2.1. Magnetic equilibrium and geometry

The magnetic geometry that we adopt is one of constant magnetic curvature, as this allows us to capture the effect of the magnetic drifts on our plasma while retaining most of the simplicity associated with conventional slab gyrokinetics (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Ivanov et al. Reference Ivanov, Schekochihin, Dorland, Field and Parra2020). We consider a domain positioned in the magnetic field of a current line at a radial distance $R$ from the central axis, and define the $\hat {\boldsymbol {x}}$ and $\hat {\boldsymbol {y}}$ directions as pointing radially outwards and parallel to the central axis, respectively, as shown in figure 1. In the context of the outboard midplane in tokamak geometry, these are the ‘radial’ and ‘poloidal’ coordinates, respectively, terms that we adopt in our later discussions. In such a geometry, the magnetic field consists of an equilibrium part that is oriented in the $\boldsymbol {b}_0 = \hat {\boldsymbol {z}}$ direction and varies radially, plus a time- and space-dependent fluctuating part:

(2.1)\begin{equation} \boldsymbol{B}(\boldsymbol{r},t) = B_0(x) \boldsymbol{b}_0 + \delta \! \boldsymbol{B}_{{\perp}}(\boldsymbol{r},t). \end{equation}

In what follows, we express the perpendicular magnetic-field fluctuations in terms of the parallel component of the magnetic vector potential:

(2.2)\begin{equation} \delta \! \boldsymbol{B}_{{\perp}}(\boldsymbol{r},t) =\boldsymbol{\nabla}\times \boldsymbol{A} ={-}\boldsymbol{b}_0 \times \boldsymbol{\nabla} A_\parallel. \end{equation}

The component of the magnetic-field fluctuations parallel to the mean field is negligible in the limit of low beta [see (A20)]. The electric field is related to the magnetic vector potential $\boldsymbol {A}$ and electrostatic potential $\phi$ by

(2.3)\begin{equation} \boldsymbol{E} (\boldsymbol{r},t) ={-} \frac{1}{c} \frac{\partial \boldsymbol{A}}{\partial t} - \boldsymbol{\nabla}\phi, \end{equation}

and is assumed to have no mean part. The equilibrium (mean) magnetic field has the scale length and radius of curvature

(2.4a,b)\begin{equation} L_B^{{-}1} ={-} \frac{1}{B_0}\frac{\mathrm{d} B_0}{\mathrm{d} x}, \quad R^{{-}1} = \left| \boldsymbol{b}_0 \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{b}_0\right|, \end{equation}

respectively, both of which are assumed to be constant across our domain. Note that for a low-beta plasma, $R = L_B$ (see Appendix A.1). We assume that the background gradient of the temperature $T_{0e}$ associated with the equilibrium distribution of the electrons also varies radially, with scale length

(2.5)\begin{equation} L_T^{{-}1} ={-} \frac{1}{T_{0e}} \frac{\mathrm{d} T_{0e}}{\mathrm{d} x}, \end{equation}

which, similarly, is assumed to be constant over the domain. The thermal speed of the electrons is then given by $v_{{\rm th}e} = \sqrt {2T_{0e}/m_e}$, where $m_e$ is the electron mass.

Figure 1. Illustration of the constant-curvature geometry, showing the domain positioned at a distance $R$ from the current axis, with the $\hat {\boldsymbol {x}}$ and $\hat {\boldsymbol {y}}$ directions pointing radially outwards and parallel to this axis, respectively. The equilibrium magnetic field is in the $\boldsymbol {b}_0$ direction. Both the equilibrium temperature $T_{0e}$ and equilibrium magnetic field $B_0$ vary radially, with their scale lengths $L_T$ and $L_B$, respectively, assumed constant across the domain.

With this local equilibrium, and adopting a low-beta ordering (see Appendix A.2), we derive evolution equations for the density ($\delta n_{e}$), parallel velocity ($u_{\parallel e }$), parallel temperature ($\delta T_{\parallel e }$) and perpendicular temperature ($\delta T_{\perp e }$) perturbations of the electrons. These equations are presented in the following sections. We assume everywhere that the electron Larmor radius $\rho _e$ is small, and so work in the drift-kinetic approximation for the electrons.

2.2. Density perturbations

The perturbed electron density satisfies the continuity equation:

(2.6)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} + \boldsymbol{\nabla}_\parallel u_{{\parallel} e} + \frac{\rho_e v_{{\rm th}e}}{2L_{B}} \frac{\partial}{\partial y} \left( \frac{\delta T_{{\parallel} e}}{T_{0e}} + \frac{\delta T_{{\perp} e}}{T_{0e}} \right) = 0. \end{equation}

This says that the density perturbation is subject to three influences: (i) advection by the $\boldsymbol {E}\times \boldsymbol {B}$ motion of the electrons:

(2.7)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} = \frac{\partial}{\partial t} + \boldsymbol{v}_E \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\perp}}, \quad \boldsymbol{v}_E = \frac{\rho_e v_{{\rm th}e}}{2} \boldsymbol{b}_0 \times \boldsymbol{\nabla}_{{\perp}} \varphi, \quad \varphi = \frac{e \phi}{T_{0e}}, \end{equation}

where $-e$ is the electron charge; (ii) compression or rarefaction due to the perturbed parallel electron flow $u_{\parallel e} \boldsymbol {b}$ along the exact magnetic field, including the perturbation of the magnetic field direction:

(2.8)\begin{equation} \boldsymbol{\nabla}_\parallel{=} \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla} = \frac{\partial}{\partial z} + \frac{\delta \! \boldsymbol{B}_{{\perp}}}{B_0} \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\perp}}, \quad \frac{\delta \! \boldsymbol{B}_{{\perp}}}{B_0} ={-} \rho_e \boldsymbol{b}_0 \times \boldsymbol{\nabla}_{{\perp}} \mathcal{A}, \quad \mathcal{A} = \frac{A_\parallel}{\rho_e B_0}; \end{equation}

and (iii) the magnetic drifts due to the finite curvature and gradient of the magnetic field. The parallel and perpendicular temperature perturbations arise from the velocity dependence of the curvature and $\boldsymbol {\nabla } B$ drifts in the gyrokinetic equation [see (A29)]. The presence of these magnetic drifts is essential for the curvature-mediated instabilities that are the focus of § 3.3 and much of § 4.

The continuity equation (2.6) is (A68) in Appendix A.5, except that in (2.6) we have set the equilibrium density gradient to zero, and ignored the magnetic-drift terms proportional to $\delta n_{e}/n_{0e}$ and $\varphi$, as they will always turn out to be smaller than the magnetic-drift terms proportional to the temperature perturbations in what follows. This is in a bid to make our equations as simple as possible, while retaining all of the relevant physics (see Appendix A.7 for further details). We shall ignore similar terms in our other equations for the perturbations, for the same reason. Cautious readers may be reassured by the fact that all of the instabilities considered in §§ 3 and 4 are derived in a limit in which this is a valid approximation.

2.3. Parallel velocity perturbations

The parallel momentum equation associated with the electrons is [see (A69)]

(2.9)\begin{equation} n_{0e} m_e \frac{\mathrm{d} u_{{\parallel} e}}{\mathrm{d} t} ={-} \boldsymbol{\nabla}_\parallel p_{{\parallel} e} - en_e E_\parallel{-} \nu_{ei}m_e u_{{\parallel} e}. \end{equation}

The three forces appearing on the right-hand side are, from right to left: (i) the collisional drag against the ions (which are assumed motionless), where $\nu _{ei}$ is the electron–ion collision frequency; (ii) the parallel electric field

(2.10)\begin{equation} E_\parallel{=} \boldsymbol{b}\boldsymbol{\cdot} \boldsymbol{E} ={-} \left( \frac{1}{c} \frac{\partial A_\parallel}{\partial t} + \boldsymbol{\nabla}_\parallel \phi \right) ={-} \left( \frac{1}{c} \frac{\mathrm{d} A_\parallel}{\mathrm{d} t} + \frac{\partial \phi}{\partial z} \right); \end{equation}

and (iii) the parallel pressure gradient, which consists of both the parallel gradient of the parallel-pressure perturbation and the projection of the equilibrium temperature gradient onto the perturbed magnetic field:

(2.11)\begin{equation} \boldsymbol{\nabla}_\parallel p_{{\parallel} e} = \boldsymbol{\nabla}_\parallel \delta p_{{\parallel} e} + n_{0e} \frac{\delta B_x}{B_0} \frac{\mathrm{d} T_{0e}}{\mathrm{d} x} = n_{0e} T_{0e} \left[ \boldsymbol{\nabla}_\parallel \left( \frac{\delta n_{e}}{n_{0e}} + \frac{\delta T_{{\parallel} e}}{T_{0e}}\right) - \frac{\rho_e}{L_T} \frac{\partial \mathcal{A}}{\partial y} \right]. \end{equation}

Since an electron flow uncompensated by an ion flow is a current, $u_{\parallel e}$ is related to $A_\parallel$ via Ampère's law [see (A41)]:

(2.12)\begin{equation} - e n_{0e} u_{{\parallel} e} = j_{{\parallel}} = \frac{c}{4{\rm \pi}} \boldsymbol{b}_0 \boldsymbol{\cdot} \left( \boldsymbol{\nabla}_{{\perp}} \times \delta \! \boldsymbol{B}_{{\perp}} \right) \quad \Rightarrow \quad u_{{\parallel} e} = v_{{\rm th}e} d_e^{2} \boldsymbol{\nabla}_{{\perp}}^{2} \mathcal{A}, \end{equation}

where $c$ is the speed of light, $d_e = c(m_e/4{\rm \pi} e^{2} n_{0e})^{1/2} = \rho _e /\sqrt {\beta _e}$ is the electron inertial scale and $\beta _e = 8{\rm \pi} n_{0e} T_{0e}/B_0^{2}$. The electron inertial scale $d_e$ is an important quantity throughout this work, as it demarcates the boundary between the electrostatic and electromagnetic regimes in the collisionless limit (see § 2.7). In the collisional limit [see (A74)], the frictional term on the right-hand side of (2.9) dominates over the electron inertial term on the left-hand side, meaning that the electron inertia can be neglected.

2.4. Temperature perturbations

The parallel temperature $T_{\parallel e} = T_{0e} + \delta T_{\parallel e}$ is advected by the local $\boldsymbol {E} \times \boldsymbol {B}$ flow and is locally increased (or decreased) by the compressional heating (or rarefaction cooling) due to $u_{\parallel e}$, as well as by the (appropriately normalised) perturbed parallel heat flux $\delta q_{\parallel e}$ [see (A70)]:

(2.13)\begin{equation} \frac{\mathrm{d} T_{{\parallel} e}}{\mathrm{d} t} = \frac{\mathrm{d} \delta T_{{\parallel} e}}{\mathrm{d} t} + \boldsymbol{v}_E \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\perp}} T_{0e} ={-} \boldsymbol{\nabla}_\parallel \frac{\delta q_{{\parallel} e}}{n_{0e} } - 2 T_{0e} \boldsymbol{\nabla}_\parallel u_{{\parallel} e} - \frac{4}{3} \nu_e\left(\delta T_{{\parallel} e} - \delta T_{{\perp} e} \right). \end{equation}

The factor of $2$ in the compressional-heating term (the second on the right-hand side) is due to the fact that we only consider the parallel (one-dimensional) motion of the electrons [perpendicular motions are formally small within our ordering; see (A22)]. The last term on the right-hand side is a consequence of our choice of collision operator [see (A55) and the subsequent discussion], and is responsible for collisional temperature isotropisation, with $\nu _e = \nu _{ee} + \nu _{ei}$, and $\nu _{ee} = \nu _{ei}/Z$ the electron–electron collision frequency ($Ze$ is the ion charge).

Similarly, the perpendicular temperature $T_{\perp e} = T_{0e} + \delta T_{\perp e}$ evolves according to [see (A71)]

(2.14)\begin{equation} \frac{\mathrm{d} T_{{\perp} e}}{\mathrm{d} t} = \frac{\mathrm{d}\delta T_{{\perp} e}}{\mathrm{d} t} + \boldsymbol{v}_E \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\perp}} T_{0e} ={-} \boldsymbol{\nabla}_\parallel \frac{\delta q_{{\perp} e}}{n_{0e}} - \frac{2}{3}\nu_e\left(\delta T_{{\perp} e} - \delta T_{{\parallel} e} \right), \end{equation}

where $\delta q_{\perp e}$ is the perturbed perpendicular heat flux. Note the absence of perpendicular compressional heating (perpendicular flows are incompressible).

The term expressing the seeding of both parallel and perpendicular temperature perturbations via the advection of the equilibrium temperature profile by the $\boldsymbol {E}\times \boldsymbol {B}$ flow becomes, after a straightforward manipulation, the familiar (electrostatic) linear drive responsible for extracting (‘electrostatic’) free energy from the equilibrium temperature gradient:

(2.15)\begin{equation} \boldsymbol{v}_E \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\perp}} T_{0e} = T_{0e} \frac{\rho_e v_{{\rm th}e}}{2 L_T} \frac{\partial \varphi}{\partial y}, \end{equation}

where $L_T$ is defined in (2.5). In order to determine the heat fluxes $\delta q_{\parallel e}$ and $\delta q_{\perp e}$, kinetic theory is needed, and so we must append to our emerging system of equations the drift-kinetic equation for electrons (see Appendix A.5), of which (2.6), (2.9), (2.13) and (2.14) are four lowest-order moments.

In the collisional limit [see (A73)], the temperature isotropisation terms in (2.13) and (2.14) are dominant, enforcing $\delta T_{\parallel e} = \delta T_{\perp e} = \delta T_{e}$ to leading order. In this limit, therefore, we no longer distinguish between parallel and perpendicular temperature perturbations, and obtain an equation for $\delta T_{e}$ from the linear combination $(1/3)$(2.13) $+(2/3)$(2.14) [see (A83) and (A90)]:

(2.16)\begin{equation} \frac{\mathrm{d} \delta T_{e}}{\mathrm{d} t} + \boldsymbol{v}_E \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\perp}} T_{0e} ={-} \frac{2}{3} \boldsymbol{\nabla}_\parallel \frac{\delta q_e}{n_{0e} } - \frac{2}{3} T_{0e} \boldsymbol{\nabla}_\parallel {u_{{\parallel} e}}, \end{equation}

where the (collisional) heat flux $\delta q_e = \delta q_{\parallel e}/2 + \delta q_{\perp e}$ can be expressed in terms of the parallel gradient of the total temperature $T_e = T_{0e} + \delta T_{e}$ along the exact magnetic field direction [see (A87)]:

(2.17)\begin{equation} \frac{\delta q_{e}}{n_{0e} T_{0e} } ={-} \frac{3}{2}\kappa \boldsymbol{\nabla}_\parallel \log T_e, \quad \kappa = \frac{5v_{{\rm th}e}^{2}}{18 \nu_e}, \end{equation}

where $\kappa$ is the electron thermal diffusivity and

(2.18)\begin{equation} \boldsymbol{\nabla}_\parallel \log T_e = \boldsymbol{\nabla}_\parallel \frac{\delta T_{e}}{T_{0e}} + \frac{\delta B_x }{B_0} \frac{1}{T_{0e}} \frac{\mathrm{d} T_{0e}}{\mathrm{d} x} = \boldsymbol{\nabla}_\parallel \frac{\delta T_{e}}{T_{0e}} - \frac{\rho_e}{L_T} \frac{\partial \mathcal{A}}{\partial y} \end{equation}

is the parallel gradient of the total electron temperature, which will prove a key quantity in what follows.

2.5. Quasineutrality

Finally, as usual, particle density is related to $\phi$ via quasineutrality, which is the route whereby ions contribute to dynamics. Since, at scales smaller than their Larmor radius $\sim \rho _i$, ions can be viewed as large motionless rings of charge, their density response is Boltzmann:

(2.19)\begin{equation} \frac{\delta n_{e}}{n_{0e}} = \frac{\delta n_i}{n_{0i}} ={-} \frac{Ze \phi}{T_{0i}} ={-} \bar{\tau}^{{-}1} \varphi, \quad \bar{\tau} = \frac{\tau}{Z}, \end{equation}

where $\tau = T_{0i}/T_{0e}$ is the ratio of the ion to electron equilibrium temperatures. The more general quasineutrality closure, for which $\bar {\tau }^{-1}$ is an operator and which includes scales comparable to the ion Larmor radius, is given in Appendix A.3, but, since we focus on scales smaller than this in our discussions, (2.19) is sufficient for our purposes.

2.6. Summary of equations

Assembling together all of the above, we end up with the following systems of equations, in the collisionless limit [see (A8) with $\nu _{ei},\nu _{ee} \rightarrow 0$]:

(2.20)\begin{align} &\frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} + \boldsymbol{\nabla}_\parallel u_{{\parallel} e} + \frac{\rho_e v_{{\rm th}e}}{2L_{B}} \frac{\partial}{\partial y} \left( \frac{\delta T_{{\parallel} e}}{T_{0e}} + \frac{\delta T_{{\perp} e}}{T_{0e}} \right) = 0, \end{align}

(2.21)\begin{align}&\frac{\mathrm{d}}{\mathrm{d} t} \left( \mathcal{A} - \frac{u_{{\parallel} e}}{v_{{\rm th}e}} \right) ={-} \frac{v_{{\rm th}e}}{2} \left[\frac{\partial \varphi}{\partial z} - \boldsymbol{\nabla}_\parallel \left( \frac{\delta n_{e}}{n_{0e}}+ \frac{\delta T_{{\parallel} e}}{T_{0e}} \right) + \frac{\rho_e}{L_T} \frac{\partial \mathcal{A}}{\partial y} \right], \end{align}

(2.22)\begin{align}&\frac{\mathrm{d} }{\mathrm{d} t} \frac{\delta T_{{\parallel} e}}{T_{0e}} + \boldsymbol{\nabla}_\parallel \left(\frac{\delta q_{{\parallel} e}}{n_{0e} T_{0e} } + 2 u_{{\parallel} e} \right) + \frac{\rho_e v_{{\rm th}e}}{2 L_T} \frac{\partial \varphi}{\partial y} =0, \end{align}

(2.23)\begin{align}&\frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta T_{{\perp} e}}{T_{0e}} + \boldsymbol{\nabla}_\parallel \frac{\delta q_{{\perp} e}}{n_{0e} T_{0e}} + \frac{\rho_e v_{{\rm th}e}}{2 L_T} \frac{\partial \varphi}{\partial y} =0, \end{align}

or, in the collisional limit [see (A74)]:

(2.24)\begin{align}& \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} + \boldsymbol{\nabla}_\parallel u_{{\parallel} e} + \frac{\rho_e v_{{\rm th}e}}{L_{B}} \frac{\partial}{\partial y} \frac{\delta T_{ e}}{T_{0e}} = 0, \end{align}

(2.25)\begin{align}&\frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} +\frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \left( \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}} + \boldsymbol{\nabla}_\parallel \log T_e \right) + \nu_{ei} \frac{u_{{\parallel} e}}{v_{{\rm th}e}}, \end{align}

(2.26)\begin{align}&\frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta T_{e}}{T_{0e}} - \kappa \boldsymbol{\nabla}_\parallel^{2} \log T_e + \frac{2}{3} \boldsymbol{\nabla}_\parallel u_{{\parallel} e} + \frac{\rho_e v_{{\rm th}e}}{2 L_{T}} \frac{\partial \varphi}{\partial y} =0, \end{align}

to which we append the field equations:

(2.27)\begin{equation} \frac{\delta n_{e}}{n_{0e}} ={-} \bar{\tau}^{{-}1} \varphi, \quad \frac{u_{{\parallel} e}}{v_{{\rm th}e}} = d_e^{2} \boldsymbol{\nabla}_{{\perp}}^{2} \mathcal{A}. \end{equation}

This system is a minimal model for describing low-beta electromagnetic plasma dynamics – whether collisionless or collisional – driven by a background ETG, and in the presence of magnetic drifts.

