2. Anisotropic pressure tensor

We wish to generalise the analysis of Hazeltine et al. (Reference Hazeltine, Mahajan and Zheng2026) to include the effects of particle trapping, because this is the main confinement mechanism both in a weakly rotating mirror device (Fitzpatrick Reference Fitzpatrick2023) and very close to the axis of a strongly rotating device (Schwartz et al. Reference Schwartz, Abel, Hassam and Romero-Talamás2024). Within the context of fluid theory, we can achieve our goal by adopting anisotropic pressure tensors for the electrons and the ions (Chew, Goldberger & Low Reference Chew, Goldberger and Low1956; Togo et al. Reference Togo, Takizuka, Reiser, Hoshino, Ibano, Li, Ogawa, Sakamoto, Ezumi and Nakashima2018; Zhao et al. Reference Zhao, Rognlien, Jarvinen and Joseph2021). Let

(2.1) \begin{align} \underline {\underline {P}}_{\,s} = p_{\parallel \,s}\,\boldsymbol{b}\,\boldsymbol{b} + p_{\perp \,s}\left (\boldsymbol{1} - \boldsymbol{ b}\,\boldsymbol{b}\right ) \end{align}

be the gyrotropic pressure tensor for species $s$ (the two species being electrons, $s= e$ , and ions, $s= i$ ). Here, $\boldsymbol{b} = \boldsymbol{B}/B$ , where $\boldsymbol{B}$ is the magnetic field strength, $\boldsymbol{1}$ is the identity tensor and $p_{\parallel \,s}$ and $p_{\perp \,s}$ are the species- $s$ parallel and perpendicular (relative to the magnetic field) pressures. In other words,

(2.2) \begin{align} p_{\parallel \,s}(\boldsymbol{r}) &= m_s\int (v_{\parallel \,s}-V_{\parallel \,s})^2\,f_s(\boldsymbol{r},\boldsymbol{v}_s)\,d^3\boldsymbol{v}_s,\\[-7pt]\nonumber \end{align}

(2.3) \begin{align} p_{\perp \,s}(\boldsymbol{r}) &= \frac {m_s}{2}\int (v_{\perp \,s}-V_{\perp \,s})^2\,f_s(\boldsymbol{r},\boldsymbol{v}_s)\,d^3\boldsymbol{v}_s, \end{align}

where $m_s$ is the species- $s$ mass, $f_s(\boldsymbol{r},\boldsymbol{v}_s)$ the (ensemble averaged) species- $s$ distribution function, where $\boldsymbol{r}$ is the position vector, and $\boldsymbol{v}_s$ the species- $s$ fluid velocity, $v_{\parallel \,s}= \boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{v}_s$ , $\boldsymbol{v}_{\perp \,s} = \boldsymbol{ v}_s - v_{\parallel \,s}\,\boldsymbol{b}$ , et cetera (Fitzpatrick Reference Fitzpatrick2023).

It is easily demonstrated that (Schwartz et al. Reference Schwartz, Abel, Hassam and Romero-Talamás2024)

(2.4) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\underline {\underline {P}}_{\,s} = \boldsymbol{\nabla }p_{\parallel \,s}+ (p_{\perp \,s}-p_{\parallel \,s})\boldsymbol{\nabla} _\parallel \ln B + \boldsymbol{\nabla} _\perp (p_{\perp \,s} - p_{\parallel \,s}) - (p_{\perp \,s}-p_{\parallel \,s})\,\boldsymbol{\kappa }_\perp , \end{align}

where $\boldsymbol{\kappa }_\perp = (\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{b}$ is the magnetic field-line curvature vector (Freidberg Reference Freidberg2014), $\boldsymbol{\nabla} _\parallel \equiv \boldsymbol{b}\,\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\nabla}$ , and $\boldsymbol{\nabla} _\perp \equiv \boldsymbol{\nabla }-\boldsymbol{\nabla} _\parallel$ . Note that $\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\kappa }_\perp =0$ .

It is helpful to introduce the concepts of species- $s$ parallel and perpendicular temperatures, via the definitions $p_{\parallel \,s} = n_s\,T_{\parallel \,s}$ and $p_{\perp \,s} = n_s\,T_{\perp \,s}$ , where $n_s$ is the species- $s$ number density. In a completely collisionless, strongly magnetised, plasma with negligible heat fluxes, we expect $T_{\perp \,s}\propto B$ and $T_{\parallel \,s}\propto n_s^{\,2}/B^{2}$ on a particular magnetic field line (Chew et al. Reference Chew, Goldberger and Low1956; Kulsrud Reference Kulsrud1983). The first scaling follows directly from the conservation of the magnetic moments of charged particles (Northrop Reference Northrop1963; Fitzpatrick Reference Fitzpatrick2023), and is fairly robust, whereas the second depends on the evolution of the parallel dynamics, and is much easier to violate because it is affected by processes that act along magnetic field lines. Generally speaking, we expect $\langle T_{\perp \,s}\rangle \gt \langle T_{\parallel \,s}\rangle$ on a given magnetic field line in a mirror device, where $\langle \boldsymbol{\cdots}\kern-1pt \rangle$ denotes an average along the field line (Post Reference Post1987; Ryutov Reference Ryutov1990). One obvious cause of this asymmetry is the conservation of the magnetic moment, which converts parallel kinetic energy into perpendicular kinetic energy as a charged particle moves along a magnetic field line into a region of greater magnetic field strength. Thus, if we imagine that $T_{\perp \,s}= T_{\parallel \,s}$ on the mid-plane of the device, where the magnetic field strength is smallest, then the conservation of magnetic moment ensures that $\langle T_{\perp \,s}\rangle \gt \langle T_{\parallel \,s}\rangle$ on a given magnetic field line. Another cause of the asymmetry is the fact that the loss cone in velocity space is almost empty (in a near-collisionless device), which ensures that $T_{\perp \,s}\gt T_{\parallel \,s}$ even on the mid-plane. Generally speaking, we expect the temperature anisotropy to be less marked for electrons than for ions, because the collisional relaxation of the anisotropy takes place much more rapidly for the former species (Fitzpatrick Reference Fitzpatrick2023). In practice, the ion temperature asymmetry is often also driven by fuelling via neutral beam injection directed perpendicular to magnetic field lines, which preferentially injects ions with larger perpendicular than parallel kinetic energies into the plasma (Endrizzi et al. Reference Endrizzi2023). There is no analogous external driving mechanism for the electron temperature anisotropy.

