2. Revision of Kotelnikov's theory for cold electrons

Before examining the effects of finite electron temperature, we reformulate Kotelnikov’s theory for cold electrons so that it solves the problem of Grad (Reference Grad1961) for the minimal-size transition layer in which trapped ions are absent. The subsequent analysis shows that the distribution (2.3) corresponds to a specific loading of the transition layer by trapped particles. In the cold electron approximation, we can completely neglect the electric field and consider the motion of ions only under the action of the Lorentz force. As in the original paper (Kotelnikov Reference Kotelnikov2020), we will assume that an isotropic Maxwellian plasma with a uniform density $n_0$ and temperature $T$ is created inside a certain radius $a$ . The magnetic field is directed along the plasma column axis and it is uniform for $r \lt a$ : $B_z(r) = B_{\mathrm{in}}$ . The transition layer lies in the region $r \gt a$ , where the plasma density and pressure gradually decrease, and the magnetic field increases to the value of the vacuum field $B_z(\infty ) = B_v$ . The problem of Grad suggests that all particles in the layer are passing ones, that is, they all come from the region $r \lt a$ . Moving in a transversely inhomogeneous and axially symmetric magnetic field $B_z = \chi ^{\prime }(r) / r$ , particles must conserve energy $\mathcal{E} = mv_{\bot }^2 / 2$ and the azimuthal generalised momentum $\mathcal{P}_\theta = mv_{\bot }r \sin (\theta - \varphi ) + e\chi (r) / c$ , where

\begin{align*} \boldsymbol{v}_{\bot } = (v_{\bot }\cos \theta , v_{\bot }\sin \theta ), \qquad \boldsymbol{r} = (r\cos \varphi , r\sin \varphi ), \end{align*}

and $\chi (r)$ is a quantity proportional to the magnetic flux

(2.1) \begin{equation} \chi (r)=\int \limits _0^r B_z(r^{\prime }) r^{\prime }\,\text{d}r^{\prime }. \end{equation}

We can use any function of the integrals of motion $\mathcal{E}$ and $\mathcal{P}_\theta$ to describe the stationary particle distribution in a transition layer. The only restriction is that it should be matched with the isotropic Maxwellian function at $r = a$ . In Kotelnikov (Reference Kotelnikov2020), it was proposed to use the function

(2.2) \begin{equation} f_i(\boldsymbol{r},\boldsymbol{v}_{\bot })=\frac {n_0}{2\pi v_T^2}e^{-\mathcal{E}/T} \mathcal{H}\left (\mathcal{P}_\theta ^{\mathrm{max}}-\mathcal{P}_\theta \right )\!, \end{equation}

which limits the range of possible values of the azimuthal momentum for each selected $v_{\bot }$ by $\mathcal{P}_\theta ^{\mathrm{max}} = mv_{\bot }a + e\chi (a) / c$ ( $\mathcal{H}$ is the Heaviside function, $v_T = \sqrt {T / m}$ is the thermal velocity). However, in a cylindrical plasma with radius $a$ , the azimuthal momentum is limited not only from above but also from below. Indeed, when the magnetic field is completely expelled from the plasma ( $B_{\mathrm{in}} = 0$ , $\chi (r \lt a) = 0$ ), the azimuthal momentum cannot be less than a value $\mathcal{P}_\theta ^{\mathrm{min}} = -mv_{\bot }a$ . For finite $B_{\mathrm{in}}$ , the function $\chi (r)$ inside a homogeneous plasma grows according to the parabolic law $\chi = B_{\mathrm{in}}r^2 / 2$ , therefore the minimum value of the azimuthal momentum corresponding to the minimum possible azimuthal velocity $-v_{\bot }$ depends non-monotonically on the radius: $\mathcal{P}_\theta ^{\mathrm{min}} = -mv_{\bot }r + e\chi (r) / c$ . For $r \lt b = v_{\bot } / (eB_{\mathrm{in}} / mc)$ the lower threshold of the azimuthal momentum $\mathcal{P}_\theta ^{\mathrm{min}}(r)$ decreases with the increase in radius, therefore in case $a \lt b$ , as before, the global minimum is reached at the largest radius $a$ of the isotropic section of plasma. If $a \gt b$ , then $\mathcal{P}_\theta ^{\mathrm{ min}}(r)$ increases with $r$ , and the global minimum of this function is achieved inside the isotropic plasma, at the radius $b$ . Thus, the lower limit on the magnitude of the azimuthal generalised momentum depends on the ratio between the plasma radius $a$ and the Larmor radius of ion $b$ , calculated from the internal magnetic field $B_{\mathrm{ in}}$

