside the exact results (3.5). Of particular note is the accuracy of the approximate expression (3.9b) for the growth rate when $k_{\|} \rho _e \gtrsim 0.6$; this suggests that (3.10) is a reasonable estimate of the peak growth rate for $|\eta _e^T| \beta _e \gtrsim 1$.

3.3.2. Oblique whistler (heat-flux) instability

Analytical expressions for the frequency and growth rate of unstable modes with an oblique wavevector at an angle to the magnetic field are more complicated than the analogous expressions for parallel modes. In Appendix J.3, we show that there are two low-frequency oblique modes, whose complex frequencies $\omega$ are given by

(3.12)\begin{equation} \omega = \frac{\varOmega_e}{\beta_e} k_{\|} \rho_e \frac{-B_{T} \pm \sqrt{B_{T}^2 + 4A_{T}C_{T}}}{2 A_{T}} , \end{equation}

where the coefficients $A_{T} = A_{T}(k_{\|} \rho _e,k_{\perp } \rho _e,\eta _e^T \beta _e)$, $B_{T} = B_{T}(k_{\|} \rho _e,k_{\perp } \rho _e,\eta _e^T \beta _e)$ and $C_{T} = C_{T}(k_{\|} \rho _e,k_{\perp } \rho _e,\eta _e^T \beta _e)$ are composed of the sums and products of the special functions defined in (2.122), and also other special functions defined in Appendix G.3. For a given wavenumber, we can use (3.12) to calculate the growth rates of any unstable oblique modes – and, in particular, demonstrate that positive growth rates are present for certain values of $\eta _e^T$. When they do exist, (3.12) suggests that they will have the typical size $\gamma \sim \varOmega _e/\beta _e \sim \eta _e^T \varOmega _e$ when $k \rho _e \sim 1$ and $\eta _e^T \beta _e \sim 1$.

For $\eta _e^T > 0$, we find that both modes (3.12) are damped; for $\eta _e^T < 0$, one mode is damped for all wavenumbers, but the other is not. Figure 4 shows the maximum (positive) growth rate $\gamma$ (normalised to $\varOmega _e/\beta _e$) of this mode at a fixed value of $\eta _e^T$, for a range of $\beta _e$. The growth rate is calculated by evaluating the imaginary part of (3.12) at a given wavenumber. For $-\eta _e^T < 1/\beta _e$, the mode of interest is damped for most wavenumbers, except for a small region of wavenumbers quasi-parallel to the magnetic field: in this region, there is a very small growth rate $\gamma \ll \varOmega _e/\beta _e$ (figure 4a). This finding is consistent with the exponentially small growth rates found for the parallel whistler modes (see (3.8)). When $-\eta _e^T \sim 1/\beta _e$, there is a marked change in behaviour: a larger region of unstable modes appears, with $\gamma \sim \varOmega _e/\beta _e$, at wavenumbers $k \rho _e \sim 1$ (figure 4b,c). The growth rate is the largest for parallel modes – but there also exist oblique modes with $k_{\bot } \lesssim k_{\|}$ whose growth rate is close to the peak growth rate. For example, for $\eta _e^T \beta _e = -4$, we find that the growth rate of the fastest-growing mode with a wavevector angle $\theta = 10^{\circ }$ is only ${\sim }2\,\%$ smaller than the fastest-growing parallel mode; for a wavevector angle $\theta = 20^{\circ }$, the reduction is by ${\sim }6\,\%$; and for $\theta = 30^{\circ }$, the reduction is by ${\sim }20\,\%$. Finally, if $-\eta _e^T \gg 1/\beta _e$, there exists a extended region of unstable modes, with $1 \lesssim k \rho _e \lesssim |\eta _e^T \beta _e|^{1/3}$, and $\gamma \sim |\eta _e^T \varOmega _e|$ (figure 4d). Again, the peak growth rate is at $k_{\perp } = 0$, but oblique modes also have a significant growth rate (for unstable modes with $\theta = 30^{\circ }$, the reduction in the largest growth rate compared with the fastest-growing parallel mode is only by ${\sim }4\,\%$). Most of the unstable modes have a non-zero real frequency: for $-\eta _e^T \sim 1/\beta _e$, $\omega \sim \gamma$ (figure 4e), while for $-\eta _e^T \gg 1/\beta _e$, $\omega \gg \gamma$ for $k \rho _e \gg 1$ (figure 4f). Note, however, that in the latter case there exists a band of wavenumbers at which there is no real frequency.

