3.1. Pressure model A1

The A1 pressure model is described in detail in Zeng & Kotelnikov (Reference Zeng and Kotelnikov2024). It is intended for modelling the transverse NBI. To facilitate comparison with other pressure models, the main properties of the A1 model are briefly summarized below, starting with the definition

(3.10)\begin{equation} \left. \begin{aligned} p_{\mathrel\perp} & = p_{0} f_k(\psi )\, \left(1-b^{2}\right),\\ p_{\|} & = p_{0} f_k(\psi )\, \left(1-b\right)^2. \end{aligned} \right\} \end{equation}

The dimensionless function $f_{k}(\psi )$ defines the radial pressure profile and is normalized so that $f_{k}(0)=1$. It was chosen in Kotelnikov et al. (Reference Kotelnikov, Zeng, Prikhodko, Yakovlev, Zhang, Chen and Yu2022b) so that the integrals over $\psi$ in the coefficients of the LoDestro equation could be calculated analytically, at least for the case of an isotropic plasma. For integer values of index $k$

(3.11)\begin{equation} f_{k}(\psi) = \begin{cases} 1 - \psi^{k}, & \text{if } 0\leq \psi \leq 1, \\ 0, & \text{otherwise}, \end{cases} \end{equation}

and for $k=\infty$

(3.12)\begin{equation} f_{\infty}(\psi) = H(1-\psi) , \end{equation}

where $H(x)=0$ for $x<0$ and $H(x)=1$ for $x>0$. In the case of oblique NBI, the integrals over $\psi$ in the coefficients of the LoDestro equations can be calculated only numerically, and therefore there is no need to restrict the class of functions $f_{k}(\psi )$, except for the comparison of the new and earlier results. Function $f_{1}$ describes the most smooth radial pressure profile. Maximal index $k=\infty$ corresponds to a stepwise profile with a sharp boundary.

The equation of transverse equilibrium (2.2) is readily resolved regarding the true magnetic field as

(3.13)\begin{equation} b(z,\psi) = \sqrt{ \frac{b_{v}^{2}(z)-2p_{0}f_{k}(\psi)}{1-2p_{0}f_{k}(\psi)}}. \end{equation}

The plasma's beta defined by (3.8) is related to parameter $p_{0}$ in (3.10) by

(3.14)\begin{equation} \beta = \frac{2 \left(R^2-1\right)p_{0}}{1 - 2p_{0}} . \end{equation}

Parameter $p_{0}$ can vary within the range

(3.15)\begin{equation} 0< p_{0}<1/2R^{2}, \end{equation}

and $\beta \to 1$ for $p_{0}\to 1/2R^{2}$. For $\beta >1$, transverse equilibrium near the median plane of the trap is impossible, at least in the paraxial approximation and within the MHD approach.Footnote ¹

Figure 2(a) shows the axial profile of the transverse pressure for the model A1 in magnetic field (2.3). The pressure peak in this model is located at the midplane of the trap and narrows as the local mirror ratio $R$ at the turning point of fast ions decreases.

Figure 2. Axial profiles of the transverse pressure in models A1 (a), A2 (b) and A3 (c) described respectively by (3.10), (3.18) and (3.28) in magnetic field (2.3) at the axis of the trap for mirror ratio $M=8$, index $q=4$ and various mirror ratios $R$ at the turning points where the pressure of hot plasma component drops to zero. The value $\beta =0.3$ for this figure is chosen so as not to exceed the threshold for excitation of the mirror and firehose instabilities for all values of the parameter $R$ indicated in the figure legend.

It seems intuitively obvious that the degree of anisotropy also increases as $R$ decreases. However, it is not so easy to give a completely satisfactory definition of the degree of anisotropy suitable for all pressure models to be considered in this paper. If the degree of anisotropy would be related to the pressure in the minimum magnetic field as

(3.16)\begin{equation} A_{1} = \frac{p_{\mathrel\perp}-p_{\|}}{p_{\mathrel\perp}+p_{\|}} , \end{equation}

then

(3.17)\begin{equation} A_{1} = {\sqrt{1-\beta}}/{R}, \end{equation}

which it is indeed inversely proportional to the parameter $R$.