(2.3) \begin{equation} \mathcal{P}_\theta ^{\mathrm{min}} = \begin{cases} -mv_{\bot }a + e\chi (a) / c, & a \lt b, \\ -mv_{\bot }b + e\chi (b) / c, & a \gt b. \end{cases} \end{equation}

Figure 1. (a) Blue stripes are the ranges of possible values of the azimuthal generalised momentum $\mathcal{P}_{\theta }=mv_{\bot }r\sin \theta +e\chi (r)/c$ at different radii for ions with two chosen values of $v_{\bot }$ , if the particles had an isotropic angular distribution everywhere (for $\chi (r)$ , the self-consistent solution at $a=10$ and $B_{\mathrm{in}}=0.01$ is used). Green and orange lines are the constraints $\mathcal{P}_\theta ^{\mathrm{max}}(a)$ and $\mathcal{P}_\theta ^{\mathrm{min}}(a)$ , which allow ions to have an isotropic distribution only in the region $r\lt a$ and separate passing particles from trapped ones in the transition layer. The regions of trapped particles that are not cut off in the Kotelnikov distribution are shaded in red (2.3). (b) Particle trajectories corresponding to the square and round dots in (a); the blue solid trajectory touches the core of the diamagnetic bubble and corresponds to particles at the boundary between the passing and trapped zones, while the orange dotted trajectory belongs to a particle trapped in the transition layer and lies entirely outside the isotropic core.

Given this additional constraint, the ion distribution function describing only passing particles in the transition layer must differ from the distribution function (2.3) as follows:

(2.4) \begin{equation} f_i(\boldsymbol{r}, \boldsymbol{v}_{\bot }) = \frac {n_0}{2\pi v_T^2} \exp \left ( \frac {v_{\bot }^2}{2v_T^2} \right ) \left [ \mathcal{H} \left (\mathcal{P}_\theta ^{\mathrm{max}} - \mathcal{P}_\theta \right ) - \mathcal{H} \left (\mathcal{P}_\theta ^{\mathrm{min}} - \mathcal{P}_\theta \right ) \right ]\!. \end{equation}

To illustrate the differences between the two distribution functions, we show figure 1. If the particles at all radii had an isotropic angular distribution, then the possible values of the azimuthal generalised momentum for two chosen $v_{\bot }$ would lie inside the ranges shown in figure 1(a) by the light blue and dark blue stripes. The constraints $\mathcal{P}_\theta ^{\mathrm{max}}$ and $\mathcal{P}_\theta ^{\mathrm{min}}$ (green and orange straight lines) leave the ion angular distribution isotropic only inside the region $r\lt a$ and separate the passing particles from those trapped in the transition layer $r\gt a$ . Kotelnikov’s distribution (2.3) cuts off trapped particles only with large positive values of $\mathcal{P}_\theta$ and leaves trapped particles in the transition layer, which fill the red regions in figure 1(a). Figure 1(b) shows that the particles with $\mathcal{P}_\theta \lt \mathcal{P}^{\mathrm{min}}_\theta$ present in the model of Kotelnikov indeed never enter the isotropic core of the diamagnetic bubble, and their trajectories lie entirely in the region $r\gt a$ . It is obvious that the population of particles trapped in the transition layer disappears if the plasma size $a$ exceeds the ion gyroradius $b$ in the field $B_{\mathrm{in}}$ .

Constraints on azimuthal momentum for each value of $v_{\bot }$ can be formulated as a restrictions on possible velocity vector directions (angle $\theta$ )

(2.5) \begin{equation} \xi _{\mathrm{min}}\lt \sin (\theta -\varphi )\lt \xi _{\mathrm{max}}, \end{equation}

where

(2.6) \begin{equation} \xi _{\mathrm{max}} = \frac {a - \bar {\chi } / v_{\bot }}{r},\qquad\qquad\qquad\quad \end{equation}

(2.7) \begin{align}\qquad\quad\qquad \xi_{\mathrm{min}} & = \left\{\begin{array}{l@{\quad}l} -\left [a + \bar {\chi } / v_{\bot }\right ] / r, & v_{\bot } \gt aB_{\mathrm{in}},\\[5pt] -\left [b + (\bar {\chi } + \varDelta \chi ) / v_{\bot }\right ] / r, & v_{\bot } \lt aB_{\mathrm{in}}, \end{array}\right.\end{align}