2.7. Flux freezing

These equations describe two broad classes of physical phenomena: electrostatic and electromagnetic, distinguished by whether the magnetic field lines are frozen into the electron flow or not. We refer to the perpendicular scale at which the transition between these two regimes occurs as the ‘flux-freezing scale’. In the collisionless limit, this scale is given by the balance between the electron inertia and the inductive parallel electric field on the left-hand side of (2.21), viz.,

(2.28)\begin{equation} k_{{\perp}} d_e \sim 1. \end{equation}

In the collisional limit, the analogous balance involves, instead of electron inertia, the resistive term – the last on the right-hand side of (2.25). However, in this limit, we always deal with perturbations for which the term in (2.25) that contains the projection of the equilibrium temperature gradient onto the perturbed magnetic field [the second part of $\boldsymbol {\nabla }_\parallel \log T_e$ written in (2.18)] is larger than $\partial \mathcal {A}/\partial t$. Therefore, it is with this term that the effect of resistivity will be usefully compared:

(2.29)\begin{equation} \omega \lesssim \omega_{*e} \equiv \frac{k_y \rho_e v_{{\rm th}e}}{2 L_T} \sim k_{{\perp}}^{2} d_e^{2} \nu_{ei}, \end{equation}

where $\omega _{*e}$ is the drift frequency associated with the ETG. For modes with $k_y \sim k_{\perp }$, the balance (2.29) can be written as

(2.30)\begin{equation} k_{{\perp}} d_e \sim \frac{\rho_e}{d_e} \frac{v_{{\rm th}e}}{ L_T \nu_{ei}} = \sqrt{\beta_e} \frac{\lambda_e}{L_T} \equiv \chi^{{-}1}, \end{equation}

where $\lambda _e = v_{{\rm th}e}/\nu _e$ is the electron mean free path. It is the scale at which $k_{\perp } d_e \chi \sim 1$ that will effectively play the role of the flux-freezing scale in the collisional limit. Note that $\chi ^{-1}\ll 1$ [see (A82)], meaning that the flux-freezing scale occurs at much longer perpendicular wavelengths than in the collisionless limit.

We refer to scales below the flux-freezing scale (2.28) or (2.30) as electrostatic scales (on which electrons are free to flow across field lines without perturbing them) and to scales above the flux-freezing scale as electromagnetic scales (on which the magnetic field is frozen into the electron flow). In Appendix C, we show that the electron flow into which the magnetic field lines are frozen on electromagnetic scales (while still remaining below the ion Larmor scale) is given by

(2.31)\begin{equation} \boldsymbol{u}_\text{eff} = \boldsymbol{v}_E - \frac{\rho_e v_{{\rm th}e}}{2} \frac{\boldsymbol{b}_0 \times \boldsymbol{\nabla} p_{{\parallel} e}}{n_{0e} T_{0e}}, \end{equation}

where $p_{\parallel e} = n_e T_{\parallel e}$, $n_e = n_{0e} + \delta n_e$ and $T_{\parallel e} = T_{0e} + \delta T_{\parallel e}$ are the total parallel pressure, density and parallel temperature, respectively; in the collisional limit, $\delta T_{\parallel e} \rightarrow \delta T_e$, as in § 2.4. The flow (2.31) is simply the part of the electron flow velocity perpendicular to the total magnetic field $\boldsymbol {B}$, comprised of the usual $\boldsymbol {E}\times \boldsymbol {B}$ drift velocity $\boldsymbol {v}_E$ [see (2.7)] and a ‘diamagnetic’ contribution coming from the electron (parallel) pressure gradient, manifest in the right-hand side of (2.21) or (2.25). This is distinct from the magnetohydrodynamic (MHD) limit (above the ion Larmor scale), in which the magnetic field is only frozen into $\boldsymbol {v}_E$ due to the dynamics being pressure balanced, a distinction that will prove important in our considerations of electromagnetic instabilities in § 4.

In what follows, all orderings introduced should be considered subsidiary to the orderings that define the collisionless and collisional limits [(A8), with $\nu _{ee},\nu _{ei} \rightarrow 0$, and (A74), respectively], and the resultant reduced equations thus to be particular limits of the collisionless [(2.20)–(2.23)] or collisional [(2.24)–(2.26)] equations.

3.1. Collisionless slab ETG

As explained above, the electrostatic limit corresponds to perpendicular scales $k_{\perp } d_e \gg 1$. If we strengthen this condition to [cf. (2.29)]

(3.1)\begin{equation} k_{{\perp}}^{2} d_e^{2} \gg \frac{\omega_{*e}}{\omega} \gtrsim 1, \end{equation}

then both $\mathcal {A}$ terms in (2.21) can be neglected in comparison with the electron inertia. Furthermore, we would like to consider the slab approximation, in which the magnetic drifts are negligible in comparison with parallel compressions. In terms of wavenumbers, this means that we assume

(3.2)\begin{equation} k_\parallel v_{{\rm th}e} \gg \omega_{de} \left( \frac{L_B}{L_T} \right)^{1/4}, \quad \omega_{de} = \frac{k_y \rho_e v_{{\rm th}e}}{2 L_B}, \end{equation}

where $\omega _{de}$ is the magnetic-drift frequency. Though not immediately obvious, it turns out that the limit (3.2) allows us to neglect the magnetic drifts in (2.20) [this follows from comparing the sizes of the last two terms in (D35) under the ordering (D34)]. Then, the perpendicular temperature perturbation (2.23) becomes decoupled from the remaining equations, leaving us with an electrostatic three-field ($\delta n_{e}$, $u_{\parallel e}$ and $\delta T_{\parallel e}$) system of the kind traditionally used to describe temperature-gradient instabilities in a slab (Cowley et al. Reference Cowley, Kulsrud and Sudan1991). The slab electron-temperature-gradient (sETG) instability (Liu Reference Liu1971; Lee et al. Reference Lee, Dong, Guzdar and Liu1987) in its most explicit, fluid form is obtained if one further assumes [see (D34)]

(3.3)\begin{equation} k_\parallel v_{{\rm th}e} \ll \omega \ll \omega_{*e}. \end{equation}

Then (2.20)–(2.22) reduce to, approximately,

(3.4)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \bar{\tau}^{{-}1} \varphi = \boldsymbol{\nabla}_\parallel u_{{\parallel} e}, \quad \frac{\mathrm{d} u_{{\parallel} e}}{\mathrm{d} t} ={-} \frac{v_{{\rm th}e}^{2}}{2} \boldsymbol{\nabla}_\parallel \frac{\delta T_{{\parallel} e}}{T_{0e}}, \quad \frac{\mathrm{d} }{\mathrm{d} t} \frac{\delta T_{{\parallel} e}}{T_{0e}} ={-} \frac{\rho_e v_{{\rm th}e}}{2L_T} \frac{\partial \varphi}{\partial y}. \end{equation}

Linearising and Fourier-transforming, we find the familiar dispersion relation [see (D36)]:

(3.5)\begin{equation} \omega^{3} ={-} \frac{k_\parallel^{2} v_{{\rm th}e}^{2} \omega_{*e} \bar{\tau}}{2} \quad \Rightarrow \quad \omega = \mathrm{sgn}(k_y) \left({-}1, \frac{1}{2} \pm {\rm i}\, \frac{\sqrt{3}}{2} \right) \left( \frac{k_\parallel^{2} v_{{\rm th}e}^{2} |\omega_{*e}| \bar{\tau}}{2}\right)^{1/3}. \end{equation}

The unstable root is the collisionless sETG.

In this limit, the instability works as follows. Suppose that a small perturbation to the parallel electron temperature is created with $k_y \neq 0$ and $k_\parallel \neq 0$, bringing the plasma from regions with higher $T_{0e}$ to those with lower $T_{0e}$ ($\delta T_{\parallel e} >0$), and vice versa ($\delta T_{\parallel e} < 0$). This temperature perturbation produces alternating hot and cold regions along the (unperturbed) magnetic field. The resulting perturbed temperature gradients drive electron flows from the hot regions to the cold regions [second equation in (3.4)], giving rise to increased electron density in the cold regions [first equation in (3.4)]. By quasineutrality, the electron-density perturbation is instantly replicated in the ion-density perturbation, and that, via Boltzmann-ion response, creates an electric field that produces a radial $\boldsymbol {E} \times \boldsymbol {B}$ drift that pushes hotter particles further into the colder region, and vice versa [third equation in (3.4)], reinforcing the initial temperature perturbation and thus completing the positive feedback loop required for the instability. This is illustrated in figure 2.

Figure 2. A cartoon illustrating the feedback mechanism of the (collisionless) sETG instability. (a) An electron-temperature perturbation with $k_y \neq 0$ and $k_\parallel \neq 0$ (red-and-blue curves) has alternating hold and cold regions along the (unperturbed) magnetic field (grey arrows), and also along $\hat {\boldsymbol {y}}$. (b) The resulting perturbed temperature gradients drive parallel electron flows $u_{\parallel e}$ (dashed arrows) from the hot regions into the cold regions, giving rise to increased electron density in the cold regions (over- and under- densities are indicated by the dark- and light-grey ellipses, respectively). (c) By quasineutrality, the electron-density perturbation gives rise to an exactly equal ion-density perturbation, and that, via Boltzmann-ion response, creates alternating electric fields $\boldsymbol {E}$ in the perpendicular plane (vertical black arrows). This produces an $\boldsymbol {E}\times \boldsymbol {B}$ drift $\boldsymbol {v}_E$ (horizontal black arrows), which pushes hotter particles further into the colder region, and vice versa, reinforcing the initial temperature perturbation and thus completing the positive feedback loop required for the instability. In this cartoon, for the sake of simplicity, we have chosen not to include the phase information between the various perturbations involved; a reader seeking such information will find it in figure 1 of Cowley et al. (Reference Cowley, Kulsrud and Sudan1991) (the equivalent picture for ITG).

The ‘fluid’ limit (3.3) is physically transparent and easy to handle, primarily because the heat flux (and thus all kinetic effects, such as Landau damping; Landau Reference Landau1946) can, in this limit, be neglected in (2.22). However, the approximation contains the seed of its own destruction: according to (3.5), perturbations with a larger $k_\parallel$ grow faster, and can only be checked by Landau damping when [see (D56)]

(3.6)\begin{equation} k_\parallel v_{{\rm th}e} \sim \omega \sim \omega_{*e}. \end{equation}

Thus, the fastest-growing collisionless sETG modes are expected to sit in this latter, kinetic regime.

3.2. Collisional slab ETG

An important difference between the collisionless and collisional limits, exemplified by the form of the collisonal heat flux (2.17), is the replacement of the parallel-streaming rate of electrons $k_\parallel v_{{\rm th}e}$ with the parallel conduction rate $(k_\parallel v_{{\rm th}e})^{2}/\nu _{ei}$. The collisional analogues of (3.1), (3.2) and (3.3) are thus [see (E19)]

(3.7)\begin{equation} k_{{\perp}}^{2} d_e^{2} \gg \frac{\omega_{*e}}{\nu_{ei}}, \quad \frac{(k_\parallel v_{{\rm th}e})^{2}}{ \nu_{ei}} \gg \omega_{de}, \quad \frac{(k_\parallel v_{{\rm th}e})^{2}}{ \nu_{ei}} \ll \omega \ll \omega_{*e}, \end{equation}

respectively, for which (2.24), (2.25) and (2.26) reduce to, approximately,

(3.8)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \bar{\tau}^{{-}1} \varphi = \boldsymbol{\nabla}_\parallel u_{{\parallel} e}, \quad \nu_{ei} u_{{\parallel} e} ={-} \frac{v_{{\rm th}e}^{2}}{2 } \boldsymbol{\nabla}_\parallel \frac{\delta T_{ e}}{T_{0e}}, \quad \frac{\mathrm{d} }{\mathrm{d} t} \frac{\delta T_{e}}{T_{0e}} ={-} \frac{\rho_e v_{{\rm th}e}}{2L_T} \frac{\partial \varphi}{\partial y}. \end{equation}

These equations are similar to (3.4), except that the parallel temperature gradient is now balanced by the electron–ion frictional force, rather than by electron inertia. The dispersion relation is [see (E20)]

(3.9)\begin{equation} \omega^{2} = {\rm i}\, \frac{k_\parallel^{2} v_{{\rm th}e}^{2}\omega_{*e} \bar{\tau}}{2\nu_{ei}} \quad \Rightarrow \quad \omega ={\pm} \frac{1-{\rm i}\,\mathrm{sgn}(k_y)}{\sqrt{2}}\left(\frac{k_\parallel^{2} v_{{\rm th}e}^{2}|\omega_{*e}| \bar{\tau}}{2\nu_{ei}} \right)^{1/2}, \end{equation}

where the unstable root is the collisional sETG. The physical mechanism of the instability is analogous to that for the collisionless sETG, except that the parallel electron flow is now determined instantaneously by the parallel temperature gradient. Similarly to the collisionless sETG, the point of maximum growth of the instability occurs when

(3.10)\begin{equation} \frac{(k_\parallel v_{{\rm th}e})^{2}}{ \nu_{ei}} \sim \omega \sim \omega_{*e}, \end{equation}

which is a balance between dissipation (through conduction, rather than Landau damping) and energy injection due to the background temperature gradient [see (E22) and the following discussion].

3.3. Curvature-mediated ETG

Both the collisionless and collisional sETG instabilities were derived assuming that the parallel wavelengths were sufficiently short for the compressional terms in (2.20) and (2.24) to be dominant in comparison with the magnetic-drift terms, while still satisfying (3.3) and (3.7). We now consider very long parallel wavelengths for which this is no longer true, ordering our frequencies as

(3.11)\begin{equation} k_\parallel v_{{\rm th}e} \ll \omega_{de} \ll \omega \ll \omega_{*e}, \quad \frac{(k_\parallel v_{{\rm th}e})^{2}}{\nu_{ei}} \ll \omega_{de} \ll \omega \ll \omega_{*e} \end{equation}

in the collisionless and collisional regimes, respectively. This, in fact, amounts to setting $k_\parallel =0$ everywhere, i.e. we are considering purely two-dimensional modes (see Appendices D.2 and E.1). In both regimes, our equations reduce to

(3.12)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \bar{\tau}^{{-}1} \varphi = \frac{\rho_e v_{{\rm th}e}}{L_B} \frac{\partial}{\partial y} \frac{\delta T_{{\parallel} e}}{T_{0e}}, \quad \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta T_{{\parallel} e}}{T_{0e}} ={-} \frac{\rho_e v_{{\rm th}e}}{2L_T} \frac{\partial \varphi}{\partial y}, \quad \frac{\delta T_{{\parallel} e}}{T_{0e}} = \frac{\delta T_{{\perp} e}}{T_{0e}}. \end{equation}

The equality between the perpendicular and parallel temperature perturbations arises in the collisionless regime because the dominant balance in both (2.22) and (2.23) is between the time derivative and the ETG injection term, which is also true in the collisional limit and with strengthened isotropisation from collisions. The dispersion relation is [see (D31) or (E14)]

(3.13)\begin{equation} \omega^{2} ={-} 2 \omega_{de} \omega_{*e}\bar{\tau} \quad \Rightarrow \quad \omega ={\pm} {\rm i}\, \left(2 \omega_{de} \omega_{*e} \bar{\tau} \right)^{1/2}, \end{equation}

which is the familiar growth rate of the curvature-mediated ETG (cETG) instability (Horton et al. Reference Horton, Hong and Tang1988). Physically, this arises due to the fact that the magnitude of the magnetic-drift velocity for a particle is proportional to its kinetic energy, and thus temperature. The presence of some temperature perturbation will cause an electron-density perturbation, as electrons in the hotter regions will drift faster than those in colder regions [first equation in (3.12)]. By quasineutrality, the electron-density perturbation gives rise to an exactly equal ion-density perturbation, and that, via Boltzmann-ion response, creates an electric field that produces an $\boldsymbol {E} \times \boldsymbol {B}$ drift which pushes hotter particles further into the colder region, and vice versa [second equation in (3.12)], completing the feedback loop required for the instability, as illustrated in figure 3. This mechanism is unaffected by collisionality; hence the cETG instability is obtained in both the collisionless and collisional limits.

Figure 3. A cartoon illustrating the feedback mechanism of the (two-dimensional) cETG instability. (a) An electron-temperature perturbation with $k_y \neq 0$ (red-and-blue curve) has alternating hot and cold regions along $\hat {\boldsymbol {y}}$. Due to the temperature dependence of the magnetic drifts $\boldsymbol {v}_{de}$, electrons in the hot regions will drift faster than those in the cold regions (red and blue arrows), creating an electron-density perturbation (over- and under-densities are indicated by the dark- and light-grey ellipses, respectively). (b) By quasineutrality, the electron-density perturbation gives rise to an exactly equal ion-density perturbation, and that, via Boltzmann-ion response, creates alternating electric fields $\boldsymbol {E}$ in the perpendicular plane (vertical black arrows). This produces an $\boldsymbol {E}\times \boldsymbol {B}$ drift $\boldsymbol {v}_E$ (horizontal black arrows), which pushes hotter particles further into the colder region, and vice versa, reinforcing the initial temperature perturbation and thus completing the positive feedback loop required for the instability.