In this paper, we shall adopt a somewhat simplistic model in which $T_{\parallel \,s}$ and $T_{\perp \,s}$ represent the average values of the species- $s$ parallel and perpendicular temperatures, respectively, on a given magnetic field line. Thus, by definition, $T_{\parallel \,s}$ and $T_{\perp \,s}$ are constant on a given field line. However, in accordance with the previous discussion, we expect $T_{\perp \,s} \gt T_{\parallel \,s}$ . Moreover, we shall allow $T_{\parallel \,s}$ and $T_{\perp \,s}$ to vary from field line to field line. It follows that

(2.5) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\underline {\underline {P}}_{\,s} &= n_s\,T_{\parallel \,s}\left [\boldsymbol{\nabla} _\parallel \ln n_s+ \gamma _s\,\boldsymbol{\nabla} _\parallel \ln B +(1+\gamma _s)\,\boldsymbol{\nabla} _\perp \ln (n_s\,T_{\parallel \,s}) +\boldsymbol{\nabla} _\perp \gamma _s -\gamma _s\,\boldsymbol{\kappa }_\perp \right ], \end{align}

where

(2.6) \begin{align} \gamma _s &= \frac {T_{\perp \,s} - T_{\parallel \,s}}{T_{\parallel \,s}} \end{align}

is a positive constant on a given magnetic field line. Note that the only two force densities that act parallel to magnetic field lines on the right-hand side of (2.5) (i.e. the first two terms within the square parenthesis) are, respectively, associated with the parallel density gradient and the fluid average of the single-particle mirror force, $-\mu _s\,\boldsymbol{\nabla }B$ , where $\mu _s= m_s v_{\perp s}^2 /2 B$ is the species- $s$ magnetic moment. Thus, if $\gamma _s$ is positive then the mirror force can maintain a parallel density gradient on a given magnetic field line in which the density is reduced in regions of strong magnetic field strength. Of course, this is the whole basis of conventional mirror confinement.

3. Fundamental equations

Within the context of ideal two-fluid theory, the complete set of equations that governs the equilibrium of a mirror device are (Fitzpatrick Reference Fitzpatrick2023)

(3.1) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{E} &= \frac {e\,(n_i-n_e)}{\epsilon _0},\\[-7pt]\nonumber \end{align}

(3.2) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{B} &= 0,\\[-7pt]\nonumber \end{align}

(3.3) \begin{align} \boldsymbol{\nabla }\times \boldsymbol{E} &= \boldsymbol{0},\\[-7pt]\nonumber \end{align}

(3.4) \begin{align} \boldsymbol{\nabla }\times \boldsymbol{B} &= \mu _0\,e\,(n_i\,\boldsymbol{V}_i-n_e\,\boldsymbol{V}_e),\\[-7pt]\nonumber \end{align}

and

(3.5) \begin{align} \boldsymbol{0 } &=\boldsymbol{V}_e \boldsymbol{\cdot }\boldsymbol{\nabla }n_e+ n_e\,\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{V}_e,\\[-7pt]\nonumber \end{align}

(3.6) \begin{align} \boldsymbol{0}&= e\,n_e\,(\boldsymbol{E}+\boldsymbol{V}_e\times \boldsymbol{B})+\boldsymbol{\nabla }\boldsymbol{\cdot }\underline {\underline {P}}_{\,e} , \end{align}

and

(3.7) \begin{align} \boldsymbol{0 } &=\boldsymbol{V}_i \boldsymbol{\cdot }\boldsymbol{\nabla }n_i+ n_i\,\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{V}_i,\\[-7pt]\nonumber \end{align}

(3.8) \begin{align} \boldsymbol{0} &= m_i\,n_i\,(\boldsymbol{V}_i\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{V}_i - e\,n_i\,(\boldsymbol{E}+ \boldsymbol{V}_i\times \boldsymbol{B}) +\boldsymbol{\nabla }\boldsymbol{\cdot }\underline {\underline {P}}_{\,i}. \end{align}

Here, $\boldsymbol{E}$ is the electric field strength, $e$ the magnitude of the electron charge, $\epsilon_0$ the electric permittivity of free space, and $\mu_0$ the magnetic permeability of free space.

Equations (3.1)–(3.4) are the steady-state Maxwell equations. Equation (3.5) is the steady-stage electron fluid continuity equation. Equation (3.6) is the steady-state, ideal equation of motion of the electron fluid, neglecting electron inertia, but incorporating an anisotropic pressure tensor. Equation (3.7) is the ion fluid continuity equation. Finally, (3.8) is the steady-state, ideal equation of motion of the ion fluid, including ion inertia, and incorporating an anisotropic pressure tensor. Note that the ions are assumed to be singly charged. The fact that we are assuming that $T_{\parallel \,s}$ and $T_{\perp \,s}$ are specified on a given magnetic field line negates the need for electron and ion energy conservation equations.

4. Normalisation

Let $L$ be a typical equilibrium variation length scale (which is assumed to be similar to the physical dimensions of the mirror), $B_0$ a typical magnetic field strength, $n_c$ a typical number density and $T_c$ a typical temperature. The mean ion thermal velocity is $v_i=(T_c/m_i)^{1/2}$ . The mean ion gyro-frequency is ${\varOmega }_{g\,i}=e\,B_0/m_i$ . The ion gyro-radius is $\rho _i= v_i/{\varOmega }_{g\,i}$ . Let $\boldsymbol{\nabla }= L^{-1}\,\hat {\boldsymbol{\nabla }}$ , $\boldsymbol{\nabla} _\parallel = L^{-1}\,\hat {\boldsymbol{\nabla} }_\parallel$ , $n_s=n_c\,\hat {n}_s$ , $T_{\perp \,s} = T_c\,\hat {T}_{\perp \,s}$ , $T_{\parallel \,s} = T_c\,\hat {T}_{\parallel \,s}$ , $\boldsymbol{V}_s= v_i\,\hat {\boldsymbol{V}}_s$ , $\boldsymbol{B}= B_0\,\hat {\boldsymbol{B}}$ and $\boldsymbol{E} = B_0\,v_i\,\hat {\boldsymbol{E}}$ . These orderings are consistent with an axisymmetric mirror that is rotating close to the ion thermal speed. Equations (3.1)–(3.8) become

