\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.

§.§.§ 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 (<ref>)
is derived. Indeed, if a Krook collision operator is assumed (as was done in section <ref> 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 (<ref>) remains the same, but
with the substitution

\begin{equation}
\tilde{\omega}_{s\|} \rightarrow {\hat{\omega}}_{s\|} \equiv \tilde{\omega}_{s\|} +
\frac{\mathrm{i}}{k_{\|} \lambda_s} \, , \label{collresconduct}
\end{equation}
in the resonant denominators (see appendix <ref>).
As for how this affects the simplifications to the dispersion relation outlined in section <ref>,
the expansion parameter in the dielectric tensor's expansion (<ref>)
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 (<ref>)
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 (<ref>)
does not depend on the size of $\omega \tau_s$.
For these microinstabilities, the effect of collisions only becomes relevant if

\begin{equation}
k_{\|} \lambda_s \lesssim 1 \, . \label{colldamp}
\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 section <ref>.
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 section <ref>); for
these instabilities, we include the effect of collisions using

Although our calculation is not formally valid when (<ref>) 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 sections <ref> and <ref>),
and thus it cannot be rigorously concluded from our work that all microinstabilities are suppressed when (<ref>)

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

\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} , \label{growthtime}
\end{equation}

where $V \sim L_V/\tau_L$ is the characteristic ion bulk-flow velocity. Considering
orderings (<ref>), it follows that $\epsilon_e \sim \mu_e^{1/2}
\epsilon_i$, and so

\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 (<ref>) becomes

\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, (<ref>) 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 (<ref>).

§.§.§ Caveats: microinstabilities in CE plasma where $\omega/k_{\|} v_{\mathrm{th}s} \not \sim \eta_s, \epsilon_s$

As mentioned in section <ref>, there are a number of important caveats to the claim that
the ordering (<ref>) 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 (<ref>) for $\Xi_s$ is in
essence a pointwise comparison at characteristic values of $\tilde{v}_s$ for which $\Xi_s$ attains its largest typical magnitude. However,
$\Xi_s$ affects the components of the conductivity tensor via
the velocity integral of its product with a complicated function of frequency and wavenumber [see (<ref>)].
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
$\Xi_s$. 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 section <ref>: for $k_{\|} \rho_s \ll 1$,
the characteristic magnitude of
the Maxwellian contribution to some components of the dielectric tensor is
$\textit{O}(\tilde{\omega}_{s\|}^2)$, as compared with the naive estimate
$\textit{O}(\tilde{\omega}_{s\|})$. This leads to certain CES microinstabilities (for example, the CE ion-shear-driven firehose instability – section <ref>)
satisfying a modified low-frequency condition

\begin{equation}
\tilde{\omega}_{s\|} \sim \epsilon_s^{1/2} \ll 1 . \label{omegascale_v2}
\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
$\textit{O}(k_{\|}/k_{\bot})$; this leads to
a CES microinstability (the CE electron-shear-driven ordinary-mode instability – see section <ref>) satisfying a modified ordering

\begin{equation}
\frac{\omega}{k_{\bot} v_{\mathrm{th}s}}\sim \epsilon_s \ll 1 . \label{omegascale_v3}
\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 (<ref>) is not applicable. Modes
of this sort are the quasi-cold plasma modes discussed in section
<ref> and appendix <ref>. 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 (<ref>)
in the low-frequency limit $\omega \ll k_{\|} v_{\mathrm{th}s}$. This approach enables the treatment of all microinstabilities
satisfying condition

\begin{equation}
\tilde{\omega}_{s\|} \sim \eta_s^{\iota_\eta} , \epsilon_s^{\iota_\epsilon} \ll 1 , \label{omegascale_new}
\end{equation}

where ${\iota_\eta}$ and ${\iota_\epsilon}$ are any fractional powers. Similarly to the discussion in section
<ref>, we claim that the microinstabilities satisfying the low-frequency condition (<ref>)
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 <ref>
– that for all plasma modes satisfying $\omega \gtrsim k_{\|} v_{\mathrm{th}s}$
and $|\Real{\; \omega}| \gg |\Imag{\; \omega}|$, 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 (<ref>) 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

§ CET (CHAPMAN-ENSKOG, TEMPERATURE-GRADIENT-DRIVEN) MICROINSTABILITIES

§.§ Form of CE distribution function

We consider first the non-Maxwellian terms of the CE distribution function arising from temperature gradients and electron-ion drifts.
Neglecting bulk-flow gradients [viz., setting $\epsilon_s = 0$ for both species – see (<ref>e,f)], the CE distribution functions (<ref>) for the electrons and ions become

f_e0(ṽ_e,ṽ_e)     =     n_e0/v_the^3 ^3/2 exp(-ṽ_e^2) {1-ṽ_e [η_e^T (ṽ_e^2-5/2)+η_e^R  ] } ,

f_i0(ṽ_i,ṽ_i)     =     n_i0/v_thi^3 ^3/2 exp(-ṽ_i^2) {1-η_i  ṽ_i (ṽ_i^2-5/2) } ,

where we have written out explicitly the electron-temperature-gradient [$\eta_e^T$, $\eta_i$ – see (<ref>a,d)] and electron-friction [$\eta_e^R$ – see (<ref>b)]
terms under the assumption that the Maxwell-Vlasov-Landau system from which these CE distribution functions were derived is governed
by a Krook collision operator.  We remind the reader that the electron-ion-drift term
[$\eta_e^u$ – see (<ref>c)]
disappears for this choice of collision operator. We also observe that the
non-Maxwellian part of the distribution functions (<ref>) have
odd parity; thus, any unstable mode with $k_{\|} > 0$ has a corresponding
unstable mode with $k_{\|} < 0$ and the signs of $\eta_e^T$, $\eta_e^R$, and $\eta_i$
reversed (see section <ref>, last paragraph).

The precise methodology that we
employ to calculate the growth rates of CET microinstabilities is described in
appendix <ref>; here, we focus on the results of those calculations.
In section <ref>, we will present the overview of the CET
stability landscape, while the microinstabilities referred to there will be
treated analytically in section <ref>.

§.§ Stability

We determine the stability (or otherwise) of the CE distribution functions of the form (<ref>a) and (<ref>b)
for different values of $\eta_e^T$, $\eta_e^R$, and $\eta_i$, the electron inertial scale
$d_e$, the electron-temperature scale length $L_T = |\nabla_{\|} \log{T_e}|^{-1}$, and for fixed electron and ion plasma betas ($\beta_e$ and $\beta_i$, respectively).
Stability calculations are carried out for particular combinations of values of $\eta_e^T$, $\eta_e^R$, $\eta_i$, $d_e$, $L_T$, $\beta_e$ and $\beta_i$ by solving for the maximum
microinstability growth rate across all wavevectors (see appendix <ref> for explanation of how this is done),
and determining whether this growth rate is positive for the microinstabilities whose wavelength is smaller
than the Coulomb mean free paths (a condition necessary for our calculation to be valid).

The results of one such stability
calculation – for a temperature-equilibrated hydrogen plasma ($\eta_e^T = \eta_i$, $\beta_i = \beta_e$) – are presented in figure <ref>.

CE-distribution-function stability map for CET microinstabilities. Exploration of the stability of the CE distribution functions
(<ref>a) and (<ref>b) for different values of small parameters $\eta_e^T$, $\eta_e^R$, and $\eta_i$, and the ratio of the electron inertial scale $d_e$ to the temperature scale length $L_T$, in a temperature-equilibrated hydrogen plasma. In this
plot, we chose $\eta_e^R = 0$ and $\eta_e^T = \eta_i $, and then show $\lambda_e/L_T = |\eta_e^T|$ with equal logarithmic spacing in the range $\left[10^{-5},10^{0}\right]$; $d_e/L_T$ is chosen with equal logarithmic spacing in the range
$\left[10^{-15},10^{0}\right]$. The total size of the grid is $400^2$. For reasons of efficiency, we calculate growth rates on a $40^2$ grid in wavenumber space with
logarithmic spacing for both parallel and perpendicular wavenumbers. In this plot, $\beta_e = \beta_i = 10^4$. a) Stable (blue) and unstable (red) regions of $\left(d_e/L_T,\lambda_e/L_T\right)$ phase space. The theoretically anticipated collisional cutoff [right – see (<ref>)] and $\beta$-stabilisation threshold (bottom)
of the CET whistler instability are shown as black dashed lines. b) Maximum normalised microinstability growth rate (red) versus $\lambda_e/L_T$ for a fixed electron inertial scale $d_e/L_T = 10^{-15}$,
along with analytically predicted maximum growth rate for the CET whistler instability in the limit $\lambda_e \beta_e/L_T  \gg 1$ [blue, see (<ref>)].
c) Parallel wavenumber of fastest-growing microinstability (red) versus $\lambda_e/L_T$ for a fixed electron inertial scale $d_e/L_T = 10^{-15}$,
along with the same quantity analytically predicted for the CET whistler instability in the limit $\lambda_e \beta_e/L_T \gg 1$ [blue, see (<ref>)]
d) Wavevector angle $\theta \equiv \tan^{-1}{(k_{\perp}/k_{\|})}$ of the fastest-growing instability over $\left(d_e/L_T,\lambda_e/L_T\right)$ parameter space.

In spite of the five-dimensional ($\eta_e^T,\eta_e^R,d_e,L_T,\beta_e$) parameter space that seemingly needs to be explored,
we can, in fact, convey the most salient information concerning the stability of the
CE distribution functions (<ref>) using plots over a two-dimensional $\left(d_e/L_T,\lambda_e/L_T\right)$
parameter space at a fixed $\beta_e$ [where we remind the reader that $\lambda_e/L_T = |\eta_e^T|$ – see (<ref>a)]. This reduction in phase-space dimensionality is
possible for two reasons. First, it transpires that the CE electron-friction
term of the form given in (<ref>a) does not drive any
microinstabilities, bur merely modifies the real frequency of
perturbations with respect to their Maxwellian frequencies (this is proven in appendix <ref>). Thus,
we can set $\eta_e^R = 0$ without qualitatively altering the stability properties of the CE distribution functions   (<ref>). Secondly, none of the salient stability thresholds applying to
CET microinstabilities depends on $d_e$ and $L_T$ separately:
one is a function of $d_e/L_T$, while another is independent of both

Figure <ref>a shows the regions of instability and stability of the CE
distribution function (<ref>) over the $\left(d_e/L_T,\lambda_e/L_T\right)$ parameter space.
The unstable region is bracketed by two thresholds. For $d_e/L_T$ below a
critical value $(d_e/L_T)_{\rm c0}$, stability is independent of $d_e/L_T$, and
only depends on the relative magnitude of $\lambda_e/L_T$ and $\beta_e$:
CET microinstabilities are quenched if $\lambda_e \beta_e/L_T \ll 1$. For
$d_e/L_T \gtrsim (d_e/L_T)_{\rm c0}$, and $\lambda_e \beta_e/L_T \gtrsim 1$,
stability is attained at fixed $\lambda_e/L_T$ for $d_e/L_T > (d_e/L_T)_{\rm c}$, where $(d_e/L_T)_{\rm c}$ increases monotonically
with $\lambda_e/L_T$. If $\lambda_e \beta_e/L_T \gtrsim 1$ and $d_e/L_T \lesssim
(d_e/L_T)_{\rm c}$, then the CE
distribution function (<ref>) is unstable.

The fastest-growing CET microinstability
is the whistler (heat-flux) instability: whistler waves driven unstable by the small anisotropy of the CE electron-temperature-gradient term (see section
That this instability with wavevector parallel to the magnetic field is indeed the
dominant microinstability is most easily ascertained by comparing simple analytic
expressions for its peak growth rate and wavevector to the equivalent quantities
recorded when performing the general stability calculation (see figures
<ref>b, <ref>c and <ref>d). The maximum microinstability growth rate matches
the analytic result (<ref>) for the CET whistler instability in the limit $\lambda_e \beta_e/L_T
\gg 1$, while the parallel wavenumber $(|k_{\|}| \rho_e)_{\rm peak}$ of the fastest-growing mode
is extremely well described by (<ref>).
In addition, figure <ref>d demonstrates that the parallel instability is indeed the fastest. The CET whistler
instability has been considered previously by a number of authors (see references in section <ref>); we note that these prior studies of this instability suggest that, nonlinearly, oblique CET whistler modes may be the more important ones, even though linearly the parallel modes are the fastest growing (see section <ref>).

The two thresholds demarcating the unstable region can then be associated with stabilisation conditions of the CET whistler instability,
each with a simple physical interpretation. The first condition is
the $\beta$-stabilisation condition of the whistler instability. It is shown in
section <ref> that when $\lambda_e \beta_e/L_T \ll 1$,
cyclotron damping on whistler modes is sufficiently strong that only quasi-parallel modes with
parallel wavenumbers $k_{\|} \rho_e \lesssim (\lambda_e \beta_e/L_T)^{1/3} \ll 1$ can be
destabilised by the anisotropy of the CE distribution function, and that the peak growth
rate $\gamma_{\rm whistler,T}$ of these unstable modes is exponentially small in $\lambda_e \beta_e/L_T$
compared to the electron Larmor frequency [see (<ref>)]:
$\gamma_{\rm whistler,T}/\Omega_e \sim \lambda_e \exp{[-(\lambda_e \beta_e/2 L_T)^{-2/3}]}/L_T $. This means that
if $\lambda_e \beta_e/L_T$ is reduced below unity, the growth rate of the CET whistler instability decreases
dramatically, and thus the instability is unable to operate effectively on timescales shorter than those over which the CE
plasma is evolving macroscopically.

The second condition is collisional stabilisation of the CET whistler
instability. Naively, it might be expected that two conditions must be satisfied
in order for the microinstability to operate: that its growth rate must satisfy $\gamma_{\rm whistler,T} \tau_{e} \gg
1$, and its characteristic wavenumber $k \lambda_{e} \gg 1$ [see (<ref>)]. Noting that for the
CET whistler instability [cf. (<ref>)],

\begin{equation}
\frac{\gamma_{\rm whistler,T} \tau_{e}}{k \lambda_{e}} = \frac{\gamma_{\rm whistler,T}}{k v_{\mathrm{th}e}} \sim
\frac{\lambda_e}{L_T} \left(  \frac{\lambda_e \beta_e}{L_T} \right)^{-1/5} \ll 1 \, ,
\end{equation}

it follows that the former condition is more restrictive. Written as a condition
on $d_e/L_T$ in terms of $\lambda_e/L_T$ [and using $\gamma_{\rm whistler,T} \sim \lambda_e \Omega_e/L_T$ – see (<ref>)], $\gamma_{\rm whistler,T} \tau_{e} \gg
1$ becomes

\begin{equation}
\frac{d_e}{L_T} \ll \beta_e^{-5/2} \left(  \frac{\lambda_e \beta_e}{L_T} \right)^2 \, ,
\end{equation}

while the condition $k \lambda_{e} \gg 1$ on the instability wavenumber $k_{\|} \rho_e \sim (\lambda_e \beta_e/L_T)^{1/5}$ [see (<ref>)] leads to

\begin{equation}
\frac{d_e}{L_T} \ll \left(\frac{d_e}{L_T}\right)_{\rm c} \equiv \beta_e^{-3/2} \left(\frac{\lambda_e \beta_e}{L_T}\right)^{6/5} \, . \label{colldampB}
\end{equation}

It is the latter that agrees well with the true result, as shown in figure
<ref>a, implying that $(d_e/L_T)_{\rm c0} = \beta_e^{-3/2}$. The (arguably surprising) result that the CET whistler instability can operate even if
$\gamma_{\rm whistler,T} \tau_{e} \lesssim 1$ is, in fact, a generic feature of
low-frequency (viz., $\omega \ll k v_{\mathrm{th}e}$) plasma instabilities (see section <ref>).
The physical instability mechanism underlying such modes can be sustained provided the time taken for thermal
particles (in this case, electrons) to cross the mode's wavelength is much shorter
than the collision time, irrespective of the mode's own frequency – in other words, $\tau_e k v_{\mathrm{th}e} = k \lambda_e \gg 1$.
We point out that the collisional-stabilisation condition of the CET whistler
instability can never be satisfied in a strongly magnetised plasma if $\lambda_e \beta_e/L_T \gtrsim 1$: this is
because its wavenumber $k$ satisfies $k^{-1} \lesssim \rho_e \ll \lambda_e$.