Given that both the collisionless and collisional sETG instabilities have maximum growth rates $\gamma _\text {max} \sim \omega _{*e}$ [see (3.6) and (3.10), respectively], the growth rate of the cETG instability is always small in comparison, at least for large temperature gradients, viz.,

(3.14)\begin{equation} \frac{\gamma_\text{max}}{\sqrt{2\omega_{de} \omega_{*e}}} \sim \left( \frac{L_B}{L_T} \right)^{1/2} \gg 1. \end{equation}

However, the sETG instabilities only exist at perpendicular scales below the flux-freezing scales (2.28) or (2.30), as they require electrons to be able to flow across field lines without perturbing them. While sETG is stabilised by magnetic tension above the flux-freezing scale, cETG is unaffected by flux freezing as it is an interchange ($k_\parallel = 0$) mode and does not trigger perpendicular magnetic-field perturbations $\delta \! \boldsymbol {B}_{\perp }$. This means that it will happily survive in the electromagnetic regime.

4.1. Isothermal curvature-mediated TAI

Previous studies of electromagnetic phenomena driven by an ETG have often assumed the electrons to be isothermal along the perturbed field line (e.g. Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Schekochihin, Kawazura & Barnes Reference Schekochihin, Kawazura and Barnes2019; Abel & Cowley Reference Abel and Cowley2013; Zielinski et al. Reference Zielinski, Smolyakov, Beyer and Benkadda2017). In our system, this assumption is valid if the thermal-conduction time dominates over all other timescales, viz., in addition to (4.9),

(4.11)\begin{equation} \xi_* \ll 1. \end{equation}

In the electrostatic regime, without the ability to have perturbations of the magnetic-field direction, adopting such a limit would simply lead to erasure of the temperature perturbation due to Landau damping or thermal conduction [see (D36) or (E20) and the following discussions], suppressing both the collisionless and collisional sETG, respectively.

The isothermal limit allows the system more leeway in the electromagnetic regime. Given (4.11), the dominant term in (4.8) is the first term on the right-hand side, meaning that, to leading order,

(4.12)\begin{equation} \boldsymbol{\nabla}_\parallel \log T_e = \boldsymbol{\nabla}_\parallel \frac{\delta T_{e}}{T_{0e}} - \frac{\rho_e}{L_T} \frac{\partial \mathcal{A}}{\partial y} = 0, \end{equation}

i.e. the temperature perturbations, rather than being zero, will always adjust to cancel the variation of the equilibrium temperature along the perturbed field line. At the next order in $\xi _*$,

(4.13)\begin{equation} \kappa \boldsymbol{\nabla}_\parallel^{3} \log T_e = \frac{\rho_e v_{{\rm th}e}}{2 L_T} \frac{\partial}{\partial y} \boldsymbol{\nabla}_\parallel \frac{\delta n_e}{n_{0e}} \quad \Rightarrow \quad \frac{|\boldsymbol{\nabla}_\parallel \log T_e |}{|\boldsymbol{\nabla}_\parallel \delta n_e/n_{0e} |} \sim \xi_* \ll 1, \end{equation}

meaning that we can neglect $\boldsymbol {\nabla }_\parallel \log T_e$ in (2.25). The $\boldsymbol {\nabla }_\parallel u_{\parallel e}$ term in (2.24) is also negligible, as can be confirmed a posteriori. Our system (2.24)–(2.26) therefore becomes

(4.14)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \frac{\delta n_{e}}{n_e} ={-}\frac{\rho_e v_{{\rm th}e}}{L_B} \frac{\partial}{\partial y} \frac{\delta T_{e}}{T_{0e}}, \quad \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_e}, \quad \boldsymbol{\nabla}_\parallel \frac{\delta T_{e}}{T_{0e}} = \frac{\rho_e}{L_T } \frac{\partial \mathcal{A}}{\partial y}, \end{equation}

where, by (2.19), $\varphi = - \bar {\tau } \delta n_{e}/n_e$. The associated dispersion relation is

(4.15)\begin{equation} \omega^{2} ={-} 2 \omega_{de} \omega_{*e} (1+ \bar{\tau}) \quad \Rightarrow \quad \omega ={\pm} {\rm i}\, \left[ 2 \omega_{de} \omega_{*e} ( 1 + \bar{\tau} ) \right]^{1/2}, \end{equation}

which looks like the familiar cETG growth rate (3.13), but enhanced by an extra order-unity contribution. In fact, this is a physically different and (as far as we know) newFootnote ¹ instability, which we refer to as the curvature-mediated thermo-Alfvénic instability (cTAI).

Physically, cTAI proceeds as follows. Suppose that a perturbation $\delta \! B_x = B_0 \rho _e \partial _y \mathcal {A}$ of the magnetic field is created, with $k_y \neq 0$ and $k_\parallel \neq 0$. According to the second equation in (4.14), such a perturbation is brought about by a radial displacement of the electron fluid associated with the velocity (2.31), which, recalling the isothermal condition (4.12), can be written as

(4.16)\begin{equation} \boldsymbol{u}_\text{eff} = \boldsymbol{v}_E - \frac{\rho_e v_{{\rm th}e}}{2} \boldsymbol{b}_0 \times \boldsymbol{\nabla} \frac{\delta n_e}{n_{0e}} ={-} \frac{\rho_e v_{{\rm th}e}}{2} \boldsymbol{b}_0 \times \boldsymbol{\nabla} \left( 1 +\bar{\tau} \right) \frac{\delta n_e}{n_{0e}}. \end{equation}

Due to the presence of the equilibrium temperature gradient, this magnetic-field perturbation will set up an apparent (parallel) variation of the equilibrium temperature along the perturbed field line, as the field line makes excursions into hot and cold regions. However, rapid thermal conduction along the field line instantaneously creates a temperature perturbation that compensates for this temperature variation, in order to enforce isothermality (4.12) [last equation in (4.14)]. This temperature perturbation will cause a parallel density gradient, as electrons in the hotter regions will curvature-drift faster than those in colder regions [first equation in (4.14)]. The resulting parallel pressure gradient must be balanced by a parallel electric field [second equation in (4.14)], whose inductive part leads to an increase in the perturbation of the magnetic field, deforming the field line further into the hot and cold regions, and in doing so completing the feedback loop required for the instability.Footnote ² This is illustrated in figure 4.

Figure 4. A cartoon illustrating the feedback mechanism of the (isothermal) cTAI. (a) A perturbation $\delta \! B_x = B_0 \rho _e \partial _y \mathcal {A}$ (solid black lines) to the equilibrium magnetic field (grey arrows, darker grey corresponding to the plane of constant $y$ containing $\delta \! B_x$, or the relevant perturbation in subsequent diagrams) is created with $k_y \neq 0$ and $k_\parallel \neq 0$ (we show half a wavelength of the mode along both $\hat {\boldsymbol {y}}$ and $\hat {\boldsymbol {z}}$). Due to the presence of the equilibrium temperature gradient, this will set up a (parallel) variation of the total temperature along the perturbed field line, as the field line makes excursions into hot and cold regions (on the left and right, respectively). However, rapid thermal conduction along the field line instantaneously creates a temperature perturbation that compensates for this temperature variation (red and blue ovals, located in the same planes of constant $y$ as $\delta \! B_x$). (b) This temperature perturbation will cause a parallel density gradient (over- and under-densities are indicated by the dark- and light-grey ellipses, respectively, lying in the planes of constant $y$ a quarter of a wavelength above those of $\delta B_x$), as electrons in hotter regions will curvature-drift faster than those in colder regions ($\boldsymbol {v}_{de}$, red and blue arrows). (c) The parallel density gradient must be balanced by a parallel electric field (black arrows, in the same planes of constant $y$ as the density perturbations), whose inductive part leads to an increase in the perturbation of the magnetic field (maroon arrows), deforming the field line further into the hot and cold regions, and in so doing completing the feedback loop for the instability. Note that the maximal rate of change of $\delta \! B_x$ occurs where the $y$-gradient of $E_\parallel$ is at a maximum, as shown.

The physical distinction between cTAI and cETG can be made obvious by the following two observations. First, unlike cETG, cTAI relies vitally on $k_{\parallel } \neq 0$ and, indeed, on $k_\parallel$ being large enough for the condition (4.11) to be satisfied – even though the growth rate (4.15) ends up being independent of $k_{\parallel }$. In § 4.2, we show that this is the peak growth rate of the instability and that it is achieved at a finite $k_{\parallel }$, while at $k_{\parallel }= 0$, the cETG growth rate (3.13) is recovered. Second, perturbations described by (4.14) can be unstable without the need for them to contain any $\boldsymbol {E}\times \boldsymbol {B}$ flows (i.e. any electrostatic potential $\varphi$) – this becomes obvious in the formal limit $\varphi = - \bar {\tau } \delta n_e/n_{0e} \rightarrow 0$ as $\bar {\tau } \rightarrow 0$ (cold ions). In contrast, the cETG growth rate (3.13) disappears in this limit. This is because cTAI extracts energy from the background temperature gradient not via $\boldsymbol {E} \times \boldsymbol {B}$ advection of said equilibrium gradient but via thermal conduction of it along the perturbed field lines. In order to complete the instability loop and reinforce the magnetic perturbation $\delta \! B_x$ required for this mechanism to work, the system only needs a perturbed density gradient. This is due to the fact that, as we discussed in § 2.7, below the ion Larmor scale, the magnetic field lines are frozen not into the $\boldsymbol {E}\times \boldsymbol {B}$ flow but into the electron flow (2.31), which involves also a ‘diamagnetic’ contribution from the electron pressure gradient – which, in the isothermal limit, consists just of the perturbed density gradient, as in (4.16). It is because of the presence of this distinct destabilisation mechanism that the cTAI growth rate (4.15) is always strictly greater than the cETG one (3.13). Thus, cTAI is not simply an ‘electromagnetic correction’ to cETG, but rather the main effect at scales above the flux-freezing scale (2.30) [or (2.28) in the collisionless limit, where, as we shall see shortly, the same instability is present]. This suggests that a purely electrostatic description of these scales is inadequate.

4.2. General TAI dispersion relation

As we have noted above, despite cTAI relying on parallel dynamics, the dispersion relation (4.15) is itself independent of $k_\parallel$. This is because we have thus far only captured the leading-order behaviour in our analysis, and further diligence is required in order to determine the details associated with the parallel dynamics. Let us give this problem the diligence that it is due, and adopt the ordering (4.9) but, for now, $\xi _{*} \sim 1$. Neglecting both the resistive term in (2.25) and the compressional term in (2.26) – since both are small under (4.9) – and determining $\boldsymbol {\nabla }_\parallel \log T_e$ in (2.25) from the balance of the two terms on the right-hand side of (4.8), viz.,

(4.17)\begin{equation} \left(\frac{\rho_e v_{{\rm th}e}}{2 L_{T}} \frac{\partial}{\partial y} - \kappa \boldsymbol{\nabla}_\parallel^{2}\right)\boldsymbol{\nabla}_\parallel \log T_e ={-}\frac{\rho_e v_{{\rm th}e}}{2 L_{T}} \frac{\partial}{\partial y} \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}}, \end{equation}

we arrive at the following dispersion relation:

(4.18)\begin{equation} \omega^{2} ={-} \left(2 \omega_{de} \omega_{*e} - \omega_\text{KAW}^{2} \right) \left( \bar{\tau} + \frac{1}{1+{\rm i}\,\xi_*} \right), \end{equation}

where $\omega _\text {KAW} = k_\parallel v_{{\rm th}e} k_{\perp } d_e /\sqrt {2}$ is the kinetic-Alfvén-wave (KAW) frequency, the physical origin of which is discussed in § 4.3. The cTAI growth rate is manifest in this expression; adopting the isothermal limit (4.11) and neglecting $\omega _\text {KAW}$, we re-obtain (4.15) to lowest order in $\xi _*$.

Though we have thus far focused on the collisional limit, it turns out that much of what we have done is directly applicable to the collisionless limit if we simply replace the parallel conduction rate with the parallel-streaming rate [see (D51)], viz., (4.18) remains valid but with

(4.19)\begin{equation} \xi_* = \frac{\sqrt{\rm \pi}}{2} \frac{\omega_{*e}}{k_{{\parallel}} v_{{\rm th}e}}. \end{equation}

The equivalent of the ordering (4.9) in the collisionless regime is [see (D45)]

(4.20)\begin{equation} \omega_{de} \ll \omega \ll \omega_{*e} \sim k_\parallel v_{{\rm th}e}, \end{equation}

and equations (4.14) are the same; note that in this regime, $\delta T_{\parallel e}=\delta T_{\perp e} = \delta T_{e}$ because both the parallel and perpendicular temperatures are constant along the field lines to leading order in $\omega /\omega _{*e}$. Furthermore, it is possible to show that (4.8) is also valid in the collisionless limit if one replaces $\delta T_{e} \rightarrow \delta T_{\parallel e}, \: -\kappa \boldsymbol {\nabla }_\parallel \log T_e \rightarrow \delta q_{\parallel e}/n_{0e} T_{0e}, \: $ $(2/3)\boldsymbol {\nabla }_\parallel ^{2} u_{\parallel e} \rightarrow 2 \boldsymbol {\nabla }_\parallel ^{2} u_{\parallel e}, \: \nu _{ei} u_{\parallel e} \rightarrow \mathrm {d} u_{\parallel e}/\mathrm {d} t$, and the heat flux must now be determined kinetically [see (D63)]. The effect is still to enforce isothermality along the field lines, but by means of parallel particle streaming, rather than collisional conduction. This means that cTAI, while being a ‘fluid’ instability, is not an intrinsically collisional one, occurring also in the collisionless, kinetic limit. Its physical picture in the collisionless limit is exactly the same as in the collisional one.

The dispersion relation (4.18) contains most of the interesting features of the TAI physics (see, however, §§ 4.3.3 and 4.4.2). The most obvious feature of (4.18) is that both the growth rate and frequency vanish when

(4.21)\begin{equation} 2 \omega_{de} \omega_{*e} = \omega_\text{KAW}^{2} \quad \Rightarrow \quad \frac{k_{{\parallel} c} L_T}{\sqrt{\beta_e}} = \left( \frac{L_T}{L_B} \right)^{1/2} \frac{k_y}{k_{{\perp}}}. \end{equation}

This corresponds to the point of transition from the curvature-dominated regime ($k_{\parallel } < k_{\parallel c}$), on which we focus in this section, to the KAW-dominated regime ($k_{\parallel } > k_{\parallel c}$), which is the subject of § 4.3.

If we extract the real and imaginary parts of (4.18), the (real) frequency $\omega _r = \text {Re}(\omega )$ and the growth rate $\gamma = \text {Im}(\omega )$ of the growing modes can be written as

(4.22)\begin{equation} \omega_r^{2} = \left|2 \omega_{de} \omega_{*e} - \omega_\text{KAW}^{2} \right| \bar{\tau} \: f_-(\xi_{*}), \quad \gamma^{2} = \left|2 \omega_{de} \omega_{*e} - \omega_\text{KAW}^{2} \right| \bar{\tau} \: f_+(\xi_{*}), \end{equation}

where

(4.23)\begin{align} f_\pm(\xi_{*}) = \frac{1}{2\bar{\tau}} \left[ \sqrt{\left( \bar{\tau} + \frac{1}{1+\xi_{*}^{2}} \right)^{2} + \frac{\xi_{*}^{2}}{(1 + \xi_{*}^{2})^{2}}} \pm \text{sgn}\left(2 \omega_{de} \omega_{*e} - \omega_\text{KAW}^{2} \right)\left( \bar{\tau} + \frac{1}{1+\xi_{*}^{2}} \right) \right]. \end{align}

The growth rate and frequency (4.22) are plotted as functions of $k_{\parallel } L_T/\sqrt {\beta _e}$ in figure 5. For $k_{\parallel } < k_{\parallel c}$, the growth rate has a maximum for some non-zero $k_{\parallel }$; it is about to turn out that this maximum corresponds to the cTAI growth rate (4.15), which was derived in the isothermal limit, $\xi _*\ll 1$. Expanding (4.23) in small $\xi _*$ to leading and subleading order, and seeking the maximum of the resultant expression with respect to $k_{\parallel }$, we find that this maximum occurs approximately at [see (F14)]

(4.24)\begin{equation} \frac{k_{{\parallel} \text{max}} L_T}{\sqrt{\beta_e}} = \left\{ \begin{array}{@{}ll} \displaystyle \left[\dfrac{\rm \pi}{64} \dfrac{3 + 4 \bar{\tau}}{(1+\bar{\tau})^{2}} \dfrac{L_T}{L_B} \right]^{1/4} \left( \dfrac{k_y^{2} d_e}{k_{{\perp}}} \right)^{1/2}, & \displaystyle \text{collisionless}, \\ \displaystyle \left[\dfrac{81}{50}\dfrac{3 + 4 \bar{\tau}}{(1+\bar{\tau})^{2}} \dfrac{L_T}{ L_B} \right]^{1/6}\left( \dfrac{k_y^{2} d_e}{k_{{\perp}}} \chi\right)^{1/3}, & \displaystyle \text{collisional}, \end{array} \right. \end{equation}

indicated by the vertical dashed lines in figure 5(a), (c); $\chi$ is defined in (2.30). Calculating the growth rate (4.22) at $k_{\parallel } = k_{\parallel \text {max}}$, one recovers (4.15) up to small corrections [see (F16)], as promised.

Figure 5. (a), (c) The growth rate and (b), (d) the real frequency of the TAI (4.22) in the isothermal limit (4.11), plotted as functions of $k_\parallel L_T/\sqrt {\beta _e}$ and normalised to the cETG growth rate (3.13) ($\bar {\tau }=1$). (a) and (b) correspond to the collisionless case, while (c) and (d) the collisional one. The vertical dashed lines in (a) and (c) are for $k_\parallel = k_{\parallel \text {max}}$ given by (4.24). The dashed lines in (b) and (d) show the isothermal KAW frequency (4.29). Both the growth rate and the real frequency vanish at the critical parallel wavenumber $k_{\parallel c}L_T/\sqrt {\beta _e} = 0.32$, given by (4.21). The insets in (a) and (c) show details of the behaviour of the growth rate for $k_\parallel > k_{\parallel c}$; the vertical dashed line in the inset of (c) is the (secondary) maximum at $k_{\parallel } = \sqrt {2} k_{\parallel c}$ discussed at the end of § 4.3. The perpendicular wavenumbers chosen in this figure are all safely below the transition wavenumber (4.26), which is $k_{\perp *} d_e = 0.71$ or $k_{\perp *} d_e \chi = 0.03$ in the collisionless or collisional cases, respectively.