(4.1) \begin{align} \left (\frac {\lambda _D}{\rho _i}\right )^2 \hat {\rho }_i\,\hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{E}}& = \hat {n}_i-\hat {n}_e,\\[-10pt]\nonumber \end{align}

(4.2) \begin{align} \hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{B}} &= 0,\\[-10pt]\nonumber \end{align}

(4.3) \begin{align} \hat {\boldsymbol{\nabla }}\times \hat {\boldsymbol{E}} &= \boldsymbol{0},\\[-10pt]\nonumber \end{align}

(4.4) \begin{align} \hat {\rho }_i\,\hat {\boldsymbol{\nabla }}\times \hat {\boldsymbol{B}} &= \beta \,(\hat {n}_i\,\hat {\boldsymbol{V}}_i-\hat { n}_e\,\hat {\boldsymbol{V}}_e), \end{align}

and

(4.5) \begin{align} 0 &= \hat {\boldsymbol{V}}_e\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\ln \hat {n}_e + \hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{V}}_e,\\[-10pt]\nonumber \end{align}

(4.6) \begin{align} \boldsymbol{0} &=\hat {\boldsymbol{E}} +\hat {\boldsymbol{V}}_e\times \hat {\boldsymbol{B}}\nonumber \\[3pt] & \quad +\hat {\rho }_i\,\hat {T}_{\parallel \,e}\big[\hat {\boldsymbol{\nabla} }_\parallel \ln \hat {n}_e+\gamma _e\,\hat {\boldsymbol{\nabla} }_\parallel \ln \hat {B} +(1+\gamma _e)\,\hat {\boldsymbol{\nabla} }_\perp \ln\! \big(\hat {n}_e\,\hat {T}_{\parallel \,e}\big) \nonumber \\ & \quad +\hat {\boldsymbol{\nabla} }_\perp \gamma _e-\gamma _e\,\boldsymbol{\kappa }_\perp \big], \end{align}

and

(4.7) \begin{align} 0 &= \hat {\boldsymbol{V}}_i\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\ln \hat {n}_i + \hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{V}}_i,\\[-10pt]\nonumber \end{align}

(4.8) \begin{align} \boldsymbol{0}&= -\hat {\boldsymbol{E}} - \hat {\boldsymbol{V}}_i\times \hat {\boldsymbol{B}}+\hat {\rho }_i\,(\hat {\boldsymbol{V}}_i\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }})\hat {\boldsymbol{V}}_i \nonumber \\ & \quad +\hat {\rho }_i\,\hat {T}_{\parallel \,i}\big[\hat {\boldsymbol{\nabla} }_\parallel \ln \hat {n}_i+\gamma _i\,\hat {\boldsymbol{\nabla} }_\parallel \ln \hat {B} +(1+\gamma _i)\,\hat {\boldsymbol{\nabla} }_\perp \ln\! \big(\hat {n}_i\,\hat {T}_{\parallel \,i}\big) \nonumber \\ & \quad +\hat {\boldsymbol{\nabla} }_\perp \gamma _i-\gamma _i\,\boldsymbol{\kappa }_\perp \big], \end{align}

where

(4.9) \begin{align} \lambda _D &= \left (\frac {\epsilon _0\,T_c}{n_c\,e^2}\right )^{1/2},\\[-10pt]\nonumber \end{align}

(4.10) \begin{align} \beta &= \frac {\mu _0\,n_c\,T_c}{B_0^{\,2}}, \end{align}

and $\hat {\rho }_i=\rho _i/L$ . Here, $\lambda _D$ is the Debye length, and $\beta$ is the ratio of the plasma thermal energy density to the magnetic energy density (Fitzpatrick Reference Fitzpatrick2023).

5. Electric and magnetic fields

Let $r$ , $\theta$ , $z$ be cylindrical coordinates. In an axisymmetric magnetic mirror

(5.1) \begin{align} \frac {\partial }{\partial \theta } \equiv 0 \end{align}

and $\hat {B}_\theta = 0$ . We can satisfy (4.2) by writing (Hazeltine et al. Reference Hazeltine, Mahajan and Zheng2026)

(5.2) \begin{align} \hat {\boldsymbol{B}} = \hat {\boldsymbol{\nabla }}\alpha \times \hat {\boldsymbol{\nabla }}\theta , \end{align}

where $\alpha (r,z)$ is a dimensionless magnetic field-line label. Thus (Hazeltine et al. Reference Hazeltine, Mahajan and Zheng2026),

(5.3) \begin{align} \hat {\boldsymbol{\nabla }}\times \hat {\boldsymbol{B}} = -(\hat {\Delta }^\ast \alpha )\,\hat {\boldsymbol{\nabla }}\theta = -\frac {\hat {\Delta }^\ast \alpha }{\hat {r}}\,\boldsymbol{e}_\theta , \end{align}

where

(5.4) \begin{align} \hat {\Delta }^\ast \equiv \frac {\partial ^2}{\partial \hat {r}^{\,2}}-\frac {1}{\hat {r}}\,\frac {\partial }{\partial \hat {r}}+ \frac {\partial ^2}{\partial \hat {z}^{\,2}}, \end{align}

and $\hat {r}=r/L$ , $\hat {z}=z/L$ . In an axisymmetric equilibrium, our previous assumptions imply that $\hat {T}_{\parallel \,s} =\hat {T}_{\parallel \,s}(\alpha )$ , $\hat {T}_{\perp \,s} =\hat {T}_{\perp \,s}(\alpha )$ and $\gamma _s=\gamma _s(\alpha )$ . Note that we could, alternatively, have assumed that $\hat {T}_{\parallel \,s} =\hat {T}_{\parallel \,s}(\alpha ,B)$ , $\hat {T}_{\perp \,s} =\hat {T}_{\perp \,s}(\alpha ,B )$ and $\gamma _s=\gamma _s(\alpha ,B )$ . Although this assumption would be more realistic, it would complicate the analysis, without significantly modifying any of the main conclusions of the paper.