(2.8) \begin{equation} \bar {\chi } = \chi (r) - \chi (a), \qquad \varDelta \chi = \chi (a) - \chi (b). \end{equation}

The quantities above are in the dimensionless form, velocities are measured in units of the light speed $c$ , spatial scales are in units of $c / \omega _{pi}$ , $\omega_{pi}=\sqrt(4\pi e^2 n_0/m_i)$ is the ion plasma frequency and magnetic field is in units of $mc\omega _{pi} / e$ . Clearly, in the region accessible to particle motion, the interval $[\xi _{\mathrm{min}}, \xi _{\mathrm{max}}]$ should overlap with $[-1, 1]$ .

The density distribution in the transition layer can be found by calculating the integral

(2.9) \begin{equation} n(r) = \int \limits _0^{\infty } {\rm d}v_{\bot } v_{\bot } \int \limits _{-\pi }^{\pi } {\rm d}\theta f_i. \end{equation}

Substitution of distribution function (2.5) into this integral gives

(2.10) \begin{align} n(r) &= \frac {1}{\pi v_T^2} \int \limits _0^{\infty } {\rm d}v_{\bot } v_{\bot } \exp \left (-\frac {v_{\bot }^2}{2v_T^2}\right ) \nonumber \\ &\quad \times \begin{cases} \arcsin \xi _{\mathrm{max}} - \arcsin \xi _{\mathrm{min}}, & v_{\bot } \gt V, \\ \pi /2 + \arcsin \xi _{\mathrm{max}}, & \bar {\chi } / (r + a) \lt v_{\bot } \lt V, \\ 0, & v_{\bot } \lt \bar {\chi } / (r + a). \end{cases} \end{align}

Here, the speed $V$ depends on the magnitude of the magnetic field in the centre of the plasma

(2.11) \begin{equation} V = \begin{cases} \bar {\chi } / (r - a), & \bar {\chi } / (r - a) \gt aB_{\mathrm{in}}, \\ B_{\mathrm{in}} \left (r - \sqrt {r^2 - 2\chi (r) / B_{\mathrm{in}}}\right )\!, & \bar {\chi } / (r - a) \lt aB_{\mathrm{in}}. \end{cases} \end{equation}

The azimuthal current density distribution can be calculated likewise

(2.12) \begin{equation} J_\varphi (r) = e\int \limits _0^{\infty } {\rm d}v_{\bot } v_{\bot } \int \limits _{-\pi }^{\pi } {\rm d}\theta v_{\bot }\sin (\theta - \varphi ) f_i. \end{equation}

In units of $en_0c$ it can be represented as

(2.13) \begin{align} J_\varphi &= \frac {1}{\pi v_T^2} \int \limits _0^{\infty } {\rm d}v_{\bot } v_{\bot }^2 \exp \left (-\frac {v_{\bot }^2}{2v_T^2}\right ) \nonumber \\&\quad \times \begin{cases} \sqrt {1 - \xi _{\mathrm{min}}^2}-\sqrt {1 - \xi _{\mathrm{max}}^2}, & v_{\bot } \gt V, \\[5pt] -\sqrt {1 - \xi _{\mathrm{max}}^2}, & \bar {\chi } / (r + a) \lt v_{\bot } \lt V, \\[5pt] 0, & v_{\bot } \lt \bar {\chi } / (r + a). \end{cases} \end{align}

Radial equilibrium of axisymmetric plasma depends on two components of the momentum flux tensor

(2.14) \begin{equation}\varPi _{rr} = \int \limits _0^{\infty } \text{d}v_{\bot } v_{\bot } \int \limits _{-\pi }^{\pi } \text{d}\theta mv_{\bot }^2 \cos ^2(\theta - \varphi ) f_i, \end{equation}

(2.15) \begin{equation} \varPi _{\varphi \varphi } = \int \limits _0^{\infty } \text{d}v_{\bot } v_{\bot } \int \limits _{-\pi }^{\pi } \text{d}\theta mv_{\bot }^2 \sin ^2(\theta - \varphi ) f_i. \end{equation}

