2. Two-dimensional, partially magnetised plasma model

As was done by Ramos et al. (Reference Ramos, Bello-Benítez and Ahedo2021), a two-dimensional, Cartesian geometry with dynamics in the plane perpendicular to the magnetic field will be assumed, as well as a single species of ions and the restriction to electrostatic modes:

(2.1) \begin{eqnarray} \boldsymbol{B} = B(z) \ \boldsymbol{e}_x , \qquad \partial / \partial x = 0 , \qquad u_{ix} = u_{ex} = 0 , \qquad \boldsymbol{ E} = - \boldsymbol{\nabla }\phi . \end{eqnarray}

Here, $\boldsymbol{u}_s$ is the macroscopic fluid velocity of the species $s$ , $\boldsymbol{E}$ and $\boldsymbol{B}$ are the electromagnetic fields, and $\phi$ is the electric potential. For an HT configuration, $x,y,z$ represent locally the radial, azimuthal and axial directions, respectively, and hence, the model concerns the axial-azimuthal dynamics. The assumed inhomogeneous but unidirectional magnetic field has non-zero curl, which we accept to carry out a fully consistent while analytically tractable drift-kinetic study of the electron linear perturbation in two-dimensional geometry. Realistically, the curl of the magnetic field is negligible in an HT and the externally produced magnetic field has curvature. The present two-dimensional work has the limitation of ignoring the contribution of such magnetic curvature as well as all the other three-dimensional effects. The plasma will be assumed to be collisionless and quasi-neutral with unit-charge ions,

(2.2) \begin{eqnarray} n_i = n_e = n . \end{eqnarray}

The ions will be treated as an unmagnetised cold fluid. The electrons will be treated as a strongly magnetised kinetic species, assuming their Larmor gyroradius to be much smaller than any characteristic length scale or mode wavelength, and their dynamics will accordingly be described with a drift-kinetic equation.

2.1. Cold ions

In a manner analogous to the analysis of Ramos et al. (Reference Ramos, Bello-Benítez and Ahedo2021) and using the same notation, the ions are described as a cold fluid,

(2.3) \begin{align} \frac {\partial n}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }(n \boldsymbol{u}_i) = 0 , \end{align}

(2.4) \begin{align} m_i \left [ \frac {\partial \boldsymbol{u}_i}{\partial t} + (\boldsymbol{u}_i \boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}_i\right ] = e \left (- \boldsymbol{\nabla }\phi + B \boldsymbol{u}_i\!\times\!\boldsymbol{e}_x \right )\!. \end{align}

The equilibrium is assumed one-dimensional, with its variables depending only on the $z$ -coordinate and denoted with a $0$ subscript. A local, linear stability analysis is carried out for small-amplitude electrostatic perturbations, with the general variables split into equilibrium and fluctuating parts as $Q(y,z,t) = Q_0(z) + Q_1(z) \exp (i k_y y + i k_z z - i \omega t)$ such that $|\partial \ln Q_1 / \partial z| \sim |\partial \ln Q_0 / \partial z| \sim L^{-1} \ll |k_y| \sim |k_z|$ . Introducing the Doppler-shifted frequency in the ion frame, $\omega _i = \omega - \boldsymbol{k} \boldsymbol{\cdot }\boldsymbol{u}_{i0}$ , and assuming $u_{iz0} \leq O(c_S)$ and $|\omega _i| \sim |\omega | \geq O(k c_S) \gg O(\omega _{ci})$ , where $c_S = (T_{e0}/m_i)^{1/2}$ is the sound speed and $\omega _{cs}= eB/m_s$ is the cyclotron frequency of species $s$ , one obtains the relation between the density and electric potential perturbations (Esipchuk & Tilinin Reference Esipchuk and Tilinin1976; Frias et al. Reference Frias, Smolyakov, Kaganovich and Raitses2012; Smolyakov et al. Reference Smolyakov, Chapurin, Frias, Koshkarov, Romadanov, Tang, Umansky, Raitses, Kaganovich and Lakhin2017; Lakhin et al. Reference Lakhin, Ilgisonis, Smolyakov, Sorokina and Marusov2018; Ramos et al. Reference Ramos, Bello-Benítez and Ahedo2021; Ripoli et al. Reference Ripoli, Ahedo and Merino2025),

(2.5) \begin{eqnarray} n_1 = \frac {e n_0 k^2}{m_i \omega _i^2} \ \phi _1 . \end{eqnarray}

2.2. Drift-kinetic electrons

For $|\omega | \ll \omega _{ce}$ , $k \rho _e \ll 1$ and the assumed two-dimensional geometry with vanishing parallel gradients, the leading-order electron distribution function, $f_e(w_\parallel , w_\perp , \boldsymbol{x}, t)$ , is independent of the gyrophase angle and satisfies the following collisionless drift-kinetic equation (Ramos Reference Ramos2008) in the reference frame of the macroscopic fluid velocity, i.e $\boldsymbol{w} = \boldsymbol{v} - \boldsymbol{u}_e(\boldsymbol{x},t)$ , where $\boldsymbol{v}$ is the velocity coordinate in the laboratory frame:

(2.6) \begin{eqnarray} \frac {\partial f_e}{\partial t} + \left (\boldsymbol{u}_E - \frac {m_e w_\perp ^2}{2eB} \boldsymbol{e}_x\!\times\!\boldsymbol{\nabla }\ln B \right ) \boldsymbol{\cdot }\frac {\partial f_e}{\partial \boldsymbol{x}} - \frac {w_\perp }{2} \left (\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}_E \right ) \frac {\partial f_e}{\partial w_\perp } = 0 . \end{eqnarray}

Here, $\boldsymbol{u}_E = B^{-1}\boldsymbol{e}_x\!\times\!\boldsymbol{\nabla }\phi$ is the electric drift velocity, so $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{ u}_E = B^{-1} \boldsymbol{e}_x \boldsymbol{\cdot }(\boldsymbol{\nabla }\ln B\!\times\!\boldsymbol{\nabla }\phi )$ . This applies to the lowest order in the small Larmor radius where the electron fluid velocity $\boldsymbol{u}_e$ can be taken as the sum of the electric drift $\boldsymbol{u}_E$ plus a diamagnetic drift.

Linearising about a spatially one-dimensional equilibrium that depends on the $z$ -coordinate,

(2.7) \begin{eqnarray} -i \omega f_{e1} \ & + & \ i k_y u_{E0} f_{e1} + \frac {ik_y}{B} \frac {\partial f_{e0}}{\partial z} \phi _1 + \frac {i k_y m_e w_\perp ^2}{2eB} \frac {\partial \ln B}{\partial z} f_{e1} \nonumber\\[4pt]& +& \ \frac {i k_y }{2B} \frac {\partial \ln B}{\partial z} w_\perp \frac {\partial f_{e0}}{\partial w_\perp } \phi _1 = 0 \end{eqnarray}

and, defining $\omega ' = \omega - k_y u_{E0}$ with $u_{E0} = - B^{-1} \partial \phi _0 / \partial z$ ,

(2.8) \begin{eqnarray} \left ( \omega ' - \frac {k_y m_e w_\perp ^2}{2eB} \frac {\partial \ln B}{\partial z} \right ) f_{e1} = \frac {k_y}{B} \left ( \frac {\partial f_{e0}}{\partial z} + \frac {1}{2} \frac {\partial \ln B}{\partial z} w_\perp \frac {\partial f_{e0}}{\partial w_\perp } \right ) \phi _1 . \end{eqnarray}