It can be proved that the A1 pressure model is stable against the mirror and firehose modes. For example, putting the first of (3.10) into (3.7) yields the inequality $b\,(1-2p_{0})>0$, which is incompatible with condition $0< p_{0}<1/2R^{2}$ within the range $0< b<1$.

3.2. Pressure model A2

The model (3.2) with index $n=2$ reduces to

(3.18)\begin{equation} \left. \begin{aligned} p_{\mathrel\perp} & = p_{0} f_k(\psi )\, b^2\left(1-b\right),\\ p_{\|} & = \frac{1}{2} p_{0} f_k(\psi )\, b\left(1-b\right)^2, \end{aligned} \right\} \end{equation}

in the region $b<1$.

In case of pressure model A1 (3.1), (2.2) reduces to a second-order polynomial equation with respect to $b$. For the model A2 the same equation is of the third order, and the model A3 at $n=3$ yields a fourth-order algebraic equation. As is well known, the solution to such equations can always be expressed in terms of radicals, which saves us from the need to solve these expressions numerically. However, the resulting formulas for a polynomial of the third (and even more so, fourth) order are extremely cumbersome. Therefore, the result of solving (2.2) is written below in a more compact form, which is introduced by Wolfram Mathematica ^©

(3.19)\begin{equation} b = \operatorname{Root}\left[ 2 {\#}^3 p+{\#}^2 ({-}2 p-1) +b_v^2 \ \& , \ 2 \right] . \end{equation}

Literally, it means the second root of the cubic equation

(3.20)\begin{equation} 2 {b}^3 p+{b}^2 ({-}2 p-1) +b_v^2 =0 \end{equation}

represented as a ‘pure function’ by the ampersand $\& $ in the first argument of the $\operatorname {Root}$ built-in utility. Parameter $p$ in this formula stands for $p_{0}f_{k}(\psi )$. The $\operatorname {Root}$ utility arranges the roots of the polynomials in an order known only to it, but in such a way that the real roots come first.

Relative plasma pressure, i.e. the parameter beta, is defined by (3.8) as the maximum of the ratio $2p_{\mathrel \perp }/b_{v}^{2}$. According to (3.4), this ratio is a decreasing function of $b$ so that the maximum is reached on the axis of the trap (where $\psi =0$) at the minimum of the vacuum field (where $b_{v}=1/R$). Hence

(3.21)\begin{equation} p_{0} = \frac{\beta}{2 (1-\beta) \left(1-\sqrt{1-\beta }/R\right)} . \end{equation}

Inversion of (3.21) yields

(3.22)\begin{align} \beta & = \operatorname{Root} \left[ 4 {\#}^3 p_{0}^2 + {\#}^2 \left( 4 p_{0}^2R^2-12 p_{0}^2+4 p_{0} R^2 + R^2 \right) \right.\nonumber\\ & \quad \left. -\,{\#} \left(8 p_{0}^2 R^2-12 p_{0}^2+4 p_{0} R^2\right) - 4 p_{0}^2 R^2+4 p_{0}^2 \& ,\ 2 \right] . \end{align}

As follows from (3.21), parameter $p_{0}$ tends to infinity as $\beta \to 1$. In fact, the mirror mode stability condition (3.7) imposes a more stringent condition

(3.23)\begin{equation} 0< p_{0}<1. \end{equation}

If $p_{0}>1$, there is no continuous solution to (2.2). This can be verified by plotting the left-hand side of the equation as a function of $b$. For $p_{0}>1$, the maximum of this function on the interval $0< b<1$ exceeds $1$, while the left-hand side of the equation is less than $1$. This means that paraxial equilibrium is impossible above the mirror instability threshold. Near this threshold,  unexpected phenomena such as equilibrium hysteresis appear. Plasma can reside in two different states, between which a transition is possible by relieving excess pressure (Kotelnikov Reference Kotelnikov2011). Signs of such a transition were found in experiments at the GDT facility (Kotelnikov, Bagryansky & Prikhodko Reference Kotelnikov, Bagryansky and Prikhodko2010). Similar structures are observed in the solar wind, see e.g. Winterhalter et al. (Reference Winterhalter, Neugebauer, Goldstein, Smith, Bame and Balogh1994), Pantellini (Reference Pantellini1998) and Jiang et al. (Reference Jiang, Verscharen, Li, Wang and Klein2022).