We can satisfy (4.3) by writing

(5.5) \begin{align} \hat {\boldsymbol{E}} = -\hat {\boldsymbol{\nabla }}\hat {\phi }, \end{align}

where $\phi = v_i\,B_0\,L\,\hat {\phi }$ is the electrostatic potential.

6. Ordering

Our fundamental ordering scheme is

(6.1) \begin{align} \lambda _D \ll \rho _i \ll L. \end{align}

In other words, the Debye length is much less than the ion gyro-radius, which, in turn, is much less than the machine dimensions. It follows that

(6.2) \begin{align} \frac {\lambda _D}{\rho _i}& \ll 1,\\[-7pt]\nonumber \end{align}

(6.3) \begin{align} \hat {\rho }_i &\ll 1. \end{align}

Let $\beta$ be order unity. Thus, all quantities appearing in (4.1)–(4.8) have leading-order contributions that are of order unity, except for $\hat {\rho }_i$ and $\lambda _D/\rho _i$ , which are both much less than unity.

Up to first order in small quantities, (4.1) gives the quasi-neutrality constraint

(6.4) \begin{align} \hat {n}_i= \hat {n}_e. \end{align}

Let

(6.5) \begin{align} \hat {\boldsymbol{V}}_i &= \hat {\boldsymbol{V}}_i^{(0)} + \hat {\rho }_i\,\hat {\boldsymbol{V}}_i^{(1)},\\[-7pt]\nonumber \end{align}

(6.6) \begin{align} \hat {\boldsymbol{V}}_e&= \hat {\boldsymbol{V}}_e^{(0)} + \hat {\rho }_i\,\hat {\boldsymbol{V}}_e^{(1)},\\[-7pt]\nonumber \end{align}

(6.7) \begin{align} \hat {\phi } &= \hat {\phi }^{(0)}+\hat {\rho }_i\,\hat {\phi }^{(1)}, \end{align}

where all superscripted quantities are order unity.

7. Lowest-order flow

To lowest order in $\hat {\rho }_i$ , (4.6), (4.8), (5.5) and (6.5)–(6.7) give

(7.1) \begin{align} \boldsymbol{0}&= -\hat {\boldsymbol{\nabla }}\hat {\phi }^{(0)}+\hat {\boldsymbol{V}}_e^{(0)}\times \hat {\boldsymbol{B}},\\[-10pt]\nonumber \end{align}

(7.2) \begin{align} \boldsymbol{0}&= -\hat {\boldsymbol{\nabla }}\hat {\phi }^{(0)}+\hat {\boldsymbol{V}}_i^{(0)}\times \hat {\boldsymbol{B}}. \end{align}

Thus, we deduce that

(7.3) \begin{align} \hat {\boldsymbol{B}}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\hat {\phi }^{(0)} = 0, \end{align}

which implies that

(7.4) \begin{align} \hat {\phi }^{(0)} = {\varPhi }(\alpha ). \end{align}

Hence,

(7.5) \begin{align} \hat {\boldsymbol{V}}_i^{(0)} = \hat {\boldsymbol{V}}_e^{(0)} = \frac {{\varPhi }'\,\hat {\boldsymbol{\nabla }}\theta }{|\hat {\boldsymbol{\nabla }}\theta |^2}= \hat {r}\,{\varPhi }'\,\boldsymbol{e}_\theta , \end{align}

where ${\varPhi }'\equiv {\rm d}{\varPhi }/{\rm d}\alpha$ , use has been made of the fact that $|\hat {\boldsymbol{\nabla }}\theta |=1/\hat {r}$ , and the zeroth-order parallel (to the magnetic field) components of $\hat {\boldsymbol{V}}_i$ and $\hat {\boldsymbol{V}}_e$ are assumed to be zero. It follows that

(7.6) \begin{align} \hat {\varOmega }_\theta \equiv \hat {\boldsymbol{V}}_i^{(0)}\boldsymbol{\cdot }\boldsymbol{e}_\theta /\hat {r} = \hat {\boldsymbol{V}}_e^{(0)}\boldsymbol{\cdot }\boldsymbol{e}_\theta /\hat {r} ={\varPhi }', \end{align}

where ${\varOmega }_\theta = \hat {\varOmega }_\theta \,v_i/L$ is the lowest-order plasma angular velocity. Thus, to lowest order, the electrostatic potential is constant along magnetic field lines, and both charge species rotate with a common angular velocity that is also constant along magnetic field lines (Hinton & Wong Reference Hinton and Wong1985). In other words, $\hat {\varOmega }_\theta =\hat {\varOmega }_\theta (\alpha )={\rm d}{\varPhi }/{\rm d}\alpha$ , which implies that the flow is iso-rotational: i.e. $\hat {\boldsymbol{B}}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}{\varOmega }_\theta =0$ (Ferraro Reference Ferraro1937; Lehnert Reference Lehnert1971; Hinton & Wong Reference Hinton and Wong1985). It also follows that

(7.7) \begin{align} \big(\hat {\boldsymbol{V}}_i^{(0)}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\big)\hat {\boldsymbol{V}}_i^{(0)} = - \hat {r}\,\hat {\varOmega }_\theta ^{\,2}\,\boldsymbol{ e}_r. \end{align}

Note that

(7.8) \begin{align} \hat {\boldsymbol{V}}_e^{(0)}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\ln \hat {n}_e=\hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{V}}_e^{(0)}= \hat {\boldsymbol{V}}_i^{(0)}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\ln \hat {n}_i=\hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{V}}_i^{(0)}=0, \end{align}

which ensures that the zeroth-order components of the electron and ion continuity equations, (4.5) and (4.7), respectively, are automatically satisfied.