Assuming a Maxwellian equilibrium distribution function,

(2.9) \begin{eqnarray} f_{e0} = n_0 \left (\frac {m_e}{2\pi T_{e0}}\right )^{3/2} \exp \left (-\frac {m_e w^2}{2 T_{e0}} \right )\!, \end{eqnarray}

with the thermal velocity defined as $v_{the} = (T_{e0}/m_e)^{1/2}$ , and introducing the definitions

(2.10) \begin{eqnarray} \omega _{*N} = - k_y \frac {T_{e0}}{eB} \ \frac {\partial \ln n_0}{\partial z} , \quad \omega _{*T} = - k_y \frac {T_{e0}}{eB} \ \frac {\partial \ln T_{e0}}{\partial z} , \quad \omega _{*B} = - k_y \frac {T_{e0}}{eB} \ \frac {\partial \ln B}{\partial z} , \qquad \end{eqnarray}

one obtains

(2.11) \begin{eqnarray} \left ( \omega ' + \frac {w_\perp ^2}{2 v_{the}^2} \omega _{*B} \right ) f_{e1} & = & - \ \frac {e n_0 \phi _1}{(2 \pi )^{3/2} T_{e0} v_{the}^3} \left [ \omega _{*N} + \left (\frac {w^2}{2 v_{the}^2} - \frac {3}{2} \right ) \omega _{*T} - \frac {w_\perp ^2}{2 v_{the}^2} \omega _{*B} \right ] \nonumber\\[12pt]&& \quad\times \exp \left ( - \frac {w^2}{2 v_{the}^2} \right )\!. \end{eqnarray}

The perturbed electron density is

(2.12) \begin{align} n_1 = 2 \pi \int_{- \infty }^\infty \text{d} w_\parallel \int _0^\infty \text{d} w_\perp \ w_\perp \ f_{e1} \end{align}

and, substituting the solution (2.11) for $f_{e1}$ ,

(2.13) \begin{align} n_1 &= - \ \frac{e n_0 \phi_1}{(2 \pi)^{1/2} T_{e0} v_{the}^3} \int_{- \infty}^\infty \textrm{d} w_\parallel \int_0^\infty \textrm{d} w_\perp \ w_\perp \ \nonumber \\& \quad \times \frac{\omega_{*N} + \left({w^2}/{2 v_{the}^2} - {3}/{2} \right) \omega_{*T} - ({w_\perp^2}/{2 v_{the}^2}) \omega_{*B}}{\omega' + ({w_\perp^2}/{2 v_{the}^2}) \omega_{*B}} \exp \left( - \frac{w^2}{2 v_{the}^2} \right)\!. \end{align}

Finally, carrying out the integral over $w_\parallel$ and calling $s=w_\perp ^2 / (2 v_{the}^2)$ ,

(2.14) \begin{eqnarray} n_1 = - \ \frac {e n_0 \phi _1}{T_{e0}} \int _0^\infty \,\mathrm{d}s \ \frac {\omega _{*N} + \omega _{*T} (s-1) - \omega _{*B} \ s}{\omega ' + \omega _{*B} \ s} \ \exp (-s), \end{eqnarray}

defined for those values of the plasma parameters and the mode frequency such that the integrand does not diverge on the integration path $(0 \lt s \lt \infty )$ and the integral exists.

3. Dispersion relation for an homogeneous magnetic field

In the special case of an homogeneous magnetic field, $\omega _{*B}=0$ and the integral of (2.14) is elementary, yielding

(3.1) \begin{eqnarray} n_1 = - \ \frac {e n_0 \phi _1}{T_{e0}} \ \frac {\omega _{*N}}{\omega '} . \end{eqnarray}

Notice that the contribution of $\omega _{*T}$ cancels out. This result coincides with the familiar one derived from fluid theory in the lowest order of the small electron Larmor radius where the electron inertia and gyroviscosity can be neglected in the momentum equation. In this limit, the electron fluid system reduces to

(3.2) \begin{eqnarray} \frac {\partial n}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }(n \boldsymbol{u}_e) = 0 , \end{eqnarray}

(3.3) \begin{eqnarray} \boldsymbol{\nabla }p_e + e n \left (- \boldsymbol{\nabla }\phi + B \boldsymbol{u}_e\!\times\!\boldsymbol{e}_x \right ) = 0 . \end{eqnarray}

Solving for the electron current,

(3.4) \begin{eqnarray} n \boldsymbol{u}_e = \frac {1}{eB} \ \boldsymbol{e}_x \times \left (en \boldsymbol{\nabla }\phi - \boldsymbol{\nabla }p_e \right ) \end{eqnarray}

and, taking into account that here $B$ is constant,

(3.5) \begin{eqnarray} \boldsymbol{\nabla }\boldsymbol{\cdot }(n \boldsymbol{u}_e) = \frac {1}{B} \ \boldsymbol{e}_x \boldsymbol{\cdot }\left ( \boldsymbol{\nabla }\phi\!\times\!\boldsymbol{\nabla }n \right)\!. \end{eqnarray}

Bringing this to the continuity equation and linearising, one gets

(3.6) \begin{eqnarray} (\omega - k_y u_{E0}) n_1 + \frac {e n_0 \omega _{*N}}{T_{e0}} \phi _1 = 0 , \end{eqnarray}

which is the same as (3.1).

Combining (2.5) and (3.1), one obtains the dispersion relation for an homogeneous magnetic field (Smolyakov et al. Reference Smolyakov, Chapurin, Frias, Koshkarov, Romadanov, Tang, Umansky, Raitses, Kaganovich and Lakhin2017)

(3.7) \begin{eqnarray} \omega _{*N} \ \omega _i^2 + k^2 c_S^2 (\omega _i - {\hat \varDelta }) = 0 , \end{eqnarray}

where

(3.8) \begin{eqnarray} {\hat \varDelta } = \omega _i - \omega ' = k_y u_{E0} - \boldsymbol{k} \boldsymbol{\cdot }\boldsymbol{u}_{i0} . \end{eqnarray}

This quadratic dispersion relation has an unstable complex root for

(3.9) \begin{eqnarray} {\hat \varDelta } \ \omega _{*N} \lt \frac{- k^2 c_S^2}{4}, \end{eqnarray}

which is the criterion for the (collisionless) S–H instability, including in $\hat \varDelta$ the ion-drift effect (Sakawa et al. Reference Sakawa, Joshi, Kaw, Chen and Jain1993). So, in the two-dimensional theory being considered here with $k_\parallel =0$ , there is no kinetic effect on the low-frequency dispersion relation if the magnetic field is homogeneous, the only instability being the fluid drift-gradient mode driven by the equilibrium density gradient. To have an instability driven by the temperature gradient, a non-zero gradient of the magnetic field is necessary.