Whilst it is the fastest-growing one (assuming $\eta_e^T \sim \eta_i$), the CET whistler instability is not the only
CET microinstability of interest. There are two other instabilities driven by the CET ion-temperature gradient
term, neither of which has previously been identified, to our knowledge: the slow (hydromagnetic) wave
instability (see section <ref>), and the long-wavelength kinetic-Alfvén wave instability (see section <ref>).
The former, whose characteristic wavenumber scale satisfies $k \rho_i \sim 1$, has a larger characteristic growth rate $\gamma_{\rm SW} \sim \lambda_i \Omega_i/L_{T_i}$
(where $L_{T_i} = |\nabla_{\|} \log{T_i}|^{-1}$ is the
scale length of the ion temperature gradient).
Similarly to the CET whistler instability, the CET slow-wave
instability has $\beta$-stabilisation and collisional-stabilisation conditions $\lambda_i \beta_i/L_{T_i} \ll
1$ and $\lambda_i \lesssim \rho_i$, respectively. Thus, unless $\lambda_i \beta_i/L_{T_i} > \lambda_e \beta_e/L_{T_e}$ (a condition equivalent to $\tau^3 L_{T_e}/L_{T_i} > Z^3$, where $\tau = T_i/T_e$), the CET
slow-wave instability only operates when the CET whistler wave instability does, but on larger, ion rather than electron, scales.
Nevertheless, the CET slow-wave instability is worth noting because,
on account of being an ion instability, it should
continue to operate even if the electron-scale CET whistler instability modifies
the underlying electron distribution function. The slow-wave instability will then be responsible for modifying the ion distribution function. We are not aware of any work on the CET slow-wave instability and, thus, on its effect on ion heat conduction.

Readers who are
interested in knowing more about the properties and growth rates of CET microinstabilities are
encouraged to continue section <ref>; those who are focused on the wider
question of the kinetic stability of the CE distribution function should jump
ahead to section <ref>.

§.§ CET microinstability classification

§.§.§ Parallel whistler (heat-flux) instability

The CET whistler instability, which has been studied previously by a
number of authors [Levinson & Eichler, 1992, Pistinner & Eichler, 1998, Gary & Li, 2000, Roberg-Clark et al., 2016, Komarov et al., 2018, Roberg-Clark et al., 2018, Roberg-Clark et al., 2018, Shaaban et al., 2019, Kuzichev et al., 2019, Drake et al., 2021],
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 $v_{\|} = \pm \Omega_e/k_{\|}$. For a positive, parallel electron heat
flux, which is driven by an anti-parallel temperature gradient ($\nabla_{\|} T_e <
0$, 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 <ref>,  and also Levinson & Eichler, 1992 and Roberg-Clark et al., 2016]
that the real frequency $\varpi$ and growth rate $\gamma$ at arbitrary $k_{\|} > 0$
are given by

ϖ/Ω_e     =      η_e^T (k_ ρ_e/4 - 1/2 k_ ρ_e)
- (η_e^T/2 + k_^3 ρ_e^3/β_e)   Z(1/k_ ρ_e) /[  Z(1/k_ ρ_e)]^2 + exp(-2/k_^2 ρ_e^2)

γ/Ω_e     =
-√()(η_e^T/2 + k_^3 ρ_e^3/β_e) /[  Z(1/k_ ρ_e)]^2 exp(1/k_^2 ρ_e^2)+ exp(-1/k_^2 ρ_e^2)

For $\eta_e^T > 0$, $\gamma < 0$, but if $\eta_e^T < 0$, then $\gamma$ is
non-negative for $k_{\|} \rho_e \leq \left(\eta_e^T \beta_e/2\right)^{1/3}$.
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 <ref> using the above formulae.

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 (<ref>a), 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
(<ref>a) and
(<ref>b), respectively.
The resulting frequencies and growth rates, when normalised as $\gamma \beta_e/\Omega_e$, 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 (<ref>a) and (<ref>b) 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 (<ref>a) and (<ref>b)
for the frequency (dashed blue) and growth rate (dashed red) in the limit $k_{\|} \rho_e \gg

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: $\Delta k_{\|} \sim k_{\|} \sim
\rho_e^{-1}$.

The expressions (<ref>a) and (<ref>b) 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 $k_{\|} \rho_e \sim \left(\eta_e^T
\beta_e\right)^{1/3} \ll 1$ under which the destabilising $\eta_e^T$ terms and the stabilising electron FLR terms are the same order, we find

ϖ    ≈    k_^2 ρ_e^2/β_e Ω_e   ,

γ    ≈
-√()/k_^2 ρ_e^2 ( η_e^T/2 + k_^3 ρ_e^3/β_e)
exp(-1/k_^2 ρ_e^2) Ω_e

The frequency corresponds to that of a whistler wave in the $k_{\|} \rho_e \ll 1$ limit [Boldyrev et al., 2013]. The fastest growth, which occurs at the wavenumber

\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 $|\eta_e^T|  \beta_e \ll 1$:

\begin{equation}
\gamma_{\rm max} \approx \frac{3 \sqrt{\upi}}{4} |\eta_e^T| \exp{\left[-\frac{2^{2/3}}{\left(|\eta_e^T|  \beta_e\right)^{2/3}}-1\right]} \Omega_e
\, . \label{whistlermode_smallk}
\end{equation}

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

ϖ    ≈    [η_e^T β_e(1/4 k_ ρ_e--2/2 k_ ρ_e)+ 2/ k_^2 ρ_e^2 ] Ω_e/β_e    ,

γ    ≈
-1/√() [η_e^T β_e(1/2-4-/2 k_^2 ρ_e^2) + k_^3 ρ_e^3 ] Ω_e/β_e

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

\begin{eqnarray}
\gamma_{\rm max} & \approx & \frac{|\eta_e^{T}|}{\sqrt{\upi}} \left\{1- \left[\frac{1}{\sqrt{\upi}}\left(\frac{4}{\upi}-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\}
\Omega_e \quad \nonumber \\
& \approx & 0.56 |\eta_e^{T}| \left[1-0.13 \left(|\eta_e^{T}|\beta_e \right)^{-2/5} \right] \Omega_e \, , \label{maxgrowthrate_heatflux}
\end{eqnarray}

at the parallel wavenumber

\begin{equation}
k_{\|} \rho_e = \left[\frac{2}{3\sqrt{\upi}}\left(\frac{4}{\upi}-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}  \, . \label{maxgrowthrate_heatflux_wavenumber}
\end{equation}

In addition, we see that the real frequency of
modes with $k_\| \rho_e \lesssim \left(|\eta_e^T|
\beta_e/{2}\right)^{1/3}$ 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 (<ref>) and (<ref>) are valid in the limits $|\eta_e^T| \beta_e \ll 1$ and $|\eta_e^T|  \beta_e \gg 1$,
and are plotted in figure <ref> alongside the exact results (<ref>). Of particular note is the accuracy of the approximate expression (<ref>b)
for the growth rate when $k_{\|} \rho_e \gtrsim 0.6$; this suggests that (<ref>)
is a reasonable estimate of the peak growth rate for $|\eta_e^T| \beta_e \gtrsim

§.§.§ 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 <ref>, we show that there are two low-frequency oblique modes, whose complex
frequencies $\omega$ are given by

\begin{equation}
\omega = \frac{\Omega_e}{\beta_e} k_{\|} \rho_e \frac{-B_{\mathrm{T}} \pm \sqrt{B_{\mathrm{T}}^2 + 4A_{\mathrm{T}}C_{\mathrm{T}}}}{2 A_{\mathrm{T}}}
\, , \label{heatflux_elec_frequency}
\end{equation}

where the coefficients $A_{\mathrm{T}} = A_{\mathrm{T}}(k_{\|} \rho_e,k_{\perp} \rho_e,\eta_e^T \beta_e)$, $B_{\mathrm{T}} = B_{\mathrm{T}}(k_{\|} \rho_e,k_{\perp}
\rho_e,\eta_e^T \beta_e)$, and $C_{\mathrm{T}} = C_{\mathrm{T}}(k_{\|} \rho_e,k_{\perp} \rho_e,\eta_e^T \beta_e)$
are composed of the sums and products of the special
functions defined in (<ref>), and also other special functions
defined in appendix <ref>.
For a given wavenumber, we can use (<ref>) 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, (<ref>) suggests that they will have the typical size
$\gamma \sim \Omega_e/\beta_e \sim \eta_e^T \Omega_e$ when $k \rho_e \sim 1$
and $\eta_e^T \beta_e \sim 1$.

For $\eta_e^T > 0$, we find that both modes (<ref>) are damped; for $\eta_e^T < 0$,
one mode is damped for all wavenumbers, but the other is not.
Figure <ref> shows the maximum (positive) growth rate $\gamma$ (normalised to $\Omega_e/\beta_e$) 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 (<ref>) at a given wavenumber.

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
(<ref>a), at arbitrary wavevectors with respect to the background
magnetic field.
The growth rates of the modes are calculated by taking the imaginary part of (<ref>), where coefficients $A_{\mathrm{T}}$, $B_{\mathrm{T}}$ and $C_{\mathrm{T}}$ 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 $\gamma \beta_e/\Omega_e$, 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 $\gamma/|\eta_e^{T}| \Omega_e$.
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 $\eta_e^{T} \beta_e = -100$.

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 $\gamma \ll \Omega_e/\beta_e$ (figure <ref>a). This finding is
consistent with the exponentially small growth rates found for the parallel
whistler modes [see (<ref>)]. When $-\eta_e^T \sim 1/\beta_e$, there is a
marked change in behaviour: a larger region of unstable modes appears, with $\gamma \sim
\Omega_e/\beta_e$, at wavenumbers $k \rho_e \sim 1$ (figures <ref>b and 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 $1 \lesssim k \rho_e \lesssim \left|\eta_e^T \beta_e\right|^{1/3}$, and $\gamma \sim |\eta_e^T \Omega_e|$ (figure <ref>d). 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 to 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 <ref>e), while for $-\eta_e^T \gg 1/\beta_e$, $\omega \gg \gamma$
for $k \rho_e \gg 1$ (figure <ref>f). Note, however, that in the latter
case there exists a band of wavenumbers at which there is no real frequency.

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 <cit.>. 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
kinetic Alfvén waves (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 ($\delta n_e \ll Z e n_{e} \varphi/T_i$, where $\varphi$ is the electrostatic potential associated with the wave), while KAWs
do not: $\delta n_e \approx Z e n_{e} {\varphi}/T_i$ [Boldyrev et al., 2013]. 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 $\delta n_e \approx -Z e n_{e} {\varphi}/T_i$ (see appendix <ref>).

For further discussion of the
physics of the whistler instability (as well as its nonlinear evolution), see Komarov et al., 2018 and the other references given at the beginning of section <ref>.

§.§.§ 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 (<ref>b) (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, 1971, Foote & Kulsrud, 1979] 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 $\omega \approx 2 \Omega_i/\beta_i$.
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 $v_{\|} = \pm \Omega_i/k_{\|}$ via gyroresonant wave-particle interactions.
For an anti-parallel ion temperature gradient (i.e., $\nabla_{\|} T_i < 0$, 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 <ref>)

ϖ/Ω_i     =      η_i (k_ ρ_i/4-1/2 k_ ρ_i)
- k_^2 ρ_i^2 [  Z(1/k_ ρ_i) + k_ ρ_i] (η_i/4 + k_ ρ_i/β_i) /[  Z(1/k_ ρ_i) + k_ ρ_i]^2 + exp(-2/k_^2 ρ_i^2)

γ/Ω_i     =     -
√() k_^2 ρ_i^2 (η_i/4 + k_ ρ_i/β_i) /[  Z(1/k_ ρ_i) + k_ ρ_i]^2 exp(1/k_^2 ρ_i^2)+ exp(-1/k_^2 ρ_i^2)

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 <ref>.

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
(<ref>b), 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
(<ref>a) and
(<ref>b), respectively.
The resulting frequencies and growth rates, when normalised as $\gamma \beta_i/\Omega_i$, 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 (<ref>) and (<ref>) 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 (<ref>a) and (<ref>b)
for the frequency (dashed blue) and growth rate (dashed red) in the limit $k_{\|} \rho_i \gg

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 (<ref>a) becomes

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

which is precisely that of the slow hydromagnetic wave, with first-order FLR corrections included [Foote & Kulsrud, 1979].
For $\eta_i < 0$ and $k_{\|} \rho_i < |\eta_i| \beta_i/4$, the growth rate (<ref>b)
is positive:

\begin{equation}
\gamma \approx - \frac{4 \sqrt{\upi}}{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)}
\Omega_i \, .   \label{slowmode_smallk_growth}
\end{equation}

The maximum growth rate (which is exponentially small in $\eta_i \beta_i/4 \ll 1$) is

\begin{equation}
\gamma_{\rm max} \approx \frac{8 \sqrt{\upi}}{|\eta_i| \beta_i^2} \exp{\left(-\frac{16}{|\eta_i|^2  \beta_i^{2}}-1\right)}
\Omega_i
\, , \label{slowmode_smallk}
\end{equation}

achieved at the parallel wavenumber

\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, $k_{\|} \rho_i \sim \left(|\eta_i| \beta_i/4\right)^{1/3} \gg 1$, we obtain

ϖ    ≈    -(η_i β_i 1-/4/k_ ρ_i -k_^2 ρ_i^2 ) Ω_i/β_i    ,

γ    ≈
-√() [η_i/4 β_i(1--3/k_^2 ρ_i^2) + k_ ρ_i ] Ω_i/β_i

The maximum positive growth rate is

\begin{equation}
\gamma_{\rm max} \approx \frac{\sqrt{\upi}}{4} \left\{1- 3\left[4 \left(\upi-3\right)\right]^{1/3} \left(|\eta_i|\beta_i \right)^{-2/3} \right\}
\Omega_i  \approx 0.44 \left[1-2.48 \left(|\eta_i|\beta_i \right)^{-2/3} \right]  |\eta_i| \Omega_i \, , \,  \label{maxgrowthrate_slowwave}
\end{equation}

realised for $\eta_i < 0$ at the parallel wavenumber

\begin{equation}
k_{\|} \rho_i \approx \left(\frac{\upi -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

The approximate expressions (<ref>), (<ref>),
and (<ref>b) for the frequency and growth rate in the
limits $k_{\|} \rho_i \ll 1$ and $k_{\|} \rho_i \gg 1$, are plotted in figure <ref>, along with the exact results (<ref>).