8. Parallel force balance

Now, from (5.2),

(8.1) \begin{align} \hat {\boldsymbol{B}}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }} = {\mathcal{J}}^{-1}\,\frac {\partial }{\partial s}, \end{align}

where $s$ is an element of normalised arc length along a magnetic field line, and ${\mathcal{J}} = (\hat {\boldsymbol{\nabla }}\alpha \times \hat {\boldsymbol{\nabla }}\theta \boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}{s})^{-1}$ is the Jacobian of the $\alpha$ , $\theta$ , $s$ coordinate system. Here, $\partial /\partial s$ is taken at constant $\alpha$ and $\theta$ .

If we take the dot product of the first-order (in $\hat {\rho }_i$ ) version of (4.6) with $\hat {\boldsymbol{B}}$ , then we get

(8.2) \begin{align} 0=\frac {\partial }{\partial s}\big[{-}\hat {\phi }^{(1)}+\hat {T}_{\parallel \,e}\,(\ln \hat {n}_e+\xi _e)\big], \end{align}

where

(8.3) \begin{align} \xi _s = \gamma _s\,\ln \hat {B}. \end{align}

It follows that (Catto, Bernstein & Tessarotto Reference Catto, Bernstein and Tessarotto1987)

(8.4) \begin{align} \hat {T}_{\parallel \,e}\,\left (\ln \hat {n}_e+\xi _e\right ) = \hat {\phi }^{(1)}+ f(\alpha ), \end{align}

where $f(\alpha )$ is arbitrary. Likewise, if we take the dot product of the first-order version of (4.8) with $\hat {\boldsymbol{B}}$ , then, making use of (7.7), we get

(8.5) \begin{align} 0=-\frac {\hat {r}\,\hat {\varOmega }_\theta ^{\,2}}{|\hat {\boldsymbol{\nabla }}\hat {r}|}\,\frac {\partial \hat {r}}{\partial s} +\frac {\partial }{\partial s}\big[\hat {\phi }^{(1)} +\hat {T}_{\parallel \,i}\,(\ln \hat {n}_i +\xi _i)\big], \end{align}

or

(8.6) \begin{align} 0=\frac {\partial }{\partial s}\!\left [-\frac {1}{2}\,\hat {r}^{\,2}\,\hat {\varOmega }_\theta ^{\,2} +\hat {\phi }^{(1)}+ \hat {T}_{\parallel \,i}\,(\ln \hat {n}_i+\xi _i)\right ], \end{align}

where use has been made of the facts that $\hat {\varOmega }_\theta =\hat {\varOmega }_\theta (\alpha )$ and $|\hat {\boldsymbol{\nabla }}\hat {r}|=1$ . It follows that (Catto et al. Reference Catto, Bernstein and Tessarotto1987)

(8.7) \begin{align} \hat {T}_{\parallel \,i}\,(\ln \hat {n}_i+\xi _i) = \frac {1}{2}\,\hat {r}^{\,2}\,\hat {\varOmega }_\theta ^{\,2}-\hat {\phi }^{(1)}+ g(\alpha ), \end{align}

where $g(\alpha )$ is arbitrary.

Let us define the field-line average operator (Hazeltine et al. Reference Hazeltine, Mahajan and Zheng2026)

(8.8) \begin{align} \langle F\rangle (\alpha ) \equiv \left .\int ^s F(\alpha ,s)\,{\rm d}s\right /\int ^s {\rm d}s, \end{align}

where the integral is taken along the whole magnetic field line whose label is $\alpha$ (i.e. from end plate to end plate), and let

(8.9) \begin{align} \tilde {F}\equiv F-\langle F\rangle . \end{align}

Without loss of generality, we can set

(8.10) \begin{align} \langle \hat {\phi }^{(1)}\rangle =0, \end{align}

which allows us to determine the two functions $f(\alpha )$ and $g(\alpha )$ .

The quasi-neutrality constraint, (6.4), can be combined with (8.4), (8.7) and (8.10) to give (Hinton & Wong Reference Hinton and Wong1985; Schwartz et al. Reference Schwartz, Abel, Hassam and Romero-Talamás2024)

(8.11) \begin{align} \hat {\phi }^{(1)}(\alpha ,s) = \hat {T}_{\parallel \,e}\,(\hat {T}_{\parallel \,e}+ \hat {T}_{\parallel \,i})^{\,-1}\left [\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}+\hat {T}_{\parallel \,i}\,(\tilde {\xi }_e-\tilde {\xi }_i)\right ], \end{align}

where (Hinton & Wong Reference Hinton and Wong1985)

(8.12) \begin{align} \hat {n}_i(\alpha ,s) &= \hat {n}_e(\alpha ,s)= \hat {n}_0(\alpha )\exp \left [(\hat {T}_{\parallel \,e}+ \hat {T}_{\parallel \,i})^{\,-1}\left (\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}-\hat {T}_{\parallel \,e}\,\tilde {\xi }_e-\hat {T}_{\parallel \,i}\,\tilde {\xi }_i\right )\right ],\\[-10pt]\nonumber \end{align}

(8.13) \begin{align} f(\alpha ) &=\hat {T}_{\parallel \,e}\left (\ln \hat {n}_0+\langle \xi _e\rangle \right ),\\[-7pt]\nonumber \end{align}

(8.14) \begin{align} g(\alpha )&= \hat {T}_{\parallel \,i}\,(\ln \hat {n}_0 + \langle \xi _i\rangle )- \frac {1}{2}\,\langle \hat {r}^{\,2}\rangle \,\hat {\varOmega }_\theta ^{\,2}. \end{align}

It follows that (Hinton & Wong Reference Hinton and Wong1985; Catto et al. Reference Catto, Bernstein and Tessarotto1987)

(8.15) \begin{align} \hat {n}_e(\alpha ,s) &= \hat {n}_0(\alpha )\,\exp \big(\hat {T}_{\parallel \,e}^{\,-1}\,\hat {\phi }^{(1)} - \tilde {\xi }_e\big),\\[-7pt]\nonumber \end{align}

(8.16) \begin{align} \hat {n}_i(\alpha ,s) &= \hat {n}_0(\alpha )\,\exp \left [\hat {T}_{\parallel \,i}^{\,-1}\left (\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}-\tilde {\phi }^{(1)}\right ) -\tilde {\xi }_i\right ]. \end{align}

According to (8.11), a non-uniform parallel electric field develops on a given magnetic field line. This field ensures that the previous two Boltzmann-like expressions yield $\hat {n}_e(\alpha ,s)=\hat {n}_i(\alpha ,s)$ , as is required by the quasi-neutrality constraint (6.4).