4. Dispersion relation for an inhomogeneous magnetic field

For $\omega _{*B} \not = 0$ , calling $\zeta = \omega ' / \omega _{*B}$ , (2.14) can be rewritten as

(4.1) \begin{eqnarray} n_1 = \frac {e n_0 \phi _1}{T_{e0} \ \omega _{*B}} \left[(\omega _{*T} - \omega _{*N}) \ g(\zeta ) - (\omega _{*T} - \omega _{*B}) \ h(\zeta ) \right] , \end{eqnarray}

where

(4.2) \begin{eqnarray} g(\zeta ) = \int _0^\infty\, \text{d}s \ \frac {\exp (-s)}{\zeta + s} \end{eqnarray}

and

(4.3) \begin{eqnarray} h(\zeta ) = \int _0^\infty \,\text{d}s \ \frac {s \ \exp (-s)}{\zeta + s} = 1 - \zeta \ g(\zeta ) . \end{eqnarray}

The function $g(\zeta )$ is real analytic, i.e. $g(\zeta ^*) = g(\zeta )^*$ , and it has a branch cut for real and negative $\zeta$ with logarithmic singularities at $\zeta =0$ and at infinity. It is related to the exponential integral function (Abramowitz & Stegun Reference Abramowitz and Stegun1965) $E_1(\zeta )$ through

(4.4) \begin{eqnarray} g(\zeta ) = \exp (\zeta ) \ E_1(\zeta ), \end{eqnarray}

where

(4.5) \begin{eqnarray} E_1(\zeta ) = \int _\zeta ^\infty \,\text{d} \sigma \ \frac {\exp (-\sigma )}{\sigma } = - \left [\ln \zeta + \gamma + \sum _{m=1}^\infty \frac {(-1)^m \zeta ^m}{m \ m!} \right ] \end{eqnarray}

and $\gamma$ is the Euler–Mascheroni constant. The derivative of $g(\zeta )$ is defined everywhere except at its singular half-line,

(4.6) \begin{eqnarray} \frac {\text{d} g(\zeta )}{\text{d} \zeta } = g(\zeta ) - \frac {1}{\zeta } . \end{eqnarray}

Taking successive derivatives, we obtain

(4.7) \begin{eqnarray} g(\zeta ) = \frac {1}{\zeta } - \frac {1}{\zeta ^2} + \frac {2}{\zeta ^3} - \frac {3!}{\zeta ^4} + \cdots + \frac {(-1)^k k!}{\zeta ^{k+1}} + \frac {\text{d}^{k+1} g(\zeta )}{\text{d} \zeta ^{k+1}}, \end{eqnarray}

which applies where the derivatives of $g(\zeta )$ exist, i.e. everywhere except for real non-positive $\zeta$ . Disregarding the residual term $d^{k+1} g(\zeta )/d \zeta ^{k+1} = O(1/|\zeta |^{k+2})$ and letting $k$ go to infinity, (4.7) can be considered as a power expansion of $g(\zeta )$ for large argument. However, this is only a non-converging asymptotic series, so care must be exercised when using it. Most crucially, it does not reproduce the singular behaviour of $g(\zeta )$ as $\zeta$ approaches a real and negative value, whose physical origin is the resonant character of the kinetic perturbation of the electron distribution function (2.13). This issue is dealt with in more detail in Appendix B, where an exponentially small correction is added in the case of $\zeta$ being close to real and negative, which recovers faithfully that singularity.

In terms of $\zeta$ , (2.5) can be rewritten as

(4.8) \begin{eqnarray} n_1 = \frac {e n_0 k^2}{m_i \omega _{*B}^2 (\zeta + {\hat \varDelta } / \omega _{*B})^2} \ \phi _1 \end{eqnarray}

and, combining (4.1), (4.3) and (4.8), we obtain the final dispersion relation

(4.9) \begin{eqnarray} \frac {k^2 c_S^2}{(\zeta + {\hat \varDelta } / \omega _{*B})^2} = \omega _{*B}\{[(\omega _{*T} - \omega _{*N}) + (\omega _{*T} - \omega _{*B}) \zeta] g(\zeta ) - (\omega _{*T} - \omega _{*B}) \} .\qquad \end{eqnarray}

So, defining the three real dimensionless parameters,

(4.10) \begin{eqnarray} \alpha = \frac {\omega _{*B}(\omega _{*N} - \omega _{*B})}{k^2 c_S^2} , \ \beta = \frac {\omega _{*B}(\omega _{*T} - \omega _{*B})}{k^2 c_S^2} , \ \delta = \frac {{\hat \varDelta }}{\omega _{*B}} = \frac {k_y u_{E0} - \boldsymbol{k} \boldsymbol{\cdot }\boldsymbol{u}_{i0}}{\omega _{*B}} ,\qquad \end{eqnarray}

the dimensionless form of the dispersion relation is

(4.11) \begin{eqnarray} \frac {1}{(\zeta + \delta )^2} + [\alpha - \beta (1+\zeta )] \ g(\zeta ) + \beta = 0 , \end{eqnarray}

whose roots are either real roots or complex conjugate pairs, since $g(\zeta )$ is real analytic. Notice that the dimensionless variables in (4.11) are independent of the magnitude $k$ of the wavevector, $\boldsymbol{k} = k_y\boldsymbol{e}_y + k_z\boldsymbol{e}_z = k(\cos \theta \boldsymbol{e}_y +\sin \theta \boldsymbol{e}_z)$ , and depend only on the propagation angle $\theta$ . As a consequence of this and the definition of $\zeta = \omega ' / \omega _{*B}$ , the dimensional complex frequency $\omega$ scales proportional to $k$ in the long-wavelength regime $k \rho _e \ll 1$ that this work considers.

5. Parametric range of instability

The dispersion relation for an inhomogeneous magnetic field, (4.9) or (4.11), is a transcendental equation whose normalised frequency roots have to be found numerically. However, the marginal stability boundary in parameter space between the region where it has complex roots (i.e. where the plasma equilibrium is unstable) and the region where it does not have any complex root (i.e. where the plasma equilibrium is stable) can be found analytically. The detailed mathematical derivation of those marginal stability conditions is given in Appendices A and B.

Figure 1. Parametric range of instability from the kinetic dispersion relation (4.11) in the $(\alpha$ , $\beta )$ plane for several representative values of $\delta$ . The white region is stable, while a colour scale indicates the normalised instability growth rate in the unstable region. The green, blue and red solid lines represent the marginal stability boundaries obtained analytically.

Figure 1 displays the analytic instability thresholds (shown as green, blue and red solid lines) and the regions of instability (indicated with a colour scale that represents the numerical growth rates) in $(\alpha , \beta )$ parametric planes at constant $\delta$ for six representative values of the latter. The agreement between the analytic and numerical results is excellent. The marginal stability boundary is divided in three sections. The lines coloured green correspond to a resonant instability threshold, where a pair of complex conjugate roots of the dispersion relation emerges at its logarithmic branch cut, with a marginal normalised real frequency $\zeta$ negative. This resonant instability threshold is given by the conditions $\beta \leq 0$ , $\beta \leq \alpha$ and $\alpha = \beta (1-\delta ) \pm \sqrt {-\beta }$ . It connects the point $(\alpha , \beta )=(0,0)$ where $\zeta \rightarrow - \infty$ , with the point $(\alpha , \beta )=(-\delta ^{-2}, -\delta ^{-2})$ where $\zeta \rightarrow 0_{-}$ . The connecting path is continuous if $\delta \lt 0$ , whereas it goes through infinity if $\delta \geq 0$ . The lines coloured blue correspond to a non-resonant or fluid-like instability threshold, where a pair of complex conjugate roots of the dispersion relation evolves from a pair of real roots that coalesce into a double root, with a marginal normalised real frequency $\zeta$ positive. The equation for this non-resonant instability threshold is given in parametric form by (A12) and (A13). It connects the point $(\alpha , \beta )=(-\delta ^{-2}, -\delta ^{-2})$ where $\zeta \rightarrow 0_{+}$ with the point $(\alpha , \beta )=(0,1)$ where $\zeta \rightarrow + \infty$ . This connecting path is continuous if $\delta \gt 0$ , whereas it goes through infinity if $\delta \leq 0$ . Finally, the segment $\alpha =0, \ 0 \leq \beta \leq 1$ , coloured red, corresponds to a resonant instability threshold where a pair of complex conjugate roots of the dispersion relation emerges at infinity. On the stable $\alpha \lt 0$ side, there is a large positive real root, while on the unstable $\alpha \gt 0$ side, a complex pair appears with large negative real part and exponentially small imaginary part.