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 <ref>):

\begin{equation}
\omega = \frac{\Omega_i}{\beta_i} k_{\|} |\rho_i| \frac{-\tilde{B}_{\mathrm{T}} \pm \sqrt{\tilde{B}_{\mathrm{T}}^2 + 4\tilde{A}_{\mathrm{T}}\tilde{C}_{\mathrm{T}}}}{2 \tilde{A}_{\mathrm{T}}}
\, , \label{heatflux_ion_frequency}
\end{equation}

where $\tilde{A}_{\mathrm{T}} = \tilde{A}_{\mathrm{T}}(k_{\|} \rho_i,k_{\perp} \rho_i,\eta_i \beta_i)$, $\tilde{B}_{\mathrm{T}} = \tilde{B}_{\mathrm{T}}(k_{\|} \rho_i,k_{\perp}
\rho_i,\eta_i \beta_i)$, and $\tilde{C}_{\mathrm{T}} = \tilde{C}_{\mathrm{T}}(k_{\|} \rho_i,k_{\perp} \rho_i,\eta_i \beta_i)$
are again sums and products of various special mathematical functions defined in (<ref>).  Investigating such modes by evaluating (<ref>) numerically
for a range of wavenumbers (see figure <ref>), 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 <ref>a); 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
<ref>b for $\eta_i \beta_i = -8$. The typical growth rate $\gamma$ satisfies $\gamma \sim
\Omega_i/\beta_i \sim \eta_i \Omega_i$, as anticipated from (<ref>).

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
(<ref>b), at arbitrary wavevectors with respect to the background
magnetic field.
The growth rates of all modes are calculated by taking the imaginary part of (<ref>), with coefficients $\tilde{A}_{\mathrm{T}}$, $\tilde{B}_{\mathrm{T}}$ and $\tilde{C}_{\mathrm{T}}$ being known functions of the wavevector (see appendix <ref>). 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 $\gamma \beta_i/\Omega_i$, 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 section <ref>.

We also observe in figure <ref>b the existence of an unstable mode at
quasi-perpendicular wavenumbers, which is discussed in section <ref>.

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 $\gamma \sim \lambda_i
\Omega_i/L_{T_i}$.
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.

§.§.§ Long-wavelength kinetic-Alfvén-wave instability

The instability observed in figure <ref>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 $v_{\|} \approx \pm \Omega_i/k_{\|}$ 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 <ref>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 <ref>) 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 (<ref>b) is

\begin{eqnarray}
\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}} \Bigg[ -\frac{\mathrm{i}\sqrt{\upi}}{2} k_{\perp} \rho_i \left(\mathcal{F}_i + \sqrt{\frac{\mu_e Z^2}{\tau}}\right)
\nonumber \\
& & \pm \sqrt{1-\frac{\upi}{4} \frac{k_{\perp}^2 \rho_i^2}{\beta_i} \bigg(\mathcal{F}_i + \sqrt{\frac{\mu_e Z^2}{\tau}}\bigg)^2 - \frac{\mathrm{i} \sqrt{\upi} \eta_i \beta_i}{4} \frac{2 \mathcal{G}_i - \mathcal{F}_i\left(1 - \mathcal{F}_i\right)}{1-\mathcal{F}_i} } \Bigg] \, ,
\label{heatfluxKAWfreq}
\end{eqnarray}

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

\begin{eqnarray}
\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] \,
, \\
\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) \,
. \label{specialfunction_quasiperp}
\end{eqnarray}

In a Maxwellian plasma (i.e., when $\eta_i = 0$), (<ref>) becomes

\begin{eqnarray}
\frac{\omega}{k_{\|} v_{\mathrm{th}i}} & = &  \frac{1}{\left(1-\mathcal{F}_i\right)^{2}} \Bigg[ -\frac{\mathrm{i}\sqrt{\upi}}{2} \frac{k_{\perp}^2 \rho_i^2}{\beta_i} \left(\mathcal{F}_i + \sqrt{\frac{\mu_e Z^2}{\tau}}\right)
\nonumber \\
&& \pm \sqrt{\frac{k_{\perp}^2 \rho_i^2}{\beta_i^2}-\frac{\upi}{4} \frac{k_{\perp}^4 \rho_i^4}{\beta_i^2} \bigg(\mathcal{F}_i + \sqrt{\frac{\mu_e Z^2}{\tau}}\bigg)^2} \Bigg] \, .
\end{eqnarray}

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., 2009, Boldyrev et al., 2013, Kunz et al., 2018].

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

\begin{equation}
\eta_i \lesssim -3.14 \left(1 + 6.5 \sqrt{\frac{\mu_e Z^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

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

Figure <ref> shows the real frequency and growth rate of such modes at three
different (negative) values of $\eta_i \beta_i$.

Quasi-perpendicular CET KAW instability.  Dispersion curves of unstable KAWs whose instability is driven by the ion-temperature-gradient term in the CE distribution function (<ref>b), for wavevectors that are almost perpendicular to the background
magnetic field (viz., $k_{\perp} \gg k_{\|}$).
The frequency (blue) and growth rates (red) of unstable modes
are calculated at (small) fixed values of $k_{\|} \rho_i$ from the real and imaginary parts of
(<ref>); the solid curves are calculated for $k_{\|} \rho_i = 0.35$, while the
dashed curves are for $k_{\|} \rho_i = 0.05$.
The resulting frequencies and growth rates, when normalised as $\gamma \beta_i/k_{\|} v_{\mathrm{th}i}$,
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 frequency (dotted blue) and growth rate (dotted red) in the limit $k_{\|} \rho_i \ll 1$, which are
calculated by taking the real and imaginary parts of (<ref>), are also plotted.

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.

Although it is possible to characterise analytically the peak growth rate of the
unstable modes (and the perpendicular wavenumber at which such growth is attained) in the limit $k_{\|} \rho_i \ll 1$
by analysing (<ref>), such estimates do not capture accurately
the behaviour of the fastest-growing modes across all wavevectors, because these
fastest-growing modes occur at finite values of $k_{\|} \rho_i$; at such values, the
dependence of the frequency and growth rate on $k_{\perp} \rho_i$ departs somewhat from (<ref>) (see figure <ref>). Instead, we
find numerically that, for $\eta_i \beta_i \lesssim -6$,

\begin{equation}
\gamma_{\rm max} \approx 0.025 |\eta_{i}| \Omega_i \quad \mathrm{at}  \quad (k_{\|}
\rho_i)_{\rm peak} \approx 0.35 \, ,
\end{equation}

independent of the specific value of either $\eta_i$ or $\beta_i$.
For values of $k_{\|} \rho_i$ that are larger than $(k_{\|} \rho_i)_{\rm peak}$, the instability is
quenched. It is clear that, in comparison to the slow-hydromagnetic wave instability, the
growth rate of the fastest-growing perpendicular modes is small [see (<ref>)]. This
difference can be attributed to the fact that, for unstable modes in
the limit $|\eta_i| \beta_i \gg 1$, $\gamma_{\rm max} \sim |\eta_i| k_{\|} \rho_i
\Omega_i$ and the value of $k_{\|} \rho_i$ at which maximum growth is achieved
is still rather small compared to unity. We conclude that the instability of
slow hydromagnetic waves that are driven by an ion temperature gradient is likely to be more
significant than the analogous instability of quasi-perpendicular/KAW

§ CES (CHAPMAN-ENSKOG, SHEAR-DRIVEN) MICROINSTABILITIES

§.§ Form of CE distribution function

Next, we consider the non-Maxwellian terms of the CE distribution arising from
bulk-flow gradients. If we set $\eta_s = 0$ for both ions and electrons (viz., neglecting both temperature gradients and electron-ion drifts),
the CE distribution functions (<ref>) for both species become

\begin{equation}
f_{s0}(v_{\|},v_{\bot}) = \frac{n_{s0}}{v_{\mathrm{th}s}^3 \upi^{3/2}} \exp \left(-\tilde{v}_{s}^2\right) \left[1 - \epsilon_s \left(\frac{v_{\|}^2}{v_{\mathrm{th}s}^2}- \frac{v_{\bot}^2}{2 v_{\mathrm{th}s}^2} \right)\right] ,
\label{CEsheardistfuncexpression}
\end{equation}

where we have again chosen the isotropic functions $C_s(\tilde{v}_s)$
to be the ones that arise from the Krook collision operator (see section <ref>).
We note that for this choice of collision operator, the constant $\mathcal{C}_s$ defined by (<ref>) is
$\mathcal{C}_s \approx 3/2$, and so the relationship (<ref>) between the CE distribution functions' pressure anisotropy $\Delta_s$
and the shear parameter $\epsilon_s$ becomes

\begin{equation}
\Delta_s = \frac{3}{2} \epsilon_s \, .  \label{pressanisop_krook}
\end{equation}

We also observe that the CE shear terms have even parity with respect to the
parallel velocity $v_{\|}$, and thus for any unstable mode with positive parallel wavenumber $k_{\|} > 0$, there is
a corresponding unstable mode with $k_{\|} < 0$.  This conclusion has the consequence
that the sign of $\epsilon_s$ [which is the same as the sign of $\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \mathsfbi{I}/3 \right) \! \boldsymbol{:} \! \mathsfbi{W}_s$, where $\mathsfbi{W}_s$ is the rate-of-strain tensor of species $s$ – see (<ref>)]
has a significant effect on possible types of CES microinstabilities. Thus, we
must consider the cases $\epsilon_s > 0$ (positive pressure anisotropy, $\Delta_s > 0$)
and $\epsilon_s < 0$ (negative pressure anisotropy, $\Delta_s < 0$) separately. For easier comparison to
previous work by other authors, we will sometimes substitute $\epsilon_s =2 \Delta_s/3$, and work in terms of $\Delta_s$.

As with the discussion of CET microinstabilities in section <ref>, in the main text, we
only present the main findings of our calculations: namely, the overview of the CES stability landscape (section <ref>), and
the analytical characterisation of CES microinstabilities with $\epsilon_s > 0$ (section <ref>) and $\epsilon_s < 0$ (section <ref>).
The methodology underlying the calculations of growth rates of CES microinstabilities
is presented in appendix <ref>.

§.§ Stability

The stability of
CE distribution functions of the form (<ref>) is determined as a function of
the parameters $\epsilon_i$, $\epsilon_e$, $d_e$, $\beta_e$, $\beta_i$, and the velocity scale length $L_V = |\left(\hat{\boldsymbol{z}} \hat{\boldsymbol{z}} - \frac{1}{3}\mathsfbi{I} \right) \! \boldsymbol{:} \! \mathsfbi{W}_i/V_i|^{-1}$ by assessing whether the maximum microinstability growth rate across all wavelengths smaller than $\lambda_e$ and $\lambda_i$
is negative or positive (see appendix <ref> for the methodology underpinning this calculation).
As with the temperature-gradient-driven instabilities, we report the
results of stability calculations that pertain to a
temperature-equilibrated hydrogen plasma; that is, the particular case in which $\beta_i = \beta_e$ and $\epsilon_e = \mu_e^{1/2}
\epsilon_i$ [where we recall that the characteristic magnitude of
the CE electron velocity-shear term in such a plasma is smaller than the analogous CE ion velocity-shear
term by a factor of $\mu_e^{1/2} = (m_e/m_i)^{1/2}$].
Because $\epsilon_i$ can take both positive and negative values (see section
<ref>), we do one stability calculation for each
case; the results of these two calculations are shown in figures <ref> and <ref>,

CE-distribution-function stability map for CES microinstabilities driven by positive pressure anisotropy. Exploration of the stability of the ion and electron CE distribution functions (<ref>)
for different positive values of small parameters $\epsilon_e$ and $\epsilon_i$ (viz., electron or ion pressure anisotropies), and the ratio of the electron inertial scale $d_e$ to the velocity scale length $L_V,$ in a temperature-equilibrated hydrogen plasma. In this
plot, we chose $\epsilon_e = \mu_e^{1/2} \epsilon_i$, and then show $\mathrm{Ma} \, \lambda_e/L_V = |\epsilon_i|$ with equal logarithmic spacing in the range $\left[10^{-5},10^{0}\right]$; $d_e/L_V$ is chosen with equal logarithmic spacing in the range
$\left[10^{-15},10^{0}\right]$. The total size of the grid is $400^2$. For reasons of efficiency, we calculate growth rates on a $40^2$ grid in wavenumber space with
logarithmic spacing for both parallel and perpendicular wavenumbers. In this plot, $\beta_e = \beta_i = 10^4$, and $\mathrm{Ma} = 1$. a) Stable (blue) and unstable (red) regions of $\left(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V\right)$ phase space. The theoretically anticipated collisional
cutoffs [right – see (<ref>)] and $\beta$-stabilisation thresholds (horizontal dashed lines) for the CES mirror and parallel transverse instabilities, respectively, are also shown. b)
Maximum normalised microinstability growth rate (red) versus $\mathrm{Ma} \, \lambda_e/L_V$ for a fixed electron inertial scale $d_e/L_V = 10^{-15}$,
along with the maximum growth rate for the mirror instability (purple) in the limit $\mathrm{Ma} \, \lambda_e \beta_e/L_V  \gg 1$ [see (<ref>)],
and for the parallel transverse instability
in the limit $\mathrm{Ma} \, \lambda_e \beta_e/L_V  \gg \mu_e^{-1/2}$ [see (<ref>), with $\theta =
c) Parallel wavenumber of the fastest-growing microinstability (red) versus $\mathrm{Ma} \, \lambda_e/L_V$ for a fixed electron inertial scale $d_e/L_V = 10^{-15}$,
along with the same quantity analytically predicted for the mirror instability (purple) in the limit $\mathrm{Ma} \, \lambda_e \beta_e/L_V \gg 1$ [see
(<ref>)], and for the parallel transverse instability (blue) in the limit $\mathrm{Ma} \, \lambda_e \beta_e/L_V  \gg \mu_e^{-1/2}$ [see
(<ref>), with $\theta = 0^{\circ}$].
d) Wavevector angle $\theta \equiv \tan^{-1}{(k_{\|}/k_{\perp})}$ of the fastest-growing instability over the $\left(d_e/L_V,\mathrm{Ma} \, \lambda_e \beta_e/L_V\right)$ parameter space.