Figure 4.

Oblique CET whistler instabilities. Maximum positive growth rates of unstable whistler modes whose instability is driven by the electron-temperature-gradient term in CE distribution function (3.1a), at arbitrary wavevectors with respect to the background magnetic field. The growth rates of the modes are calculated by taking the imaginary part of (3.12), where coefficients $A_{T}$, $B_{T}$ and $C_{T}$ are known functions of the wavevector. The growth rates are calculated on a $400^2$ grid, with equal logarithmic spacing in both perpendicular and parallel directions between the minimum and maximum wavenumbers. The resulting growth rates, when normalised as $\gamma \beta _e/\varOmega _e$, are functions of the dimensionless quantity $\eta _e^{T} \beta _e$; (a) $\eta _e^{T} \beta _e = -0.5$, (b) $\eta _e^{T} \beta _e = -4$. (c) Same as (b) but with normalisation $\gamma /|\eta _e^{T}| \varOmega _e$. (d) Same as (c), but with $\eta _e^{T} \beta _e = -100$. (e) Ratio of growth rate to absolute value of real frequency for unstable modes for $\eta _e^{T} \beta _e = -4$. ( f) Same as (e), but with $\eta _e^{T} \beta _e = -100$.

In summary, we have (re-)established that the fastest-growing modes of the CET whistler instability are parallel to the magnetic field; however, we have shown semi-analytically (a novel result of this work) that the growth of oblique perturbations can be almost as large. This result is of some significance, because it has been argued that oblique whistler modes are necessary for the instability to scatter heat-carrying electrons efficiently (see, e.g. Komarov et al. Reference Komarov, Schekochihin, Churazov and Spitkovsky2018). It was proposed previously that such modes could arise from modifications to the CET electron-temperature-gradient terms induced by the unstable parallel whistler modes rendering the oblique modes the fastest-growing ones; our calculations suggest that it would only a require a small change to the CET whistler growth rates for this to be realised.

As a further aside, we observe that in a plasma with sufficiently high plasma $\beta _e$, these oblique modes are in fact closer in nature to KAWs than to whistler waves. Whistler waves are characterised as having effectively immobile ions ($\omega \gg k_{\bot } v_{\mathrm {th}i}$), while KAWs have warm ions ($\omega \ll k_{\bot } v_{\mathrm {th}i}$); as a consequence, whistler waves have a negligible density perturbation ($\delta n_e \ll Z_i e n_{e} \varphi /T_i$, where $\varphi$ is the electrostatic potential associated with the wave), while KAWs do not: $\delta n_e \approx Z_i e n_{e} {\varphi }/T_i$ (Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013). In a $\beta _e \sim 1$ plasma for $k_{\bot } \gtrsim k_{\|}$, the real frequency of whistler modes satisfies $\omega /k_{\bot } v_{\mathrm {th}i} \sim k_{\|} \rho _i/\beta _e \sim k_{\|} \rho _i$; thus, we conclude from our above considerations that the two waves must operate in different regions of wavenumber space, viz. $k_{\|} \rho _i \ll 1$, $k_{\bot } \rho _i > 1$ for KAWs, and $k_{\|} \rho _i \gg 1$ for whistlers. However, for $\beta _e \gtrsim \mu _e^{-1/2}$ (where $\mu _e = m_e/m_i$) and $k_{\bot } \sim k_{\|} \gg \rho _i^{-1}$, the frequency of whistler waves is too low for $\omega \gg k_{\bot } v_{\mathrm {th}i}$ to be satisfied whilst also maintaining $k_{\|} \rho _e \ll 1$. Instead, the ions participate in the wave mechanism, and $\delta n_e \approx -Z_i e n_{e} {\varphi }/T_i$ (see Appendix H.2).