Represented in units of $n_0mc^2$ it can be written as

(2.16) \begin{equation} \varPi _{rr,\varphi \varphi } = \frac {1}{2\pi v_T^2} \int \limits _0^{\infty } \text{d}v_{\bot } v_{\bot }^3 \exp \left (-\frac {v_{\bot }^2}{2v_T^2}\right ) \begin{cases} \varPi _1, & v_{\bot } \gt V, \\ \varPi _2, & \bar {\chi } / (r + a) \lt v_{\bot } \lt V, \\ 0, & v_{\bot } \lt \bar {\chi } / (r + a), \end{cases} \end{equation}

where

(2.17) \begin{align} \varPi _1 = \arcsin \xi _{\mathrm{max}} - \arcsin \xi _{\mathrm{min}} \pm \left (\xi _{\mathrm{max}}\sqrt {1 - \xi _{\mathrm{max}}^2} - \xi _{\mathrm{min}}\sqrt {1 - \xi _{\mathrm{min}}^2}\right )\!, \end{align}

(2.18) \begin{align} \varPi _2 = \arcsin \xi _{\mathrm{max}} + \frac {\pi }{2} \pm \xi _{\mathrm{max}}\sqrt {1 - \xi _{\mathrm{max}}^2} \end{align}

(top sign for $\varPi _{rr}$ , bottom sign for $\varPi _{\varphi \varphi }$ ). It can be seen that quantities of $\varPi _{rr}$ , $\varPi _{\varphi \varphi }$ and $J_\varphi$ calculated this way do satisfy the radial equilibrium equation

(2.19) \begin{equation} -\frac {\partial \varPi _{rr}}{\partial r} - \frac {\varPi _{rr} - \varPi _{\varphi \varphi }}{r} + J_\varphi B_z = 0. \end{equation}

Figure 2. Comparison of the Kotelnikov (Reference Kotelnikov2020) theory (a, c, e, g) and its modified version (b, d, f, h) at $B_{\mathrm{ in}}=0.1$ ( $T_i = 10$ keV, $T_e = 0$ ): equilibrium profiles of the plasma density, azimuthal current density and magnetic field at $a = 10$ (a, b) and at $a = 1.5$ (e, f); profiles of $\varPi _{rr}$ , $\varPi _{\varphi \varphi }$ , $\varDelta \varPi$ and $B^2 / 2$ at $a = 10$ (c, d) and at $a = 1.5$ (g, h).

Substituting the current created by the ions into the Maxwell’s equation $-B_z^{\prime } = J_\varphi$ , we obtain an ordinary differential equation for determining the magnetic flux profile $\bar {\chi }(r)$ in the transition layer

(2.20) \begin{equation} \frac {\text{d}}{\text{d}r}\left (\frac {1}{r}\frac {\text{d} \bar {\chi }}{\text{d}r}\right ) = -J_\varphi , \end{equation}

that should be solved with the boundary conditions: $\bar {\chi }(a) = 0$ , $\bar {\chi }^{\prime }(a) = aB_{\mathrm{in}}$ .

Figure 3. Comparison of the Kotelnikov (Reference Kotelnikov2020) theory (a, c) and its modified version (b, d) with almost complete displacement of the magnetic field $B_{\mathrm{in}} = 0.01$ ( $T_i = 10$ keV, $T_e = 0$ ): equilibrium profiles of plasma density, azimuthal current density and magnetic field (a, b) and the corresponding pressure balance (c, d).

Figure 2 shows how strongly the additional constraint on the azimuthal momentum we have introduced affects the equilibrium profiles of the plasma and magnetic field. The graphs in figures 2(a), 2(c), 2(e) and 2(g) were calculated using the Kotelnikov theory, in which the azimuthal generalised momentum is limited only from above, and figures 2(b), 2(d), 2(f) and 2(h) correspond to solutions of (2.21) with two constraints $\mathcal{P}^{\mathrm{min}}_\theta \lt \mathcal{P}_\theta \lt \mathcal{P}^{\mathrm{max}}_\theta$ . Let us first find the equilibrium in the case when a proton plasma with a temperature of $T=10$ keV and a sufficiently large radius of $a = 10$ reduces the magnetic field in its volume to a value of $B_{\mathrm{in}} = 0.1$ (figures 2 a–d). It is evident from figures 2(a) and 2(b) that the differences between the two solutions in this regime are insignificant. The barely noticeable decrease in the vacuum magnetic field compared with Kotelnikov’s prediction is due to a slightly larger difference between the radial and azimuthal components of the momentum flux $\varPi _{rr}$ and $\varPi _{\varphi \varphi }$ , i.e. the stronger non-gyrotropy of the plasma. As discussed earlier (Kotelnikov Reference Kotelnikov2020; Timofeev et al. Reference Timofeev, Kurshakov and Berendeev2024), the non-gyrotropic effect results in the vacuum magnetic field pressure being less than the sum of the pressures inside the plasma $\varPi _{rr}(0) + B_{\mathrm{in}}^2 / 2$ by