CE-distribution-function stability map for CES microinstabilities driven by negative pressure anisotropy. Same as figure <ref>, but for negative values of the small parameters $\epsilon_e$ and $\epsilon_i$. a) Stable (blue) and unstable (red) regions of $\left(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V\right)$ phase space. The theoretically anticipated collisional
cutoffs [right – see (<ref>)] for the CES firehose and oblique transverse instabilities, respectively, and the $\beta$-stabilisation thresholds (horizontal dashed lines)
for the CES firehose, CES electron-scale-transition (EST) and whisper instabilities are also shown. b)
Maximum normalised microinstability growth rate (red) versus $\mathrm{Ma} \, \lambda_e/L_V$ for a fixed electron inertial scale $d_e/L_V = 10^{-15}$,
along with analytically predicted maximum growth rate for the firehose instability (purple) [see (<ref>)],
for the EST instability (green) in the limit $\mu_e^{-1/2} \beta_e^{-5/7} \gg \mathrm{Ma} \, \lambda_e/L_V \gg \mu_e^{-1/2} \beta_e^{-1}$ [see (<ref>)]
for the whisper instability (yellow) in the limit $\mu_e^{-1/2} \beta_e^{-1/3} \gg \mathrm{Ma} \, \lambda_e/L_V \gg \mu_e^{-1/2}
\beta_e^{-5/7}$ [see (<ref>)], and for the oblique transverse instability (blue) in the limit $\mathrm{Ma} \, \lambda_e/L_V \gg \mu_e^{-1/2} \beta_e^{-1}$
[see (<ref>)].
c) Same as b), but for the parallel wavenumber of the fastest-growing microinstability. The analytical predictions of this quantity for the firehose instability (purple) [see (<ref>)],
for the EST instability (green) [see (<ref>b)],
and for the whisper instability (yellow) [see (<ref>b)], respectively, are also shown. d) Same as b), but for the perpendicular wavenumber of the fastest-growing microinstability. The analytical predictions of this quantity for the firehose instability (purple) [see (<ref>)],
for the EST instability (green) [see (<ref>a)],
and for the whisper instability (yellow) [see (<ref>a)], are also shown.

The key characteristics of the stability of the CE distribution function
(<ref>) for ions and electrons can be shown using
plots over a two-dimensional $(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V)$ parameter space at fixed $\beta_e$
and $\mathrm{Ma}$ – we remind the reader that $\mathrm{Ma} \, \lambda_e/L_V = |\epsilon_i|$, and that
the Mach number $\mathrm{Ma}$ is assumed to satisfy $\mathrm{Ma} \lesssim 1$  – as opposed to the five-dimensional $(\epsilon_i,d_e,L_V,\beta_e,\mathrm{Ma})$ parameter space that might naively be anticipated, because the two relevant stability thresholds are not independent
functions of $d_e$, $\mathrm{Ma}$, and $L_V$.

The regions of stability presented in figure <ref>a for $\epsilon_i > 0$ (viz., for shear flows that drive
positive pressure anisotropy) and in figure <ref>a for $\epsilon_i < 0$ (viz., for shear flows driving
negative pressure anisotropy), respectively, are broadly similar
to the region of stability for CET microinstabilities described in section
<ref> (and shown in figure <ref>a), but with one crucial difference. Once again, for $d_e/L_V$ less than a critical value $\left(d_e/L_V\right)_{\rm c0}$, stability is
independent of $d_e/L_V$, and there are no instabilities for $\mathrm{Ma} \, \lambda_e \beta_e/L_V \ll
1$; for $d_e/L_V \gtrsim \left(d_e/L_V\right)_{\rm c0}$ and $\mathrm{Ma} \, \lambda_e \beta_e/L_V
> 1$, stability is guaranteed if (and only if) $d_e/L_V > \left(d_e/L_V\right)_{\rm c}$ at fixed $\mathrm{Ma} \, \lambda_e/L_V$, where $\left(d_e/L_V\right)_{\rm c}$
is a monotonically increasing function of $\mathrm{Ma} \, \lambda_e/L_V$.
As before, these two bounding thresholds correspond to the
$\beta$-stabilisation conditions and collisional stabilisation conditions,
respectively, of CES microinstabilities. However, the dependence of $\left(d_e/L_V\right)_{\rm c}$ on
$\mathrm{Ma} \, \lambda_e/L_V$ is more complicated than the analogous relationship
between $\left(d_e/L_T\right)_{\rm c}$ and
$\mathrm{Ma} \, \lambda_e/L_T$ that was presented in figure <ref>a. Namely,
if $\mathrm{Ma} \, \lambda_e/L_T \gtrsim \beta_e^{-1} \mu_e^{-1/2}$, then
$\left(d_e/L_V\right)_{\rm c}$ suddenly shifts towards a larger value, with the
subsequent (power-law) relationship between $\left(d_e/L_V\right)_{\rm c}$ and $\mathrm{Ma} \, \lambda_e/L_V$
being distinct from the analogous relationship when $\mathrm{Ma} \, \lambda_e/L_T \lesssim \beta_e^{-1}
\mu_e^{-1/2}$. This behaviour is the result of a feature of the unstable region that is present for CES but not CET microinstabilities: different instabilities being dominant in different regions of
the $(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V)$ parameter space.
As we will see, this arises because CES microinstabilities on ion scales have less stringent $\beta$-stabilisation thresholds
than those on electron scales. Although their regions of stability are
qualitatively similar, the types of
microinstabilities that arise when $\epsilon_i > 0$ or $\epsilon_i < 0$ are quite different, so we now discuss each case in turn.

§.§.§ Positive pressure anisotropy

For $\epsilon_i > 0$ and $0.5 \mu_e^{-1/2} \beta_e^{-1} \gtrsim \mathrm{Ma} \, \lambda_e/L_V \gg \beta_e^{-1}$,
the fastest-growing CES microinstability is the mirror instability:
that is, a non-propagating, compressible slow mode on ion scales that is destabilised by positive
ion pressure anisotropy. For $\mathrm{Ma} \, \lambda_e \beta_e/L_V \gtrsim 0.5
\mu_e^{-1/2}$, a faster-growing CES microinstability emerges on electron Larmor scales, driven
by positive electron pressure anisotropy: the whistler (electron-cyclotron)
instability. For fixed $\beta_i$,
the CES mirror instability can operate at smaller values of $\mathrm{Ma} \, \lambda_e/L_V$
than the CES whistler instability, because the mirror-instability threshold
$\Delta_i \beta_i = 3 \mathrm{Ma} \, \lambda_e \beta_i/2 L_V \geq 1$ (see section <ref>)
is a less stringent condition on $\mathrm{Ma} \, \lambda_e/L_V$ for fixed $\beta_e$ than the threshold $\Delta_e \beta_e = 3 \mu_e^{1/2} \mathrm{Ma} \, \lambda_e \beta_i/2 L_V \gtrsim
0.5$ of the CES whistler instability (see section <ref>).
On the other hand, once $\mathrm{Ma} \, \lambda_e \beta_e/L_V \gtrsim 0.5 \mu_e^{-1/2}$,
the maximum growth rate of the CES mirror instability $\gamma_{\rm mirr} \sim \Delta_i \Omega_i$
is much smaller than that of the CES whistler instability: $\gamma_{\rm whistler,S} \sim \Delta_e \Omega_e \sim \mu_e^{-1/2} \Delta_i \Omega_i
\gg \Delta_i \Omega_i$. For $\mathrm{Ma} \, \lambda_e \beta_e/L_V \gg
\mu_e^{-1/2}$, in addition to unstable whistler modes, modes on sub-electron-Larmor scales
are also destabilised: this is the parallel transverse instability, a microinstability that is
essentially unmagnetised ($k \rho_i \gg 1$) in character.
When it can operate, the CES parallel transverse instability has a much larger growth rate than the unstable electron-Larmor-scale whistler waves, $\gamma_{\rm trans} \sim \Delta_e \left(\Delta_e \beta\right)^{1/2} \Omega_e \gg
\gamma_{\rm whist} \sim \Delta_e \Omega_e$, so if $\mathrm{Ma} \, \lambda_e \beta_e/L_V \gg
\mu_e^{-1/2}$, the transverse instability dominates.

Numerical evidence for the dominance of the CES mirror instability when $\mu_e^{-1/2} \gg \mathrm{Ma} \, \lambda_e/L_V \gg 1$, and then the CES parallel transverse
instability when $\mathrm{Ma} \, \lambda_e/L_V \gg \mu_e^{-1/2}$, can be produced by isolating the maximum growth rate, the parallel wavenumber and the wavevector angle associated with peak
growth for the unstable regions of the $(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V)$ parameter space. Figure
<ref>b shows that, for fixed $d_e/L_V$ and a range of $\mathrm{Ma} \, \lambda_e/L_V$,
the peak microinstability growth rate is a reasonable match for that
of the mirror instability [viz., (<ref>)] for $0.5 \mu_e^{-1/2} \beta_e^{-1} \gtrsim \mathrm{Ma} \, \lambda_e/L_V \gg \beta_e^{-1}$,
and a good match for the parallel transverse instability [viz., (<ref>)] for $\mathrm{Ma} \, \lambda_e/L_V \gtrsim \mu_e^{-1/2} \beta_e^{-1}$.
Figure <ref>c demonstrates that, for $\mu_e^{-1/2} \beta_e^{-1} \gtrsim \mathrm{Ma} \, \lambda_e/L_V \gg \beta_e^{-1}$, the (non-dimensionalised) parallel wavenumber $(k_{\|} \rho_e)_{\rm peak}$ of peak growth satisfies $(k_{\|} \rho_e)_{\rm peak} \sim \mu_e^{-1/2}$, in agreement with the expected parallel wavenumber of the fastest-growing mirror modes [see (<ref>)]. At $\mathrm{Ma} \, \lambda_e/L_V \sim \mu_e^{-1/2}
\beta_e^{-1}$, there is a dramatic shift in $(k_{\|} \rho_e)_{\rm peak}$ to a value $(k_{\|} \rho_e)_{\rm peak} \gtrsim 1$ that agrees with the expected parallel wavenumber of the parallel transverse instability [see (<ref>)]. As for the peak-growth wavevector angle (figure <ref>d), for $\beta_e^{-1} \lesssim \mathrm{Ma} \, \lambda_e/L_V \lesssim \mu_e^{-1/2}
\beta_e^{-1}$, the dominant instability is oblique (as would be expected for the mirror
instability), while for $\mathrm{Ma} \, \lambda_e/L_V \gtrsim 0.5 \mu_e^{-1/2} \beta_e^{-1}$, it is parallel (implying that the CES whistler/parallel transverse instability dominates). We conclude that the mirror instability
is indeed dominant when $0.5 \mu_e^{-1/2} \beta_e^{-1} \gtrsim \mathrm{Ma} \, \lambda_e/L_V \gg \beta_e^{-1}$,
and the parallel transverse instability when $\mathrm{Ma} \, \lambda_e/L_V \gg \mu_e^{-1/2} \beta_e^{-1}$.

§.§.§ Negative pressure anisotropy

Now considering the case when $\epsilon_i < 0$, i.e., the case of negative
pressure anisotropy, the only CES microinstability that operates when
$\mu_e^{-1/2} \beta_e^{-1} \gtrsim \mathrm{Ma} \, \lambda_e/L_V \gg \beta_e^{-1}$
is the firehose instability: the destabilisation of Alfvén waves by ion pressure
anisotropies $\Delta_i \lesssim -1/\beta_i$[In the limit of wavelengths much larger than the ion Larmor radius, the firehose instability
threshold is well known to be $\Delta_i = (\Delta_i)_{\rm c} < -2/\beta_i$. However, for plasmas whose ion species have either a CE distribution function or a bi-Maxwellian distribution,
the instability threshold for oblique ion-Larmor-scale firehose modes is somewhat less stringent: see section
If $\mathrm{Ma} \, \lambda_e/L_V \gtrsim \mu_e^{-1/2} \beta_e^{-1}$, several
electron-scale CES microinstabilities arise, all of which tend to have larger growth rates than the firehose
instability. The first of these to develop (at $\mathrm{Ma} \, \lambda_e/L_V \sim \mu_e^{-1/2} \beta_e^{-1}$)
is the oblique electron firehose instability: the destabilisation of oblique kinetic-Alfvén waves by
negative electron pressure anisotropy. For
$\mu_e^{-1/2} \beta_e^{-1} \lesssim  \mathrm{Ma} \, \lambda_e/L_V \lesssim  \mu_e^{-1/2}
\beta_e^{-5/7}$, the electron-scale-transition (EST) instability begins to operate; this is a non-propagating
quasi-perpendicular mode on electron Larmor scales ($k_{\perp} \rho_e \sim 1 \gg k_{\|} \rho_e$), which,
while damped in a Maxwellian plasma, is unstable for sufficiently negative
electron pressure anisotropies, and grows more rapidly than the oblique electron firehose instability. For
$\mu_e^{-1/2} \beta_e^{-5/7} \lesssim \mathrm{Ma} \, \lambda_e/L_V \lesssim \mu_e^{-1/2}
\beta_e^{-1/3}$, the EST instability is surpassed by the
whisper instability: the instability of a newly discovered propagating wave in a Maxwellian plasma (a whisper wave) whose
perpendicular wavelength is on sub-electron-Larmor scales ($k_{\perp} \rho_e \gg 1$), but
whose parallel wavelength is above the electron-Larmor scale ($k_{\|} \rho_e <
1$). Finally, when $\mathrm{Ma} \, \lambda_e/L_V \gtrsim \mu_e^{-1/2}
\beta_e^{-1/3}$, the oblique transverse instability comes to predominate;
unlike either the oblique electron firehose, the EST, or whisper instabilities,
it is unmagnetised in nature (like its parallel relative). Of these four
instabilities, the oblique electron firehose and transverse instabilities have been
identified previously (see references in sections <ref> and <ref>, respectively), but not the EST or whisper instabilities.

We support these claims (in an analogous manner to the $\epsilon_i > 0$ case) by calculating the growth rate of the dominant microinstabilities for given points
in the $(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V)$ parameter space.
Figure <ref>b shows the maximum growth rate for a
fixed value of $d_e/L_V$. For $\mu_e^{-1/2} \beta_e^{-1} \gtrsim \mathrm{Ma} \, \lambda_e/L_V \gg \beta_e^{-1}$,
the peak growth rate follows the analytical prediction for the ion firehose instability, $\gamma_{\rm fire} \sim |\Delta_i|^{1/2} \Omega_i/\sqrt{\log{1/|\Delta_i|}}$,
when $\Delta_i \ll -2/\beta_i$ [see (<ref>)].
For $\mathrm{Ma} \, \lambda_e/L_V \gtrsim \mu_e^{-1/2} \beta_e^{-1}$, the peak growth rate becomes much greater than
$\gamma_{\rm fire}$; for $\beta_e^{-5/7} \gtrsim \mu_e^{1/2} \mathrm{Ma} \, \lambda_e/L_V \gg
\beta_e^{-1}$, it instead matches that of the EST instability, $\gamma_{\mathrm{EST}} \sim |\Delta_e| \left(|\Delta_e|
\beta_e\right)^{3/2}
\Omega_e/\sqrt{\log{|\Delta_e| \beta_e}}$ [see (<ref>)], where we remind the reader that
$|\Delta_e| = 3 \mu_e^{1/2} \mathrm{Ma} \, \lambda_e/2 L_V$. For $\mu_e^{1/2} \mathrm{Ma} \, \lambda_e/L_V \gg
\beta_e^{-5/7}$, the observed growth rate agrees with an analytical prediction for the whisper instability,
$\gamma_{\mathrm{whisp}} \sim |\Delta_e|^{1/2} \left(|\Delta_e| \beta_e\right)^{1/4}
\Omega_e/\sqrt{\log{|\Delta_e| \beta_e}}$ [see (<ref>)].
Finally, because of the value of $\beta_e$ chosen for this numerical example, the condition
$\mathrm{Ma} \, \lambda_e/L_V \gtrsim \mu_e^{-1/2} \beta_e^{-1/3}$ under which
the oblique transverse instability dominates is never met for $\mathrm{Ma} \, \lambda_e/L_V \ll
1$, and thus the numerically measured growth rate of the dominant CES microinstability is larger than the tranverse instability's peak growth rate $\gamma_{\rm trans} \sim |\Delta_e| \left(|\Delta_e| \beta_e\right)^{1/2} \Omega_e$ [see (<ref>)] for
the entire range of $\mathrm{Ma} \, \lambda_e/L_V$ that we show in figure <ref>b, (blue line) .