In practice, it turned out that the upper limit of the interval (3.23) is sometimes inaccessible to the shooting method when solving the LoDestro equation (A2). The solution of this equation with initial conditions (A13) and (A14) on the left boundary $z=0$ of the domain of definition of the Sturm–Liouville problem (see Appendix A) sometimes went to infinity, before reaching the right boundary $z=z_{E}$. Most often this happened if the parameter $p_{0}$ was close enough to $p_{0}=1$.

The threshold value of parameter $p_{0}$, above which the mirror instability is excited, will be further written as a function

(3.24)\begin{equation} p_{\text{mm}}(R) = 1, \end{equation}

where the subscript ‘mm’ stands for ‘mirror mode’. The beta limit is obtained from (3.22) by substituting $p_{0}=1$

(3.25)\begin{equation} \beta_{\text{mm}}(R) = \operatorname{Root} \left[ 4 {\#}^3 + {\#}^2 \left(9 R^2-12\right) + {\#} \left(12-12 R^2\right)+4 R^2-4 \ \& ,\ 2 \right] . \end{equation}

The firehose instability threshold is described by more cumbersome formulas

(3.26)\begin{gather} p_{\text{fh}}(R) = \operatorname{Root} \left[ {\#}^3 \left( 4 R^4 + 4 R^2\right) + {\#}^2 \left(R^4-30 R^2-27\right) - 24 {\#} R^2 -4 R^2 \ \& , \ j \right], \end{gather}

(3.27)\begin{gather}\beta_{\text{fh}}(R) = \operatorname{Root} \left[ {\#}^3 + {\#}^2\left(R^2-9\right) + 24 {\#} - 16 \ \& , \ j \right], \end{gather}

where $j=1$ if $1< R<\sqrt {9+6 \sqrt {3}}$ and $j=3$ if $R>\sqrt {9+6 \sqrt {3}}$; the subscript ‘fh’ is shorthand for ‘firehose’ mode.

The stability zone bounded by the conditions $\beta <\beta _{\textrm {mm}}(R)$ and $\beta <\beta _{\textrm {fh}}(R)$ is shown in figure 3(a). Its width shrinks to zero both in the $R\to 1$ limit and in the $R\to \infty$ limit. The maximum width corresponds to the point of intersection of the curves $\beta =\beta _{\textrm {mm}}(R)$ and $\beta =\beta _{\textrm {fh}}(R)$, where $\beta = 0.6202$, $R=3.358$. At the same time, it is worth noting that, at the maximum, these functions are equal to $\beta _{\textrm {mm}}(\infty )=2/3$ and $\beta _{\textrm {fh}}(1)=0.9126$.

Figure 3. Threshold of mirror and firehose instabilities in an anisotropic plasma with oblique NBI within the framework of models A2 (a) and A3 (b). Unstable areas are shaded. The same shading scheme is used below in the stability maps without further reminder.

Limitation of the stability zone in an anisotropic plasma by the threshold of mirror instability was taken into account earlier by Kesner et al. (Kesner Reference Kesner1985; Li et al. Reference Li, Kesner and LoDestro1987b), but the firehose instability did not attract the attention of these authors.

Exceeding the threshold of mirror instability within the framework of the applicability of the LoDestro equation is impossible, since this is equivalent to a violation of the equilibrium configuration of the plasma in the paraxial approximation, which is indirectly confirmed by experiments in a gas-dynamic trap (Kotelnikov et al. Reference Kotelnikov, Bagryansky and Prikhodko2010). The PEK code refused to continue computing if $p_{0}>p_{\textrm {mm}}$. However, with a few tweaks, it works above the firehose instability threshold. In some cases, it finds a solution in the interval $p_{\textrm {fh}}(R)< p_{0}< p_{\textrm {mm}}(R)$.