Figure 2 displays the same information, in $(\alpha , \delta )$ parametric planes at constant $\beta$ . For $\beta \gt 1$ , all the instability thresholds are of the non-resonant type. For $0 \leq \beta \leq 1$ , we have non-resonant and root-at-infinity thresholds. For $\beta \lt 0$ , we have both the non-resonant and the resonant with root at the branch cut types of thresholds, the latter appearing now as two parallel straight half-lines that leave a stable band in between. These two parallel lines start at the points $(\alpha , \delta ) = (\beta , \pm 1/\sqrt {-\beta })$ and have slope $-1/\beta$ , so the width of the stable band shrinks to zero both as $\beta \rightarrow 0_-$ when the lines merge with the $\alpha =0$ axis and as $\beta \rightarrow -\infty$ when the lines merge with the $\delta = 0$ axis.

Figure 2. Parametric range of instability from the kinetic dispersion relation (4.11) in the $(\alpha$ , $\delta )$ plane for several representative values of $\beta$ . The white region is stable, while a colour scale indicates the normalised instability growth rate in the unstable region. The green, blue and red solid lines represent the marginal stability boundaries obtained analytically.

In these $(\alpha , \delta )$ planes, we see that the instability largely fills the first, second and fourth quadrants. The third quadrant is stable for positive $\beta$ , but develops an increasingly large region of instability as $\beta$ becomes increasingly negative. Recalling the definitions (2.10) and (4.10), $\alpha$ and $\beta$ are positive when the gradients of $n_0/B$ and $T_{e0}/B$ , respectively, point in the direction of the gradient of $B$ . Also, if we assume for simplicity $|k_y u_{E0}| \gg |\boldsymbol{k} \boldsymbol{\cdot }\boldsymbol{u}_{i0}|$ as is often the case, $\delta$ is positive when the applied equilibrium electric field points opposite to the direction of the gradient of $B$ . Therefore, the second and fourth quadrants, $\alpha \delta \lt 0$ , correspond to the gradient of $n_0/B$ pointing in the direction of the applied electric field, which is the condition for the classic Simon–Hoh instability. Accordingly, we may regard the instability in those second and fourth quadrants as a generalisation of the S–H instability, now modified in a kinetically consistent manner by the effects of the gradients of the magnetic field and the electron temperature. These kinetic modifications are significant and the classic form of the S–H instability threshold in (3.9), $\alpha \delta \simeq -1/4$ , is observed only in the narrow parametric region characterised by $|\beta | \ll 1, \ |\alpha | \sim |\delta |^{-1} \ll 1$ and $\alpha \lt 0$ that will be studied in the next section.

The instability in the first and third quadrants is unconventional, as it happens when the gradient of $n_0/B$ points opposite to the applied electric field. This instability, not of S–H character, becomes generally stronger as the magnitude of $\beta$ increases, indicating a drive due to the electron temperature gradient. In the first quadrant, where the gradient of $n_0/B$ is in the direction of the gradient of $B$ , we observe a relatively weak resonant instability with a stable band if $\beta \lt 0$ (i.e. if the gradient of $T_{e0}/B$ is opposite to the gradients of $n_0/B$ and $B$ ), but a more robust instability if $\beta$ is positive and increasing (i.e. if the gradient of $T_{e0}/B$ is increasingly large in the direction of the gradients of $n_0/B$ and $B$ ). The third quadrant, where the gradient of $n_0/B$ is opposite to the gradient of $B$ , is stable if $\beta \gt 0$ (i.e. if the gradient of $T_{e0}/B$ is opposite to the gradient of $n_0/B$ ). However, a strong non-resonant instability, clearly driven by the temperature gradient, appears in the third quadrant if $\beta \lt 0$ (i.e. if the gradient of $T_{e0}/B$ is in the direction of the gradient of $n_0/B$ ), with a condition for instability of the form $\beta \lt \beta _c(\alpha , \delta ) \lt 0$ . In summary, a robust instability, not of S–H character since it happens in the first or third quadrant where the gradient of $n_0/B$ points opposite to the applied electric field, is observed when the gradient of $T_{e0}/B$ is in the direction of the gradient of $n_0/B$ . It is triggered or becomes stronger as the magnitude of the electron temperature gradient increases, so it can be categorised as a type of ETG mode.

6. Weak magnetic inhomogeneity and comparison with the Simon–Hoh instability criterion

The kinetic analysis of §§ 4 and 5 necessitates a non-zero gradient of the magnetic field. This gives rise to a resonant contribution to the perturbation of the electron density in (2.14) that, for a weak magnetic gradient, comes from particles of high perpendicular velocity whose number is exponentially small, at the tail of their distribution function. Nevertheless, even if its quantitative effect may be small, the kinetic resonance is present no matter how small the magnetic gradient is, meaning a qualitative difference with the situation where the magnetic field is strictly constant and the resonance does not exist. Therefore, the approach, as the magnetic gradient tends to zero, of the kinetic result of §§ 4 and 5 towards the constant magnetic field result of § 3 (i.e. towards the classic S–H instability) is a non-trivial issue worth studying.

To investigate this, we consider a regime of weak magnetic inhomogeneity characterised by a magnetic gradient scale length $L_B = |\partial \ln B / \partial z|^{-1}$ much larger than the ion sound gyroradius $\rho _S = c_S / \omega _{ci}$ . Thus, we introduce the small expansion parameter

(6.1) \begin{eqnarray} \varepsilon = \frac {|\omega _{*B}|}{kc_S} = \frac {|k_y| \rho _S}{k L_B} \ll 1 . \end{eqnarray}

In addition, we assume the maximal ordering for the other variables,

(6.2) \begin{eqnarray} |\omega _{*N}| \sim |\omega _{*T}| \sim |{\hat \varDelta }| \sim |\omega '| \sim k c_S , \end{eqnarray}

which implies $|\alpha | \sim |\beta | \sim \varepsilon$ and $|\delta | \sim |\zeta | \sim \varepsilon ^{-1}$ . Then, we use the large-argument asymptotic expansion (4.7) to approximate the function $g(\zeta )$ . Keeping the first two significant orders, our kinetic dispersion relation (4.11) reduces to

(6.3) \begin{eqnarray} (\zeta + \delta )^{-2} + \alpha (\zeta ^{-1} - \zeta ^{-2}) - \beta \ \zeta ^{-2} = O(|\alpha \zeta |^{-3}) + O(|\beta \zeta |^{-3}), \end{eqnarray}

or, equivalently,

(6.4) \begin{eqnarray} \alpha (\zeta + \delta )^2 + \zeta - (\alpha + \beta ) (\zeta + \delta )^2 \zeta ^{-1} = O(\varepsilon ) . \end{eqnarray}

This is a cubic equation for $\zeta$ that can be solved perturbatively. As a first approximation, we equate to zero the leading terms of this equation, which are its first two terms of order $\varepsilon ^{-1}$ , to get the leading-order solution

(6.5) \begin{eqnarray} \zeta _0 = \frac {1}{2\alpha } \big[\!-(1+ 2 \alpha \delta ) \pm \sqrt {1+4 \alpha \delta } \big] . \end{eqnarray}