A further confirmation that the most important microinstabilities are those that we have explicitly identified is obtained by
calculating the parallel
and perpendicular wavenumbers associated with
the dominant microinstability. Figures <ref>c and <ref>d show that, for $\beta_e^{-1} \ll \mathrm{Ma} \, \lambda_e/L_V \ll \mu_e^{-1/2}
\beta_e^{-1}$, $(k_\| \rho_e)_{\rm{peak}} \sim (k_\perp \rho_e)_{\rm{peak}} \sim
\mu_e^{1/2}$. These values of $(k_\| \rho_e)_{\rm{peak}}$ are consistent with the properties of the fastest-growing
unstable firehose modes (see sections <ref> and <ref>), whose parallel wavenumber (approximately) satisfies $(k_\| \rho_i)_{\rm{peak}} \sim 1/\sqrt{\log{1/|\Delta_i|}}$ when $\Delta_i \ll -2/\beta_{i}$ [see (<ref>)], and whose wavevector angle is
$\theta_{\mathrm{peak}} \approx 39^{\rm{o}}$.
At $\mathrm{Ma} \, \lambda_e/L_V \sim \mu_e^{-1/2} \beta_e^{-1}$,
the magnitudes of the parallel and perpendicular wavenumbers changes abruptly, to $(k_\| \rho_e)_{\rm{peak}} \sim (k_\perp \rho_e)_{\rm{peak}}
\sim 1$; this is in line with expectations from the onset of the oblique electron firehose instability
when $|\Delta_e| \beta_e \sim 1$. For $\mathrm{Ma} \, \lambda_e/L_V \gg
\beta_e^{-1}$ ($|\Delta_e| \beta_e \gg 1$), the parallel scale of the fastest-growing mode
remains above electron Larmor scales [$(k_\| \rho_e)_{\rm{peak}}  <
1$], while $(k_\perp \rho_e)_{\rm{peak}}$ increases monotonically
above unity. Both findings match theoretical expectations
concerning the evolution of the parallel and perpendicular wavenumbers of the EST and whisper instabilities
as functions of increasing $|\Delta_e| \beta_e$, and analytic formulae for these
quantities are in reasonable agreement with the numerical results (see sections <ref> and  <ref>).

§.§.§ Collisional stabilisation

For both $\epsilon_i > 0$ and $\epsilon_i < 0$, the shift in
$\left(d_e/L_V\right)_{\rm c}$ at $\mathrm{Ma} \, \lambda_e/L_V \sim \mu_e^{-1/2} \beta_e^{-1}$ observed in
figures <ref>a and <ref>a can be explained in terms of the ion-scale and electron-scale microinstabilities
having distinct collisional-stabilisation conditions of the form (<ref>) (viz., $k \lambda_e \sim k \lambda_i \lesssim 1$), with the condition on the
ion-scale instabilities being more restrictive.
The wavenumbers $k_{\rm mirr}$ and $k_{\rm fire}$ at which
maximal growth of the ion mirror and firehose instabilities occurs satisfy $k_{\rm mirr} \rho_i \sim 1$ and $k_{\rm fire} \rho_i \lesssim 1$, respectively, for
$\mathrm{Ma} \, \lambda_e \beta_e/L_V \gg 1$,
leading to the collisional-stabilisation condition

\begin{equation}
\frac{\lambda_e}{L_V} \lesssim \frac{\rho_i}{L_V} \sim \mu_e^{-1/2} \beta_e^{1/2} \frac{d_e}{L_V} \,  .  \label{mirror_coll_stab}
\end{equation}

For the electron-scale microinstabilities, the parallel and the oblique transverse
instabilities have the largest (common) wavenumber of all such instabilities
that operate when $\epsilon_i > 0$ and $\epsilon_i < 0$, respectively, and
so provide the most demanding collisional-stabilisation conditions.
For both transverse instabilities, the wavenumber at which peak growth occurs for the
satisfies $k_{\rm trans}  \rho_e \sim (\mu_e^{1/2} \mathrm{Ma} \, \lambda_e \beta_e/L_V)^{1/2}$
[see (<ref>)], which in turn can be rearranged to give
the collisional-stabilisation condition

\begin{equation}
\frac{\lambda_e}{L_V} \lesssim  \mathrm{Ma}^{-1/3} \mu_e^{-1/6} \left(\frac{d_e}{L_V}\right)^{2/3} \, .  \label{transverse_coll_stab}
\end{equation}

Bringing these results together, we find

\begin{equation}
\left(\frac{d_e}{L_V}\right)_{\rm c} = \left\{
\begin{array}{ll}
\mu_e^{1/2} \beta_e^{-1/2} {\lambda_e}/{L_V} , & \beta_e^{-1} \ll \mathrm{Ma} \, {\lambda_e}/{L_V} < \mu_e^{-1/2} \beta_e^{-1},  \\[2pt]
\mu_e^{1/4} \mathrm{Ma}^{1/2} \left({\lambda_e}/{L_V}\right)^{3/2},  & \mathrm{Ma} \, {\lambda_e}/{L_V} \gtrsim \mu_e^{-1/2}
\beta_e^{-1},
\end{array} \right. \label{de_LVc}
\end{equation}

with $\left(d_e/L_V\right)_{\rm c0} = \mu_e^{1/2} \beta_e^{-3/2}$.
This matches
asymptotically the numerical results shown in figures <ref>a and <ref>a.
These findings confirm that,
once again, the relevant collisional-stabilisation condition for the microinstabilities with
wavenumber $k$ is $k \lambda_e  = k \lambda_i\ll 1$
[viz., (<ref>)], as opposed to the more restrictive conditions $\gamma \tau_i \gg 1$ and $\gamma \tau_e \gg 1$ on the CES ion-scale and electron-scale instabilities, respectively.
Similary to the collisional-stabilisation condition on the CET whistler instability (see section <ref>), we note that the collisional-stabilisation condition on any of these
microinstabilities can never actually be satisfied in a strongly magnetised plasma, because $k \lambda_i \gtrsim \lambda_i/\rho_i \gg
1$ for the ion-scale instabilities, and $k \lambda_e \gtrsim \lambda_e/\rho_e \gg 1$ for the electron-scale instabilities.

§.§.§ Outline of the rest of this section

Further discussion about the properties and growth rates of CES microinstabilities
with $\epsilon_s > 0$ (viz., those driven by positive pressure anisotropy) can be found in
section <ref>, with the mirror, whistler and transverse
instabilities discussed in sections <ref>, <ref> and
<ref>, respectively. In addition to these,
there is another instability (the electron mirror instability) that can be driven by positive pressure
anisotropy of CE distribution functions that we note in passing: it consists in KAWs driven unstable by the CE electron-shear term, and to some extent by the ion-shear term (section <ref>).
The electron mirror instability does not appear
to be the fastest-growing CES microinstability anywhere in the $(d_e/L_V,\mathrm{Ma} \, \lambda_e/L_V)$ parameter
space; since the instability is subdominant to two other electron-scale instabilities (the whistler and transverse
instabilities), this would seem to imply that the instability is comparatively less important.

CES microinstabilities with $\epsilon_s < 0$ (viz., those driven by negative pressure
anisotropy) are explored in section <ref>. The firehose
instability is overviewed in section
<ref>, with then four subclasses of the instability (parallel, oblique, critical-line, and sub-ion-Larmor-scale) considered in sections <ref>, <ref>, <ref>, and<ref>. The oblique electron firehose instability is discussed in section
<ref>, the EST instability in section
<ref>, the oblique transverse instability in section
<ref>, and the whisper instability in section
<ref>. We identify two additional CES microinstabilities
which are never the fastest-growing microinstability in any unstable region:
the parallel electron firehose instability (section
<ref>), which (in spite of its name) has a different
underlying physical mechanism than the oblique electron firehose, and the
ordinary-mode instability (section <ref>), which only
operates at very high $\beta_e$ ($\beta_e \gtrsim |\Delta_e|^{-3}$), and is only characteristically
distinct from the oblique transverse instability in a regime in which
it is slower growing.

Readers who do not wish to dwell on specific CES
microinstabilities should proceed directly to section <ref>.

§.§ CES microinstability classification: positive pressure anisotropy ($\epsilon_i > 0$)

§.§.§ Mirror instability

The CES mirror instability consists in the destabilisation of compressive slow modes
by a sufficiently large positive
ion pressure anisotropy associated with the ion-shear term of the ion CE
distribution function. In a high-$\beta$ plasma with Maxwellian ion and electron distribution functions, the slow mode – which is one of the two plasma
modes which exist at oblique wavevector angles $\theta \gtrsim \beta_i^{-1/4}$ (the other being the shear Alfvén
wave), and consists of a perturbation to the magnetic field's strength – is non-propagating, being subject to strong Barnes' (equivalently, transit-time)
damping [Barnes, 1966]. This damping is the result of Landau-resonant interactions
between the slow mode and co-moving ions with $v_{\|} = \omega/k_{\|}$; since,
for a distribution function that decreases monotonically with $v_{\|} > 0$, there
are more ions with $v_{\|} < \omega/k_{\|}$ than with $v_{\|} > \omega/k_{\|}$,
there is a net transfer of free energy from the slow modes to the ions (as a particle acceleration
process, this is sometimes called betatron acceleration). However, in a plasma
with $\Delta_i > 0$, there is an increase in the relative number of ions with large pitch angles
in the troughs of the slow mode's magnetic-field strength perturbation,
giving rise to excess perpendicular pressure. When $\Delta_i > 1/\beta_i$, this excess pressure overbalances the
magnetic pressure, leading to the mirror instability. In CE plasma with $0 < \Delta_i \beta_i -1 \ll 1$, only quasi-perpendicular long-wavelength
mirror modes ($k_{\|} \rho_i \ll k_{\perp} \rho_i \ll 1$) are destabilised; for
larger values of $\Delta_i$, a broad range of slow modes (including ion-Larmor-scale ones)
become unstable. Chronologically, the earliest discussions of the
mirror instability in pressure-anisotropic plasmas are due to Parker, 1958 and Hasegawa, 1969.
Southwood & Kivelson, 1993 provide a detailed and lucid discussion of the linear physics of the mirror instability <cit.>; various analytical [Pokhotelov et al., 2008, Rincon et al., 2015] and
numerical [Hellinger et al., 2009, Kunz et al., 2014, Riquelme et al., 2015, Melville et al., 2016]
studies investigating its nonlinear evolution have also been carried out.

The CES mirror instability can be characterised analytically – and simple expressions
derived for the maximum growth rate and the wavevector at which that growth is attained – in the limit of
marginal instability. First, we define the threshold parameter $\Gamma_i \equiv \beta_i \Delta - 1$,
where $\Delta \equiv \Delta_i + \Delta_e = (1+\mu_e^{1/2})\Delta_i$,
and assume that $\Gamma_i \ll 1$. It can then be shown (see appendix <ref>) that under the orderings

\begin{equation}
k_{\|} \rho_i  \sim k_{\bot}^2 \rho_i^2 \sim \Gamma_i \ll 1 \, , \quad  \frac{\gamma}{\Omega_i} \sim \frac{\Gamma_i^2}{\beta_i}
\ll 1
\, ,
\label{mirror_order_marg}
\end{equation}

the mirror modes have a growth rate given by

\begin{equation}
\frac{\gamma}{\Omega_i} = \frac{k_{\|} \rho_i}{\sqrt{\upi} \beta_i}  \left(\Gamma_i -\frac{3}{2} \frac{k_{\|}^2}{k_{\bot}^2}-\frac{3}{4} k_{\bot}^2 \rho_i^2 \right)
\, . \label{mirrorgrowthrate_marg}
\end{equation}

This is the same result as the growth rate of the mirror instability in a
bi-Maxwellian plasma, with
(the anticipated) threshold $\Gamma_i > 0$ [Hellinger, 2007].
The peak growth rate $\gamma_{\mathrm{max}}$
is then given by

\begin{equation}
\gamma_{\mathrm{max}} = \frac{ \Gamma_i^2}{6 \sqrt{2 \upi} \beta_i} \Omega_i \, ,
\end{equation}

achieved at the wavenumber

\begin{equation}
(k_{\|} \rho_i)_{\mathrm{peak}} =
\frac{\Gamma_i}{3\sqrt{2}}    \, , \quad (k_{\bot} \rho_i)_{\mathrm{peak}} = \frac{\Gamma_i^{1/2}}{\sqrt{3}}\, . \label{mirror_marg_kval}
\end{equation}

This recovers the results of Hellinger, 2007.

Figure <ref> illustrates the accuracy of the above predictions for $\gamma$ (and therefore $\gamma_{\rm max}$),
$(k_{\|} \rho_i)_{\mathrm{peak}}$ and $(k_{\bot} \rho_i)_{\mathrm{peak}}$ by
comparing them with the equivalent values obtained numerically using the general method outlined in appendix
<ref> for a particular value of $\Gamma_i \ll 1$.

Mirror instability at $\Gamma_i = \Delta \beta_i -1 \ll 1$. a) Growth rates of unstable mirror modes resulting from the CE ion-shear term in the CE distribution function (<ref>) for $\Gamma_i = 0.04 \ll 1$ ($\Delta \beta_i = 1.04$).
The growth rates of all modes are calculated using the approach outlined in appendix <ref>. 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 $\gamma \beta_i/\Omega_i$, are functions of the dimensionless quantity $\Delta \beta_i$.
The dashed white lines indicate the analytical prediction (<ref>)
for the wavenumber at which peak growth is achieved.
b) The mirror mode's growth rate (solid line) as a function of $k_{\|} \rho_i$ with $k_{\bot} \rho_i = \Gamma_i^{1/2}/\sqrt{3}$ (top), and as a function of $k_{\perp} \rho_i$ with $k_{\|} \rho_i = \Gamma_i/3\sqrt{2}$ (bottom). The
dashed lines show the analytical prediction (<ref>) for these quantities.

The wavenumber dependence of the numerically determined growth rate
(see figure <ref>a) corroborates that, close to marginality, the unstable
mirror modes are quasi-perpendicular; more quantitatively, the values of
$k_{\|} \rho_i$ and $k_{\bot} \rho_i$
at which peak growth is obtained numerically match (<ref>).
Furthermore, the growth rate (<ref>)
agrees well with the numerical result when plotted as a function of $k_{\|} \rho_i$ with fixed $k_{\bot} \rho_i$,
and also as a function of $k_{\bot} \rho_i$ with fixed $k_{\|} \rho_i$ (figure

In contrast, for finite $\Gamma_i \gtrsim 1$, simple expressions for $\gamma_{\rm max}$,
$(k_{\|} \rho_i)_{\mathrm{peak}}$, and $(k_{\bot} \rho_i)_{\mathrm{peak}}$ are
challenging to derive analytically. Our numerical calculations indicate that, when $\Gamma_i \sim
1$, a broad range of (purely growing) oblique modes becomes unstable, with
maximum growth rate $\gamma_{\rm max} \sim \Omega_i/\beta_i \sim \Delta
\Omega_i$ attained when $k_{\|} \rho_i \lesssim k_{\perp} \rho_i \sim 1$
(figure <ref>a).