This solution, however, is only valid so long as it remains in the isothermal limit (4.11). Evaluating $\xi _*$ at $k_{\parallel } =k_{\parallel \text {max}}$, we find, defining $\alpha =1,2$ in the collisionless and collisional limits, respectively [see (F15)],

(4.25)\begin{equation} \xi_* \left(k_{{\parallel} \text{max}}\right) \sim \frac{k_{{\parallel} \text{max}}}{k_{{\parallel} c}} \sim \left( \frac{k_{{\perp}}}{k_{{\perp} *}} \right)^{1/(1+\alpha)} \ll 1 \end{equation}

provided that $k_{\perp } \ll k_{\perp *}$, where $k_{\perp *}$ is the perpendicular wavenumber at which $\xi _*(k_{\parallel c} ) \sim 1$, viz.,

(4.26)\begin{equation} k_{{\perp} *} d_e = \left\{ \begin{array}{@{}ll} \displaystyle \dfrac{4}{\sqrt{\rm \pi}} \left( \dfrac{L_{T}}{L_B} \right)^{1/2}, & \displaystyle \text{collisionless},\\ \displaystyle \dfrac{5}{9} \dfrac{L_{T}}{L_B} \chi^{{-}1}, & \displaystyle \text{collisional}, \end{array} \right. \quad \Rightarrow \quad \xi_*(k_{{\parallel} c}) = \left\{ \begin{array}{@{}ll} \displaystyle \dfrac{k_{{\perp}}}{k_{{\perp} *}}, & \displaystyle \text{collisionless}, \\ \displaystyle \dfrac{k_{{\perp}}^{2}}{k_{{\perp} *} k_y}, & \displaystyle \text{collisional}. \end{array} \right. \end{equation}

Thus, the isothermal regime is valid at sufficiently long perpendicular wavelengths. At $k_{\perp } >k_{\perp *}$, a different, isobaric regime takes over, which is considered in § 4.4.

Lastly, we note that, for $k_{\parallel } < k_{\parallel c}$, the magnitude of the real frequency is vanishingly small when compared with the growth rate: expanding both the growth rate and the real frequency in (4.22) for $\xi _* \ll 1$, we find

(4.27)\begin{equation} \frac{\omega_r^{2}}{\gamma^{2}} \approx \frac{\xi_*^{2}}{4(1+\bar{\tau})^{2}} \ll 1. \end{equation}

Thus, cTAI is, like cETG, an (almost) purely growing mode; this is distinct from the case of sETG, whose frequency and growth rate are comparable at the latter's maximum (see § 3).

4.3. Isothermal KAWs and slab TAI

4.3.1. Isothermal KAWs

If $k_{\perp } \ll k_{\perp }^{*}$, i.e. $\xi _*(k_{\parallel c} ) \ll 1$, then the isothermal approximation (4.12) continues to be satisfied at $k_{\parallel } > k_{\parallel c}$, but the effects of the magnetic drifts become negligible for $k_{\parallel } \gg k_{\parallel c}$. In this region, our system (2.24)–(2.26) becomes, approximately,

(4.28)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} ={-} v_{{\rm th}e} \boldsymbol{\nabla}_\parallel d_e^{2} \boldsymbol{\nabla}_{{\perp}}^{2} \mathcal{A}, \quad \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}}, \quad \varphi ={-} \bar{\tau} \frac{\delta n_{e}}{n_{0e}}. \end{equation}

These equations are also valid in the collisionless limit [there is no intrinsically collisional physics in (4.28), as the resistive term in (2.25) is negligible under the ordering (4.9)]. We recognise these as the equations of electron reduced MHD (see Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009 or Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013), which describe the dynamics, linear and nonlinear, of kinetic Alfvén waves (KAWs). Indeed, linearising and Fourier transforming (4.28), we find the dispersion relation

(4.29)\begin{equation} \omega^{2} = k_\parallel^{2} v_{{\rm th}e}^{2} k_{{\perp}}^{2} d_e^{2} \frac{1+\bar{\tau}}{2} = \omega_\text{KAW}^{2} \left(1 + \bar{\tau} \right). \end{equation}

These are the familiar (isothermal) KAWs that arise in homogeneous systems (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009, Reference Schekochihin, Kawazura and Barnes2019; Zocco & Schekochihin Reference Zocco and Schekochihin2011; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013; Passot, Sulem & Tassi Reference Passot, Sulem and Tassi2017). The physics of these waves is as follows. Suppose that a density perturbation $\delta n_{e}/n_{0e} = -\bar {\tau }^{-1} \varphi$ with $k_{\parallel } \neq 0$ is created. This gives rise to a parallel pressure gradient, which manifests itself as a parallel (perturbed) density gradient, as any parallel temperature variation is instantaneously ironed out by rapid parallel streaming or thermal conduction. This parallel pressure gradient must be balanced by the parallel electric field [second equation in (4.28)], whose inductive part, through Ampère's law (2.27), leads to a parallel current. But a parallel current is a parallel electron flow, which leads to compressional rarefaction along the field that opposes the original density perturbation [first equation in (4.28)]. This is also the reason for the reduction of the cTAI growth rate at $k_\parallel > k_{\parallel \text {max}}$ and its vanishing at $k_{\parallel } = k_{\parallel c}$ (see figure 5a,c): the parallel compression that provides the restoring force for the KAW perturbations increases as $k_\parallel$ increases, weakening the instability mechanism of the cTAI described in § 4.1.

4.3.2. Isothermal slab TAI

Remarkably, however, it turns out that isothermal KAWs, at $k_{\parallel } > k_{\parallel c}$, are still unstable in the presence of an equilibrium ETG: expanding (4.18) or (4.22) for $\xi _* \ll 1$ at $k_{\parallel } \gg k_{\parallel c}$ (the latter in order to drop the $\omega _{de}$ effects), we find

(4.30)\begin{equation} \omega_r^{2} \approx \omega_\text{KAW}^{2} (1+\bar{\tau}),\quad \gamma^{2} \approx \omega_\text{KAW}^{2} \frac{\xi_*^{2}}{4(1+\bar{\tau})}. \end{equation}

By analogy with sETG, we henceforth refer to this KAW-dominated TAI as the sTAI; it was our original motivation for calling the instability ‘thermo-Alfvénic’.Footnote ³

The precise mechanism by which the isothermal sTAI operates is somewhat subtler than that for cTAI, relying on the fact that the isothermal condition (4.12) that led to (4.29) is, in fact, only approximately satisfied. Indeed, $\boldsymbol {\nabla }_\parallel \log T_e$ is determined, in the collisional limit, at next order in $\xi _*$ by (4.13) which, linearising and Fourier transforming, can be written as

(4.31)\begin{equation} \left(\boldsymbol{\nabla}_\parallel \log T_e \right)_{\boldsymbol{k}} ={-}{\rm i}\, \xi_* \left(\boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}} \right)_{\boldsymbol{k}}. \end{equation}

This means that a small but finite parallel gradient of temperature effectively introduces a correction to the parallel density gradient in (4.28) that is ${\rm \pi} /2$ out of phase with the contribution that enables the isothermal KAWs. This gives rise to the instability (4.30) in both the collisional limit and, it turns out, the collisionless one, where (4.31) also holds but with $\xi _*$ given by (4.19) [see (D74)]. Restoring finite parallel temperature gradients in (4.28), we have

(4.32)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} ={-}v_{{\rm th}e} \boldsymbol{\nabla}_\parallel d_e^{2} \boldsymbol{\nabla}_{{\perp}}^{2} \mathcal{A}, \quad \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}} + \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \log T_e, \end{equation}

with dispersion relation

(4.33)\begin{equation} \omega^{2} = \omega_\text{KAW}^{2} (1+ \bar{\tau} - {\rm i}\, \xi_*) \quad \Rightarrow \quad \omega \approx{\pm} \omega_\text{KAW}\left( \sqrt{1+\bar{\tau}} -\frac{{\rm i}\,\xi_*}{2 \sqrt{1+\bar{\tau}}} + \cdots \right), \end{equation}

whose real and imaginary parts are exactly the frequency and growth rate (4.30).

If we restore the magnetic-drift terms in the density equation, we find, in the collisionless limit, that the sTAI growth rate increases from zero at $k_{\parallel } = k_{\parallel c}$ to a finite, $k_{\parallel }$-independent limit (4.30) at $k_{\parallel } \gg k_{\parallel c}$, viz.,

(4.34)\begin{equation} \gamma \rightarrow \frac{1}{4} \sqrt{\frac{\rm \pi}{2(1+\bar{\tau})}} k_{{\perp}} d_e \omega_{*e} = \sqrt{\frac{\omega_{de} \omega_{*e}}{2(1+\bar{\tau})}} \frac{k_{{\perp}}}{k_{{\perp} *}}, \end{equation}

where $k_{\perp *}$ is given by (4.26) (see figure 5a, inset). As we shall see shortly in § 4.3.3, this value only persists up to a certain $k_\parallel$ where sTAI is stabilised by compressional heating, which was neglected in (4.18). In the collisional limit, $\gamma \rightarrow 0$ as $k_{\parallel } \rightarrow \infty$ (also, in fact, shown to go to $\gamma <0$ in § 4.3.3). The growth rate has a maximum at $k_\parallel = \sqrt {2} k_{\parallel c}$, which is shown by the vertical dashed line in the inset of figure 5(c). The growth rate at this maximum is

(4.35)\begin{equation} \gamma = \frac{k_{{\perp}}^{2} d_e^{2} v_{{\rm th}e}^{2}}{8 \sqrt{2(1+\bar{\tau})} \kappa} \sqrt{\frac{\omega_{*e}}{\omega_{de}}} = \frac{1}{2} \sqrt{\frac{\omega_{de} \omega_{*e}}{2(1+\bar{\tau})}} \frac{k_{{\perp}}^{2}}{k_{{\perp} *} k_y} . \end{equation}

These results are derived at the end of Appendix F.1. Both the maximum growth rates (4.34) and (4.35) are manifestly much smaller than the maximum growth rate of cTAI (4.15) as long as $k_{\perp } \ll k_{\perp *}$, i.e. as long as the isothermal approximation, in which all of these results have been derived in the first place, is valid.

Thus, at long perpendicular wavelengths ($k_{\perp } \ll k_{\perp *}$), the dominant instability is cTAI, reaching its maximum growth rate (4.15) at the parallel wavenumber (4.24).

4.3.3. Stabilisation of isothermal slab TAI

The sTAI growth rates do not, in fact, stay positive to infinite parallel wavenumbers. The instability is eventually quenched by the compressional-heating term in the temperature equation [(2.22) or (2.26) in the collisionless or collisional limits, respectively] that begins to compete with the TAI drive.

To show this, let us consider the collisional limit and, instead of (4.9), the ordering

(4.36)\begin{equation} (k_{{\perp}} d_e)^{2} \nu_{ei} \ll \omega \sim \omega_{*e} \ll \kappa k_\parallel^{2}. \end{equation}

In this limit, the system is still isothermal to leading order in $\xi _* \ll 1$, but now we must also retain the compressional heating term in (4.8) to determine $\boldsymbol {\nabla }_\parallel \log T_e$ at next order: instead of (4.13), we have, therefore,

(4.37)\begin{equation} \kappa \boldsymbol{\nabla}_\parallel^{3} \log T_e = \frac{\rho_e v_{{\rm th}e}}{2L_T} \frac{\partial}{\partial y} \boldsymbol{\nabla}_\parallel \frac{\delta n_e}{n_{0e}} + \frac{2}{3} \boldsymbol{\nabla}_\parallel^{2} u_{{\parallel} e}. \end{equation}

Furthermore, it turns out that we must also retain the resistive term in (2.25) at this order as it will end up making a contribution of the same order as the second term in (4.37). Thus, the second equation in (4.32) is replaced by

(4.38)\begin{equation} \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}} + \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \log T_e + \nu_{ei} \frac{u_{{\parallel} e}}{v_{{\rm th}e}}. \end{equation}

Combining (4.37) and (4.38) with the density equation, still the same as in (4.32), we obtain the following dispersion relation:

(4.39)\begin{equation} \omega^{2} - \omega_\text{KAW}^{2}(1+\bar{\tau} - {\rm i}\,\xi_*) ={-} {\rm i}\, \left(\frac{2}{3}+a \right) \frac{\omega}{\kappa k_\parallel^{2}} \omega_\text{KAW}^{2}, \end{equation}

where $a$ is a numerical constant of order unity [see (E17)]. This is the same as (4.33) apart from the right-hand side, previously neglected. At the stability boundary, the frequency $\omega$ must be purely real, and both the real and imaginary parts of (4.39) must vanish individually, giving [cf. (E36)]

(4.40)\begin{equation} \omega^{2} = \omega_\text{KAW}^{2} (1+\bar{\tau}), \quad \omega ={-} \frac{\omega_{*e}}{a + 2/3} \quad \Rightarrow \quad \mp \omega_{\text{KAW}} \sqrt{1+\bar{\tau}} = \frac{\omega_{*e}}{a + 2/3}. \end{equation}

For $k_y \sim k_{\perp }$, (4.40) are lines of constant $k_\parallel$ in wavenumber space, limiting the isothermal sTAI at large parallel wavenumbers:

(4.41)\begin{equation} \frac{k_\parallel L_{T}}{\sqrt{\beta_e}} ={\pm} \frac{1}{(a + 2/3)\sqrt{2(1+\bar{\tau})}} \frac{k_y}{k_{{\perp}}}. \end{equation}

This stabilisation of the isothermal sTAI was not captured in the TAI dispersion relation (4.18) because the ordering (4.9) did not formally allow frequencies comparable to the drift-wave frequency, required by (4.40).

In the collisionless limit, we also find that the sTAI is stabilised above a line of constant $k_\parallel$ [cf. (D78)]:

(4.42)\begin{equation} \omega_\text{KAW} \sim \omega_{*e}, \end{equation}

due again to the competition between the compressional heating in the equation for the parallel temperature (2.22) and the TAI drive. In Appendix D.7.2, we detail a collisionless calculation analogous to that performed above in the collisional limit, but the latter is sufficient here for illustrating the physics underlying the stabilisation mechanism. In both cases, the stabilisation of sTAI does not appear in figure 5 (and, later, in figure 6) because the orderings (4.9) or (4.20) that lead to the TAI dispersion relation (4.18) do not formally allow this stabilisation; instead, readers will find it in figures 11(c) and 14(c) in the collisionless and collisional limits, respectively, where solutions of a more precise dispersion relation are shown.

Figure 6. (a), (c) The growth rate and (b), (d) the real frequency of the TAI (4.22) in the isobaric limit (4.43), plotted as functions of $k_\parallel L_T/\sqrt {\beta _e}$ and normalised to the cETG growth rate (3.13) ($\bar {\tau }=1$). (a) and (b) correspond to the collisionless case, while (c) and (d) the collisional one. The vertical dashed line in (c) is for $k_{\parallel \text {max}}$ given by (4.54). The dashed and dotted lines in (b) and (d) are the isothermal (4.29) and isobaric (4.49) KAW frequencies, respectively. Both the growth rate and the real frequency vanish at the critical parallel wavenumber $k_{\parallel c}L_T/\sqrt {\beta _e} = 0.1$, given by (4.21). The perpendicular wavenumbers chosen in this figure are all safely above the transition wavenumber (4.26), which is $k_{\perp *} d_e = 0.23$ or $k_{\perp *} d_e \chi = 0.003$ in the collisionless or collisional cases, respectively. The parallel wavenumber corresponding to the transition between the isobaric and isothermal regimes at a fixed $k_y$ (viz., for $\xi _* \sim 1$) is given by $k_\parallel L_B/\sqrt {\beta _e} = 0.35$ or $0.36$ in the collisionless or collisional cases, respectively. We chose a very large value of $L_B/L_T$ to show the asymptotic regimes clearly.

Though useful for delineating the precise regions of instability in the $(k_{\perp }, k_\parallel )$ space, this stabilisation of the isothermal sTAI is of secondary importance because it is cTAI that is the dominant instability at long perpendicular wavelengths ($k_{\perp } \ll k_{\perp *}$), which was the main conclusion of § 4.3.2.

4.4. Isobaric limit

Let us now consider what happens in the opposite limit of short perpendicular wavenlengths, $k_{\perp } \gg k_{\perp }^{*}$, corresponding to thermal conduction (or its collisionless analogue, parallel streaming) being weak in comparison with the $\omega _{*e}$ driving, viz.,

(4.43)\begin{equation} \xi_{*} \gg 1. \end{equation}

Assuming this in addition to (4.9) or (4.20), we find that the dominant term in (4.8) is the second term on the right-hand side, meaning that, to leading order,

(4.44)\begin{equation} \boldsymbol{\nabla}_\parallel \log p_e = \boldsymbol{\nabla}_\parallel \log T_e + \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}} = 0. \end{equation}

This is the isobaric limit, in which the total pressure is constant along the perturbed field lines, rather than just the total temperature. That is, the temperature perturbation has to adjust to cancel not just the variation of the equilibrium temperature along the perturbed field line, as was the case in the isothermal limit, but now also the variation of the perturbed density. At next order in $\xi _*$, from (4.8), we have

(4.45)\begin{equation} \frac{\rho_e v_{{\rm th}e}}{2L_T} \frac{\partial}{\partial y}\boldsymbol{\nabla}_\parallel \log p_e ={-}\kappa \boldsymbol{\nabla}_\parallel^{3} \frac{\delta n_{e}}{n_{0e}} \quad \Rightarrow \quad \frac{|\boldsymbol{\nabla}_\parallel \log p_e |}{|\boldsymbol{\nabla}_\parallel \delta n_e/n_{0e} |} \sim \frac{1}{\xi_*} \ll 1, \end{equation}

so we can now neglect the entire right-hand side of (2.25), reducing the latter equation to $E_{\parallel } = 0$. For $k_{\parallel } \ll k_{\parallel c}$, i.e. neglecting the KAW restoring force, our system (2.24)–(2.26) therefore becomes

(4.46)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} ={-}\frac{\rho_e v_{{\rm th}e}}{L_B} \frac{\partial}{\partial y} \frac{\delta T_{e}}{T_{0e}}, \quad \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = 0, \quad \boldsymbol{\nabla}_\parallel \frac{\delta T_{e}}{T_{0e}} = \frac{\rho_e}{L_T } \frac{\partial \mathcal{A}}{\partial y} - \boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}}, \end{equation}

where $\varphi = - \bar {\tau } \delta n_{e}/n_e$. As with the isothermal cTAI, these equations remain valid in the collisionless limit, because $\delta T_{\parallel e} = \delta T_{\perp e} = \delta T_e$ to leading order in $\omega /\omega _{*e}$.