For further discussion of the physics of the whistler instability (as well as its nonlinear evolution), see Komarov et al. (Reference Komarov, Schekochihin, Churazov and Spitkovsky2018) and the other references given at the beginning of § 3.3.1.

3.3.3. Slow-(hydromagnetic)-wave instability

Although parallel ion heat fluxes in a classical, collisional plasma are typically much weaker than electron heat fluxes, they can still act as a free-energy source for instabilities, by introducing anisotropy to the ion distribution function (3.1b) (i.e. the CE ion-temperature-gradient term). Furthermore, anisotropy in the ion distribution function can enable the instability of plasma modes that are not destabilised by the CE electron-temperature-gradient term. This exact situation is realised in the CET slow-hydromagnetic-wave instability, in which a sufficiently large CET ion-temperature-gradient term counteracts the effect of ion-cyclotron damping on slow hydromagnetic waves. The slow hydromagnetic wave (or slow wave) (Rogister Reference Rogister1971; Foote & Kulsrud Reference Foote and Kulsrud1979) is the left-hand-polarised quasi-parallel electromagnetic mode in high-$\beta$ plasma; it exists for parallel wavenumbers $k_{\|}$ that satisfy $\beta _i^{-1/2} \ll k_{\|} \rho _i \lesssim 1$, and has a characteristic frequency $\omega \approx 2 \varOmega _i/\beta _i$. To the authors’ knowledge, no instability of the slow wave due to the ion heat flux has previously been reported. The instability's mechanism is analogous to the CET whistler instability: the slow waves drain energy from ions with parallel velocities $v_{\|} = \pm \varOmega _i/k_{\|}$ via gyroresonant wave–particle interactions. For an anti-parallel ion-temperature gradient (i.e. $\boldsymbol {\nabla }_{\|} T_i < 0$, so $\eta _i < 0$), slow waves propagating down the temperature gradient are destabilised, while those propagating up the temperature gradient are not.

As before, the slow-wave instability is most easily characterised in the subsidiary limit $k_{\bot } \rho _i \rightarrow 0$ ($k = k_{\|}$). Under the ordering $k_{\|} \rho _i \sim 1$, the real frequency $\varpi$ and growth rate $\gamma$ are given by (see Appendix J.4.1)

(3.13a)$$\begin{align} \frac{\varpi}{\varOmega_i} &= \eta_i \left(\frac{k_{\|} \rho_i}{4}-\frac{1}{2 k_\| \rho_i}\right) - \frac{k_{\|}^2 \rho_i^2 \left[{\rm Re}{\,Z\!\left({1}/{k_\| \rho_i}\right)} + k_{\|} \rho_i\right] \left(\eta_i/4 + k_\| \rho_i/\beta_i\right) }{\left[{\rm Re}{\,Z\!\left({1}/{k_\| \rho_i}\right)} + k_{\|} \rho_i\right]^2 + {\rm \pi}\exp{\left(-{2}/{k_\|^2 \rho_i^2}\right)}} , \end{align}$$

(3.13b)$$\begin{align}\frac{\gamma}{\varOmega_i} &={-} \frac{\sqrt{\rm \pi} k_{\|}^2 \rho_i^2 \left(\eta_i/4 + k_\| \rho_i/\beta_i\right) }{\left[{\rm Re}{\,Z\!\left({1}/{k_\| \rho_i}\right)} + k_{\|} \rho_i\right]^2 \exp{\left({1}/{k_\|^2 \rho_i^2}\right)}+ {\rm \pi}\exp{\left(-{1}/{k_\|^2 \rho_i^2}\right)}} . \end{align}$$

The CET electron-temperature-gradient term does not appear because its contributions to the frequency and growth rate are much smaller than the equivalent contributions of the CET ion-temperature-gradient term at $k_{\|} \rho _i \sim 1$. Plots of $\varpi = \varpi (k_{\|})$ and $\gamma = \gamma (k_{\|})$ for different values of $\eta _i \beta _i < 0$ are shown in figure 5.