(2.21) \begin{equation} \varDelta \varPi = \int \limits _0^r \frac {\varPi _{rr} - \varPi _{\varphi \varphi }}{r^{\prime }} \text{d}r^{\prime }, \end{equation}

which follows directly from the integration of (2.20). To evaluate the role of the various components in this pressure balance, we show in figure 2(c, d, g, h) the profiles of $\varPi _{rr}$ and $B^2 / 2 + \varDelta \varPi$ . Their sum indeed remains constant at all radii

(2.22) \begin{equation} \varPi _{rr} + \frac {B^2}{2} + \varDelta \varPi = \varPi _{rr}(0) + \frac {B_{\mathrm{in}}^2}{2}. \end{equation}

A stronger difference between the two theories arises when a small plasma radius $a = 1.5$ , comparable to the Larmor radius of an ion in a vacuum field, is considered (figure 2 e–f). In this case, the modified theory predicts a wider transition layer and a several times lower vacuum magnetic field, which is associated with a much stronger plasma non-gyrotropy than it is in (Kotelnikov Reference Kotelnikov2020) ( $\varDelta \varPi$ in figure 2(h) not only ceases to be a small addition, but becomes noticeably larger than $B^2 / 2$ ). In other words, to be able to displace the magnetic field from a plasma of a sufficiently small radius, the plasma pressure must significantly exceed the pressure of the vacuum magnetic field. This conclusion may be important for the interpretation of experiments at the CAT facility (Bagryansky Reference Bagryansky2016), in which the plasma size is comparable to the gyroradius of fast ions arising due to neutral injection.

A similar effect of increasing the non-gyrotropic addition and decreasing the vacuum magnetic field is also observed when the magnetic field $B_{\mathrm{in}}$ inside the isotropic plasma is decreased. Figure 3 shows that the almost complete displacement of the field ( $B_{\mathrm{in}} = 0.01$ ) even at the large radius $a = 10$ leads to an increase in the contribution of $\varDelta \varPi$ and a decrease in the contribution of $B^2/2$ to the overall pressure balance.

3. Accounting for the finite electron temperature

Let us now consider the case of a finite temperature of electrons. Let both ions and electrons have isotropic Maxwellian distributions inside the radius $a$ with the same densities and different temperatures $T_i$ and $T_e$ . Unlike the case of cold electrons, now the propagation of particles into the transition layer is affected not only by an increasing magnetic field $B_z(r)$ , but also a radial electric field $E_r = -\partial \phi / \partial r$ . The latter occurs due to separation of charges produced by a large difference in the Larmor radii of electrons and ions. The ion distribution function in the transition layer ( $r \gt a$ ) has the same form as in (2.5) with the only difference that, instead of a velocity module, the value $\mathcal{E} = mv_{\bot }^2(r) / 2 + e\phi (r)$ is now conserved. Taking into account the change in the particle velocity module at radius $r$ , compared with its value $v_{\bot }$ in the isotropic section, the azimuthal generalised momentum is given by

(3.1) \begin{equation} \mathcal{P}_{\theta i} = m_i r \sqrt {v_{\bot }^2 - \frac {2e\phi (r)}{m_i}}\sin (\theta - \varphi ) + e\chi (r) / c. \end{equation}

Since $\phi = 0$ for $r \lt a$ , the maximum and minimum values of this momentum ( $\mathcal{P}_\theta ^{\mathrm{max}}$ and $\mathcal{P}_\theta ^{\mathrm{min}}$ ) remain unchanged. Then the restrictions on the possible values of $\sin (\theta - \varphi )$ in the presence of an electric field in the layer are modified as follows:

(3.2) \begin{equation} \xi _{\mathrm{i,max}} = \frac {a - \bar {\chi }/v_{\bot }}{r\sqrt {1 - \varPhi /v_{\bot }^2}},\qquad\qquad\quad \end{equation}