In (4.46), the temperature perturbation is determined from the third equation, which is simply the isobaric condition (4.44). However, given the ordering (4.9), the correction to $\delta T_{e}$ due to the density perturbation is small, viz., $\delta n_{e}/n_{0e} \sim (\omega _{de}/\omega ) \delta T_{e}/T_{0e}$, which follows from the first equation in (4.46). That is, to leading order, there is no difference between the isothermal and isobaric conditions when it comes to determining the temperature perturbation. Hence, the associated dispersion relation is

(4.47)\begin{equation} \omega^{2} ={-} 2 \omega_{de} \omega_{*e} \bar{\tau} \quad \Rightarrow \quad \omega ={\pm} {\rm i} \left(2 \omega_{de} \omega_{*e} \bar{\tau}\right)^{1/2}. \end{equation}

Analysing – and plotting, in figure 6 – the dispersion relation (4.22) in the isobaric regime, both collisional and collisionless, we find that the maximum of the growth rate in the region $k_{\parallel } < k_{\parallel c}$ occurs at $k_{\parallel } = 0$, i.e. it is, in fact, the two-dimensional cETG mode that has the fastest growth. At finite $k_{\parallel }$, it is weakened by the presence of the restoring force associated with KAWs, reaching $\gamma = 0$ at $k_{\parallel } = k_{\parallel c}$ – this is evident in figure 6(a), (c).

The dispersion relation (4.47) is identical to the cETG dispersion relation (3.13). This is because the second equation in (4.46) is simply $E_\parallel = 0$, implying that the magnetic field is now frozen into the $\boldsymbol {E}\times \boldsymbol {B}$ flow, as are the temperature perturbations [see (2.31), wherein the second term vanishes in the isobaric limit]. This is distinct from the case of the isothermal cTAI introduced in § 4.1, where the magnetic field was frozen into a different velocity field from the temperature perturbations, viz., the mean electron flow (4.16). As a result, unlike in the isothermal case, there is no enhancement of the cETG growth rate by the TAI mechanism in the isobaric regime: (4.47) can simply be regarded as an extension of the familiar cETG into the electromagnetic regime. However, physically, the isobaric cTAI is not an interchange mode, since it involves $k_\parallel \neq 0$. Its mechanism is similar to its isothermal cousin (figure 4), except the balance along the perturbed field is of pressure rather than temperature. It may therefore be appropriate to regard it as an electron-scale extension of MHD-like modes, such as the KBM – indeed, the condition $E_\parallel = 0$, which is a direct consequence of pressure balance (4.44), is often invoked as a signature of such modes (Snyder & Hammett Reference Snyder and Hammett2001b; Kotschenreuther et al. Reference Kotschenreuther, Liu, Hatch, Mahajan, Zheng, Diallo, Groebner, Hillesheim and Maggi2019).

4.4.1. Isobaric slab TAI

For $k_\parallel \gg k_{\parallel c}$, and still assuming (4.43), our system (2.24)–(2.26) becomes, approximately,

(4.48)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} ={-} v_{{\rm th}e} \boldsymbol{\nabla}_\parallel d_e^{2} \boldsymbol{\nabla}_{{\perp}}^{2} \mathcal{A}, \quad \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = 0, \quad \varphi ={-} \bar{\tau} \frac{\delta n_{e}}{n_{0e}}. \end{equation}

As with the isothermal KAW, these equations are also valid in the collisionless limit. These equations are similar to (4.28), except that the parallel electric field is now zero because the parallel gradient of the perturbed pressure vanishes. This new system describes the dynamics of isobaric KAWs – so called because they obey (4.44). Their dispersion relation is

(4.49)\begin{equation} \omega^{2} = k_{{\parallel}}^{2} v_{{\rm th}e}^{2} k_{{\perp}}^{2} d_e^{2} \frac{\bar{\tau}}{2} = \omega_\text{KAW}^{2} \bar{\tau}. \end{equation}

These isobaric KAWs, which arise in strongly driven systems (large $\omega _{*e}$), work in a similar fashion to their isothermal cousins described at the beginning of § 4.3, except the inductive part of the parallel electric field now creates a magnetic perturbation and, therefore, a parallel current, from the electrostatic part of the parallel electric field, rather than from a combination of the latter and the parallel pressure gradient.

Like the isothermal KAWs, the isobaric KAWs are unstable to sTAI: expanding (4.18) or (4.22) for $\xi _* \gg 1$ at $k_{\parallel } \gg k_{\parallel c}$, we find

(4.50)\begin{equation} \omega_r^{2} = \omega_\text{KAW}^{2} \bar{\tau}, \quad \gamma^{2} = \omega_\text{KAW}^{2} \frac{1}{4\bar{\tau} \xi_*^{2}}, \end{equation}

which is (4.49) once again but with a small, but finite, growth rate. In a similar fashion to the isothermal sTAI described in § 4.3, the instability arises due to the fact that the isobaric condition (4.44) that led to (4.48) is, in fact, only approximately satisfied. In the collisional limit, $\boldsymbol {\nabla }_\parallel \log p_e$ is determined at next order in $\xi _*^{-1}$ by (4.45), which, linearising and Fourier transforming, can be written as

(4.51)\begin{equation} \left(\boldsymbol{\nabla}_\parallel \log p_e \right)_{\boldsymbol{k}} = \frac{1}{{\rm i}\,\xi_*} \left(\boldsymbol{\nabla}_\parallel \frac{\delta n_{e}}{n_{0e}} \right)_{\boldsymbol{k}}. \end{equation}

This means that there will be a term in the second equation in (4.48) that is ${\rm \pi} /2$ out of phase with the electrostatic contribution to the parallel electric field that enables the isobaric KAWs. The result is the instability (4.50), which exists also in the collisionless limit, but with $\xi _*$ given by (4.19) [see (D74)]. Indeed, restoring finite pressure gradients in (4.48), we have

(4.52)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} \frac{\delta n_{e}}{n_{0e}} ={-} v_{{\rm th}e} \boldsymbol{\nabla}_\parallel d_e^{2} \boldsymbol{\nabla}_{{\perp}}^{2} \mathcal{A}, \quad \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \log p_e, \end{equation}

leading to the dispersion relation

(4.53)\begin{equation} \omega^{2} = \omega_\text{KAW}^{2} \left( \bar{\tau} + \frac{1}{{\rm i}\,\xi_*} \right) \quad \Rightarrow \quad \omega \approx{\pm} \omega_\text{KAW} \left( \sqrt{\bar{\tau}} - \frac{{\rm i}}{2\sqrt{\bar{\tau}} \xi_*} + \cdots \right), \end{equation}

whose real and imaginary parts are exactly the frequency and growth rate (4.50).

As $k_\parallel$ is increased, the isobaric limit (4.43) must eventually break down and be replaced by the isothermal limit (4.11). This means that there will be a transition between isobaric and isothermal KAWs, and the associated limits of sTAI, occurring, clearly, at $\xi _* \sim 1$. In the collisionless limit, the growth rate once again asymptotes to a constant value as $k_{\parallel } \rightarrow \infty$ ($\xi _* \rightarrow 0$) – this is just the isothermal limit (4.34) except now, since $k_{\perp } \gg k_{\perp *}$, this growth rate is large in comparison with the cETG growth rate achieved at $k_\parallel = 0$ (see figure 6a, noting the normalisation). Note that $\gamma \sim \omega _\text {KAW}$ near the transition $\xi _* \sim 1$ (i.e. at $k_\parallel \sim \omega _{*e}/v_{{\rm th}e}$), but $\gamma \ll \omega _\text {KAW}$ as $k_{\parallel } \rightarrow \infty$. In the collisional limit, there is peak growth at $\xi _* \sim 1$, or

(4.54)\begin{equation} k_{{\parallel} \text{max}} \sim \sqrt{\frac{\omega_{*e}}{\kappa}} \sim \frac{1}{v_{{\rm th}e}} \sqrt{\omega_{*e} \nu_e}. \end{equation}

Determining the precise prefactor, which depends only on $\bar {\tau }$ and is, thus, order unity, is only possible numerically, but is, at any rate, inessential. The growth rate at this wavenumber is

(4.55)\begin{equation} \gamma \sim \omega_\text{KAW} \sim k_{{\perp}} d_e \sqrt{\omega_{*e} \nu_e}. \end{equation}

Again, this growth rate is large in comparison with the cETG peak growth rate at $k_{\parallel } = 0$:

(4.56)\begin{equation} \frac{\gamma}{\sqrt{2\omega_{de} \omega_{*e}}} \sim k_{{\perp}} d_e \sqrt{\frac{\nu_e}{\omega_{de}}} \sim \left( \frac{k_{{\perp}}}{k_{{\perp} *}} \right)^{1/2} \gg 1. \end{equation}

Figure 6 illustrates all of this behaviour. We remind the reader that at large $k_\parallel$ (i.e. in the deep isothermal regime), the instability is quenched by compressional heating in both collisional and collisionless limits (see § 4.3.3).

Thus, the isobaric ($k_{\perp } \gg k_{\perp *}$) regime of the TAI is quite different from the isothermal one: the dominant instability is again electromagnetic, rather than electrostatic, but it is the slab TAI – an instability of KAWs reaching peak growth at the parallel wavenumber where the relevant parallel timescale --- either the parallel-streaming or thermal-conduction rate in the collisionless or collisional regimes, respectively --- is comparable to $\omega _{*e}$. It must be appreciated, of course, that this behaviour only occurs in a relatively narrow interval of perpendicular wavelengths satisfying $k_{\perp *} \ll k_{\perp } \ll d_e^{-1}$ (or $\ll d_e^{-1} \chi ^{-1}$ in the collisional regime). For $k_{\perp } d_e \gtrsim 1$ (or $\chi ^{-1}$ in the collisional regime), it is replaced by the electrostatic instabilities described in § 3.

4.4.2. Stabilisation of isobaric slab TAI

As was the case with the isothermal sTAI, the isobaric sTAI is also stabilised within a certain region of wavenumber space, this time due to the effects of finite resistivity, or finite electron inertia, in the parallel momentum equation – (2.21) and (2.25) in the collisionless and collisional limits, respectively.

To work out this stabilisation, we once again consider the collisional limit and, instead of (4.9), the ordering

(4.57)\begin{equation} (k_{{\perp}} d_e)^{2} \nu_{ei} \sim \omega \sim \kappa k_\parallel^{2} \ll \omega_{*e}. \end{equation}

A direct consequence of this ordering is that one has to retain the resistive term in the leading-order parallel momentum equation, viz., the second equation in (4.52), coming from (2.25), is replaced with

(4.58)\begin{equation} \frac{\mathrm{d} \mathcal{A}}{\mathrm{d} t} + \frac{v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial z} = \frac{v_{{\rm th}e}}{2} \boldsymbol{\nabla}_\parallel \log p_e + \nu_{ei} \frac{u_{{\parallel} e}}{v_{{\rm th}e}}. \end{equation}

This means that, instead of the system being isobaric to leading order in $\xi _* \gg 1$, the parallel pressure gradient now balances the electron–ion frictional force:

(4.59)\begin{equation} \boldsymbol{\nabla}_\parallel \log p_e + \frac{2\nu_{ei} u_{{\parallel} e}}{v_{{\rm th}e}^{2}} = 0. \end{equation}

This is obvious from (4.8) in the limit (4.57). To next order, we must now retain both the time derivative of $\boldsymbol {\nabla }_\parallel \log T_e$ and the compressional heating term in (4.8):

(4.60)\begin{equation} \frac{\rho_e v_{{\rm th}e}}{2L_T} \frac{\partial}{\partial y} \left( \boldsymbol{\nabla}_\parallel \log p_e + \frac{2\nu_{ei} u_{{\parallel} e}}{v_{{\rm th}e}^{2}} \right) = \left( \frac{\mathrm{d}}{\mathrm{d} t} - \kappa \boldsymbol{\nabla}_\parallel^{2} \right) \left( \boldsymbol{\nabla}_\parallel \frac{\delta n_e}{n_{0e}} + \frac{2\nu_{ei} u_{{\parallel} e}}{v_{{\rm th}e}^{2}} \right) - \frac{2}{3}\boldsymbol{\nabla}_\parallel^{2} u_{{\parallel} e}. \end{equation}

Combining (4.58), (4.60) and the density equation from (4.52), we find the dispersion relation

(4.61)\begin{equation} \omega^{2} - \omega_\text{KAW}^{2} \left(\bar{\tau} + \frac{1}{{\rm i}\, \xi_*} \right) ={-} \frac{1}{{\rm i}\, \xi_*} \frac{(k_{{\perp}} d_e)^{2} \nu_{ei}}{\kappa k_\parallel^{2}} \omega^{2} - \frac{1}{\xi_*} \left( \frac{5}{3} + a \right) \frac{\omega}{\kappa k_\parallel^{2} } \omega_\text{KAW}^{2}, \end{equation}

where $a$ is the same numerical constant as in (4.39) [see (E17)]. This is the same as (4.53), apart from the right-hand side, previously neglected. The second term on the right-hand side simply leads to a small, in $\xi _*\ll 1$, modification of the (real) frequency, and so can be neglected.

As usual, at the stability boundary, the frequency $\omega$ must be purely real, and both the real and imaginary parts of (4.61) must vanish individually, giving [cf. (E38)]

(4.62)\begin{equation} \omega^{2} =\omega_\text{KAW}^{2} \bar{\tau}, \quad \frac{(k_{{\perp}} d_e)^{2} \nu_{ei}}{\kappa k_\parallel^{2}} \omega^{2} = \omega_\text{KAW}^{2} \quad \Rightarrow \quad \frac{(k_{{\perp}} d_e)^{2} \nu_{ei}}{\kappa k_\parallel^{2}} = \frac{1}{\bar{\tau}}. \end{equation}

This is a line $k_\parallel \propto k_{\perp }$ in wavenumber space; moving from small to large parallel wavenumbers, there is a sliver of stability around this line, above which (viz., towards higher $k_\parallel$) the isobaric sTAI grows again to its peak at $\xi _* \sim 1$: see figures 14(a) and 15(a) in Appendix E, where the stability boundary is worked out exactly. As with the case of the isothermal sTAI, this stabilisation was not captured by the general TAI dispersion relation (4.18) because the ordering (4.9) did not formally allow frequencies comparable to both the heat-conduction and the resistive-dissipation rates, required by (4.62).

In the collisionless limit, we find that the isobaric sTAI is stabilised at the flux-freezing scale (2.28) [cf. (D84)]:

(4.63)\begin{equation} k_{{\perp}} d_e \sim 1. \end{equation}

This is not via a mechanism analogous to the collisional case, as there are no resistive effects in the collisionless limit, but is instead due to the effect of finite electron inertia appearing in the parallel-momentum equation (2.21) (see Appendix D.7.3).

The stabilisation of the isobaric sTAI is somewhat more relevant than the stabilisation of the isothermal sTAI (§ 4.3.3), owing to the fact that the isobaric sTAI is the dominant instability for $k_{\perp *} \ll k_{\perp } \ll d_e^{-1}$ (or $\ll d_e^{-1} \chi ^{-1}$ in the collisional regime). However, we shall discover in § 6.3.1 that the isobaric sTAI contributes only an order-unity amount to the turbulent energy injection – rather than introducing significant qualitative differences – and so the (linear) stabilisation thereof appears to be of little consequence in the nonlinear context.

5.1. Collisionless limit

Let us first focus on the collisionless limit. At electrostatic scales $k_{\perp } \gtrsim d_e^{-1}$ [i.e. below the flux-freezing scale (2.28)], we have both the sETG and cETG instabilities. The transition between these two instabilities occurs when their growth rates are comparable, viz.,

(5.1)\begin{equation} (k_\parallel^{2} v_{{\rm th}e}^{2} \omega_{*e} )^{1/3} \sim (\omega_{de} \omega_{*e})^{1/2} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e}} \sim \left( \frac{L_T}{L_B} \right)^{3/4} k_y d_e. \end{equation}

The sETG instability begins to be quenched by Landau damping when its growth rate becomes comparable to the parallel-streaming rate:

(5.2)\begin{equation} (k_\parallel^{2} v_{{\rm th}e}^{2} \omega_{*e} )^{1/3} \sim k_\parallel v_{{\rm th}e} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e}} \sim k_y d_e. \end{equation}

Note that this is the same line as that corresponding to the maximum of the sETG growth rate, viz., $k_\parallel v_{{\rm th}e} \sim \omega _{*e}$. However, it must be stressed that this is only true asymptotically, as is evident from figures 11(a) and 12(a). Furthermore, careful analysis of collisionless dispersion relation reveals that the sETG instability is also effectively stabilised – with only exponentially small growth rates remaining – around the flux-freezing scale [see (D43) and the surrounding discussion]. This ‘fluid’ stabilisation occurs when its growth rate becomes comparable to the KAW frequency:

(5.3)\begin{equation} (k_\parallel^{2} v_{{\rm th}e}^{2} \omega_{*e} )^{1/3} \sim \omega_\text{KAW} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e}} \sim (k_{{\perp}} d_e)^{{-}2}. \end{equation}

For $k_{\perp *} \lesssim k_{\perp } \lesssim d_e^{-1}$, the dominant instability is the isobaric sTAI, which is separated from the cETG instability by $k_\parallel = k_{\parallel c}$. The cETG instability in this perpendicular-wavenumber range, and for $k_\parallel \lesssim k_{\parallel c}$, can also be thought of as either the isobaric version of cTAI or the electron version of KBM (see § 4.4). The isobaric sTAI instability at $k_\parallel \gtrsim k_{\parallel c}$ is stabilised around the flux-freezing scale $k_{\perp } d_e \sim 1$ [see (4.63)]. The area bounded by the lines $k_{\perp } d_e \sim 1$, $k_\parallel = k_{\parallel c}$ and (5.3) thus contains only exponentially small growth rates that would be quenched by the effects of finite dissipation in any real physical system.

For $k_{\perp } \lesssim k_{\perp *}$, the cETG (or isobaric cTAI) instability is superseded by the isothermal cTAI, which is now the dominant instability, and is separated from sTAI along the horizontal line $k_\parallel = k_{\parallel c}$.