Then, we solve

(6.6) \begin{eqnarray} \alpha (\zeta + \delta )^2 + \zeta - (\alpha + \beta )(\zeta _0 + \delta )^2 \zeta _0^{-1} = 0 , \end{eqnarray}

which simplifies to

(6.7) \begin{eqnarray} \alpha (\zeta + \delta )^2 + \zeta + (\alpha + \beta ) / \alpha = 0 \end{eqnarray}

and yields

(6.8) \begin{eqnarray} \zeta = \frac {1}{2\alpha } \big[\!-(1+ 2 \alpha \delta ) \pm \sqrt {1+4 (\alpha \delta - \alpha - \beta )} \big] . \end{eqnarray}

This gives the condition for instability

(6.9) \begin{eqnarray} \alpha (\delta - 1) - \beta \lt - 1/4 \end{eqnarray}

or, in terms of dimensional parameters,

(6.10) \begin{eqnarray} ({\hat \varDelta } - \omega _{*B}) \ (\omega _{*N} - \omega _{*B}) - \omega _{*B} \ (\omega _{*T} - \omega _{*B}) \lt - k^2 c_S^2/4 . \end{eqnarray}

For the unstable range of these parameters, the growth rate of the instability is

(6.11) \begin{align} {\rm Im} \omega = \frac {k c_S}{|\omega _{*N} - \omega _{*B}|} \bigl [\omega _{*B} \ (\omega _{*T} - \omega _{*B})-({\hat \varDelta } - \omega _{*B}) \ (\omega _{*N} - \omega _{*B}) - k^2 c_S^2/4 \bigr ]^{1/2} . \end{align}

In the limit $\omega _{*B} \rightarrow 0$ , (6.10) becomes the same as (3.9) and the growth rate (6.11) becomes the same as the corresponding growth rate of order $kc_S$ , so our kinetic theory recovers the S–H instability criterion for an homogeneous magnetic field. Moreover, (6.10) and (6.11) provide a kinetically based extension of such a criterion that includes the gradients of the magnetic field and the electron temperature, albeit in a perturbative sense for small $\omega _{*B} / k \rho _S$ . This kinetically based extension should be contrasted with other extensions of the S–H criterion (Frias et al. Reference Frias, Smolyakov, Kaganovich and Raitses2012; Smolyakov et al. Reference Smolyakov, Chapurin, Frias, Koshkarov, Romadanov, Tang, Umansky, Raitses, Kaganovich and Lakhin2017; Ramos et al. Reference Ramos, Bello-Benítez and Ahedo2021; Boeuf & Smolyakov Reference Boeuf and Smolyakov2023; Ripoli et al. Reference Ripoli, Ahedo and Merino2025) derived within the fluid theory framework under various working hypotheses.

The regime of weak magnetic inhomogeneity under consideration has another region of instability in the parametric domain characterised by the alternative orderings

(6.12) \begin{eqnarray} |\omega _{*N}| \sim \varepsilon ^{1/2} kc_S , \qquad |\omega _{*T}| \sim |{\hat \varDelta }| \sim k c_S , \end{eqnarray}

where a frequency solution of the order $|\omega '| \sim \varepsilon ^{1/2} kc_S$ is found. This implies $|\alpha | \sim \varepsilon ^{3/2}, \ |\beta | \sim \varepsilon , \ |\delta | \sim \varepsilon ^{-1}$ and $|\zeta | \sim \varepsilon ^{-1/2}$ , so the large-argument approximation for $g(\zeta )$ can still be used. Bringing these alternative orderings to (6.4), the leading terms are now of order $\varepsilon ^{-1/2}$ and give

(6.13) \begin{eqnarray} \alpha \delta ^2 + \zeta - \beta \delta ^2 \zeta ^{-1} = 0 \,. \end{eqnarray}

This has the solution

(6.14) \begin{align} \zeta = \frac {1}{2} \Bigl (-\alpha \delta ^2 \pm \delta \sqrt {\alpha ^2 \delta ^2 + 4 \beta } \Bigr ), \end{align}

with the condition for instability

(6.15) \begin{eqnarray} \beta \lt - \alpha ^2 \delta ^2 / 4 \lt 0 . \end{eqnarray}

In terms of dimensional parameters, taking into account the assumptions (6.12) and the fact that here we are keeping only the leading significant order, the instability condition is

(6.16) \begin{eqnarray} \omega _{*B} \ \omega _{*T} \ k^2 c_S^2 \lt - \ {\hat \varDelta }^2 \ \omega _{*N}^2 / 4 , \end{eqnarray}

with growth rate in its unstable parameter range

(6.17) \begin{eqnarray} {\rm Im} \omega = \frac {|{\hat \varDelta }|}{2k^2 c_S^2} \bigl (-{\hat \varDelta } ^2 \ \omega _{*N}^2 - 4 \ \omega _{*B} \ \omega _{*T} \ k^2 c_S^2 \bigr )^{1/2} . \end{eqnarray}

This instability disappears in the constant magnetic field limit $\omega _{*B} \rightarrow 0$ but, for $\omega _{*B} \neq 0$ and parameters satisfying (6.16), it exists with a weak growth rate ${\rm Im} \omega \sim |kc_s \omega _{*B}|^{1/2}$ .

Figure 3. Stability diagram in the $(\alpha , \delta )$ plane for two small constant values of $\beta$ . The white region is stable, while a colour scale indicates the normalised instability growth rate in the unstable region. The green, blue and red solid lines are the exact marginal stability boundaries obtained analytically from (4.11). The orange dashed lines are the marginal stability boundaries from (6.9).

All the above-mentioned perturbative results necessitate a qualification because they rely on the large-argument asymptotic expansion (4.7) for $g(\zeta )$ . As was pointed out earlier and is discussed in detail in Appendix B, the asymptotic expansion (4.7) must be modified with the addition of a small imaginary term if $\zeta$ is close to a real negative value, to represent faithfully the branch cut brought about by the kinetic resonance. The marginally stable value of $\zeta$ on the lower branch of the hyperbola that bounds the condition (6.9), i.e. in the $\alpha \gt 0, \ \delta -1 \lt 0$ quadrant of the $(\alpha , \delta )$ plane, is negative. Therefore, unlike the upper branch with $\alpha \lt 0$ and $\delta -1 \gt 0$ where the marginal $\zeta$ is positive, the lower branch of the hyperbola that bounds (6.9) need not represent properly the exact kinetic instability threshold. The same thing happens with the weaker instability in (6.14), (6.15), whose marginal $\zeta$ is positive only on the branches of the instability threshold located in the $\alpha \lt 0$ half-plane. For $\zeta$ close to real and negative, the magnitude of the imaginary correction to the asymptotic representation of $g(\zeta )$ is exponentially small and tends to zero as $\varepsilon = |\omega _{*B}| / (k c_S) \rightarrow 0$ : $\ |{\rm Im} g(\zeta )| = \pi \exp (- |{\rm Re} \zeta |)$ with $|{\rm Re} \zeta | \sim \varepsilon ^{-1}$ or $|{\rm Re} \zeta | \sim \varepsilon ^{-1/2}$ . This causes only an infinitesimal modification of the instability growth rates that approach continuously the corresponding values for a constant magnetic field at $\omega _{*B} = 0$ . However, the locus of marginal stability in parameter space experiences a finite modification and, as $\omega _{*B} \rightarrow 0$ , does not approach continuously the marginal stability boundary for a constant magnetic field given by (6.10) with $\omega _{*B}=0$ , except for a section that approaches its upper hyperbola branch of positive marginal $\zeta$ . This is illustrated by figure 3 that shows the stability diagrams in $(\alpha , \delta )$ planes for two small constant values of $\beta = \pm 0.04$ and with a scaling of the axes appropriate to the regime of small $\alpha$ and large $\delta$ . The green, blue and red solid lines are the exact instability thresholds from the complete dispersion relation (4.11), while the orange dashed lines are the perturbative thresholds given by (6.9). The magnitude of the normalised instability growth rate is indicated with a colour scale in the unstable region. As expected, the perturbative upper branch of the hyperbola (6.9) approximates very well the non-resonant part of the exact instability threshold where a strong instability with growth rate of the order of $kc_S$ arises. However, the lower branch of (6.9) does not reproduce the remainder of the marginal stability boundary that is determined either by the non-resonant onset of a weaker instability of the type (6.14)–(6.17) (observed for $\beta \lt 0$ and $\alpha \lt 0$ ) or by resonant kinetic effects that require taking into account the branch cut of the function $g(\zeta )$ . Nevertheless, this lower branch of (6.9) does represent accurately the sharp transition from low to high growth rates within the region of instability.