(3.3) \begin{equation} \;\qquad\qquad\xi _{\mathrm{i,min}} = \begin{cases} -\dfrac {a + \bar {\chi }/v_{\bot }}{r\sqrt {1 - \varPhi / v_{\bot }^2}}, & v_{\bot } \gt aB_{\mathrm{ in}}, \\[12pt] -\dfrac {b_i + (\bar {\chi } + \varDelta \chi ) / v_{\bot }}{r\sqrt {1 - \varPhi / v_{\bot }^2}}, & v_{\bot } \lt aB_{\mathrm{ in}}, \end{cases} \end{equation}

where $\varPhi = 2e\phi / (m_ic^2)$ is a dimensionless electric potential. Thus, the ion density and current density can be written as

(3.4) \begin{align} n_i(r) &= \frac {1}{\pi v_{Ti}^2} \int \limits _{\sqrt {\varPhi }}^{\infty } \text{d}v_{\bot } v_{\bot } \exp \left (-\frac {v_{\bot }^2}{2v_{Ti}^2}\right )\nonumber \\ &\quad \times \begin{cases} \arcsin \xi _{\mathrm{i,max}} - \arcsin \xi _{\mathrm{i,min}}, &|\xi _{\mathrm{i,max}}|, |\xi _{\mathrm{i,min}}| \lt 1 \\[6pt] \pi /2 + \arcsin \xi _{\mathrm{i,max}}, &|\xi _{\mathrm{i,max}}| \lt 1, \xi _{\mathrm{i,min}} \lt -1 \\[6pt] \pi /2 - \arcsin \xi _{\mathrm{i,min}}, &\xi _{\mathrm{i,max}} \gt 1, |\xi _{\mathrm{i,min}}| \lt 1 \\[6pt] 0, &|\xi _{\mathrm{i,max}}|, |\xi _{\mathrm{i,min}}| \gt 1, \end{cases} \end{align}

(3.5) \begin{align} J_{\varphi }^i &= \frac {1}{\pi v_{Ti}^2} \int \limits _{\sqrt {\varPhi }}^{\infty } \text{d}v_{\bot } v_{\bot }\sqrt {v_{\bot }^2 - \varPhi } \exp \left (-\frac {v_{\bot }^2}{2v_{Ti}^2}\right ) \nonumber \\[10pt] & \quad \times \left [\sqrt {1 - \xi _{\mathrm{i,min}}^2}\mathcal{H}(1 - \xi _{\mathrm{i,min}}^2) - \sqrt {1 - \xi _{\mathrm{ i,max}}^2}\mathcal{H}\big(1 - \xi _{\mathrm{i,max}}^2\big)\right ]\!. \end{align}

For electrons, the generalised azimuthal momentum can be obtained from the ion form by replacing the mass and sign of the electric charge

(3.6) \begin{equation} \mathcal{P}_{\theta e} = m_e r \sqrt {v_{\bot }^2 + \frac {2e\phi (r)}{m_e}}\sin (\theta -\varphi )-e\chi (r)/c. \end{equation}

Using the analogous definitions of the maximum and minimum values of this momentum inside an isotropic region of plasma, we obtain the following constraints on the electron velocity angles for $r \gt a$ :

(3.7) \begin{equation} \xi _{\mathrm{e,min}} \lt \sin (\theta -\varphi ) \lt \xi _{\mathrm{e,max}}, \end{equation}

where

(3.8) \begin{align} \xi _{\mathrm{e,min}} = -\frac {a - \mu \bar {\chi } / v_{\bot }}{r\sqrt {1 + \mu \varPhi / v_{\bot }^2}},\qquad\qquad \end{align}

(3.9) \begin{align}\qquad\qquad\; \xi _{\mathrm{e,max}} = \begin{cases} \dfrac {a + \mu \bar {\chi } / v_{\bot }}{r\sqrt {1 + \mu \varPhi / v_{\bot }^2}}, & v_{\bot } \gt a\mu B_{\mathrm{in}},\\[20pt] \dfrac {b_e + \mu (\bar {\chi } + \varDelta \chi ) / v_{\bot }}{r\sqrt {1 + \mu \varPhi / v_{\bot }^2}}, & v_{\bot } \lt a\mu B_{\mathrm{in}}, \end{cases} \end{align}

(3.10) \begin{eqnarray} b_e = v_{\bot } / (\mu B_{\mathrm{in}}), \qquad \mu = m_i / m_e. \end{eqnarray}