Equation (8.12) can be rewritten in the form

(8.17) \begin{align} & \hat {n}_i(\alpha ,s) = \hat {n}_e(\alpha ,s)\nonumber \\[3pt] & \quad = \hat {n}_0(\alpha )\exp \left \{-(\hat {T}_{\parallel \,e}+\hat {T}_{\parallel \,i})^{\,-1}\,\left [({\Delta }\hat {T}_{e}+{\Delta }\hat {T}_i)(\ln \hat {B}-\langle \ln \hat {B}\rangle )+ \frac {1}{2}\,\hat {\varOmega }_\theta ^{\,2}(\langle \hat {r}^{\,2}\rangle -\hat {r}^{\,2}) \right ]\right \}\!,\nonumber\\[5pt] \end{align}

where ${\Delta }\hat {T}_s =\hat {T}_{\perp \,s} - \hat {T}_{\parallel \,s}$ . By examining this equation, we can appreciate the two possible types of plasma confinement in an axisymmetric mirror device. Conventional particle trapping (i.e. the terms involving ${\Delta }\hat {T}_s$ ) reduces the relative particle density on those parts of a magnetic field line where the magnetic field strength is greater than average (Endrizzi et al. Reference Endrizzi2023). Centrifugal confinement, on the other hand, (i.e. the term involving $\hat {\varOmega }_\theta ^{\,2}$ ) reduces the particle density on those parts of a magnetic field line that are closer than average to the magnetic axis (Schwartz et al. Reference Schwartz, Abel, Hassam and Romero-Talamás2024).

Note, finally, that in the absence of plasma rotation, equation (8.17) can be written

(8.18) \begin{align} \hat {n}_i(\alpha ,B) = \hat {n}_e(\alpha ,B)= \hat {n}_0(\alpha )\left (\frac {\bar {B}}{B}\right )^\zeta , \end{align}

where

(8.19) \begin{align} \zeta (\alpha ) &= \frac {{\Delta }\hat {T}_{e}+{\Delta }\hat {T}_i}{\hat {T}_{\parallel \,e}+\hat {T}_{\parallel \,i}},\\[-10pt]\nonumber \end{align}

(8.20) \begin{align} \bar {B}(\alpha ) &= \exp (\langle \ln B\rangle ). \end{align}

The idea that, in a non-rotating mirror plasma, $\hat {n}_s=\hat {n}_s(\alpha ,B)$ , which also implies that $p_{\parallel \,s}= p_{\parallel \,s}(\alpha ,B)$ and $p_{\perp \,s}= p_{\perp \,s}(\alpha ,B)$ , and, moreover, that the functional relation between the parallel and perpendicular pressures and the magnetic field- strength is a power law, is quite an old one (Taylor Reference Taylor1963; Grad Reference Grad1967; Lindvall et al. Reference Lindvall, Holmberg, Bendzt, Scheffel and Lundberg2025).

9. First-order flows

The first-order version of (4.6) yields

(9.1) \begin{align} \hat {\boldsymbol{V}}_e^{(1)}\times \hat {\boldsymbol{B}} &=\boldsymbol{\chi }_{\perp \,e}, \end{align}

where

(9.2) \begin{align} \boldsymbol{\chi }_{\perp \,e}&=-\hat {T}_{\parallel \,e}\big\{(1+\gamma _e)\,\hat {\boldsymbol{\nabla} }_\perp \big[\!\ln (\hat {n}_0\,\hat {T}_{\parallel \,e})-\tilde {\xi }_e\big] +\gamma _e\,\hat {\boldsymbol{\nabla} }_\perp \big(\hat {T}_{\parallel \,e}^{\,-1}\,\hat {\phi }^{(1)}\big) \nonumber \\[3pt] & \quad + \hat {\boldsymbol{\nabla} }_\perp \gamma _e -\gamma _e\,\boldsymbol{\kappa }_\perp \big\}, \end{align}

and use has been made of (8.11), (8.12) and (8.15). Thus, we obtain

(9.3) \begin{align} \hat {\boldsymbol{V}}_e^{(1)}= \frac {\hat {\boldsymbol{B}}\times \boldsymbol{\chi }_{\perp \,e}}{\hat {B}^{\,2}}+ \hat {V}_{\parallel \,e}^{(1)}\,\boldsymbol{b}, \end{align}

which reduces to

(9.4) \begin{align} \hat {\boldsymbol{V}}_e^{(1)}= \hat {r}\,\hat {\varOmega }_{\theta \,e}^{(1)}\,\boldsymbol{e}_\theta + \hat {V}_{\parallel \,e}^{(1)}\,\boldsymbol{b}, \end{align}

where

(9.5) \begin{align} \hat {\varOmega }_{\theta \,e}^{(1)}(\alpha ,s) = \frac {\hat {\boldsymbol{\nabla }}\alpha \boldsymbol{\cdot }\boldsymbol{\chi }_{\perp \,e}}{|\hat {\boldsymbol{\nabla }} \alpha |^2}. \end{align}

The first-order version of (4.5) gives

(9.6) \begin{align} 0 = \hat {\boldsymbol{V}}_e^{(1)}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\ln \,\hat {n}_e+\hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{V}}_e^{(1)}, \end{align}

which reduces to

(9.7) \begin{align} \hat {\boldsymbol{B}}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\left [\ln \left (\frac {\hat {V}_{\parallel \,e}^{(1)}\,\hat {n}_e}{\hat {B}}\right )\right ]= 0. \end{align}

Making use of (8.12), we deduce that

(9.8) \begin{align} \frac {\hat {V}_{\parallel \,e}^{(1)}}{\hat {B}} = \hat {V}_{\parallel \,e\,0}^{(1)}(\alpha )\exp \left [- \big(\hat {T}_{\parallel \,e}+ \hat {T}_{\parallel \,i}\big)^{\,-1}\left (\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}-\hat {T}_{\parallel \,e}\,\tilde {\xi }_e-\hat {T}_{\parallel \,i}\,\tilde {\xi }_i\right ) \right ], \end{align}

where $\hat {V}_{\parallel \,e\,0}^{(1)}(\alpha )$ is arbitrary. Note that the parallel electron fluid velocity, $\hat {V}_{\parallel \,e}^{(1)}$ , cannot change sign on a given magnetic field line. In this paper, we restrict attention to equilibria without imposed end-to-end particle fluxes. Thus, we set $\hat {V}_{\parallel \,e\,0}^{(1)}(\alpha )=0$ on all field lines, which implies that $\hat {V}_{\parallel \,e}^{(1)}=0$ on all field lines.