7. Correspondence with fluid theory

The classic S–H analysis, as well as other works on the stability of $E\!\times\!B$ plasmas against drift-gradient modes (Frias et al. Reference Frias, Smolyakov, Kaganovich and Raitses2012; Smolyakov et al. Reference Smolyakov, Chapurin, Frias, Koshkarov, Romadanov, Tang, Umansky, Raitses, Kaganovich and Lakhin2017; Ramos et al. Reference Ramos, Bello-Benítez and Ahedo2021; Boeuf & Smolyakov Reference Boeuf and Smolyakov2023; Ripoli et al. Reference Ripoli, Ahedo and Merino2025), have followed the macroscopic fluid approach. It is therefore worthwhile to review explicitly the relationship between the kinetic model adopted in the present work and the corresponding fluid model. The electron system of macroscopic fluid equations that is derived under the present assumptions, namely absence of dissipation, curl-free electric field, lowest order in the small Larmor radius $(k \rho _e \ll 1)$ , and two-dimensional perpendicular geometry with zero parallel gradients, zero magnetic curvature and zero parallel flows, is

(7.1) \begin{eqnarray} \frac {\partial n}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }(n \boldsymbol{u}_e) = 0 , \end{eqnarray}

(7.2) \begin{eqnarray} \boldsymbol{u}_e = \frac {1}{eB} \ \boldsymbol{e}_x \times \left ( e \boldsymbol{\nabla }\phi - \frac {1}{n} \boldsymbol{\nabla }p_e \right ), \end{eqnarray}

(7.3) \begin{eqnarray} \frac {\partial p_e}{\partial t} + \boldsymbol{u}_e \boldsymbol{\cdot }\boldsymbol{\nabla }p_e + 2 p_e \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}_e + \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{q}_e = 0 , \end{eqnarray}

(7.4) \begin{eqnarray} \boldsymbol{q}_e = \frac {1}{eB} \ \boldsymbol{e}_x \times \left (\frac {2 p_e}{n} \boldsymbol{\nabla }p_e - \boldsymbol{\nabla }r_e \right)\!. \end{eqnarray}

Here, the relevant odd moments of the electron distribution function, namely the perpendicular macroscopic fluid velocity $\boldsymbol{u}_e$ and the perpendicular flux of perpendicular heat $\boldsymbol{q}_e$ , are given explicitly by the algebraic (7.2) and (7.4). The relevant even moments are the density

(7.5) \begin{eqnarray} n = \int\, \text{d}^3 \boldsymbol{w} \ f_e , \end{eqnarray}

the perpendicular pressure

(7.6) \begin{eqnarray} p_e = \frac {m_e}{2} \int \,\text{d}^3 \boldsymbol{w} \ w_\perp ^2 \ f_e \end{eqnarray}

and the perpendicular fourth-rank moment

(7.7) \begin{eqnarray} r_e = \frac {m_e^2}{4} \int \,\text{d}^3 \boldsymbol{w} \ w_\perp ^4 \ f_e . \end{eqnarray}

Notice that the parallel pressure does not enter the problem and is completely inconsequential, because the divergence of the pressure tensor is just equal to the (perpendicular) gradient of the perpendicular pressure due to the geometrical assumptions of zero parallel gradients and zero magnetic curvature. The same thing happens with the other two fourth-rank gyrotropic moments. This fluid system provides the dynamical evolution in (7.1) and (7.3) for $n$ and $p_e$ , but does not specify $r_e$ , so it is not closed.

If the magnetic field is constant, the contribution of $p_e$ to $\boldsymbol{\nabla }\boldsymbol{\cdot }(n \boldsymbol{u}_e)$ and the contribution of $r_e$ to $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{q}_e$ vanish. So, in this special case, the fluid system becomes closed and, as shown in § 3, the result derived from the fluid theory coincides with the kinetic result.

For a general inhomogeneous magnetic field, substituting (7.2) and (7.4) in (7.1) and (7.3) and carrying out the algebra, one obtains

(7.8) \begin{eqnarray} \frac {\partial n}{\partial t} + \frac {1}{B} \ \boldsymbol{e}_x \boldsymbol{\cdot }\left[ \boldsymbol{\nabla }\phi \times \boldsymbol{\nabla }n + \boldsymbol{\nabla }\ln B \times \left( n \boldsymbol{\nabla }\phi - \frac {1}{e} \boldsymbol{\nabla }p_e \right) \right] = 0 \end{eqnarray}

and

(7.9) \begin{eqnarray} \frac {\partial p_e}{\partial t} + \frac {1}{B} \ \boldsymbol{e}_x \boldsymbol{\cdot }\left [ \boldsymbol{\nabla }\phi \times \boldsymbol{\nabla }p_e + \boldsymbol{\nabla }\ln B \times \left ( 2 p_e \boldsymbol{\nabla }\phi - \frac {1}{e} \boldsymbol{\nabla }r_e \right ) \right ] = 0 . \end{eqnarray}

Therefore, the linearised continuity equation is

(7.10) \begin{eqnarray} - i \omega n_1 + \frac {i k_y}{B} \left [ \frac {\partial n_0}{\partial z} \phi _1 - \frac {\partial \phi _0}{\partial z} n_1 - \frac {\partial \ln B}{\partial z} \left ( n_0 \phi _1 - \frac {p_{e1}}{e} \right )\right ] = 0 \end{eqnarray}

or

(7.11) \begin{eqnarray} \omega ' n_1 + \omega _{*B} \frac {p_{e1}}{T_{e0}} + (\omega _{*N} - \omega _{*B}) \frac {e n_0 \phi _1 }{T_{e0}} = 0 , \end{eqnarray}

and the linearised pressure equation is

(7.12) \begin{eqnarray} - i \omega p_{e1} + \frac {i k_y}{B} \left [ \frac {\partial p_{e0}}{\partial z} \phi _1 - \frac {\partial \phi _0}{\partial z} p_{e1} - \frac {\partial \ln B}{\partial z} \left ( 2 p_{e0} \phi _1 - \frac {r_{e1}}{e} \right )\right ] = 0 \end{eqnarray}

or

(7.13) \begin{eqnarray} \omega ' p_{e1} + \omega _{*B} \frac {r_{e1}}{T_{e0}} + (\omega _{*N} + \omega _{*T} - 2 \omega _{*B}) e n_0 \phi _1 = 0 . \end{eqnarray}