The sTAI growth rate is cut off at large parallel wavenumbers due to the effect of parallel compression [see (4.42)], i.e. when

(5.4)\begin{equation} \omega_\text{KAW} \sim \omega_{*e} \quad \Rightarrow \quad \frac{ k_\parallel L_T}{\sqrt{\beta_e}} \sim 1. \end{equation}

This is all illustrated in figure 7, where the solid line shows the location of the peak growth rate at each $k_y$ – following, at $k_{\perp } \lesssim k_{\perp *}$, the peak growth of the isothermal cTAI (4.24), and at $k_{\perp } \geqslant k_{\perp *}$, the boundary $\xi _* \sim 1$ between the isothermal and isobaric regimes. The increase of the growth rate with $k_y$ is unchecked in the drift-kinetic approximation that we have adopted, and requires the damping effects associated with the finite Larmor radius of the electrons to be taken into account; this will introduce some ultraviolet cutoff in perpendicular wavenumbers. At the largest scales, we must eventually encounter ion dynamics, but the effects that this may have are outside the scope of this paper. All of these modes are, of course, limited by the finite parallel system size $L_\parallel$, meaning that the smallest accessible parallel wavenumber is $k_\parallel \sim L_{\parallel }^{-1}$.

Figure 7. Collisionless modes in the $(k_{\perp }, k_\parallel )$ plane, where the axes are plotted on logarithmic scales. The dotted lines are the asymptotic boundaries between the various modes, with the shaded regions indicating stability. The stable region and the stability boundary are derived and plotted in a more quantitative way in Appendix D.6 (see figure 12a). At electrostatic scales (i.e. those below the flux-freezing scale, $k_{\perp } d_e >1$), the cETG (3.13) transitions into the sETG (3.5) along the boundary (5.1), while the sETG is damped by parallel streaming above (5.2). ‘Fluid’ stabilisation of the sETG occurs along (5.3). At electromagnetic scales (i.e. those above the flux-freezing scale, $k_{\perp } d_e <1$), sTAI (4.50) is stabilised along $k_{\perp } d_e \sim 1$, meaning that the region enclosed by the lines $k_{\perp } d_e \sim 1$, $k_\parallel = k_{\parallel c}$ and (5.3) contains only exponentially small growth rates, and can thus effectively be considered stable [note that $k_{\parallel c} L_T/\sqrt {\beta _e} =(L_T/L_B)^{1/2}$; see (4.21)]. The cETG transitions into the cTAI (4.15) along $k_{\perp } = k_{\perp *}$, with $k_{\perp *}$ defined in (4.26). The cTAI is separated from sTAI by the horizontal line $k_\parallel = k_{\parallel c}$, while sTAI is stabilised by compressional heating at the horizontal line given by (5.4), transitioning into purely oscilliatory (isothermal) KAWs (4.29). Electron finite-Larmor-radius effects eventually provide an ultraviolet cutoff at large perpendicular wavenumbers $k_{\perp } d_e$, though this is outside the range of validity of our drift-kinetic approximation. Note that the transition to ion-scale physics at small perpendicular wavenumbers $k_{\perp } \rho _i \lesssim 1$ lies outside our adiabatic-ion approximation. The solid black line indicates the location of the maximum growth rate at each fixed $k_{\perp }$, while the solid dots are the (possible) locations of the energy-containing scale(s) (see § 6). The dotted vertical lines indicate the location in $k_{\perp }$ of figures 5 and 6, which show the isothermal and isobaric regimes, respectively.

5.2. Collisional limit

The picture is qualitatively similar in the collisional limit, except the transition between the electrostatic and electromagnetic regimes is modified, as discussed in § 2.7. At electrostatic scales $k_{\perp } \gtrsim d_e^{-1} \chi ^{-1}$ [i.e. those below the flux-freezing scale (2.30)], we once again have both the (collisional) sETG and cETG instabilites, whose growth rates become comparable when

(5.5)\begin{equation} \left( \frac{k_\parallel^{2} v_{{\rm th}e}^{2} \omega_{*e}}{\nu_{ei}} \right)^{1/2} \sim \left( \omega_{de} \omega_{*e} \right)^{1/2} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e}} \sim \left( \frac{L_T}{L_B} \right)^{1/2} (k_y d_e \chi)^{1/2}. \end{equation}

The sETG instability is now quenched by thermal conduction at

(5.6)\begin{equation} \left( \frac{k_\parallel^{2} v_{{\rm th}e}^{2} \omega_{*e}}{\nu_{ei}} \right)^{1/2} \sim \kappa k_\parallel^{2} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e} } \sim (k_y d_e \chi)^{1/2}. \end{equation}

Note that this is the same line as that corresponding to the maximum of the collisional sETG growth rate, viz., $(k_\parallel v_{{\rm th}e})^{2}/\nu _{ei} \sim \omega _{*e}$. As in the collisionless case, this is, of course, only true asymptotically (see figures 14a and 15a).

For $k_{\perp *} \lesssim k_{\perp } \lesssim d_e^{-1} \chi ^{-1}$, the dominant instability is once again the isobaric sTAI, separated from cETG by $k_\parallel = k_{\parallel c}$. As in the collisionless limit, the cETG instability in this perpendicular-wavenumber range, and for $k_\parallel \lesssim k_{\parallel c}$, can also be thought of as either the isobaric version of cTAI or the electron version of KBM (see § 4.4). The isobaric sTAI is stabilised due to the effects of finite resistivity along the line [see (4.62)]

(5.7)\begin{equation} \kappa k_\parallel^{2} \sim (k_{{\perp}} d_e)^{2} \nu_{ei} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e}} \sim k_{{\perp}} d_e \chi. \end{equation}

For $k_{\perp } \lesssim k_{\perp *}$, the cETG (or isobaric cTAI) instability is superseded by the isothermal cTAI, which is once again the dominant instability, and is separated from the isothermal sTAI by $k_\parallel = k_{\parallel c}$. As in the collisionless case, the isothermal sTAI is cut off at large parallel wavenumbers due to the effects of parallel compression [see (4.40)], viz.,

(5.8)\begin{equation} \omega_\text{KAW} \sim \omega_{*e} \quad \Rightarrow \quad \frac{k_\parallel L_T}{\sqrt{\beta_e}} \sim 1. \end{equation}

This is all illustrated in figure 8, where the solid line again shows the location of the fastest growth for each $k_y$. As in the collisionless case, modes are stabilised at large perpendicular numbers, this time by perpendicular electron viscosity, and limited by the parallel system size for small parallel wavenumbers. However, they are now also limited at large parallel wavenumbers by the mean free path $\lambda _e$, at which the collisional approximation breaks down. This means that the maximum parallel wavenumber allowed in this collisional limit is $k_\parallel \sim \lambda _e^{-1}$.

Figure 8. Collisional modes in the $(k_{\perp }, k_\parallel )$ plane, where the axes are plotted on logarithmic scales. The dotted lines are the asymptotic boundaries between the various modes, with the shaded regions indicating stability. The stable region and the stability boundary are derived and plotted in a more quantitative way in Appendix E.4 (see figure 15a). At electrostatic scales (i.e. those below the flux-freezing scale, $k_{\perp } d_e \chi >1$), the cETG (3.13) transitions into the (collisional) sETG (3.9) along the boundary (5.5). The sETG is damped by parallel heat conduction above (5.6). At electromagnetic scales (i.e. those above the flux-freezing scale, $k_{\perp } d_e \chi <1$), the sTAI (4.50) is stabilised by the effects of finite resistivity along (5.7), while cETG transitions into the cTAI (4.15) at $k_{\perp } = k_{\perp *}$, with $k_{\perp *}$ defined in (4.26). The cTAI is separated from sTAI by the horizontal line $k_\parallel = k_{\parallel c}$ [note that $k_{\parallel c} L_T/\sqrt {\beta _e} =(L_T/L_B)^{1/2}$; see (4.21)], while the sTAI is stabilised by compressional heating at the horizontal line given by (5.8), transitioning into purely oscilliatory (isothermal) KAWs (4.29). Perpendicular electron viscosity will eventually provide an ultraviolet cutoff for these modes at large perpendicular wavenumbers $k_{\perp } d_e$, though this is outside the range of validity of our drift-kinetic approximation. As in figure 7, the ion-scale range $k_{\perp } \rho _i\lesssim 1$ is left outside our considerations. The solid black line indicates the location of maximum growth at each fixed $k_{\perp }$, while the solid dots are (possible) locations of the energy-containing scale(s) (see § 6). The dotted vertical lines indicate the location in $k_{\perp }$ of figures 5 and 6, which show the isothermal and isobaric regimes, respectively.

All of the boundaries between modes derived in this section are, of course, only asymptotic illustrations, and do not quantitatively reproduce, for example, the exact stability boundaries in wavenumber space (which are derived in Appendices D.6 and E.4). However, given that the arguments of the following section rely on scaling estimates, rather than quantitative relationships between parameters, the illustrations of the layout of wavenumber space provided by figures 7 and 8 will be sufficient for our purposes.

6.1. Free energy

Magnetised plasma systems containing small perturbations around a Maxwellian equilibrium nonlinearly conserve free energy, which is a quadratic norm of the magnetic perturbations and the perturbations of the distribution functions of both ions and electrons away from the Maxwellian. In the system that we are considering, the (normalised) free energy takes the form

(6.1)\begin{equation} \frac{W}{n_{0e} T_{0e}} = \int \frac{\mathrm{d}^{3} \boldsymbol{r}}{V} \: \left(\frac{\varphi \bar{\tau}^{{-}1} \varphi}{2} + \left| d_e \boldsymbol{\nabla}_{{\perp}} \mathcal{A} \right|^{2} + \frac{1}{2}\frac{\delta n_{e}^{2}}{n_{0e}^{2}} + \frac{u_{{\parallel} e}^{2}}{v_{{\rm th}e}^{2}} + \frac{1}{4} \frac{\delta T_{{\parallel} e}^{2}}{T_{0e}^{2}} + \frac{1}{2} \frac{\delta T_{{\perp} e}^{2}}{T_{0e}^{2}} + \cdots\right). \end{equation}

The ‘$\cdots$’ stand for the squares of further moments of the perturbed distribution function (such as the parallel and perpendicular heat fluxes $\delta q_{\parallel e}, \delta q_{\perp e}$, etc.). The derivation of (6.1) can be found in Appendix B.1. In the collisional limit, these further moments of the perturbed distribution function are negligible, and (6.1) becomes [see (B8)]

(6.2)\begin{equation} \frac{W}{n_{0e} T_{0e}} = \int \frac{\mathrm{d}^{3} \boldsymbol{r}}{V} \: \left(\frac{\varphi \bar{\tau}^{{-}1} \varphi}{2} + \left| d_e \boldsymbol{\nabla}_{{\perp}} \mathcal{A} \right|^{2} + \frac{1}{2}\frac{\delta n_{e}^{2}}{n_{0e}^{2}} + \frac{3}{4} \frac{\delta T_{ e}^{2}}{T_{0e}^{2}} \right). \end{equation}

The free energy is a nonlinear invariant, i.e. it is conserved by nonlinear interactions (Abel et al. Reference Abel, Plunk, Wang, Barnes, Cowley, Dorland and Schekochihin2013), but can be injected into the system by equilibrium gradients, and is dissipated by collisions; even when these are small, they are always eventually accessed via phase-mixing of the distribution function towards small velocity scales and nonlinear interactions towards small spatial scales.

In view of this, the time evolution of the free energy (6.1) can be written as [see (B22)]

(6.3)\begin{equation} \frac{1}{n_{0e} T_{0e}} \frac{\mathrm{d} W}{\mathrm{d} t}= \varepsilon - D, \end{equation}

where $D$ stands for the collisional dissipation [see (B11) and (B18)] and $\varepsilon$ is the injection rate due, in our system, to the ETG [see (B14) and (B19)]:

(6.4)\begin{equation} \varepsilon = \frac{1}{L_T} \int \frac{\mathrm{d}^{3} \boldsymbol{r}}{V}\left\{ \begin{array}{@{}ll} \displaystyle \left( \dfrac{1}{2} \dfrac{\delta T_{{\parallel} e}}{T_{0e}} + \dfrac{\delta T_{{\perp} e}}{T_{0e}} \right) v_{Ex} + \dfrac{\frac{1}{2} \delta q_{{\parallel} e} + \delta q_{{\perp} e}}{n_{0e} T_{0e}} \dfrac{\delta \! B_x}{B_0}, & \displaystyle \text{collisionless},\\ \displaystyle \dfrac{3}{2} \dfrac{\delta T_{e}}{T_{0e}} v_{Ex} + \dfrac{\delta q_e}{n_{0e} T_{0e}} \dfrac{\delta \! B_x}{B_0}, & \displaystyle \text{collisional}, \end{array} \right. \end{equation}

where

(6.5)\begin{equation} v_{Ex} ={-} \frac{\rho_e v_{{\rm th}e}}{2} \frac{\partial \varphi}{\partial y}, \quad \frac{\delta \! B_x}{B_0} = \rho_e \frac{\partial \mathcal{A}}{\partial y}, \quad \frac{\delta q_e}{n_{0e} T_{0e}} ={-} \frac{3}{2} \kappa \boldsymbol{\nabla}_\parallel \log T_e. \end{equation}

The expression multiplying $1/L_T$ is the ‘turbulent’ heat flux due to the energy transport by the $\boldsymbol {E}\times \boldsymbol {B}$ flows and to the heat fluxes along the perturbed field lines. The first term in (6.4) is the energy injection by ETG (§ 3), the second that by TAI (§ 4). Evidently, the latter is only present in the electromagnetic regime, when perturbations of the magnetic-field direction are allowed.

Free energy is normally the quantity whose cascade from large (injection) to small (dissipation) scales determines the properties of a plasma's turbulent state (see Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Plunk, Quataert and Tatsuno2008, Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009, and references therein). Temperature-gradient-driven turbulence is no exception (Barnes et al. Reference Barnes, Parra and Schekochihin2011), and so we devote the remainder of this section to working out at what scales and to what saturated amplitudes the ETG–TAI injection (6.4) will drive turbulent fluctuations.

6.2. Electrostatic turbulence

6.2.1. Collisionless slab ETG turbulence

Following Barnes et al. (Reference Barnes, Parra and Schekochihin2011), we conjecture that our fully developed electrostatic turbulence always organises itself into a state wherein there is a local cascade of the free energy (6.1) that carries the injected power $\varepsilon$ from the outer scale, through some putative ‘inertial range’, to the dissipation scale. The outer scale is something that we will have to determine, while the dissipation scale will be near $k_{\perp } \rho _e \sim 1$, and so outside the range of validity of our drift-kinetic approximation.

The perpendicular nonlinearity in our equations is the advection of fluctuations by the fluctuating $\boldsymbol {E}\times \boldsymbol {B}$ flows. Therefore, we take the nonlinear turnover time associated with such a cascade to be the nonlinear $\boldsymbol {E}\times \boldsymbol {B}$ advection rate:

(6.6)\begin{equation} t_\text{nl}^{{-}1} \sim k_{{\perp}} v_E \sim \rho_e v_{{\rm th}e} k_{{\perp}}^{2} \bar{\varphi} \sim \varOmega_e (k_{{\perp}} \rho_e)^{2} \bar{\varphi}. \end{equation}

Here and in what follows, $\bar {\varphi }$ refers to the characteristic amplitude of the electrostatic potential at the scale $k_{\perp }^{-1}$, rather than to the Fourier transform of the field. More formally, we take $\bar {\varphi }$ to be defined by

(6.7)\begin{equation} \bar{\varphi}^{2} = \int_{k_{{\perp}}}^{\infty} \,\mathrm{d} k_{{\perp}}' \: E_{{\perp}}^{\varphi}(k_{{\perp}}'), \quad E_{{\perp}}^{\varphi}(k_{{\perp}}) \equiv 2{\rm \pi} k_{{\perp}} \int_{-\infty}^{\infty} \,\mathrm{d} k_\parallel \left\langle|\varphi_{\boldsymbol{k}}|^{2} \right\rangle, \end{equation}

where $E_{\perp }^{\varphi }(k_{\perp })$ is the one-dimensional perpendicular energy spectrum, $\varphi _{\boldsymbol {k}}$ the spatial Fourier transform of the potential and the angle brackets denote an ensemble average. Perturbations of other quantities, such as the velocity, parallel temperature, magnetic field, etc., will similarly be taken to refer to their characteristic amplitude at a given perpendicular scale.

Assuming that any possible anisotropy in the perpendicular plane can be neglected,Footnote ⁴ a Kolmogorov-style constant-flux argument leads to the scaling of the amplitudes in the inertial range:

(6.8)\begin{equation} \frac{\bar{\tau}^{{-}1}\bar{\varphi}^{2}}{ t_{\text{nl}}} \sim \varepsilon = \text{const.} \quad \Rightarrow \quad \bar{\varphi} \sim \left( \frac{\varepsilon}{\varOmega_e} \right)^{1/3} \left(k_{{\perp}} \rho_e \right)^{{-}2/3}. \end{equation}

The scaling (6.8) translates into the following one-dimensional spectrum:

(6.9)\begin{equation} E_{{\perp}}^{\varphi}(k_{{\perp}}) \sim \frac{\bar{\varphi}^{2}}{k_{{\perp}}} \propto k_{{\perp}}^{{-}7/3}, \end{equation}

the same as was obtained, using a similar argument, and confirmed numerically, by Barnes et al. (Reference Barnes, Parra and Schekochihin2011) for electrostatic, gyrokinetic ITG turbulence. In making this argument, we have assumed that the free-energy density at a given scale $k_{\perp }^{-1}$ can be adequately represented by the first term in the integrand of (6.1), i.e. that all the other fields whose squares contribute to the free energy are either small or comparable to $\varphi$, but never dominant in comparison with it. Whether this is true will depend on the nature of the turbulent fluctuations supported by the system in any given part of the $(k_{\perp }, k_\parallel )$ space through which the cascade might be taking free energy on its journey towards dissipation. Let us specialise to the region of the wavenumber space (marked ‘sETG’ in figure 7) where the fluctuations are collisionless, electrostatic drift waves described by (3.4). From the first two equations of (3.4),Footnote ⁵

(6.10)\begin{equation} \bar{\tau}^{{-}1} \bar{\varphi} \sim \frac{k_\parallel v_{{\rm th}e}}{\omega} \frac{\bar{u}_{{\parallel} e}}{v_{{\rm th}e}} \sim \left( \frac{k_\parallel v_{{\rm th}e}}{\omega} \right)^{2} \frac{\delta \bar{T}_{{\parallel} e}}{T_{0e}}, \end{equation}

where evidently we ought to estimate $\omega \sim t_\text {nl}^{-1}$. Then, all three fluctuating fields do indeed have the same size and the same scaling if we posit

(6.11)\begin{equation} t_\text{nl}^{{-}1} \sim k_\parallel v_{{\rm th}e}. \end{equation}

This is a statement of critical balance, whereby the characteristic time associated with propagation along the field lines is assumed comparable to the nonlinear advection rate $t_\text {nl}^{-1}$ at each perpendicular scale $k_{\perp }^{-1}$. Barnes et al. (Reference Barnes, Parra and Schekochihin2011) justified this by the standard causality argument borrowed from MHD turbulence (Goldreich & Sridhar Reference Goldreich and Sridhar1995, Reference Goldreich and Sridhar1997; Boldyrev Reference Boldyrev2005; Nazarenko & Schekochihin Reference Nazarenko and Schekochihin2011): two points along the field line can only remain correlated with one another if information can propagate between them faster than they are decorrelated by the nonlinearity. We have taken the rate of information propagation along the field lines to be $k_\parallel v_{{\rm th}e}$; a somewhat involved reasoning is needed to explain why this should work even though $k_\parallel v_{{\rm th}e}$ is the rate of phase-mixing (which, in the linear theory, is expected to give rise to Landau damping) rather than of wave propagation, and why Landau damping is ineffective in the nonlinear state (see Schekochihin et al. Reference Schekochihin, Parker, Highcock, Dellar, Dorland and Hammett2016; Adkins & Schekochihin Reference Adkins and Schekochihin2018).