The first-order version of (4.8) gives

(9.9) \begin{align} \hat {\boldsymbol{V}}_i^{(1)}\times \hat {\boldsymbol{B}} &=\boldsymbol{\chi }_{\perp \,i}, \end{align}

where

(9.10) \begin{align} \boldsymbol{\chi }_{\perp \,i}& = \hat {T}_{\parallel \,i}\big\{(1+\gamma _i)\,\hat {\boldsymbol{\nabla} }_\perp \big[\!\ln (\hat {n}_0\,\hat {T}_{\parallel \,i})-\tilde {\xi }_i\big] -\gamma _i\,\hat {\boldsymbol{\nabla} }_\perp \big(\hat {T}_{\parallel \,i}^{\,-1}\,\hat {\phi }^{(1)}\big) \nonumber \\[3pt] & \quad + \hat {\boldsymbol{\nabla} }_\perp \gamma _i -\gamma _i\,\boldsymbol{\kappa }_\perp \big\} \nonumber \\[3pt] & \quad -\frac {1}{2}\,\hat {\varOmega }_{\theta }^{\,2}\,\hat {\boldsymbol{\nabla} }_\perp \langle \hat {r}^{\,2}\rangle + \frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\boldsymbol{\nabla} }_\perp \hat {\varOmega }_\theta ^{\,2} + \gamma _i\,\hat {\boldsymbol{\nabla} }_\perp \!\left (\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}\right ) -\gamma _i\,\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}\,\hat {\boldsymbol{\nabla} }_\perp \ln \hat {T}_{\parallel \,i}, \end{align}

and use has been made of (7.7), (8.11), (8.12) and (8.16). Thus, we obtain

(9.11) \begin{align} \hat {\boldsymbol{V}}_i^{(1)}= \frac {\hat {\boldsymbol{B}}\times \boldsymbol{\chi }_{\perp \,i}}{\hat {B}^{\,2}}+ \hat {V}_{\parallel \,i}^{(1)}\,\boldsymbol{b}, \end{align}

which reduces to

(9.12) \begin{align} \hat {\boldsymbol{V}}_i^{(1)}= \hat {r}\,\hat {\varOmega }_{\theta \,i}^{(1)}\,\boldsymbol{e}_\theta + \hat {V}_{\parallel \,i}^{(1)}\,\boldsymbol{b}, \end{align}

where

(9.13) \begin{align} \hat {\varOmega }_{\theta \,i}^{(1)}(\alpha ,s) = \frac {\hat {\boldsymbol{\nabla }}\alpha \boldsymbol{\cdot }\boldsymbol{\chi }_{\perp \,i}}{|\hat {\boldsymbol{\nabla }} \alpha |^2}. \end{align}

The first-order version of (4.7) gives

(9.14) \begin{align} 0 = \hat {\boldsymbol{V}}_i^{(1)}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\ln \,\hat {n}_i+\hat {\boldsymbol{\nabla }}\boldsymbol{\cdot }\hat {\boldsymbol{V}}_i^{(1)}, \end{align}

which reduces to

(9.15) \begin{align} \hat {\boldsymbol{B}}\boldsymbol{\cdot }\hat {\boldsymbol{\nabla }}\left [\ln \left (\frac {\hat {V}_{\parallel \,i}^{(1)}\,\hat {n}_i}{\hat {B}}\right )\right ]= 0. \end{align}

By analogy with (9.7) and (9.8), we deduce that

(9.16) \begin{align} \frac {\hat {V}_{\parallel \,i}^{(1)}}{\hat {B}}=\hat {V}_{\parallel \,i\,0}^{(1)}(\alpha )\exp \left [ - (\hat {T}_{\parallel \,e}+ \hat {T}_{\parallel \,i})^{-1}\left (\frac {1}{2}\,\widetilde {\hat {r}^{\,2}}\,\hat {\varOmega }_\theta ^{\,2}-\hat {T}_{\parallel \,e}\,\tilde {\xi }_e-\hat {T}_{\parallel \,i}\,\tilde {\xi }_i\right ) \right ]. \end{align}

As before, because we are restricting our attention to equilibria without imposed end-to-end particle fluxes, we set $\hat {V}_{\parallel \,i}^{(1)}=0$ on all field lines.

It follows from the previous analysis that the ion and electron fluid velocities can be written

(9.17) \begin{align} \hat {\boldsymbol{V}}_e &= \hat {r}\,\hat {\varOmega }_e\,\boldsymbol{e}_\theta,\\[-10pt]\nonumber \end{align}

(9.18) \begin{align} \hat {\boldsymbol{V}}_i &= \hat {r}\,\hat {\varOmega }_i\,\boldsymbol{e}_\theta .\end{align}

In other words, the ions and electron fluids both circulate about the $\theta$ -axis with (normalised) angular velocities, $\hat {\varOmega }_e$ and $\hat {\varOmega }_i$ , respectively, where

(9.19) \begin{align} \hat {\varOmega }_e(\alpha ,s) &= \hat {\varOmega }_\theta (\alpha ) +\hat {\rho }_i\,\hat {\varOmega }_{\theta \,e}^{(1)}(\alpha ,s),\\[-10pt]\nonumber \end{align}

(9.20) \begin{align} \hat {\varOmega }_i(\alpha ,s)&=\hat {\varOmega }_\theta (\alpha )+ \hat {\rho }_i\,\hat {\varOmega }_{\theta \,i}^{(1)}(\alpha ,s). \end{align}