Figure 5.

Parallel CET slow-hydromagnetic-wave instability. Dispersion curves of slow hydromagnetic waves whose instability is driven by the ion-temperature-gradient term in the CE distribution function (3.1b), for wavevectors co-parallel with the background magnetic field (viz. $\boldsymbol {k} = k_{\|} \hat {\boldsymbol {z}}$). The frequency (solid blue) and growth rates (solid red) of the modes are calculated using (3.13a) and (3.13b), respectively. The resulting frequencies and growth rates, when normalised as $\gamma \beta _i/\varOmega _i$, are functions of the dimensionless quantity $\eta _i \beta _i$; we show the dispersion curves for three different values of $\eta _i \beta _i$. The approximations (3.14) and (3.15) for the frequency (dotted blue) and growth rate (dotted red) in the limit $k_{\|} \rho _i \ll 1$ are also plotted, as are the approximations (3.18a) and (3.18b) for the frequency (dashed blue) and growth rate (dashed red) in the limit $k_{\|} \rho _i \gg 1$.

As with the CET whistler instability, we can derive simple expressions for the peak growth rate (and the wavenumber associated with that growth rate) in subsidiary limits.

First, ordering $k_{\|} \rho _i \sim |\eta _i| \beta _i/4 \ll 1$ so that the destabilising $\eta _i$ terms and the stabilising ion FLR terms are the same order, we find that the real frequency (3.13a) becomes

(3.14)\begin{equation} \varpi \approx \frac{2 \varOmega_i}{\beta_i} \left(1-\frac{1}{4} k_{\|} \rho_i \eta_i \beta_i -\frac{3}{2} k_{\|}^2 \rho_i^2 \right) , \end{equation}

which is precisely that of the slow hydromagnetic wave, with first-order FLR corrections included (Foote & Kulsrud Reference Foote and Kulsrud1979). For $\eta _i < 0$ and $k_{\|} \rho _i < |\eta _i| \beta _i/4$, the growth rate (3.13b) is positive:

(3.15)\begin{equation} \gamma \approx{-} \frac{4 \sqrt{\rm \pi}}{k_\|^4 \rho_i^4} \left(\frac{\eta_i}{4} + \frac{k_{\|} \rho_i}{\beta_i}\right)\exp{\left(-\frac{1}{k_\|^2 \rho_i^2}\right)} \varOmega_i . \end{equation}

The maximum growth rate (which is exponentially small in $\eta _i \beta _i/4 \ll 1$) is

(3.16)\begin{equation} \gamma_{\rm max} \approx \frac{8 \sqrt{\rm \pi}}{|\eta_i| \beta_i^2} \exp{\left(-\frac{16}{|\eta_i|^2 \beta_i^{2}}-1\right)} \varOmega_i , \end{equation}

achieved at the parallel wavenumber

(3.17)\begin{equation} k_{\|} \rho_i \approx \frac{|\eta_i| \beta_i}{4} -\frac{|\eta_i|^3 \beta_i^3}{128} . \end{equation}

In the opposite limit, $k_{\|} \rho _i \sim (|\eta _i| \beta _i/4)^{1/3} \gg 1$, we obtain

(3.18a)$$\begin{align} \varpi &\approx{-}\left(\eta_i \beta_i \frac{1-{\rm \pi}/4}{k_\| \rho_i} -k_\|^2 \rho_i^2 \right) \frac{\varOmega_i}{\beta_i} , \end{align}$$

(3.18b)$$\begin{align}\gamma &\approx{-}\sqrt{\rm \pi} \left[\frac{\eta_i}{4} \beta_i\left(1-\frac{{\rm \pi}-3}{k_{\|}^2 \rho_i^2}\right) + k_{\|} \rho_i \right] \frac{\varOmega_i}{\beta_i} . \end{align}$$