Combining (6.6), (6.8) and (6.11), we find

(6.12)\begin{equation} k_\parallel v_{{\rm th}e} \sim t_\text{nl}^{{-}1} \sim \varOmega_e \left( \frac{\varepsilon}{\varOmega_e} \right)^{1/3} (k_{{\perp}} \rho_e)^{4/3}. \end{equation}

By comparison, for the most unstable sETG modes, (3.6) gives us

(6.13)\begin{equation} k_\parallel v_{{\rm th}e} \sim \omega_{*e} \sim k_y \rho_e \frac{v_{{\rm th}e}}{L_T}. \end{equation}

These modes grow at a rate $\omega _{*e} \propto k_y$. This means that the nonlinear interactions must overwhelm the linear instability in the inertial range.Footnote ⁶ The outer scale, i.e. the scale that limits the inertial range on the infrared side and at which the free energy is effectively injected, is then the scale at which the nonlinear cascade rate and the rate of maximum growth of the instability are comparable: balancing (6.12) and (6.13), we get

(6.14)\begin{equation} \varOmega_e (k_{{\perp}}^{o} \rho_e)^{2} \bar{\varphi}^{o} \sim k_\parallel^{o} v_{{\rm th}e} \sim \omega_{*e}^{o} \quad \Rightarrow \quad \bar{\varphi}^{o} \sim (k_{{\perp}}^{o} L_T)^{{-}1}, \quad k_y^{o} \rho_e \sim k_\parallel^{o} L_T, \end{equation}

where the superscript ‘$o$’ refers to quantities at the outer scale.

Now, in order to determine $k_{\perp }^{o}$, we need a further constraint. Barnes et al. (Reference Barnes, Parra and Schekochihin2011) found it by conjecturing that $k_\parallel ^{o}$ in (6.14) would be set by the parallel system size $L_\parallel$ (the connection length $\sim {\rm \pi}q L_B$, in the case of tokamaks). This was the only reasonable choice because there was no lower cutoff in $k_{\perp }$ of the (electrostatic) ITG-unstable modes. This is not, however, the case in our model of the sETG instability, which is stabilised at the flux-freezing scale (2.28), i.e. at $k_{\perp } d_e \sim 1$. It appears to be a general rule, confirmed by numerical simulations (F.I. Parra & M.A. Barnes, personal communication 2012), that the outer scale is, in fact, determined by the smallest possible $k_y \rho _e$ or the smallest possible $k_\parallel L_T$, whichever is larger. Putting this within the visual context of figure 7, the outer scale is set either by $k_\parallel ^{o} \sim L_\parallel ^{-1}$ or by $k_{\perp }^{o} \sim d_e^{-1}$, whichever is encountered first when moving along the solid black line from the ultraviolet cutoff towards larger scales. The former possibility, $k_\parallel ^{o} \sim L_\parallel ^{-1}$, is realised when $L_\parallel \ll L_T/\sqrt {\beta _e}$, and the latter, $k_{\perp }^{o} \sim d_e^{-1}$, otherwise. Thus,

(6.15)\begin{equation} k_{{\perp}}^{o} d_e \sim \frac{k_\parallel^{o} L_T}{\sqrt{\beta_e}} \sim \left\{ \begin{array}{@{}ll} \displaystyle \dfrac{L_T}{L_\parallel \sqrt{\beta_e}}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle 1, & \displaystyle \dfrac{L_B}{L_T} \gtrsim \dfrac{L_B}{L_\parallel \sqrt{\beta_e}}. \end{array} \right. \end{equation}

We have inserted the normalisation of the temperature gradient to $L_B$ for future convenience.

Let us now estimate the energy flux that is injected by sETG at the outer scale (6.15): considering the first term in the expression for the energy flux (6.4) (the second, involving finite perturbations to the magnetic field, is negligible in the electrostatic regime) and ignoring any possibility of a non-order-unity contribution from phase factors, we have

(6.16)\begin{equation} \varepsilon \sim \omega_{*e}^{o} \bar{\varphi}^{o} \frac{\delta \bar{T}_{{\parallel} e}^{o}}{T_{0e}} \sim \frac{v_{{\rm th}e} \rho_e^{2}}{L_T^{3} \sqrt{\beta_e}}(k_{{\perp}}^{o} d_e)^{{-}1}, \end{equation}

where we have used $\delta \bar {T}_{\parallel e}^{o}/T_{0e} \sim \bar {\varphi }^{o}$ and (6.14). This quantity is directly related to the turbulent heat flux: combining (6.16) with (6.15), we get

(6.17)\begin{equation} Q^{\text{sETG}} \sim n_{0e}T_{0e} \varepsilon L_T \sim Q_\text{gB}\left\{ \begin{array}{@{}ll} \displaystyle \left( \dfrac{L_\parallel}{L_B} \right)\left( \dfrac{L_B}{L_T} \right)^{3}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle \dfrac{1}{\sqrt{\beta_e}} \left( \dfrac{L_B}{L_{T}} \right)^{2}, & \displaystyle \dfrac{L_B}{L_T} \gtrsim \dfrac{L_B}{L_\parallel \sqrt{\beta_e}}, \end{array} \right. \end{equation}

where the ‘gyro-Bohm’ flux is $Q_\text {gB} = n_{0e} T_{0e} v_{{\rm th}e} (\rho _e/L_B )^{2}$. Note that the $L_B/L_T$ scaling in (6.17) is only valid for sufficiently large $L_B/L_T$ as our analysis ignores any finite critical temperature gradients associated with the sETG instability (see Appendix E.1.2). The first expression in (6.17) is the same scaling as that obtained by Barnes et al. (Reference Barnes, Parra and Schekochihin2011), but this time for electrostatic turbulence driven by an ETG.Footnote ⁷ In the formal limit of $\beta _e \rightarrow 0$, this is the only possible outcome because the second inequality in (6.17) can never be satisfied. At finite $\beta _e$, however, in the sense in which it is allowed by our ordering and for sufficiently large temperature gradients, we obtain a different, less steep scaling of the turbulent heat flux, given by the second expression in (6.17).

Whether the scaling (6.17) is relevant in our system depends on the dominant energy injection therein being from the electrostatic sETG drive at $k_{\perp } d_e \gtrsim 1$. That is, in fact, far from guaranteed if $L_\parallel > L_T/\sqrt {\beta _e}$, i.e. if sufficiently small $k_\parallel$ are allowed for the electromagnetic instabilities to matter – and so for the outer scale to be located at even larger scales along the thick black line in figure 7. Another reason why we must consider the electromagnetic part of the wavenumber space is to do with the cETG instability. At $k_{\perp } d_e \gtrsim 1$, its growth rate is always small in comparison with the with sETG [for the large $L_B/L_T$ that we are considering here; see (3.14)], but it is a two-dimensional mode, so it is not stabilised at $k_{\perp } d_e \sim 1$ (it does not bend magnetic fields) and there is no reason to assume that it cannot provide the dominant energy injection at some large scale $k_{\perp } d_e \ll 1$. There, it competes with TAI (§ 4), so we have to examine the TAI turbulence alongside the cETG one.

These topics are, of course, the raison d’être of this work and we shall tackle them in § 6.3, but first we wish, for the sake of completeness, to work out the collisional version of sETG turbulence – an impatient reader can skip this.

6.2.2. Collisional slab ETG turbulence

For collisional sETG turbulence, the argument proceeds similarly to that of § 6.2.1. Instead of (6.10), we now have, in view of (3.8),

(6.18)\begin{equation} \bar{\tau}^{{-}1} \bar{\varphi} \sim \frac{k_\parallel v_{{\rm th}e}}{\omega} \frac{\bar{u}_{{\parallel} e}}{v_{{\rm th}e}} \sim \frac{(k_\parallel v_{{\rm th}e})^{2}}{\omega \nu_{ei}} \frac{\delta \bar{T}_e}{T_{0e}} \sim \frac{\delta \bar{T}_e}{T_{0e}}, \end{equation}

where we assume that all frequencies, including the nonlinear rate (6.6), are now comparable to the rate of parallel thermal conduction [instead of the parallel-streaming rate; see (3.10)]:

(6.19)\begin{equation} t_\text{nl}^{{-}1}\sim \omega \sim \frac{(k_\parallel v_{{\rm th}e})^{2}}{\nu_{ei}}. \end{equation}

This condition now replaces (6.11) as the ‘critical-balance’ conjecture, whereby the parallel scale of the perturbations is determined in terms of their perpendicular scale. Note that, since now $\bar {u}_{\parallel e}/v_{{\rm th}e} \ll \bar {\varphi }$, it is still reasonable to estimate the free-energy density by $\sim \bar {\tau }^{-1} \bar {\varphi }^{2}$.

At the outer scale, using (3.10) and (6.19), we find, analogously to (6.14),

(6.20)\begin{equation} \varOmega_e (k_{{\perp}}^{o} \rho_e )^{2} \bar{\varphi}^{o} \sim \frac{( k_\parallel^{o} v_{{\rm th}e} )^{2}}{\nu_{ei}} \sim \omega_{*e}^{o} \quad \Rightarrow \quad \bar{\varphi}^{o} \sim (k_{{\perp}}^{o} L_T)^{{-}1}, \quad k_y^{o} \rho_e \sim (k_\parallel^{o})^{2} L_T \lambda_e. \end{equation}

Note that the relationship between the parallel and perpendicular outer scales can be recast as

(6.21)\begin{equation} \frac{k_\parallel^{o} L_T}{\sqrt{\beta_e}} \sim \left(k_y^{o} d_e \chi\right)^{1/2}, \quad \chi \equiv \frac{L_T}{\lambda_e \sqrt{\beta_e}}, \end{equation}

where $\chi$ is defined as in (2.30).

By analogous logic to the collisionless sETG case, the outer scale can be set either by the parallel system size or by the flux-freezing scale (2.30), $k_{\perp } d_e \chi \sim 1$, depending on which is encountered first by the thick black line in figure 8 when descending towards larger scales. The result is

(6.22)\begin{equation} k_{{\perp}}^{o} d_e \chi \sim \left\{ \begin{array}{@{}ll} \displaystyle \left( \dfrac{L_T}{L_\parallel \sqrt{\beta_e}} \right)^{2}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle 1, & \displaystyle \dfrac{L_B}{L_T} \gtrsim \dfrac{L_B}{L_\parallel \sqrt{\beta_e}}. \end{array} \right. \end{equation}

In view of (6.20), the energy flux is again given by (6.16), which, with the substitution of (6.22), becomes

(6.23)\begin{equation} \varepsilon \sim \frac{v_{{\rm th}e} \rho_e^{2}}{L_T^{3} \sqrt{\beta_e}} \chi \left\{\begin{array}{@{}ll} \displaystyle \left( \dfrac{L_\parallel \sqrt{\beta_e}}{L_T} \right)^{2}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle 1, & \displaystyle \dfrac{L_B}{L_T} \gtrsim \dfrac{L_B}{L_\parallel \sqrt{\beta_e}}. \end{array} \right. \end{equation}

Therefore, finally, the turbulent heat flux is

(6.24)\begin{equation} Q^{\text{sETG}}_\nu\sim Q_\text{gB} \left\{ \begin{array}{@{}ll} \displaystyle \left(\dfrac{L_\parallel}{L_B} \right)^{2} \left( \dfrac{L_B}{\lambda_e} \right) \left( \dfrac{L_B}{L_T} \right)^{3}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle \dfrac{1}{\beta_e} \left( \dfrac{L_B}{\lambda_e} \right) \left( \dfrac{L_B}{L_T} \right), & \displaystyle \dfrac{L_B}{L_T} \gtrsim \dfrac{L_B}{L_\parallel \sqrt{\beta_e}}. \end{array} \right. \end{equation}

These are the collisional analogues of the scalings (6.17), and are both proportional to the electron collision frequency ($\propto \lambda _e^{-1}$). Such a scaling of turbulent heat flux with collisionality was identified by Colyer et al. (Reference Colyer, Schekochihin, Parra, Roach, Barnes, Ghim and Dorland2017) from their simulations of electrostatic ETG turbulence, though their argument relied on consideration of the dynamics of zonal flows within their electron-scale system, and so the comparison is superficial.

6.3. Electromagnetic turbulence

6.3.1. Kinetic-Alfvén-wave-dominated, slab TAI turbulence

On the large-scale side of the flux-freezing scales (2.28) and (2.30), for $k_{\perp *} \lesssim k_{\perp } \lesssim d_e^{-1}$ (or $d_e^{-1} \chi ^{-1}$ in the collisional limit), the dominant instability is the isobaric sTAI (see § 4.4), an instability of KAWs. KAW turbulence has been studied quite extensively, both numerically (Cho & Lazarian Reference Cho and Lazarian2004, Reference Cho and Lazarian2009; Howes et al. Reference Howes, Tenbarge, Dorland, Quataert, Schekochihin, Numata and Tatsuno2011; Boldyrev & Perez Reference Boldyrev and Perez2012; Meyrand & Galtier Reference Meyrand and Galtier2013; Told et al. Reference Told, Jenko, TenBarge, Howes and Hammett2015; Franci et al. Reference Franci, Landi, Verdini, Matteini and Hellinger2018; Grošelj et al. Reference Grošelj, Mallet, Loureiro and Jenko2018, Reference Grošelj, Chen, Mallet, Samtaney, Schneider and Jenko2019) and observationally (Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Sahraoui et al. Reference Sahraoui, Goldstein, Belmont, Canu and Rezeau2010; Chen et al. Reference Chen, Boldyrev, Xia and Perez2013), in the context of the ‘kinetic-range’ free-energy cascade in the solar wind (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013; Passot et al. Reference Passot, Sulem and Tassi2017). The theory of this cascade proceeds along the same lines as the theory of any critically balanced cascade in a wave-supporting anisotropic medium (Nazarenko & Schekochihin Reference Nazarenko and Schekochihin2011) and leads again to a $k_{\perp }^{-7/3}$ energy spectrum (Cho & Lazarian Reference Cho and Lazarian2004; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) or, with some modifications, to a $k_{\perp }^{-8/3}$ one (Boldyrev & Perez Reference Boldyrev and Perez2012; Meyrand & Galtier Reference Meyrand and Galtier2013), which appears to be closer to what is observed.

Ignoring the latter nuance, it is easy to see that the re-emergence of the $k_{\perp }^{-7/3}$ spectrum is unsurprising, as the arguments of § 6.2 that led to (6.8) and (6.9) are unchanged for KAWs. What is changed, however, is the linear propagation rate that must be used in the critical-balance conjecture: the parallel scale $k_\parallel ^{-1}$ of a perturbation is now the distance that an (isobaric) KAW can travel in one nonlinear time, so, from (4.49), we have, instead of (6.11) or (6.19),

(6.25)\begin{equation} \omega_{\text{KAW}} \sim k_\parallel v_{{\rm th}e} k_{{\perp}} d_e \sim t_\text{nl}^{{-}1}, \end{equation}

where $t_\text {nl}$ is still given by (6.6).Footnote ⁸

This is the standard argument of KAW turbulence theory (see references cited above), which, however, was developed for situations in which energy arrived to sub-Lamor scales from larger scales (i.e. from $k_{\perp } \rho _i < 1$) and cascaded down to smaller scales – as indeed it typically does in space-physical and astrophysical contexts. In contrast, here we are dealing with an energy source in the form of an ETG-driven instability, the isobaric sTAI, which operates most vigorously at the smallest electromagnetic scales. Indeed, as we saw at the end of § 4.4.1, for a given $k_{\perp } d_e$, the sTAI growth rate peaks at $\xi _* \sim 1$, and is of the order of the KAW frequency $\omega _\text {KAW}$ at that scale. This gives

(6.26)\begin{equation} \gamma \sim \omega_\text{KAW} \sim \left\{ \begin{array}{@{}ll} \displaystyle \omega_{*e} k_{{\perp}} d_e \sim \dfrac{v_{{\rm th}e}}{L_T \sqrt{\beta_e}} (k_{{\perp}} \rho_e)^{2}, & \displaystyle \text{collisionless},\\ \displaystyle \sqrt{\omega_{*e} \nu_e} k_{{\perp}} d_e \sim \dfrac{v_{{\rm th}e}}{\sqrt{L_T \lambda_e \beta_e}} \left( k_{{\perp}} \rho_e \right)^{3/2}, & \displaystyle \text{collisional}, \end{array} \right. \end{equation}

where we used $k_{\parallel } \sim \omega _{*e}/v_{{\rm th}e}$ and $k_\parallel \sim (\omega _{*e}/\kappa )^{1/2} \sim (\omega _{*e} \nu _e)^{1/2}/v_{{\rm th}e}$ for the collisionless and collisional estimates, respectively. Comparing (6.26) with (6.12), we see that, in both cases, the instability growth rate increases faster with $k_{\perp }$ than the nonlinear cascade rate $t_\text {nl}^{-1} \propto k_{\perp }^{4/3}$. It is intuitively obvious that these two rates reach parity at the flux-freezing scale, $k_{\perp } d_e \sim 1$ or $k_{\perp } d_e \chi \sim 1$, in the collisionless and collisional limits, respectively. This can be formally confirmed by a calculation analogous to the one in § 6.2. Thus, the dominant injection occurs at the small-scale end of the putative ‘inertial range’. In the absence of any inverse cascade, there is nothing to push the energy towards larger scales. This means that the balances (6.8), (6.12) and (6.25) are not, in fact, realised for KAW turbulence driven by the isobaric sTAI.