10. Grad–Shafranov equation

Equation (4.4) yields the Grad–Shafranov equation

(10.1) \begin{align} \hat {\Delta }^\ast \alpha = \beta \,\hat {r}^{\,2}\,\hat {n}_0\,\exp \big(\hat {T}_{\parallel \,e}^{\,-1}\,\hat {\phi }^{(1)} - \tilde {\xi }_e\big)\big[{\varOmega }_{\theta \,e}^{(1)} - {\varOmega }_{\theta \,i}^{(1)}\big], \end{align}

where use has been made of (5.3), (8.15) and (9.17)–(9.20). The previous equation describes how the $\theta$ -directed plasma current that flows in the plasma modifies the structure of the magnetic field. It is clear from our previous expressions for ${\varOmega }_{\theta \,e}^{(1)}$ and ${\varOmega }_{\theta \,i}^{(1)}$ that the current has a diamagnetic component driven by perpendicular pressure gradients, a component driven by temperature anisotropy, a component driven by the lowest-order plasma rotation and a component driven by magnetic field-line curvature.

11. Summary and discussion

We have investigated the equilibrium of a rotating magnetic mirror using ideal two-fluid theory. Our main results are as follows. Equation (7.4) states that the lowest-order electrostatic potential is constant on a particular magnetic field line. We expect this potential to be determined by the spatial boundary conditions. Equation (8.11) states how the first-order electrostatic potential, which is needed to ensure quasi-neutrality, varies along a particular field line due to centrifugal and temperature anisotropy effects. Equation (8.17) specifies how the common number density of the two charge species varies along magnetic field lines, due to centrifugal and temperature anisotropy effects. According to (9.19) and (9.20), the fact that the lowest-order electrostatic potential is constant on a given magnetic field line implies that both charge species rotate at a common angular velocity that is also constant on magnetic field lines. To higher order, the two charge species have slightly different angular velocities, due to diamagnetic, centrifugal, temperature anisotropy and magnetic field-line curvature effects. Moreover, these higher-order corrections are not necessarily constant on magnetic field lines. Finally, (10.1) can be solved to determine the structure of the equilibrium magnetic field, once the vacuum magnetic field, as well as the arbitrary functions $\hat {n}_0(\alpha )$ , $\hat {\varOmega }_\theta (\alpha )$ , $\hat {T}_{\parallel \,e}(\alpha )$ , $\hat {T}_{\perp \,e}(\alpha )$ , $\hat {T}_{\parallel \,i}(\alpha )$ and $\hat {T}_{\perp \,i}(\alpha )$ , are specified.

The ordering assumptions which underlie our analysis apply as long as the ion gyro-radius is much less than the size of the machine (which is assumed to be the typical length scale on which equilibrium quantities vary), and as long as first-order terms in our expressions for the electrostatic potential and the angular velocities of the two charge species remain much smaller than zeroth-order terms. This is the case provided

(11.1) \begin{align} \hat {\varOmega }_\theta \ll \frac {1}{\hat {\rho }_i}, \end{align}

or

(11.2) \begin{align} {\varOmega }_\theta \ll {\varOmega }_{g\,i}. \end{align}

Obviously, it would not be possible to confine ions in the mirror if the ion gyro-radius were greater than, or of order, the machine size. Likewise, it seem completely impractical to spin the plasma at an angular velocity, ${\varOmega }_\theta$ , that is greater than, or of order, the ion gyro-frequency, ${\varOmega }_{g\,i}$ . (For one thing, spinning a plasma this rapidly would lead to the complete breakdown of the guiding-centre theory which is the whole basis for ion confinement in a mirror.) Thus, our orderings apply to any practical mirror machine with sonic (or even supersonic, provided that the criterion (11.2) is not violated) rotation.

The main question addressed in this paper is whether the standard result, (ultimately) due to Ferraro, that the plasma angular velocity is approximately constant along individual equilibrium magnetic field lines in a rotating axisymmetric magnetic mirror machine, continues to hold when the rotation becomes sonic or supersonic. Equation (7.4) shows that, provided our ordering assumptions are satisfied, the lowest-order electrostatic potential in a rotating magnetic mirror machine is constant on magnetic field lines, which implies, from (7.6), that the lowest-order plasma angular velocity is also constant on magnetic field lines, in accordance with Ferraro (Reference Ferraro1937). Our analysis can be reconciled with that of Hazeltine et al. (Reference Hazeltine, Mahajan and Zheng2026) by noting that the solution in the latter paper that is termed ‘subsonic’, which is similar to the solution described in this paper, actually applies to all plasma angular velocities that are significantly less than the ion gyro-frequency (which implies that it also applies to moderately supersonic rotation), whereas the solution that is termed ‘sonic’ only applies to impractically high rotation levels such that the plasma angular velocity is similar to the ion gyro-frequency.

Of course, an axisymmetric mirror confinement device differs fundamentally from a toroidal confinement system because all of the magnetic field lines that confine the plasma eventually encounter the two conducting end plates that bookend the device. We expect very thin plasma sheaths (whose thicknesses are of order the Debye length) to form on the inward facing surfaces of the end plates. However, we would also expect the vast difference in scale between the thicknesses of the sheaths and the dimensions of the main plasma to allow the sheath solutions to effectively decouple from the main equilibrium solution. In other words, the sheath solutions should only depend on the local properties of the main equilibrium solution at the two end plates. As is well known, an electrostatic potential difference develops across a sheath. Moreover, this potential difference, which is easily calculated, depends, amongst other things, on the electron temperature and the ion sound speed (Reimann Reference Riemann1991). Thus, in principle, by electrically biasing insulated segments of the two end plates, it should be possible to bias individual magnetic field lines within the main plasma (Endrizzi et al. Reference Endrizzi2023). In other words, it should be possible to adjust the lowest-order electrostatic potential, ${\varPhi }(\alpha )$ , in the main plasma, and, thereby, to vary the lowest-order plasma angular velocity, ${\varOmega }_\theta (\alpha )\propto {\rm d}{\varPhi }/{\rm d}\alpha$ (Endrizzi et al. Reference Endrizzi2023).