The maximum positive growth rate is

(3.19)\begin{align} \gamma_{\rm max} \approx \frac{\sqrt{\rm \pi}}{4} \left\{1- 3\left[4 \left({\rm \pi}-3\right)\right]^{1/3} \left(|\eta_i|\beta_i \right)^{{-}2/3} \right\} |\eta_i| \varOmega_i \approx 0.44 \left[1-2.48 \left(|\eta_i|\beta_i \right)^{{-}2/3} \right] |\eta_i| \varOmega_i , \end{align}

realised for $\eta _i < 0$ at the parallel wavenumber

(3.20)\begin{equation} k_{\|} \rho_i \approx \left(\frac{{\rm \pi} -3}{2}\right) ^{1/3} \left(|\eta_i|\beta_i\right)^{1/3} \approx 0.41 \left(|\eta_i|\beta_i\right)^{1/3} . \end{equation}

We note that, in contrast to the CET whistler instability, the real frequency of the fastest-growing unstable mode is smaller than its growth rate: $\omega _{\rm peak}/\gamma _{\rm max} \approx 0.36 (|\eta _i|\beta _i)^{-1/3}$.

The approximate expressions (3.14), (3.15), (3.18a) and (3.18b) for the frequency and growth rate in the limits $k_{\|} \rho _i \ll 1$ and $k_{\|} \rho _i \gg 1$, are plotted in figure 5, along with the exact results (3.13).

As with the CET whistler instability, a general expression for the complex frequency of oblique ion CET instabilities can be derived in the form (see Appendix J.4)

(3.21)\begin{equation} \omega = \frac{\varOmega_i}{\beta_i} k_{\|} |\rho_i| \frac{-\tilde{B}_{T} \pm \sqrt{\tilde{B}_{T}^2 + 4\tilde{A}_{T}\tilde{C}_{T}}}{2 \tilde{A}_{T}} , \end{equation}

where $\tilde {A}_{T} = \tilde {A}_{T}(k_{\|} \rho _i,k_{\perp } \rho _i,\eta _i \beta _i)$, $\tilde {B}_{T} = \tilde {B}_{T}(k_{\|} \rho _i,k_{\perp } \rho _i,\eta _i \beta _i)$ and $\tilde {C}_{T} = \tilde {C}_{T}(k_{\|} \rho _i,k_{\perp } \rho _i,\eta _i \beta _i)$ are again sums and products of various special mathematical functions defined in (2.122). Investigating such modes by evaluating (3.21) numerically for a range of wavenumbers (see figure 6), we find that, for $\eta _i < 0$, there is one mode that is always damped and one that can be unstable. For $-\eta _i \lesssim 4/\beta _i$, the unstable modes are restricted to quasi-parallel modes (see figure 6a); for $-\eta _i \gtrsim 4/\beta _i$, there is a much broader spectrum of unstable modes (including oblique ones). The positive growth rates of the unstable mode are shown in figure 6(b) for $\eta _i \beta _i = -8$. The typical growth rate $\gamma$ satisfies $\gamma \sim \varOmega _i/\beta _i \sim \eta _i \varOmega _i$, as anticipated from (3.21). We also observe in figure 6(b) the existence of an unstable mode at quasi-perpendicular wavenumbers, which is discussed in § 3.3.4.

Figure 6.

Oblique CET ion-Larmor-scale instabilities. Maximum positive growth rates of unstable ion-Larmor-scale modes whose instability is driven by the CE ion-temperature-gradient term in the CE distribution function (3.1b), at arbitrary wavevectors with respect to the background magnetic field. The growth rates of all modes are calculated by taking the imaginary part of (3.21), with coefficients $\tilde {A}_{T}$, $\tilde {B}_{T}$ and $\tilde {C}_{T}$ being known functions of the wavevector (see Appendix J.4). The growth rates are calculated on a $400^2$ grid, with logarithmic spacing in both perpendicular and parallel directions between the minimum and maximum wavenumber magnitudes. The resulting growth rates, when normalised as $\gamma \beta _i/\varOmega _i$, are functions of $\eta _i \beta _i$; (a) $\eta _i \beta _i = -2.5$, (b) $\eta _i \beta _i = -8$. The unstable $k_{\|} \rho _i \ll k_{\perp } \rho _i \sim 1$ modes appearing in (b) are dealt with in § 3.3.4.