Mirror instability at $\Gamma_i = \Delta \beta_i -1 \sim 1$. a) Growth rates of unstable mirror modes resulting from the CE ion-shear term in the CE distribution function (<ref>) for $\Gamma_i = 1$ ($\Delta \beta_i = 2$).
The growth rates of all modes are calculated in the same way as in figure <ref>. The dashed white lines indicate the analytic prediction (<ref>)
for the parallel/perpendicular wavenumber at which peak growth is achieved, while the dotted line
indicates the analytical prediction (<ref>) for the perpendicular
wavenumber above which
long-wavelength ($k_{\|} \rho_i \lesssim k_{\perp} \rho_i \ll 1$) mirror modes
become unstable.
b) The mirror mode's growth rate (solid line) as a function of $k_{\|} \rho_i$ with $k_{\bot} \rho_i = \Gamma_i^{1/2}/\sqrt{3}$ (top), and as a function of $k_{\perp} \rho_i$ with $k_{\|} \rho_i = \Gamma_i/3\sqrt{2}$ (bottom). The
dashed lines show the analytical prediction (<ref>) for this quantity.

Therefore, asymptotic expansions that treat $k_{\bot} \rho_i$ and $k_{\|} \rho_i$
as small or large cannot be used to derive simplified expressions for the growth
rate of the fastest-growing mirror modes. While the expressions (<ref>) for the wavenumber of peak growth derived
in the case of near-marginality remain
qualitatively correct, they are no longer quantitatively accurate; the same
conclusion applies to the expression (<ref>) for the growth rate
when $k_{\|} \rho_i \sim k_{\perp} \rho_i \sim 1$ (figure
<ref>b). That being said, an expression similar to
(<ref>) can be derived (see appendix <ref>) for long-wavelength unstable mirror
that satisfy the ordering

\begin{equation}
k_{\|} \rho_i  \sim k_{\bot} \rho_i  \ll 1 \, , \quad  \frac{\gamma}{\Omega_i} \sim \frac{k_{\|} \rho_i}{\beta_i} \sim {\Delta} k_{\|} \rho_i
\ll 1
\, .
\label{mirror_order}
\end{equation}

This expression is

\begin{equation}
\frac{\gamma}{\Omega_i} = \frac{k_{\|} \rho_i}{\sqrt{\upi} \beta_i}  \left(\Gamma_i -\frac{\Gamma_i + 3}{2} \frac{k_{\|}^2}{k_{\bot}^2}\right)
\, . \label{mirrorgrowthrate}
\end{equation}

It implies that all such modes with

\begin{equation}
k_{\bot} > \left(\frac{3+\Gamma_i}{2 \Gamma_i}\right)^{1/2} k_{\|} \label{mirror_angleinstab}
\end{equation}

will be unstable, a prediction that is consistent with the unstable region
observed in figure <ref>a.

When $\Gamma_i \gg 1$, but $\Gamma_i < (m_i/m_e)^{1/2}$,
the region of $(k_{\|},k_{\bot})$ space in which mirror modes are unstable is qualitatively similar
to the $\Gamma_i \sim 1$ case, albeit more extended (figure

Mirror instability at $\Gamma_i = \Delta \beta_i \gg 1$. a) Growth rates of unstable mirror modes resulting from the CE ion-shear term in the CE distribution function (<ref>) for $\Gamma_i = 29 \gg 1$ ($\Delta \beta_i = 30$).
The growth rates of all modes are calculated in the same way as in figure <ref>.
The dot-dashed white lines indicate the parallel/perpendicular wavenumbers (<ref>) at which peak growth is achieved,
while the dotted line indicates the analytical prediction (<ref>) for the perpendicular
wavenumber above which
long-wavelength ($k_{\|} \rho_i \lesssim k_{\perp} \rho_i \ll 1$) mirror modes
become unstable.
b) Normalised maximum positive growth rate $\gamma_{\rm max}/\Delta \Omega_i$ (solid red line) of the unstable mirror mode as
a function of $\Delta \beta_i$ along with the parallel (solid blue line) and perpendicular (solid yellow line) wavenumbers, $(k_{\|} \rho_i)_{\mathrm{peak}}$ and $(k_{\perp} \rho_i)_{\mathrm{peak}}$ respectively, at
which that growth is attained. The analytical prediction (<ref>) of $\gamma_{\rm max}$ for marginally unstable modes,
as well as the analogous predictions (<ref>) for $(k_{\|} \rho_i)_{\mathrm{peak}}$ and $(k_{\perp} \rho_i)_{\mathrm{peak}}$, are shown as dashed lines.

We find that in this limit, the maximum growth rate
$\gamma_{\mathrm{max}}$ becomes
directly proportional to $\Delta$ (see figure
<ref>b), in contrast to the marginal case (<ref>):

\begin{equation}
\gamma_{\mathrm{max}} \approx 0.2 \Delta \Omega_i \, . \label{mirrorgrowth}
\end{equation}

This growth is attained at parallel and perpendicular wavenumbers

\begin{equation}
(k_{\bot} \rho_i)_{\mathrm{peak}} \approx 1.2 \, , \quad (k_{\|} \rho_i)_{\mathrm{peak}} \approx
0.7 \, , \label{mirrorgrowth_kval}
\end{equation}

which depend only weakly on $\Delta \beta_i$.

Some understanding of these results can be derived by considering the dispersion
relation of mirror modes on sub-ion Larmor scales. Adopting the ordering

\begin{equation}
k_{\|} \rho_i \sim k_{\perp} \rho_i \sim (\Delta_i \beta_i)^{1/2} \gg 1 , \quad
\frac{\gamma}{\Omega_i} \sim \Delta_i \, , \label{order_subionmirror}
\end{equation}

while assuming that $\Delta_i \beta_i \ll \mu_e^{-1/2}$,
one finds (see appendix <ref>) that

\begin{equation}
\frac{\gamma}{\Omega_i} \approx \frac{k_\|}{k} \sqrt{\left(\frac{k^2 \rho_i^2}{\beta_i} - \Delta_i \frac{k_{\|}^2-k_{\bot}^2}{k^2} \right) \left(\Delta_i \frac{k_{\|}^2}{k^2} - \frac{k^2 \rho_i^2}{\beta_i} \right)} \, .
\label{ionmirror_subionscale}
\end{equation}

This can be re-written in terms of the wavevector angle $\theta = \tan^{-1}{(k_{\perp}/k_{\|})}$ as

\begin{equation}
\frac{\gamma}{\Omega_i} \approx \cos{\theta} \sqrt{\left[\frac{k^2 \rho_i^2}{\beta_i} - \Delta_i \left(\cos^2{\theta}-\sin^2{\theta}\right) \right] \left(\Delta_i \cos^2{\theta} - \frac{k^2 \rho_i^2}{\beta_i} \right)} \, .
\label{ionmirror_subionscale_theta}
\end{equation}

Analysing this expression leads to three conclusions. First, for $\theta > 45^{\circ}$, there is an
instability at all wavenumbers satisfying $k \rho_i < (\Delta_i \beta_i)^{1/2}
\cos{\theta}$, explaining the expansion of the unstable region of $(k_{\|},k_{\bot})$-space
with increasing $\Delta_i \beta_i$. For $\theta \leq 45^{\circ}$, growth only occurs over a more limited range
of wavenumbers $\sqrt{\cos^2{\theta}-\sin^2{\theta}} < k \rho_i/(\Delta_i \beta_i)^{1/2} <
\cos{\theta}$. Secondly, growth in this limit is maximised when $k \rho_i \ll (\Delta_i
\beta_i)^{1/2}$, with the maximal growth rate

\begin{equation}
\gamma_{\rm max} = \frac{1}{3 \sqrt{3}} \Delta_i \Omega_i  \approx 0.19 \Delta_i \Omega_i \label{mirrorgrowth_largek}
\end{equation}

attained at $\cos{\theta} = 1/\sqrt{3}$ ($\theta \approx 55^{\circ}$). This expression for $\gamma_{\rm max}$
is (surprisingly) close to the numerically measured peak growth rate
(<ref>). For $k \rho_i \sim (\Delta_i
\beta_i)^{1/2}$, the maximum growth rate is smaller than
(<ref>) by an order-unity factor. Finally, when $k \rho_i \gg (\Delta_i \beta_i)^{1/2}$, viz.,
in a wavenumber regime where there are no unstable mirror modes, (<ref>)
becomes imaginary, implying that the modes have a real frequency given by

\begin{equation}
\omega \approx \pm k_\| k_{\bot} \rho_e \frac{\Omega_e}{\beta_i} \, . \label{KAWdisprel_mirror}
\end{equation}

This is the dispersion relation of kinetic Alfvén waves (KAWs) in a high-$\beta$
plasma[We note that (<ref>) is also the same dispersion relation as that of oblique whistler waves <cit.>.
However, as was discussed in section <ref>, in a
high-$\beta$ plasma ($\beta_e \gg \mu_e^{-1/2}$), the small frequency ($\omega \ll k_{\|} v_{\mathrm{th}i}$) of
perturbations prohibits all but parallel perturbations from not interacting significantly with
the ions, and thus we believe that the modes are more accurately identified as KAWs.].
In short, at $\Delta_i \beta_i \gg 1$, KAWs are also destabilised by positive ion pressure
anisotropy in addition to longer-wavelength mirror modes. We note that KAWs can also be destabilised by positive electron anisotropy, but the characteristic wavelength of such modes is preferentially comparable to electron Larmor scales (see section <ref>).

§.§.§ Whistler instability

The CES whistler instability arises when
the free energy associated with positive electron-pressure anisotropy $\Delta_e$ of the
electron CE distribution function destabilises whistler waves, overwhelming both the electron cyclotron damping (which
is the dominant stabilisation mechanism for whistler
waves with $k_{\|} \rho_e \sim 1$) and the Landau damping due to the ion species (the domininant
stabilisation mechanism for waves with $k_{\|} \rho_e \ll 1$).
In the special case of static ions, electron cyclotron damping can
be overcome by a positive electron-pressure anisotropy of any magnitude for whistler waves
with sufficiently long wavelengths.
Retaining mobile ions, the instability operates only if $\Delta_e$
exceeds a threshold of order $(\Delta_e)_{\rm c} \sim \beta_e^{-1}$.
When $\Delta_e > (\Delta_e)_{\rm c}$,
gyroresonant interactions between electrons with $v_{\|} = \pm \Omega_e/k_{\|}$
and whistler waves allow for free energy to pass from the former to the latter,
and so an increasingly broad spectrum of unstable parallel and oblique modes emerges on
electron Larmor scales.
The analogue of this instability in a bi-Maxwellian plasma
was found by Kennel & Petschek, 1966, and it has since been studied numerically in moderately high-$\beta$ plasma ($\beta_e \sim 1$-$10$) by
several authors <cit.>.

Similarly to the CET whistler instability, the simplest characterisation
of the CES whistler instability is for unstable parallel whistler modes
(viz., $k \approx k_{\|}$). Assuming that these modes satisfy the

\begin{equation}
\tilde{\omega}_{e\|} = \frac{\omega}{k_{\|} v_{\mathrm{th}e}} \sim \Delta_e \sim \frac{1}{\beta_e} \, , k_{\|} \rho_e \sim 1 ,
\label{electronWeibel_order}
\end{equation}

it can be shown (see appendix <ref>) that their real frequency $\varpi$ and growth rate $\gamma$

ϖβ_e/Ω_e     =      ±Δ_e β_e
±k_ ρ_e [Δ_e β_e (1+μ_e^1/2)- k_^2 ρ_e^2]   Z(1/k_ ρ_e) /[  Z(1/k_ ρ_e)]^2 + exp(-2/k_^2 ρ_e^2)

γβ_e/Ω_e      =
k_ ρ_e [exp(-1/k_^2 ρ_e^2)+μ_e^1/2](Δ_e β_e- k_^2 ρ_e^2) +μ_e^1/2 Δ_e β_e   Z(1/k_ ρ_e) /[  Z(1/k_ ρ_e)]^2/√() + √() exp(-2/k_^2 ρ_e^2)

where the terms proportional to $\mu_e^{1/2}$ are associated with the ion species[Formally, these terms are $\textit{O}(\mu_e^{1/2})$ under our assumed ordering,
and so should be dropped. However, because of the exponential dependence of the other damping/growth terms on $k_{\|} \rho_e$,
these terms play an important role for moderate values of $k_{\|} \rho_e$, viz. $\mu_e^{1/2} \exp{\left({1}/{k_\|^2 \rho_e^2}\right)} \geq 1$ for $k_\| \rho_e \leq \sqrt{2}/\sqrt{\log{m_i/m_e}} \approx
0.5$, so we retain them.]. In the limit $\mu_e \rightarrow 0$,
formally there is always instability provided $\Delta_e \beta_e > 0$; however,
for a hydrogen plasma ($\mu_e \approx 1/1836$), it can be shown numerically that the
numerator of (<ref>b) only becomes positive (over a narrow interval
of parallel wavenumbers around $k_{\|} \rho_e \approx 0.60$) for $\Delta_e \beta_e >
0.56$. The
dispersion curves $\varpi(k_{\|})$ and $\gamma(k_{\|})$ of the unstable whistler waves in
a hydrogen plasma for three different values of $\Delta_e \beta_e$
that are above the necessary value for instability are shown in
figure <ref>.

Parallel CES whistler instability.  Dispersion curves of unstable whistler modes whose instability is driven by the electron-shear term in CE distribution function
(<ref>), for wavevectors that are co-parallel with the background
magnetic field (viz., $\boldsymbol{k} = k_{\|} \hat{\boldsymbol{z}}$).
The frequency (solid blue) and growth rate (solid red) of the modes are calculated using
(<ref>a) and
(<ref>b), respectively.
The resulting frequencies and growth rates, when normalised as $\gamma \beta_e/\Omega_e$, are functions of the dimensionless quantity $\Delta_e \beta_e$; we show the dispersion curves for three different values of $\Delta_e \beta_e$.
The approximations (<ref>a) and (<ref>b) 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 (<ref>a) and (<ref>b)
for the frequency (dashed blue) and growth rate (dashed red) in the limit $k_{\|} \rho_e \gg

When $\Delta_e \beta_e \gtrsim 1$, the growth rate is postive
for a range $\Delta k_{\|} \sim \rho_e^{-1}$ around $k_\| \rho_e \sim
1$, attaining a characteristic magnitude $\gamma \sim \varpi \sim \Omega_e/\beta_e$.

As before, we characterise the growth rate for various
values of $\Delta_e \beta_e$ by taking subsidiary limits.
First, for $\Delta_e \beta_e \ll 1$, a necessary (though not always sufficient) condition
for positive growth is $k_\| \rho_e < (\Delta_e \beta_e)^{1/2} \ll 1$.
We therefore expand (<ref>) in $k_\| \rho_e \sim (\Delta_e \beta_e)^{1/2} \ll
1$, finding that

ϖ    ≈    k_^2 ρ_e^2/β_e Ω_e

γ    ≈
√()/k_ ρ_e {exp(-1/k_^2 ρ_e^2) (Δ_e - k_^2 ρ_e^2/β_e) -μ_e^1/2 k_^2 ρ_e^2/β_e } Ω_e .