With the presently derived drift-kinetic solution for the perturbed electron distribution function, we obtained the kinetic expression (2.14) for the perturbation of the density. We can also obtain the following kinetic expressions for the perturbations of the pressure and the fourth-rank moment:

(7.14) \begin{align} p_{e1} = - \ e n_0 \phi _1 \int _0^\infty \,\text{d}s \ \frac {s[\omega _{*N} + \omega _{*T} (s-1) - \omega _{*B} \ s]}{\omega ' + \omega _{*B} \ s} \ \exp (-s), \end{align}

(7.15) \begin{align} r_{e1} = - \ e n_0 T_{e0} \phi _1 \int _0^\infty\, \text{d}s \ \frac {s^2 [\omega _{*N} + \omega _{*T} (s-1) - \omega _{*B} \ s]}{\omega ' + \omega _{*B} \ s} \ \exp (-s) . \end{align}

It is now straightforward to verify that (2.14), (7.14) and (7.15) satisfy identically the macroscopic fluid equations (7.11) and (7.13). So, we verify explicitly that the kinetic description of the present work and the fluid description (7.1)–(7.4) are consistent with each other, but the fluid system must include the contribution of the fourth-rank moment $r_e$ to the perpendicular heat flux. As is well known, the proper specification of $r_e$ cannot be completed with macroscopic arguments alone; hence, the inability of an exclusively macroscopic fluid theory to reproduce all the kinetic results.

If one wants to continue further within the fluid approach, some more or less arbitrary hypothesis has to be adopted to specify $r_e$ and close the system. One such possible closure ansatz is to assume that $r_e$ is a function of the lower even moments $n$ and $p_e$ . If there are no fundamental constants available with dimensions of energy or length, dimensional analysis requires that the functional dependence be $r_e = \lambda p_e^2 / n$ , with $\lambda$ a dimensionless constant. Under this assumption, the linear perturbation of $r_e$ is

(7.16) \begin{eqnarray} r_{e1} = \lambda \big(2 T_{e0} p_{e1} - T_{e0}^2 n_1\big) . \end{eqnarray}

Substituting this in (7.13) and eliminating $p_{e1}$ between (7.11) and (7.13), one arrives at the following relationship between the perturbations of the density and the electric potential:

(7.17) \begin{eqnarray} n_1 = \frac {e n_0 \phi _1}{T_{e0} \ \omega _{*B}} \left[(\omega _{*T} - \omega _{*N}) \ g_F \left(\frac {\omega '}{\omega _{*B}} \right) - (\omega _{*T} - \omega _{*B}) \ h_F \left(\frac {\omega '}{\omega _{*B}} \right) \right]\!, \end{eqnarray}

where

(7.18) \begin{eqnarray} g_F(\zeta ) = \frac {\zeta + 2 \lambda -1}{\zeta ^2 + 2 \lambda \zeta + \lambda } \end{eqnarray}

and

(7.19) \begin{eqnarray} h_F(\zeta ) = \frac {\zeta + 2 \lambda -2}{\zeta ^2 + 2 \lambda \zeta + \lambda } . \end{eqnarray}

We observe that (7.17) has the same form as the kinetic equation (4.1), only with the functions $g_F(\zeta )$ and $h_F(\zeta )$ in (7.18) and (7.19) substituting for the kinetic $g(\zeta )$ in (4.2) and $h(\zeta )$ in (4.3). The similarity could be made stronger if the functions $g_F(\zeta )$ and $h_F(\zeta )$ satisfied the same relationship as their kinetic counterparts, $h_F(\zeta ) = 1 - \zeta \ g_F(\zeta )$ . This is accomplished with the choice of the parameter $\lambda = 2$ . With this choice, the fluid dispersion relation

(7.20) \begin{align} \frac {k^2 c_S^2 \ \omega _{*B}}{(\omega ' + {\hat \varDelta })^2} = \left[(\omega _{*T} - \omega _{*N}) + (\omega _{*T} - \omega _{*B}) \left(\frac {\omega '}{\omega _{*B}} \right) \right] g_F\left(\frac {\omega '}{\omega _{*B}} \right) - (\omega _{*T} - \omega _{*B}), \end{align}

mimics the kinetic one in (4.9), the only difference being that the function

(7.21) \begin{eqnarray} g_F(\zeta ) = \frac {\zeta + 3}{\zeta ^2 + 4 \zeta + 2} \end{eqnarray}

replaces the kinetic $g(\zeta )$ . Moreover, the asymptotic behaviour of this $g_F(\zeta )$ for $|\zeta | \gg 1$ is

(7.22) \begin{eqnarray} g_F(\zeta ) = \frac {1}{\zeta } - \frac {1}{\zeta ^2} + \frac {2}{\zeta ^3} + O \left( \frac {1}{|\zeta |^4} \right)\!, \end{eqnarray}

which matches the first three terms of the corresponding asymptotic expansion (4.7) of the kinetic $g(\zeta )$ . This means that, if the analysis of (6.1)–(6.17) for weak magnetic inhomogeneity is carried out based on this fluid model, one obtains exactly the same results, because the only information about the function $g(\zeta )$ used there was its asymptotic expansion for $|\zeta | \gg 1$ to the third term. So, the fluid model (7.1)–(7.4) with the $r_e = 2 p_e^2 / n$ closure affords a derivation of the temperature-gradient-dependent correction (6.10) to the S–H criterion, strictly within a fluid formalism. This closure corresponds to evaluating the perpendicular fourth rank moment $r_e(n, p_e)$ on a two-temperature bi-Maxwellian distribution function with density $n$ and perpendicular pressure $p_e$ , which yields the following expression for the perpendicular flux of perpendicular heat:

(7.23) \begin{eqnarray} \boldsymbol{q}_e = -\frac {2 p_e}{eB} \ \boldsymbol{e}_x \times \boldsymbol{\nabla }\left (\frac {p_e}{n} \right)\!. \end{eqnarray}

Substituting for $g_F(\omega '/\omega _{*B})$ , the fluid dispersion relation (7.20) can also be written as

(7.24) \begin{align} \frac {k^2 c_S^2}{(\omega ' + {\hat \varDelta })^2} = \frac {\omega '(\omega _{*B}-\omega _{*N}) + \omega _{*B}(\omega _{*T}-3\omega _{*N}+2 \omega _{*B})}{\omega '^2+4 \omega ' \omega _{*B}+2\omega _{*B}^2} . \end{align}

Figure 4. Functions $g_R(\zeta _R) = {\rm Re} g(\zeta _R)$ (solid red), $g_F(\zeta _R)$ (dash-dotted blue) and $g_{A3}(\zeta _R)$ (dashed orange) for real argument $\zeta _R$ . The function $g_R$ is the real part of the kinetic dispersion relation function $g$ in (4.4) and (4.5), $g_F$ is the fluid dispersion relation function (7.18) and $g_{A3}$ is their 3-term asymptotic approximation for large argument (7.22).