In summary, an ion-temperature gradient can destabilise ion-Larmor-scale, slow hydromagnetic waves via a similar mechanism to an electron-temperature gradient destabilising electron-Larmor-scale whistler waves. If $\beta _i \gg L_{T_i}/\lambda _i$, the characteristic growth rate of these modes is $\gamma \sim \lambda _i \varOmega _i/L_{T_i}$. Unstable modes whose wavevector is parallel to $\boldsymbol {B}_0$ grow most rapidly, although the growth rate of (moderately) oblique modes is only somewhat smaller. While the CET whistler instability is faster growing than the CET slow-wave instability, both modes grow much more quickly than characteristic hydrodynamic time scales in a strongly magnetised plasma. In any conceivable saturation mechanism, the electron mode will adjust the electron heat flux, and the ion mode the ion heat flux. Thus, it seems likely that understanding the evolution (and ultimately, the saturation) of both instabilities would be necessary to model correctly the heat transport in a classical, collisional plasma that falls foul of the $\beta$-stabilisation condition.

3.3.4. Long-wavelength kinetic-Alfvén-wave instability

The instability observed in figure 6(b) at wavevectors satisfying $k_{\|} \rho _i \ll k_{\perp } \rho _i \sim 1$ is different in nature to the slow-hydromagnetic-wave instability: it is an ion-temperature-gradient-driven instability of long-wavelength KAWs. Like the CET slow-wave instability, it operates on account of resonant wave–particle interactions that allow free energy to be drained from the anisotropy of the ion distribution function, which itself arises from the ion-temperature gradient. However, the gyroresonances $v_{\|} \approx \pm \varOmega _i/k_{\|}$ operate inefficiently for modes with $k_{\|} \rho _i \ll 1$ in a CE plasma, because there are comparatively few particles with $v_{\|} \gg v_{\mathrm {th}i}$; the dominant resonance is instead the Landau resonance $v_{\|} = \omega /k_{\|}$. More specifically, KAWs with $k_{\perp } \rho _i \gtrsim 1$, which are usually subject to strong Landau and Barnes damping (that is, the damping rate of the waves is comparable to their real frequency), can be destabilised if the (ion) plasma beta is sufficiently large: $\beta _i \gtrsim L_{T_i}/\lambda _i$. In figure 6(b), the peak growth rate of the CET KAW instability is smaller than that of the CET slow-hydromagnetic-wave instability by an order of magnitude; as will be shown below, this is, in fact, a generic feature of the instability.

Similarly to quasi-parallel unstable modes, quasi-perpendicular ones such as unstable KAWs can be characterised analytically, allowing for a simple identification of unstable modes and their peak growth rates. It can be shown (see Appendix J.4.2) that, in the limit $k_{\|} \rho _i \ll 1$, $k_\perp \rho _i \sim 1$, the complex frequency of the low-frequency ($\omega \ll k_{\|} v_{\mathrm {th}i}$) modes in a plasma whose ion distribution function is (3.1b) is

(3.22)\begin{align} \frac{\omega}{k_{\|} v_{\mathrm{th}i}} & = \frac{\eta_i \mathcal{G}_i}{2 \left(1-\mathcal{F}_i\right)} + \frac{k_{{\perp}} \rho_i}{\beta_i \left(1-\mathcal{F}_i\right)^{2}} \left[ -\frac{\mathrm{i}\sqrt{\rm \pi}}{2} k_{{\perp}} \rho_i \left(\mathcal{F}_i + \sqrt{\frac{\mu_e Z_i^2}{\tau}}\right)\right. \nonumber\\ & \left.\quad \pm \sqrt{1-\frac{\rm \pi}{4} \frac{k_{{\perp}}^2 \rho_i^2}{\beta_i} \bigg(\mathcal{F}_i + \sqrt{\frac{\mu_e Z_i^2}{\tau}}\bigg)^2 - \frac{\mathrm{i} \sqrt{\rm \pi} \eta_i \beta_i}{4} \frac{2 \mathcal{G}_i - \mathcal{F}_i\left(1 - \mathcal{F}_i\right)}{1-\mathcal{F}_i} } \right] , \end{align}

where $\mathcal {F}_i \equiv \mathcal {F}(k_{\perp } \rho _i)$, $\mathcal {G}_i \equiv \mathcal {G}(k_{\perp } \rho _i)$ and