The density and current density of electrons is written by analogy with (3.4) and (3.5)

(3.11) \begin{align} n_e(r) &= \frac {1}{\pi v_{Te}^2} \int \limits _{0}^{\infty } {\rm d}v_{\bot } v_{\bot } \exp \left (-\frac {v_{\bot }^2}{2v_{Te}^2}\right ) \nonumber \\ &\quad \times\begin{cases} \arcsin \xi _{\mathrm{e,max}} - \arcsin \xi _{\mathrm{e,min}}, &|\xi _{\mathrm{e,max}}|, |\xi _{\mathrm{e,min}}| \lt 1 \\ \pi /2 + \arcsin \xi _{\mathrm{e,max}}, &|\xi _{\mathrm{e,max}}| \lt 1, \xi _{\mathrm{e,min}} \lt -1 \\ \pi /2 - \arcsin \xi _{\mathrm{e,min}}, &\xi _{\mathrm{e,max}} \gt 1, |\xi _{\mathrm{e,min}}| \lt 1 \\ 0, &|\xi _{\mathrm{e,max}}|, |\xi _{\mathrm{e,min}}| \gt 1, \end{cases} \end{align}

(3.12) \begin{align} J_{\varphi }^e &= -\frac {1}{\pi v_{Te}^2} \int \limits _{0}^{\infty } {\rm d}v_{\bot } v_{\bot }\sqrt {v_{\bot }^2 + \mu \varPhi } \exp \left (-\frac {v_{\bot }^2}{2v_{Te}^2}\right ) \nonumber \\& \quad \times \left [\sqrt {1 - \xi _{\mathrm{e,min}}^2}\mathcal{H}\big(1 - \xi _{\mathrm{e,min}}^2\big) - \sqrt {1 - \xi _{\mathrm{ e,max}}^2}\mathcal{H}\big(1 - \xi _{\mathrm{e,max}}^2\big)\right ]\!. \end{align}

Given the densities and currents, the magnetic field potential $\bar {\chi }(r)$ is found from the equation

(3.13) \begin{equation} \frac {\rm d}{{\rm d}r}\left (\frac {1}{r}\frac {{\rm d} \bar {\chi }}{{\rm d}r}\right ) = -J_{\varphi }^e-J_{\varphi }^i. \end{equation}

To find the electric field potential $\varPhi (r)$ , the Poisson equation has to be solved

(3.14) \begin{equation} \frac {1}{r}\frac {\rm d}{{\rm d}r}\left (r\frac {{\rm d}\varPhi }{{\rm d}r}\right ) = 2\left (n_e-n_i\right )\!, \end{equation}

with the boundary conditions: $\varPhi (a) = 0$ , $\varPhi ^{\prime }(a) = 0$ .

4. Solution with no trapped electrons

Figure 4. Solution without trapped electrons in the layer, demonstrating unlimited electric field growth at a finite electron temperature in the central plasma ( $T_e \approx 45$ eV, $T_i \approx 8.44$ keV, $a = 1.5$ , $B_{\textrm{in}} = 0.01$ , $\mu = 1836$ ): (top) radial profiles of the electron and ion density and the electric field profile, (bottom) ion and electron azimuthal currents.

The numerical solution obtained in the limit of the complete absence of trapped electrons in the transition layer is shown in figure 4 ( $v_{Ti} = 0.003$ , $\nu = v_{Ti} / v_{T_e} = 0.32$ , $B_{\mathrm{in}} = 0.01$ , $a = 1.5$ ). It is seen that, if the plasma is quasi-neutral at $r \lt a$ , then in the transition layer, outside the region accessible to electron motion, this quasi-neutrality breaks at some radius. This leads to the electric field growth and to unlimited acceleration of ions, providing a very slow decrease in the radial profiles of their density and current. This problem was first recognised in the work of Paskievici (Reference Paskievici1962), where a similar non-physical solution was obtained for a flat transition layer. For a cylindrical plasma, a similar solution in the form of an infinitely extended charged column is realised for a stationary distribution function of the form $f \propto \exp (\mathcal{E} - F(\mathcal{P}_\theta ))$ (Shustov et al. Reference Shustov, Artemyev, Vasko and Yushkov2016).