One way to gauge how well the closed fluid model (7.1)–(7.3), (7.23) approximates the exact kinetic theory is to compare the graphs of the fluid $g_F(\zeta )$ function (7.21) and the real part of the kinetic $g(\zeta )$ function, $g_R(\zeta ) ={\rm Re} g(\zeta )$ , for a real argument $\zeta = \zeta _R$ ranging from $-\infty$ to $+\infty$ (with the real part of the kinetic $g(\zeta _R)$ defined by continuity when its argument is negative). Such a plot is shown in figure 4 that displays also the function $g_{A3}(\zeta _R)$ equal to the asymptotic series (7.22) truncated at its third term. When $\zeta _R \gt 0$ , we see that $g_F(\zeta _R)$ approximates $g_R(\zeta _R)$ very well for $\zeta _R \gt 1$ and fairly well for $\zeta _R$ between 0 and 1, whereas $g_{A3}(\zeta _R)$ does a good approximation only for larger values of $\zeta _R$ . However, both $g_F(\zeta _R)$ and $g_{A3}(\zeta _R)$ are clearly inadequate when $\zeta _R \lt 0$ (except at the large- $|\zeta _R|$ tail), in addition to being unable to reproduce the discontinuous imaginary part that $g(\zeta )$ has when $\zeta \rightarrow \zeta _R \lt 0$ . If anything, $g_F(\zeta _R)$ appears to be the worse approximation there because of its two poles that are likely to give rise to spurious results. Hence, the closed fluid model (7.1)–(7.3), (7.23) is expected to approximate well the non-resonant kinetic instability thresholds where the normalised marginal frequency $\zeta$ is positive and not too close to $0$ . However, the fluid instability threshold conditions whose normalised marginal frequency is negative are questionable, since a realistic kinetic treatment would yield different results determined by the effect of the phase space resonance felt when $\zeta$ is close to real and negative that the fluid description cannot reproduce.

Figure 5. Instability growth rates from kinetic and fluid dispersion relations for $\omega _{*N} - \omega _{*B} = {\hat \varDelta } = 2kc_S$ and eight different values of $\omega _{*B}/kc_S$ . The red lines represent the solution of the kinetic equation (4.9) and the blue lines are obtained from the fluid equation (7.24).

Figure 6. Doppler-shifted real part of the complex frequencies of the unstable modes in figure 5. The red lines represent the solution of the kinetic equation (4.9) and the blue lines are obtained from the fluid equation (7.24).

A numerical comparison between solutions of the kinetic dispersion relation (4.9) and the fluid dispersion relation (7.24) is presented in figures 5 and 6. Here, all the frequencies have been normalised to $kc_S$ and all the dimensionless ratios used are directly related to the physical quantities of interest. We have considered one choice of fixed values for the gradient of $n_0/B$ and the relative drift between species, given by the fixed parameters $\omega _{*N}-\omega _{*B} = \hat \varDelta = 2kc_S$ . With this choice, the gradient of $n_0/B$ is opposite to the applied electric field and $\alpha \delta =4$ , so the configuration is well stable with regard the Simon–Hoh criterion. The instabilities found are therefore of the ETG kind and we concentrate in these modes and their dependence on the gradients of $T_{e0}/B$ and $B$ . The instability growth rates are shown in figure 5 as functions of $(\omega _{*T} - \omega _{*B})/kc_S = -\cos \theta \ \rho _S \ \partial \ln (T_{e0}/B)/\partial z$ for eight representative values of $\omega _{*B}/kc_S = -\cos \theta \ \rho _S \ \partial \ln B/\partial z$ , where $\theta$ was defined as the wavevector propagation angle with $\cos \theta = \pm 1$ for azimuthal propagation. So, given that we have chosen a positive $(\omega _N - \omega _{*B})/kc_S$ , positive values of $(\omega _{*T} - \omega _{*B})/kc_S$ and $\omega _{*B}/kc_S$ correspond respectively to the gradients of $T_{e0}/B$ and $B$ pointing in the direction of the gradient of $n_0/B$ . The real part of the unstable mode frequencies, ${\rm Re} \omega '$ Doppler-shifted to the frame of the electric drift, is similarly given in figure 6. These numerical results show ETG-like instabilities driven by the combination of the gradients of $T_{e0}/B$ and $B$ , in accordance with the discussion of the previous sections. The observed ${\rm Re} \omega '/kc_S$ are always negative for the kinetic modes, therefore, $\zeta _R = {\rm Re} \omega '/\omega _{*B}$ has the sign opposite to that of $\omega _{*B}/kc_S$ , and the modes have non-resonant character for $\omega _{*B}/kc_S \lt 0$ and resonant character for $\omega _{*B}/kc_S \gt 0$ . This is an expected feature since the sign of $\zeta _R$ tends in most cases to be opposite to the sign of $\delta$ , which here has the same sign as $\omega _{*B}/kc_S$ because we have chosen $\hat \varDelta /kc_S \gt 0$ . The non-resonant instabilities found for $\omega _{*B}/kc_S \lt 0$ (i.e. when the gradient of $B$ is opposite to the gradient of $n_0/B$ ) are the robust temperature-gradient-driven modes in the third quadrants of the diagrams of figure 2 that necessitate the gradient of $T_{e0}/B$ to be in the direction of the gradient of $n_0/B$ and above a certain threshold in magnitude. The resonant instabilities found for $\omega _{*B}/kc_S \gt 0$ (i.e. when the gradient of $B$ is in the direction of the gradient of $n_0/B$ ) are unstable modes in the first quadrants of the diagrams of figure 2. These first-quadrant instabilities can exist when the gradient of $T_{e0}/B$ has either sign, but they are robust again when it points in the direction of the gradient of $n_0/B$ . The fluid dispersion relation provides a fairly good approximation to the kinetic non-resonant results with $\omega _{*B}/kc_S \lt 0$ , but performs poorly when $\omega _{*B}/kc_S \gt 0$ and the kinetic modes have a resonant character.

The closed fluid model under discussion, defined by our (7.1)–(7.3), (7.23), is akin to the one presented by Frias et al. (Reference Frias, Smolyakov, Kaganovich and Raitses2012). One difference is that Frias et al. (Reference Frias, Smolyakov, Kaganovich and Raitses2012) use the evolution equation for a scalar mean pressure (which involves the perpendicular flux of total heat), instead of our perpendicular pressure equation. This implies different numerical coefficients from those in our (7.3) and (7.23). Such a model assumes that the perturbation of the pressure remains isotropic, which is not consistent with collisionless kinetic theory. The other difference is that Frias et al. (Reference Frias, Smolyakov, Kaganovich and Raitses2012) consider a magnetic field configuration without curl, thus necessitating magnetic curvature. If one stays within the framework of zeroth-Larmor-radius-order fluid theory with isotropic pressure, this can be accommodated in a manner that simply amounts to modify the definition of the magnetic-gradient drift frequency by a factor $2$ . The resulting dispersion relation (written using our notation and sign conventions) is

(7.25) \begin{eqnarray} \frac {k^2 c_S^2}{(\omega ' + {\hat \varDelta })^2} = \frac {\omega '[(2\omega _{*B})-\omega _{*N}] + (2\omega _{*B})[\omega _{*T}-{{7}/{3}}\omega _{*N}+{{5}/{3}} (2\omega _{*B})]}{\omega '^2+{{10}/{3}} \omega ' (2\omega _{*B})+{{5}/{3}} (2\omega _{*B})^2} , \end{eqnarray}

which differs from (7.24) only in its numerical coefficients. Applying this isotropic pressure model to the same straight magnetic field configuration that the present work considers, means omitting the factor $2$ that multiplies $\omega _{*B}$ in (7.25). In this case, the numerical coefficients of (7.25) become close to those of (7.24) and the two dispersion relations yield very similar solutions. However, unlike (7.24), (7.25) cannot be expressed in the kinetically related form (7.20) with some other $g_F(\omega ' / \omega _{*B})$ function, whether the factor $2$ in $\omega _{*B}$ is included or not.