(3.23)$$\begin{align} \mathcal{F}(\alpha) &\equiv \exp{\left(-\frac{\alpha^2}{2}\right)} \left[{I}_{0}\left(\frac{\alpha^2}{2}\right) - {I}_{1}\left(\frac{\alpha^2}{2}\right)\right] , \end{align}$$

(3.24)$$\begin{align}\mathcal{G}(\alpha) &\equiv 2 \alpha^2 \mathcal{F}(\alpha)-\exp{\left(-\frac{\alpha^2}{2}\right)} {I}_{1}\left(\frac{\alpha^2}{2}\right) . \end{align}$$

In a Maxwellian plasma (i.e. when $\eta _i = 0$), (3.22) becomes

(3.25)\begin{align} \frac{\omega}{k_{\|} v_{\mathrm{th}i}} & = \frac{1}{\left(1-\mathcal{F}_i\right)^{2}} \left[ -\frac{\mathrm{i}\sqrt{\rm \pi}}{2} \frac{k_{{\perp}}^2 \rho_i^2}{\beta_i} \left(\mathcal{F}_i + \sqrt{\frac{\mu_e Z_i^2}{\tau}}\right)\right. \nonumber\\ &\quad \left.\pm \sqrt{\frac{k_{{\perp}}^2 \rho_i^2}{\beta_i^2}-\frac{\rm \pi}{4} \frac{k_{{\perp}}^4 \rho_i^4}{\beta_i^2} \left(\mathcal{F}_i + \sqrt{\frac{\mu_e Z_i^2}{\tau}}\right)^2} \right] . \end{align}

In the subsidiary limit $k_{\perp } \rho _i \gg 1$, we recover $\omega \approx \pm k_{\|} v_{\mathrm {th}i} k_{\perp } \rho _i/\beta _i$, which is the well-known dispersion relation of a KAW (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013; Kunz et al. Reference Kunz, Abel, Klein and Schekochihin2018).

For $\eta _i \neq 0$, we find that, for modes with a positive propagation direction with respect to the background magnetic field (viz. $k_{\|} > 0$), there is an instability provided

(3.26)\begin{equation} \eta_i \lesssim -3.14 \left(1 + 6.5 \sqrt{\frac{\mu_e Z_i^2}{\tau}}\right) \beta_i^{{-}1} , \end{equation}

with the perpendicular wavenumber $k_{\perp } \rho _i$ of the fastest-growing unstable mode at fixed $k_{\|}$ just beyond this threshold being approximately given by

(3.27)\begin{equation} k_{{\perp}} \rho_i \approx 1.77 \left(1 - 3.4 \sqrt{\frac{\mu_e Z_i^2}{\tau}}\right) . \end{equation}

Figure 7 shows the real frequency and growth rate of such modes at three different (negative) values of $\eta _i \beta _i$. As $\eta _i$ is decreased beyond the threshold, modes over an increasingly large range of perpendicular wavenumbers are destabilised at both super- and sub-ion-Larmor scales. Indeed, in the limit $|\eta _i| \beta _i \gg 1$, the peak growth rate $\gamma _{\rm max}$ (for a fixed $k_{\|}$) occurs at a perpendicular wavenumber $k_{\perp } \rho _i < 1$, which decreases as $|\eta _i| \beta _i$ increases. Such modes are, in fact, no longer well described physically as KAWs; their analogues in a Maxwellian plasma are Barnes-damped, non-propagating slow modes.