In order to predict the power injected by sTAI, and the associated contribution to the turbulent heat flux, we resurrect the argument that, for sETG, we tossed aside in footnote 6. We conjecture that the sTAI instability dominates the energy injection at each scale, and the energy thus injected is removed to the next smaller scale by the nonlinearity, at a rate $t_\text {nl}^{-1}$; we confirm a posteriori that this is a consistent scheme. The resulting balance gives us, using (6.6) and (6.26),

(6.27)\begin{equation} t_\text{nl}^{{-}1} \sim \varOmega_e (k_{{\perp}} \rho_e)^{2} \bar{\varphi} \sim \gamma \quad \Rightarrow \quad \bar{\varphi} \sim \left\{ \begin{array}{@{}ll} \displaystyle \dfrac{d_e}{L_T}, & \displaystyle \text{collisionless}, \\ \displaystyle \dfrac{d_e}{\sqrt{L_T \lambda_e}} (k_{{\perp}} \rho_e)^{{-}1/2}, & \displaystyle \text{collisional}, \end{array} \right. \end{equation}

and $\delta \! \bar {B}_{\perp }/B_0 \sim k_{\perp } \rho _e \bar {\mathcal {A}} \sim (\rho _e/d_e) \bar {\varphi }$ [where we have used $k_{\perp } d_e \bar {\mathcal {A}} \sim \bar {\varphi }$, which follows from the first equation in (4.48) with $\omega \sim \omega _{\text {KAW}}$]. The corresponding energy spectra (6.7) are $\propto k_{\perp }^{-1}$ and $\propto k_{\perp }^{-2}$ in the collisionless and collisional regimes, respectively. The injected power is

(6.28)\begin{equation} \gamma \bar{\varphi}^{2} \sim \frac{v_{{\rm th}e} \rho_e^{2}}{L_T^{3} \sqrt{\beta_e}} \left\{ \begin{array}{@{}ll} \displaystyle (k_{{\perp}} d_e)^{2}, & \displaystyle \text{collisionless}, \\ \displaystyle (k_{{\perp}} d_e )^{1/2} \chi^{3/2}, & \displaystyle \text{collisional}, \end{array} \right. \end{equation}

where $\chi$ is defined in (2.30) or (6.21). This means that, at each scale, the energy that arrives from larger scales can be ignored in comparison with the energy injected locally by sTAI – unlike for the sETG cascade, this scale-by-scale injection scheme is consistent for ‘sTAI turbulence’.

It is clear from (6.28) that the injected power is dominated by the flux-freezing scale (2.28) or (2.30), where it reaches parity with the power injected by sETG, (6.16) or (6.23), and where also the sTAI approximation breaks down and sETG takes over. Thus, the turbulent heat flux due to the sTAI turbulence is given by the same expression as that for the sETG turbulence at sufficiently large temperature gradients – the second expressions in (6.17) and (6.24). The only effect of sTAI is to equip the sETG turbulence spectrum (6.9) with an electromagnetic tail at long wavelengths – scaling as $k_{\perp }^{-1}$ and $k_{\perp }^{-2}$ in the collisionless and collisional cases, respectively – but without changing by more than an order-unity amount its ability to transport energy.Footnote ⁹

6.3.2. Curvature-mediated TAI turbulence

At $k_{\perp } \lesssim k_{\perp *}$, the isothermal cTAI replaces the isobaric sTAI as the dominant instability. Since the nonlinear cascading is still done by the $\boldsymbol {E}\times \boldsymbol {B}$ flows, the nonlinear time is still given by (6.6). However, how to work out the ‘inertial-range’ scalings for this cascade is not obvious: since the real frequency is vanishingly small in comparison to the growth rate at the cTAI maximum [see (4.27)], there is no obvious analogue of the ‘critical balance’ conjectures (6.11) or (6.25); indeed, it is not even a given that the cascade will be local in wavenumber space. We shall not be deterred by this uncertainty, as we can, in fact, still calculate the injected free-energy flux (6.4) by considering solely the fluctuations at the injection scale; we shall then propose a way of determining what that scale is, and hence calculate the turbulent heat flux.

First, let us assume that the dominant free-energy injection will occur at the wavenumbers (4.24), where the cTAI growth rate is largest, and given by (4.15):

(6.29)\begin{equation} \gamma \sim \frac{k_y^{o} \rho_e v_{{\rm th}e}}{\sqrt{L_B L_T} }. \end{equation}

Unlike for the electrostatic modes, the second, ‘electromagnetic’ term in (6.4) – involving energy transport due to heat flux along perturbed field lines – must contribute to the energy injection by cTAI. Let us estimate its size at the outer scale. The third equation in (4.14) gives us

(6.30)\begin{equation} \frac{\delta\! \bar{B}_x^{o}}{B_0} \sim k_y^{o}\rho_e \bar{\mathcal{A}} \sim k_{{\parallel}}^{o} L_T \frac{\delta \bar{T}_e^{o}}{T_{0e}}. \end{equation}

Recalling (2.17), we estimate the size of the perturbed heat flux in the collisional limit from (4.31):

(6.31)\begin{equation} \frac{\delta \bar{q}_e^{o}}{n_{0e} T_{0e} } \sim \kappa\boldsymbol{\nabla}_\parallel \log \bar{T}_e^{o} \sim \kappa \xi_*^{o} k_\parallel^{o} \frac{\delta \! \bar{n}_e^{o}}{n_{0e}} \sim \frac{\omega_{*e}^{o} }{ k_\parallel^{o}} \bar{\varphi}^{o}. \end{equation}

Analogously, in the collisionless limit, we find that [see (D71) and what follows it]

(6.32)\begin{equation} \frac{\delta \bar{q}_{{\parallel} e}^{o}}{n_{0e} T_{0e}} \sim \frac{\delta \bar{q}_{{\perp} e}^{o}}{n_{0e} T_{0e}} \sim \xi_*^{o} \frac{\delta \! \bar{n}_e^{o}}{n_{0e}} \sim \frac{\omega_{*e}^{o} }{ k_\parallel^{o}} \bar{\varphi}^{o}. \end{equation}

Thus, in both limits, the electromagnetic contribution to the free-energy injection can be written, at the outer scale, as

(6.33)\begin{equation} \varepsilon \sim \frac{1}{L_T} \frac{\delta \bar{q}_e^{o}}{n_{0e} T_{0e}} \frac{\delta \bar{B}_x^{o}}{B_0} \sim \omega_{*e}^{o} \bar{\varphi}^{o} \frac{\delta \bar{T}_e^{o}}{T_{0e}}, \end{equation}

meaning that it is comparable to the first term in (6.4), the electrostatic contribution due to energy transport by the $\boldsymbol {E}\times \boldsymbol {B}$ flow.

The potential at the outer scale can once again be estimated from the balance of the nonlinear time (6.6) with the growth rate (6.29):

(6.34)\begin{equation} \rho_e v_{{\rm th}e} (k_{{\perp}}^{o})^{2} \bar{\varphi} \sim \gamma \quad \Rightarrow \quad \bar{\varphi}^{o} \sim \frac{1}{k_{{\perp}}^{o}\sqrt{L_BL_T} }, \end{equation}

while the temperature perturbations can be related to $\varphi ^{o}$ via the first equation in (4.14):

(6.35)\begin{equation} \frac{\delta \bar{T}_e^{o}}{T_{0e}} \sim \frac{\gamma}{\omega_{de}^{o}} \frac{\delta \bar{n}_{e}^{o}}{n_{0e}}\sim \left( \frac{L_B}{L_T} \right)^{1/2} \bar{\varphi}^{o} \sim (k_{{\perp}}^{o} L_T)^{{-}1}. \end{equation}

Therefore, the injected energy flux (6.33) is

(6.36)\begin{equation} \varepsilon \sim \frac{v_{{\rm th}e} \rho_e^{2}}{L_T^{3} \sqrt{\beta_e}} \left( \frac{L_T}{ L_B} \right)^{1/2} (k_{{\perp}}^{o} d_e)^{{-}1}. \end{equation}

We must now determine $k_{\perp }^{o}$. We conjecture that, like in sETG turbulence, the nonlinear interaction rate in cTAI turbulence will increase faster with $k_{\perp }$ than the growth rate (6.29), $\gamma \propto k_y$. This would certainly be the case if the cascade were local, wherein the Kolmogorov-style argument leading to (6.8) applied (in which case $t_\text {nl}^{-1} \propto k_{\perp }^{4/3}$ again). Then $k_{\perp }^{o}$ will be the smallest that it can be. Since it is related to $k_\parallel ^{o}$ via (4.24) (corresponding to the maximum growth rate), viz.,

(6.37)\begin{equation} \frac{k_{{\parallel}}^{o} L_T}{\sqrt{\beta_e}} \sim \left\{ \begin{array}{@{}ll} \displaystyle \left( \dfrac{L_T}{ L_B} \right)^{1/4} (k_{{\perp}}^{o} d_e)^{1/2}, & \displaystyle \text{collisionless},\\ \displaystyle \left( \dfrac{L_T}{ L_B} \right)^{1/6} (k_{{\perp}}^{o} d_e \chi)^{1/3}, & \displaystyle \text{collisional}, \end{array} \right. \end{equation}

we can treat this expression as the analogue of the last expression in (6.14) or (6.20). As we did in our treatment of sETG turbulence in §§ 6.2.1 and 6.2.2, we now posit that the parallel outer scale of cTAI turbulence will be set by the system's parallel size, $k_\parallel ^{o} \sim L_\parallel ^{-1}$. Then, from (6.37),

(6.38)\begin{equation} k_{{\perp}}^{o} d_e \sim \left\{ \begin{array}{@{}ll} \displaystyle \dfrac{L_T^{3/2} L_B^{1/2} }{\beta_e L_\parallel^{2}}, & \displaystyle \text{collisionless},\\ \displaystyle \dfrac{L_T^{3/2} L_B^{1/2} \lambda_e }{\beta_e L_\parallel^{3}}, & \displaystyle \text{collisional}. \end{array} \right. \end{equation}

This, of course, assumes that there is no dynamics at larger scales that can set the perpendicular outer scale. We discuss the constraints set by this assumption in § 7.2.1. Using (6.38) in (6.36), we can estimate the heat flux due to cTAI turbulence:

(6.39)\begin{equation} Q^{\text{cTAI}} \sim n_{0e} T_{0e} \varepsilon L_T \sim Q_\text{gB} \left\{ \begin{array}{@{}ll} \displaystyle \sqrt{\beta_e}\left( \dfrac{L_\parallel}{L_B} \right)^{2} \left( \dfrac{L_B}{L_{T}} \right)^{3}, & \displaystyle \text{collisionless},\\ \displaystyle \sqrt{\beta_e}\left( \dfrac{L_\parallel}{L_B} \right)^{3} \left(\dfrac{L_B}{\lambda_e} \right)\left( \dfrac{L_B}{L_{T}} \right)^{3}, & \displaystyle \text{collisional}. \end{array} \right. \end{equation}

In order for this construction to be valid, $L_\parallel$ must be large enough for $k_\parallel ^{o} \sim L_\parallel ^{-1} \lesssim k_{\parallel c}$, the latter given by (4.21) – otherwise the system cannot access the cTAI regime in the first place. The condition for this is

(6.40)\begin{equation} \frac{k_\parallel^{o} L_T}{\sqrt{\beta_e}} \lesssim \left(\frac{L_T}{L_B} \right)^{1/2} \quad \Leftrightarrow \quad \frac{L_B}{L_T} \gtrsim \left( \frac{L_B}{L_\parallel \sqrt{\beta_e}} \right)^{2}. \end{equation}

Thus, cTAI turbulence is relevant for temperature gradients that are even larger than those needed to access the sETG and sTAI regimes described by (6.17) and (6.24). By comparing the heat fluxes (6.39) with the second expressions in (6.17) and (6.24), it is not hard to ascertain that the cTAI fluxes are larger than the sETG–sTAI ones as long as (6.40) is satisfied.

6.4. Summary of turbulent regimes

In §§ 6.2 and 6.3, we found scaling estimates for the turbulent heat fluxes arising from sETG, sTAI and cTAI in both the collisionless and collisional limits. Which of these scalings is realised is determined by the size of the ETG, $L_T$, for given values of $L_\parallel$, $L_B$ and $\beta _e$. There are three distinct regimes. For

(6.41)\begin{equation} k_{{\perp}}^{o} d_e \sim \frac{L_T}{L_\parallel \sqrt{\beta_e}} \gg 1 \quad \Leftrightarrow \quad \frac{L_B}{L_T} \ll \frac{L_B}{L_\parallel \sqrt{\beta_e}}, \end{equation}

the system contains only electrostatic (perpendicular) scales, and the heat flux will simply be that arising from sETG turbulence, given by the first expressions in (6.17) and (6.24) in the collisionless and collisional limits, respectively. For

(6.42)\begin{equation} k_{{\parallel} c} \lesssim \frac{1}{L_\parallel} \lesssim\frac{\sqrt{\beta_e}}{L_T} \quad \Leftrightarrow \quad \frac{L_B}{L_\parallel \sqrt{\beta_e}} \lesssim\frac{L_B}{L_T} \lesssim \left(\frac{L_B}{L_\parallel \sqrt{\beta_e}}\right)^{2}, \end{equation}

the system can access electromagnetic (perpendicular) scales, with the (isobaric) sTAI and stable KAW being added to the collection of possible modes. However, we showed in § 6.3.1 that the only effect of the sTAI was to equip the sETG turbulent spectrum with an electromagnetic tail at longer wavelengths, with at most an order-unity enhancement of the turbulent heat flux. This heat flux is still the same as that arising from the sETG turbulence, but with the outer scaled fixed at the flux-freezing scale – it is given by the second expressions in (6.17) and (6.24). Finally, for

(6.43)\begin{equation} \frac{1}{L_\parallel} \lesssim k_{{\parallel} c} \quad \Leftrightarrow \quad \frac{L_B}{L_T} \gtrsim \left(\frac{L_B}{L_\parallel \sqrt{\beta_e}}\right)^{2}, \end{equation}

the system has a large enough parallel size to activate cTAI. The resultant turbulent heat flux, given by (6.39), dominates over that due to the sETG and sTAI.

To summarise, we can write the turbulent heat flux in the collisionless limit as

(6.44)\begin{equation} Q \sim Q^{\text{gB}} \left\{ \begin{array}{@{}ll} \displaystyle \left( \dfrac{L_\parallel}{L_B} \right)\left( \dfrac{L_B}{L_T} \right)^{3}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle \dfrac{1}{\sqrt{\beta_e}} \left( \dfrac{L_B}{L_{T}} \right)^{2}, & \displaystyle \dfrac{L_B}{L_\parallel \sqrt{\beta_e}} \lesssim \dfrac{L_B}{L_T} \lesssim \left( \dfrac{L_B}{L_\parallel \sqrt{\beta_e}} \right)^{2},\\ \displaystyle \sqrt{\beta_e} \left( \dfrac{L_\parallel}{L_B} \right)^{2} \left( \dfrac{L_B}{L_{T}} \right)^{3}, & \displaystyle \dfrac{L_B}{L_T} \gtrsim \left(\dfrac{L_B}{L_\parallel \sqrt{\beta_e}} \right)^{2}, \end{array} \right. \end{equation}

or, in the collisional limit, as

(6.45)\begin{equation} Q_\nu \sim Q^{\text{gB}} \left\{ \begin{array}{@{}ll} \displaystyle \left(\dfrac{L_\parallel}{L_B} \right)^{2} \left( \dfrac{L_B}{\lambda_e} \right) \left( \dfrac{L_B}{L_T} \right)^{3}, & \displaystyle \dfrac{L_B}{L_T} \ll \dfrac{L_B}{L_\parallel \sqrt{\beta_e}},\\ \displaystyle \dfrac{1}{\beta_e} \left( \dfrac{L_B}{\lambda_e} \right) \left( \dfrac{L_B}{L_T} \right), & \displaystyle \dfrac{L_B}{L_\parallel \sqrt{\beta_e}} \lesssim \dfrac{L_B}{L_T} \lesssim \left( \dfrac{L_B}{L_\parallel \sqrt{\beta_e}} \right)^{2},\\ \displaystyle \sqrt{\beta_e} \left( \dfrac{L_\parallel}{L_B} \right)^{3} \left(\dfrac{L_B}{\lambda_e} \right)\left( \dfrac{L_B}{L_{T}} \right)^{3}, & \displaystyle \displaystyle \dfrac{L_B}{L_T} \gtrsim \left(\dfrac{L_B}{L_\parallel \sqrt{\beta_e}} \right)^{2}. \end{array} \right. \end{equation}

Notably, this implies that the effect of increasing $\beta _e$ (or increasing $L_\parallel /L_B \sim {\rm \pi}q$, as in a tokamak edge) is first to make the electron heat transport less stiff, as flux freezing pins down the ETG injection scale, and then to stiffen it back again, as cTAI takes over. This is sketched in figure 9. A striking (and perhaps disturbing) feature of these results is the discontinuity in the collisional turbulent heat flux around the transition between the sTAI- and cTAI-dominated regimes, described by the last two expressions in (6.45). Comparing these, it is easy to see that the latter is larger than the former for

(6.46)\begin{equation} \frac{L_B}{L_T} \gtrsim \left(\frac{L_B}{L_\parallel \sqrt{\beta_e}} \right)^{3/2}. \end{equation}

This condition is obviously met before the parallel system size is large enough in order to activate the cTAI, meaning that the sTAI regime must persist – despite it supporting a notionally lower flux than that predicted by the cTAI scaling – until the inequality in (6.43) is satisfied, at which point the cTAI takes over, leading to the discontinuity. Whether this and the other simple ‘twiddle-algebra’ considerations that led to (6.44) and (6.45) survive the encounter with quantitative reality is left for future numerical investigations to determine.

Figure 9. The scaling of the turbulent heat flux with $L_B/L_T$ in the (a) collisionless and (b) collisional limits. As $L_B/L_T$ is increased, the electron transport initially becomes less stiff, as flux freezing pins down the ETG injection scale, after which it stiffens again as cTAI takes over.