To make the solution physically meaningful, the above-mentioned works assumed the presence of a background cold plasma, the influx of electrons from which compensates the excess of the ion positive charge. A similar situation should be realised in mirror traps. The plasma in such open systems leans on the end conducting walls of the plasma receivers. If a deficit of electrons occurs on the magnetic field line running along the periphery of the plasma at the central part of the trap, the missing charge will be compensated for by electrons from the wall, which are cold in the transverse direction. Thus, in one part of the transition layer, the quasi-neutrality should be maintained by hot electrons coming from the central part of the plasma column, and in another part – by cold electrons from the wall.

5. Analysis of solutions with trapped cold electrons at different hot population temperatures

Figure 5. Solutions with finite electron temperature $T_e / T_i = 5.3 \times10^{-3}$ ( $\nu = 0.32$ ) (right column), $T_e / T_i = 0.53$ ( $\nu = 0.032$ ) (centre column) and $T_e / T_i = 1.36$ ( $\nu = 0.02$ ) (right column) in the presence of trapped cold electrons ( $T_i = 8.44$ keV ( $v_{Ti} = 0.003$ ), $a = 1.5$ , $B_{\mathrm{in}} = 0.01$ and $\mu = 1836$ ). Radial profiles of potentials and fields $\varPhi$ , $\bar {\chi }$ , $E_r$ and $B_z$ (upper row), of various plasma components‘ azimuthal currents (third row), their profiles of the density (second row) and pressure profiles (lower row).

Now, let us consider the modification of the system of equations to be solved as follows. In those sections of the transition layer where the density of hot electrons exceeds the density of ions, the old (3.13) and (3.14) are solved. In the places where $n_i \gt n_e$ , we will add cold electrons with a density of $n_{ce}(r) = n_i - n_e$ to completely neutralise the space charge. Note that the excess of electron density in the initial part of the transition layer is very small, so the electric potential there can be found from the quasi-neutrality condition. In this case, the equation for the magnetic flux function should be completed with the azimuthal current created by cold electrons due to their drift in the crossed fields $J_{\varphi }^{ce} = n_{ce}E_r/B_z$ .

Solving the system of equations modified this way, we obtain the following result for the plasma density, current and pressure profiles (figure 5). Let us find out how these profiles change with the increase in electron temperature at a small bubble radius $a = 1.5$ and almost complete displacement of the magnetic field $B_{\mathrm{in}} = 0.01$ . While the electron temperature is negligibly small compared with the ion temperature ( $T_e \ll T_i$ , $\nu = v_{Ti} / v_{Te} = 0.32$ ) (left column of plots in figure 5), we reproduce the same profiles that were obtained in § 2. The only difference is the appearance of an electron current, the density of which is strongly localised at the inner boundary of the transition layer and significantly exceeds the ion current density. Despite such local dominance, the total magnetic pressure jump caused by the electron current is proportional to the small electron pressure and is almost unnoticeable against the background of the field change that occurs at the ion scale due to the ion current. It is seen that this regime is still characterised by strong non-gyrotropy of the ion component, therefore due to large $\varDelta \varPi _i$ the vacuum magnetic field pressure remains several times lower than the plasma pressure at the centre of the column, which corresponds to a very significant excess of the MagnetoHydroDynamic (MHD) limit for beta ( $\beta = 2\varPi _{rr}(0) / B_v^2 \gt 1$ ). If the electron temperature is increased to values comparable with ion temperature ( $T_e \approx 0.53 T_i$ or $\nu = 0.032$ in the central column of the figure 5), the two-scale structure of the transition layer becomes clearly visible. Now, the electron pressure becomes large enough, enabling the diamagnetic current at the inner boundary of the sheet to produce a noticeable jump in the magnetic field on the scale of several electron gyroradii (the width of the current sheet in the figure is shown in electron gyroradii $\rho _{\ast }$ and $\rho _v$ , calculated from the magnetic field values $B_{\ast }$ and $B_v$ ). The rest of this jump is produced by the ion current flowing through a more extended region having a width of the order of ion gyroradius. It is also seen that the non-gyrotropic addition to the ion momentum flux $\varDelta \varPi _i$ is significantly reduced in this case. Increasing the electron temperature further (right column of images in figure 5), the jump in the magnetic field produced by electrons is increased and the role of the non-gyrotropic effect is reduced, bringing the beta of the confined plasma closer to the MHD limit $\beta \approx 1$ . We also note that, in the limit of unimpeded inflow of cold electrons, the resulting electric field is so small that the electron drift current $\boldsymbol{ E} \times \boldsymbol{B}$ is negligible compared with the diamagnetic one.