Similarly to what we showed in section <ref> for the CET whistler instability,
we have once again found unstable
whistler waves. For comparison's sake, the approximate expressions (<ref>)
are plotted in figure <ref> in addition
to their exact analogues (<ref>); it is clear that
there is reasonable agreement for a moderately small value of $\Delta_e
\beta_e$, but that the approximations become less accurate for  $k_\| \rho_e \gtrsim
0.5$ and $\Delta_e \beta_e > 1$.

In the limit $\mu_e \rightarrow 0$, the expression (<ref>b)
for the growth rate is very similar to that of the whistler (electron-cyclotron) instability in a plasma with a bi-Maxwellian
distribution and positive electron pressure anisotropy [Davidson, 1983]. In this case, whistler modes with $k_\| \rho_e < (\Delta_e \beta_e)^{1/2}$
are always unstable, although the growth rate
of such modes is exponentially small in $\Delta_e \beta_e \ll 1$ as compared to
the frequency (<ref>a), and so $\gamma \ll \varpi \sim \Omega_e/\beta_e$.
By contrast, with small but finite $\mu_e = m_e/m_i$, it can be shown analytically that,
for (<ref>b) to be positive, $\Delta_e > (\Delta_e)_{\rm c}$, where

\begin{eqnarray}
(\Delta_e)_{\rm c} & = & \frac{1}{\beta_e W_{\rm Lam}\left[\mu_e^{-1/2}\exp{(-1)}\right]}
\nonumber\\
& \approx &  \frac{1}{\beta_e} \frac{1}{\log{(\mu_e^{-1/2})}-1-\log{[\log{(\mu_e^{-1/2})}-1}]}  \, .
\label{CETwhist_thresh}
\end{eqnarray}

Here, $W_{\rm Lam}(x)$ denotes the Lambert W function [Corless et al., 1996].
Unstable modes first develop around $(k_{\|} \rho_e)_c = (\Delta_e)_{\rm c}^{1/2} /[(\Delta_e)_{\rm c}+1/\beta_e]^{1/2}$. In a hydrogen plasma, this gives $(\Delta_e)_{\rm c} \approx 0.49/\beta_e$ and $(k_{\|} \rho_e)_c \approx
0.57$, which are similar to the instability threshold and wavenumber, respectively, determined numerically if $\gamma$ is computed for
arbitrary values of $k_{\|} \rho_e$; the small discrepancy is
due to the finite value of $k_{\|}  \rho_e$ at which instability
first emerges. Formally, $(\Delta_e)_{\rm c} \rightarrow 0$ as $\mu_e \rightarrow 0$, but
the limit converges only logarithmically in $\mu_e$, suggesting that in an
actual plasma, the CES whistler instability will generically have a threshold at a finite value of $\Delta_e \beta_e$.

Let us now turn to the opposite subsidiary limit $\Delta_e \beta_e \gg 1$. We find from (<ref>b) that maximal growth
occurs at $k_\| \rho_e \sim (\Delta_e \beta)^{1/2} \gg 1$:

ϖ    ≈    1/[Δ_e (-2)+ k_^2 ρ_e^2/β_e] Ω_e

γ    ≈
k_ ρ_e/√() (Δ_e - k_^2 ρ_e^2/β_e) Ω_e   .

Alongside $k_\| \rho_e \ll 1$ approximations, these approximations are plotted in figure
<ref>, and agree well with the numerical results for $\Delta_e \beta_e \gtrsim 3$
and $k_\| \rho_e \gtrsim 2$. The maximum growth rate

\begin{equation}
\gamma_{\mathrm{max}} = \frac{2}{3 \sqrt{3 \upi}} \Delta_e (\Delta_e \beta_e)^{1/2} \Omega_e \approx 0.22 \Delta_e (\Delta_e \beta_e)^{1/2} \Omega_e \label{CESwhistler_maxgrowth}
\end{equation}

is attained at the parallel wavenumber

\begin{equation}
(k_\| \rho_e)_{\mathrm{peak}} = (\Delta_e \beta_e/3)^{1/2}. \label{CESwhistler_maxgrowth_kval}
\end{equation}

A notable feature of the CES whistler instability in this subsidary limit is that the fastest-growing
modes are on sub-electron-Larmor scales; thus, such modes are arguably better
conceptualised not as whistler modes, but as unstable, unmagnetised plasma modes (see section <ref>).

Similarly to the CET whistler instability, analytical expressions for the frequency and growth rate of
unstable modes that have an oblique wavevector angle are much less simple
that the analogous expressions for parallel whistler modes. It can be shown (see appendix <ref>) that
the complex frequency of such modes is given by

\begin{equation}
\omega = \frac{\Omega_e}{\beta_e} k_{\|} \rho_e \frac{- \mathrm{i} B_{\mathrm{S}} \pm \sqrt{-B_{\mathrm{S}}^2 + 4A_{\mathrm{S}}C_{\mathrm{S}}}}{2 A_{\mathrm{S}}}
\, , \label{shear_frequency}
\end{equation}

where the functions $A_{\mathrm{S}} = A_{\mathrm{S}}(k_{\|} \rho_e,k_{\perp} \rho_e,\Delta_e \beta_e)$, $B_{\mathrm{S}} = B_{\mathrm{S}}(k_{\|} \rho_e,k_{\perp}
\rho_e,\Delta_e \beta_e)$, and $C_{\mathrm{S}} = C_{\mathrm{S}}(k_{\|} \rho_e,k_{\perp} \rho_e,\Delta_e \beta_e)$
are composed of the sums and products of special mathematical functions. When $\Delta_e \beta_e \sim 1$,
(<ref>) implies that if there is an instability, its growth rate will be of order $\gamma \sim
\Omega_e/\beta_e$ at $k_{\|} \rho_e, k_{\perp} \rho_e \sim 1$.

To confirm this expectation, in figure <ref> we plot the maximum growth rate (obtained numerically) of oblique modes across the $(k_{\|},k_{\perp})$-plane
for two of the values of $\Delta_e \beta_e$ used in figure <ref>.

Oblique unstable modes at $\Delta_e \beta_e \sim 1$: a) $\Delta_e \beta_e = 0.75$.
b) $\Delta_e \beta_e = 3$. Maximum positive growth rates of linear perturbations
resulting from CE ion- and electron-shear terms in the CE distribution function (<ref>) for $\Delta_e \beta_e \sim 1$.
Here, a temperature-equilibrated hydrogen plasma is considered, viz. $\Delta_e = \mu_e^{1/2} \Delta_i$, and $\beta_i = \beta_e$.
The growth rates of all modes are calculated using the approach outlined in appendix <ref>. The growth rates are
calculated on a $400^2$ grid, with logarithmic spacing between wavenumbers in
both perpendicular and parallel directions.
The resulting growth rates, when normalised as $\gamma \beta_e/\Omega_e$, are functions of $\Delta_e \beta_e$, or, equivalently, $\epsilon_e \beta_e$. The vertical dashed lines indicate $k_{\|} \rho_i = 1$ and $k_{\|} \rho_e =
1$, respectively, while the horizontal ones indicate $k_{\perp} \rho_i = 1$ and $k_{\perp} \rho_e =

For $\Delta_e \beta_e$ not far beyond the threshold of the CES whistler
instability (figure <ref>a), the unstable modes are quasi-parallel
and have growth rates $\gamma \ll \Omega_e/\beta_e$
(cf. figure <ref>, left panel).
For $\Delta_e \beta_e \gtrsim
1$, a broader spectrum of wavenumbers becomes unstable (figure
<ref>b). The parallel mode remains the fastest growing in this case;
however, oblique modes with $k_{\perp} \lesssim k_{\|}/2$ also have growth rates of comparable magnitude: e.g., the fastest-growing mode with wavevector angle $\theta = 10^{\circ}$ has $\gamma_{\rm max}/\gamma_{\rm max}(k_{\perp} = 0) \approx
0.93$, and for a wavevector angle $\theta = 10^{\circ}$, $\gamma_{\rm max}/\gamma_{\rm max}(k_{\perp} = 0) \approx
0.76$. For more oblique angles, the growth rate is reduced significantly: e.g., for $\theta = 30^{\circ}$, $\gamma_{\rm max}/\gamma_{\rm max}(k_{\perp} = 0) \approx
0.22$. Thus, we conclude that a spectrum of oblique modes in addition to parallel ones is
indeed destabilised, with $\gamma \sim \Omega_e/\beta_e \lesssim \gamma(k_{\perp} = 0)$.

We note that, in addition to oblique CES whistler modes, whose characteristic wavenumber domain is $k_{\perp} \rho_e \lesssim k_{\|} \rho_i \sim 1$, we observe two other
unstable modes in figure <ref>a with different characteristic values of $k_{\|}$ and $k_{\perp}$. The first of
these, which exists on ion scales, is the CES mirror instability, which we already discussed in section
<ref>. The second is the CES electron mirror instability – we shall consider this instability
in section <ref>.

§.§.§ Parallel transverse instability

As was shown in section <ref>, in the limit $\Delta_e \beta_e \gg 1$, the fastest-growing
CES microinstability is essentially unmagnetised, and is a variant of the so-called transverse
instability [Kahn, 1962, Kahn, 1964, Albright, 1970]. This instability is also sometimes referred to as the resonant (electron) Weibel instability,
or the Weibel instability at small anisotropy [Weibel, 1959, Fried, 1959]. Both the linear theory of
this instability and its physical mechanism have been explored extensively for
bi-Maxwellian plasmas <cit.>, and various
studies (both analytical and numerical) of its nonlinear evolution have also been performed [Albright, 1970, Davidson et al., 1972, Lemons et al., 1979, Califano et al., 1998, Califano et al., 2002, Kato, 2005, Pokhotelov & Amariutei, 2011, Ruyer et al., 2015].
For the small anisotropy case that is relevant to CE
plasma, the mechanism of the instability is somewhat subtle, involving both
non-resonant and Landau-resonant wave-particle interactions. In a Maxwellian plasma, transverse
modes are non-propagating
and Landau-damped by electrons
with velocities $v \approx \omega/k_{\|}$. However, this damping can be
reversed by the free energy associated with positive electron-pressure anisotropy at
wavenumbers that satisfy $k d_e \lesssim \Delta_e^{1/2}$; the
electron Landau damping increases more rapidly with $k$ than the instability's drive, which in turn sets
the wavenumber at which peak growth occurs. The requirement for the corresponding scale to be well below
the electron Larmor scale – and thus for the plasma to be quasi-unmagnetised with respect to the transverse modes –
sets the restriction $\Delta_e \beta_e \gg 1$ on the instability's operation.
In general, transverse modes whose wavevectors are co-parallel to the velocity-space direction along
which the temperature is smallest are the fastest growing; in the case of a CE electron distribution function of the form (<ref>) with $\Delta_e > 0$, these modes' wavevectors
are parallel to the magnetic field. However, a broad spectrum of
oblique transverse modes is also destabilised when $\Delta_e > 0$.

To characterise the transverse instability's growth analytically, we first assume $\Delta_e \beta_e \gg
1$, and then take directly the unmagnetised limit of the full CES dispersion relation
(see appendix <ref>) under the orderings

\begin{equation}
k_{\bot} \rho_e \sim k_{\|} \rho_e \sim \left(\Delta_e \beta_e\right)^{1/2} \gg 1 \, , \quad  \tilde{\omega}_{e\|} = \frac{\omega}{k_{\|} v_{\mathrm{th}e}}  \sim \Delta_e \, .
\end{equation}

We obtain two non-propagating modes (real frequency $\varpi = 0$) that have growth rates

γ_1     =     k v_the/√() (Δ_e k_^2-k_^2/k^2 - k^2 ρ_e^2/β_e)   ,

γ_2     =     k v_the/√() (Δ_e k_^2/k^2 - k^2 ρ_e^2/β_e)   .

For $\Delta_e > 0$, the growth rate of the second mode is always positive
and larger than that of the first mode; the first mode only has a positive growth rate provided $k_{\bot} < k_{\|}$. Now
taking the subsidiary limit $k_{\|} \rho_e \gg k_{\bot} \rho_e \gg 1$, we find that
both roots have the same growth rate:

\begin{equation}
\gamma \approx \frac{k_{\|} v_{\mathrm{th}e}}{\sqrt{\upi}} \left(\Delta_e - \frac{k_{\|}^2 \rho_e^2}{\beta}\right)
\, , \label{partransverse_growthrate}
\end{equation}

which is identical to (<ref>b). We note by comparison with (<ref>a) that
the unmagnetised limit fails to recover the non-zero real frequencies of the $k_{\|} \rho_e \gg 1$
whistler modes; this is because the
ratio of these modes' real frequency $\varpi$ to their growth rate $\gamma$ is $\varpi/\gamma \sim 1/k_{\|}
\rho_e \ll 1$.

The maximum growth rate $\gamma_{\mathrm{max}}$ of the second mode (<ref>b) for an oblique wavevector with angle $\theta$ is

\begin{equation}
\gamma_{\mathrm{max}} = \frac{2}{3 \sqrt{3 \upi}} \cos^{3}{\theta} \, \Delta_e (\Delta_e \beta_e)^{1/2} \Omega_e  ,\label{transverse_maxgrowth}
\end{equation}

attained at the (total) wavenumber

\begin{equation}
(k \rho_e)_{\mathrm{peak}} = \cos{\theta} \, (\Delta_e \beta_e/3)^{1/2}. \label{transverse_maxgrowth_kval}
\end{equation}

The parallel and perpendicular wavenumbers of this maximum growth are then

\begin{equation}
(k_{\|} \rho_e)_{\mathrm{peak}} = \cos^2{\theta} \, (\Delta_e \beta_e/3)^{1/2} , \quad (k_{\bot} \rho_e)_{\mathrm{peak}} = \cos{\theta} \sin{\theta} \, (\Delta_e \beta_e/3)^{1/2} . \label{transverse_maxgrowth_kpl_pp_val}
\end{equation}

In the special case of parallel modes ($\theta = 0^{\circ}$), this recovers the peak growth rate (<ref>)
of the CES whistler instability at $k_{\|} $ in the limit $\Delta_e \beta_e \gg 1$.

In figure <ref>, we demonstrate that the fastest-growing unstable modes in the limit $\Delta_e \beta_e \gg 1$ are indeed transverse
ones. This figure shows the numerically determined growth rate
as a function of $k_{\|}$ and $k_{\perp})$, for a particular large value of $\Delta_e \beta_e$.

Oblique unstable modes at $\Delta_e \beta_e \gg 1$. a) Maximum positive growth rates of linear perturbations
resulting from CE ion- and electron-shear terms in the CE distribution function (<ref>) for $\Delta_e \beta = 100$.
Here, a temperature-equilibrated hydrogen plasma is considered, viz., $\Delta_e = \mu_e^{1/2} \Delta_i$ and $\beta_i = \beta_e$.
The growth rates of all modes are calculated in the same way as figure <ref>. The vertical dashed line
indicates the value of $k_{\|} \rho_e$ at which maximum growth of the parallel transverse instability is attained [see (<ref>)], while the horizontal one indicates $k_{\perp} \rho_e =
b) The transverse mode's growth rate (solid line) as a function of $k_{\|} \rho_e$ with $k_{\bot} \rho_e = 1$ (top), and as a function of $k_{\perp} \rho_e$ with $k_{\|} \rho_e =
(\Delta_e \beta_e/3)^{1/2}$ (bottom). The
dashed lines show the analytical prediction (<ref>b) for this quantity.

A broad range of sub-electron-Larmor scale modes are unstable (figure <ref>a), with the
parallel wavenumber of the fastest-growing ones closely agreeing with the
analytical prediction (<ref>). The analytical expression (<ref>b) for
the transverse instability's growth rate also agrees well with the numerical result as a function
of both $k_{\|}$ and $k_{\perp}$ (figure <ref>b).

§.§.§ Electron mirror instability

The oblique microinstability evident in figure <ref>b at sub-ion-Larmor scales
is the CES electron mirror instability: the destablisation of KAWs
by excess perpendicular electron pressure (viz., $\Delta_e > 0$) associated with the CE electron-shear term.
The instability <cit.> is
perhaps confusingly named, given that its physical mechanism is rather
different to that of the (ion-scale) mirror instability: non-resonant
interactions between the anisotropic distribution of electrons and the KAWs causes the restoring force
underpinning the latter's characteristic oscillation to be negated if $\Delta_e > 1/\beta_e$.
The electron mirror instability has been extensively explored in $\beta_e \sim 1$
plasma <cit.>;
in plasmas with $\beta_e \gg 1$, it has been analytically characterised and its
physical mechanism elucidated in the quasi-perpendicular ($k_{\|} \ll k_{\perp}$) limit
of gyrokinetics [Kunz et al., 2018]. Here, we find that once its marginality
condition ($\Delta_e = 1/\beta_e$) is surpassed sufficiently, oblique modes
with $k_{\|} \lesssim k_{\perp}$ are also destabilised.

As with the mirror instability, a simple analytic characterisation of the CES electron mirror instability can be
performed in the case of marginal instability. We define
the marginality parameter $\Gamma_e \equiv \Delta_e \beta_e -1$, and adopt the

\begin{equation}
k_{\perp}^2 \rho_e^2 \sim  k_{\|} \rho_e \sim \tilde{\omega}_{e\|} \beta_e \sim \Gamma_e
\ll 1 ,
\label{KAW_ords_marg}
\end{equation}

with the additional assumption that $\Gamma_e \gg \mu_e^{1/2}$ in order that
the effect of ion pressure anisotropy can be neglected. Then, it can be
shown (see appendix <ref>) that the growth rate is

\begin{equation}
\frac{\gamma}{\Omega_e} = \frac{k_{\|} \rho_e}{\beta_e} \left[-\frac{3\sqrt{\upi}}{4} k_{\perp}^2 \rho_e^2+ \sqrt{\frac{3}{2} \Gamma_e k_{\perp}^2 \rho_e^2 -\frac{9}{4} k_{\|}^2 \rho_e^2 + \frac{9}{16}\left(\upi-2\right) k_{\perp}^4 \rho_e^4 }
\right]
. \label{electronmirror_growthrate}
\end{equation}

It follows that the maximum growth rate is

\begin{equation}
\gamma_{\rm max} = \frac{\left[\upi-8+\sqrt{\upi\left(16+\upi\right)}\right]^{3/2}}{48\left(\upi-2\right)} \left[ \sqrt{\frac{\upi+4+\sqrt{\upi\left(16+\upi\right)}}{\upi-8+\sqrt{\upi\left(16+\upi\right)}}}-\sqrt{\frac{\upi}{\upi-2}}\right] \approx 0.055 \frac{\Gamma_e^2}{\beta_e} , \label{CESelectronmirror_peakgrowthrate}
\end{equation}

attained at

(k_ ρ_e)_peak     =     √(-8+√((16+))/36(-2)) Γ_e ≈0.27 Γ_e,

(k_⊥ρ_e)_peak     =     √(-8+√((16+))/6(-2)) Γ_e^1/2 ≈0.65 Γ_e^1/2.

Figure <ref> demonstrates that these predictions are accurate
by comparing them to numerical results for a particular (small) value of $\Gamma_e$.

Electron mirror instability at $\Gamma_e = \Delta_e \beta_e -1 \ll 1$. a) Growth rates of unstable electron mirror modes associated with the CE distribution function (<ref>) for $\Gamma_e= 1/3$ ($\Delta_e \beta_e = 4/3$).
The growth rates of all modes are calculated in the same way as figure <ref>.
The dashed white lines indicate the analytical prediction (<ref>)
for the parallel/perpendicular wavenumber at which peak growth is achieved.
b) Plot of the electron mirror mode's growth rate (solid line) as a function of $k_{\|} \rho_e$ with $k_{\bot} \rho_e = 0.65 \Gamma_e^{1/2}$ (top), and as a function of $k_{\perp} \rho_e$ with $k_{\|} \rho_e = 0.27 \Gamma_e$ (bottom). The
dashed lines show the analytical prediction (<ref>) for this quantity.

More specifically, figure <ref>a shows that the location in the
$(k_{\|},k_{\perp})$ plane at which the maximum growth of the electron mirror instability is attained
closely matches the analytical prediction
(<ref>), while figure <ref>b confirms that the
wavenumber dependence of the growth rate agrees with (<ref>) for $k_{\perp} \rho_e \gtrsim
\mu_e^{1/4}$. We note that, in addition to the electron mirror, another instability operating at smaller characteristic values
of $k_{\perp} \rho_e$ is evident in figure <ref>. These are the $k_{\perp} \rho_i \gtrsim 1$ mirror modes
driven unstable by the CE ion-shear term that were discussed in section
<ref>; for $1 \ll k \rho_i \ll \mu_e^{-1/4}$, the ion-pressure anisotropy associated
with the CE ion-shear terms remains a greater free-energy source for KAW instabilities
than the CE electron-shear term, even when $\Delta_e > 1/\beta_e$.

For $\Gamma_e \gtrsim 1$, our near-marginal theory anticipates that peak growth
occurs at electron Larmor scales ($k_{\|} \rho_e \lesssim k_{\|} \rho_e \sim 1$), with $\gamma_{\rm max} \sim
\Omega_e/\beta_e$. These expectations are indeed realised numerically, as shown in figure <ref> (see also
figure <ref>).

Electron mirror instability at $\Gamma_e = \Delta_e \beta_e -1 \sim 1$. a) Growth rates of unstable electron mirror modes associated with the CE distribution function (<ref>) for $\Gamma_e = 1$ ($\Delta_e \beta_e = 2$).
The growth rates of all modes are calculated in the same way as figure <ref>.
The dashed white lines indicate the analytical prediction (<ref>)
for the parallel/perpendicular wavenumber at which peak growth is achieved, while the dotted line
indicates the analytical prediction (<ref>) for the total
wavenumber below which oblique
long-wavelength ($k_{\|} \rho_e < k_{\perp} \rho_e \ll 1$) electron mirror modes
become unstable.
b) The electron mirror mode's growth rate (solid line) as a function of $k_{\|} \rho_e$ with $k_{\bot} \rho_e = 0.65 \Gamma_e^{1/2}$ (top), and as a function of $k_{\perp} \rho_e$ with $k_{\|} \rho_e = 0.27 \Gamma_e$ (bottom). The
dashed lines show the analytical prediction (<ref>) for this quantity.

The expression (<ref>) for the growth rate as a function of wavenumber that was derived in the case of $\Gamma_e \ll 1$ remains qualitatively – but not quantitatively –
accurate (see figure <ref>b). Figure <ref> shows that a similar conclusion holds for the expression (<ref>)
for the peak growth rate, and also for the expressions (<ref>a) and (<ref>b) of the parallel and perpendicular wavenumbers at
which that growth occurs.

The maximum growth of the electron mirror instability. The maximum normalised growth rate $\gamma_e \beta_e/\Omega_e$ (left panel, solid red line) of unstable electron mirror modes as
a function of $\Delta_e \beta_e$, as well as parallel (middle panel, solid blue line) and perpendicular (right panel, solid yellow line) wavenumbers, $(k_{\|} \rho_e)_{\mathrm{peak}}$ and $(k_{\perp} \rho_e)_{\mathrm{peak}}$, respectively, at
which that growth is attained. The analytical prediction (<ref>) of $\gamma_{\rm max}$ for marginally unstable electron mirror modes,
as well as the analogous predictions (<ref>) for $(k_{\|} \rho_e)_{\mathrm{peak}}$ and $(k_{\perp} \rho_e)_{\mathrm{peak}}$, are shown as dashed
lines. The dotted lines are the maximum growth rate and (parallel) wavenumber of peak growth
for the CET parallel whistler instability (see section <ref>) as functions of $\Delta_e \beta_e$.

To confirm our prior claim in section <ref> that the CES parallel whistler
instability is faster growing than the electron mirror instability, we show the
former's numerically computed growth rate on figure <ref> (left panel); as
it approaches the asymptotic value (<ref>) that is valid in the
limit $\Delta_e \beta_e \gg 1$, we observe that the electron mirror's growth
rate is a factor of ${\sim}3$ smaller (cf. figure <ref>a). The
parallel wavenumber at which peak growth of the whistler instability occurs is
also larger than the analogous quantity for the electron mirror by an order-unity factor.

While we cannot derive a simple analytic expression for the growth rate of the dominant
electron mirror modes when $\Gamma_e \gtrsim 1$, we can calculate this quantity for long-wavelength (viz., $k \rho_e \ll 1$) modes. For this calculation,
we assume that $k \rho_e \sim \mu_e^{1/4} \ll 1$, $k_{\bot} \sim k_{\|}$, and
the ordering

\begin{equation}
\tilde{\omega}_{e\|}  = \frac{\omega}{k_{\|} v_{\mathrm{th}e}}  \sim \frac{k \rho_e}{\beta_e}  \sim |\Delta_e| k \rho_e \, .
\label{KAW_ords}
\end{equation}

Under these assumptions, we obtain (see appendix <ref>) two modes whose complex frequencies
$\omega$ are given by

\begin{eqnarray}
\omega & \approx & \pm k_\| \rho_e \Omega_e \Bigg\{\left[\frac{1}{\beta_e} + \Delta_e\left(\frac{1}{2}- \mu_e^{1/2} \frac{k_{\|}^2 \rho_e^2- k_{\bot}^2 \rho_e^2}{k^4 \rho_e^4}\right) \right] \nonumber\\
&& \qquad \qquad \times \left[ \frac{k^2 \rho_e^2}{\beta_e} - \Delta_e \left(k_{\bot}^2 \rho_e^2 + \mu_e^{1/2} \frac{k_{\|}^2}{k^2} - \frac{1}{2} k_{\|}^2 \rho_e^2 \right) \right]\Bigg\}^{1/2} .
\label{obliqueinstab_freq_gen}
\end{eqnarray}

The terms proportional to $\mu_e^{1/2} \Delta_e$ are associated
with the CE ion-shear term, which plays a non-negligible role for $k \rho_e \lesssim
\mu_e^{1/4}$. In the subsidiary limit $k \rho_e \ll \mu_e^{1/4}$, (<ref>)
becomes the dispersion relation (<ref>) obtained in section <ref> for
unstable mirror modes in the limit $\Delta_i \beta_i \gg 1$.
In the opposite subsidiary limit $k \rho_e \gg \mu_e^{1/4}$ (but $k \rho_e \ll 1$),
(<ref>) simplifies to

\begin{eqnarray}
\omega & \approx & \pm k_\| \rho_e \Omega_e \sqrt{\left(\frac{1}{\beta_e} + \frac{\Delta_e}{2}\right)\left[\frac{k^2 \rho_e^2}{\beta_e} - \Delta_e \left(k_{\bot}^2 \rho_e^2 - \frac{1}{2} k_{\|}^2 \rho_e^2\right) \right]} \, .
\label{obliqueinstab_freq_elec}
\end{eqnarray}

For $k_{\|} \ll k_{\perp}$, this recovers the high-$\beta$ limit of the
dispersion relation for unstable KAWs previously derived in the gyrokinetic
calculations of Kunz et al., 2018; our calculations show that this dispersion
relation also applies to oblique ($k_{\|} \lesssim k_{\perp}$) electron mirror modes. For $\Delta_e > 0$, we (as expected) have an unstable root if and only if

\begin{equation}
\Delta_e > \frac{1}{\beta_e} , \label{threshold_electronmirror}
\end{equation}

with the unstable mode's growth rate being

\begin{eqnarray}
\gamma & \approx & k_\| \rho_e \Omega_e \sqrt{\left(\frac{1}{\beta_e} + \frac{\Delta_e}{2}\right)\left[\Delta_e \left(k_{\bot}^2 \rho_e^2 - \frac{1}{2} k_{\|}^2 \rho_e^2\right) - \frac{k^2 \rho_e^2}{\beta_e} \right]} \, .
\label{obliqueinstab_growth_elec}
\end{eqnarray}

We can now provide an analytical demonstration that
a broad spectrum of electron mirror modes is unstable if $\Gamma_e \gtrsim 1$.
It follows directly from (<ref>) that instability arises for all
modes with $k_{\perp} > k_{\|}$ if the following constraint on the total wavenumber $k$ is satisfied:

\begin{equation}
k \rho_i < \sqrt{\frac{2 \mu_e^{1/2} \left(\Gamma_e+1\right) \cos^2{\theta}}{\left(\Gamma_e+3\right) \cos^2{\theta}-2 \Gamma_e \sin^2{\theta}}}
\, , \label{elecmirror_instabbound}
\end{equation}

where $\theta = \tan^{-1}{(k_{\perp}/k_{\|})}$ is, as normal, the
wavevector angle. The validity of this bound is illustrated in figure <ref>a.
(<ref>) is particularly simple to interpret in the
subsidiary limit $k \rho_e \gg \mu_e^{1/4}$, yielding a lower bound on $\theta$ alone:

\begin{equation}
\theta > \tan^{-1}{\sqrt{\frac{\Gamma_e+3}{2 \Gamma_e}}} . \label{elecmirror_angleinstab}
\end{equation}

For $\Gamma_e \ll 1$ (but $\Gamma_e > 0$), this implies that the only unstable electron mirror modes are
quasi-perpendicular, as anticipated from our calculations pertaining to the marginal state of the
instability. On the other hand, for $\Gamma_e \gtrsim 1$, modes with a
wide range of wavevector angles will
be destabilised.

§.§ CES microinstability classification: negative pressure anisotropy $(\epsilon_i < 0)$

§.§.§ Firehose instability

The best-known instability to be triggered by either negative ion or electron pressure anisotropy
associated with the CE ion- and electron-shear terms, respectively, is the CES firehose
instability. The linear theory of the firehose (or garden-hose) instability in high-$\beta$ plasma, the first
studies of which were completed over half a century ago [Rosenbluth, 1956, Parker, 1958, Chandrasekhar et al., 1958, Vedenov & Sagdeev, 1958],
has previously been explored in the contexts of plasmas with bi-Maxwellian distributions <cit.>,
CE distributions <cit.>, and even characterisations that are
independent of the ion distribution function <cit.>.
Its physical mechanism is well established: negative pressure anisotropies
reduce the elasticity of magnetic-field lines that gives rise to Alfvén waves, and can completely
reverse it when $\Delta_i$ is negative enough. The long-wavelength `fluid'
firehose instability (whose mechanism is independent of the particular ion distribution
function) is non-resonant in nature; however, resonant damping mechanisms such
as Barnes damping or cyclotron damping play an important role in regulating
the growth of modes on scales comparable to the ion-Larmor scale, and thereby
set the scale of peak firehose growth. Beyond linear theory, nonlinear analytical
studies of the parallel firehose instability in high-$\beta$ plasma have been
completed <cit.>, as well as numerical
ones <cit.>.