There are a number of theoretical papers (Kotelnikov et al. Reference Kotelnikov, Bagryansky and Prikhodko2010; Kotelnikov Reference Kotelnikov2011; Beklemishev Reference Beklemishev2016) that predict the stabilization of mirror instability in a non-uniform magnetic field. According to these works, when the mirror instability threshold is exceeded, a region appears in the plasma near the trap axis, where something like a magnetic field jump is formed, and non-paraxiality effects smooth out this jump. It is not yet clear at the moment whether the LoDestro equation can be modified to extend its range of applicability to equilibria with narrow non-paraxial jumps inside the plasma column.

The condition (3.6) guarantees the stability of small-scale perturbations of the firehose type, which, if the parameter $p_{0}$ slightly exceeds $p_{\text{fh}}(R)$, can become unstable near the axis of the plasma column. It is expected that such perturbations are not critical for the rigid ballooning mode.

Figure 4 confirms the above statement that the relation $\beta (z) = 2p_{\mathrel \perp }/B_{v}^{2}$, which has the meaning of a local beta, has a global maximum in the median plane of the trap. Moreover, $\beta (z)$ decreases monotonically in a monotonically increasing magnetic field. However, kinetic calculations of the distribution function of sloshing ions using various numerical codes, in particular the DOL code (Yurov, Prikhodko & Tsidulko Reference Yurov, Prikhodko and Tsidulko2016), show that, if the condition (3.5) is violated, the $\beta (z)$ dependence may turn out to be non-monotonic and a local peak appears on the $\beta (z)$ graph near fast ion turning point.

Figure 4. Axial profile of the local beta $\beta (z) = 2p_{\mathrel \perp }/B_{v}^{2}$ in the magnetic field (2.3) for mirror ratio $M=8$, index $q=4$ and various mirror ratios $R$ at the turning points where the pressure of the hot plasma component drops to zero: (a) transversal NBI, pressure model A1; (b) oblique NBI, model A2; (c) oblique NBI, model A3. The values of the $\beta$ parameter for each $R$ value indicated on the graphs are chosen to be equal to the smallest of the two stability thresholds for the mirror and firehose instabilities.

3.3. Pressure model A3

In the model A3, the pressure functions have the form

(3.28)\begin{equation} \left. \begin{aligned} p_{\mathrel\perp} & = p_{0} f_k(\psi )\, b^2\left(1-b\right)^{2},\\ p_{\|} & = \tfrac{1}{3}\, p_{0} f_k(\psi )\, b\left(1-b\right)^3 . \end{aligned} \right\} \end{equation}

In at least one respect, the A3 model is more realistic than the other anisotropic pressure models discussed so far. The point is that in models A1 and A2 the derivative ${\partial p_{\mathrel \perp }}/{\partial b}$ suffers a discontinuity at $b=1$. It should be expected that the actual pressure distribution in a mirror trap should not have such a discontinuity, since it will inevitably be smoothed out by Coulomb collisions of plasma particles. At the discontinuity point, the derivative ${\partial b}/{\partial z}$ tends to infinity, as on the threshold of mirror instability. In the A3 model, the derivative ${\partial b}/{\partial z}$ also tends to infinity as it approaches the mirror instability threshold, but this happens at some distance from the turning point $b=1$, namely, at $b=3/4$.

Comparing figures 2(a), 2(b) and 2(c), which show the axial transverse pressure profiles for models A1, A2 and A3, respectively, one can see that, in the latter panel, the pressure profile is smoother near the turning point. Another difference is that, for the same values of the parameter $R$, the maximum pressure in the right panel is shifted closer to the centre of the trap compared with the central panel.

It was noted in Kotelnikov et al. (Reference Kotelnikov, Prikhodko and Yakovlev2023) and Zeng & Kotelnikov (Reference Zeng and Kotelnikov2024) that the parameter $R$ characterizes the degree of plasma anisotropy in the A1 model. According to (3.17), the degree of anisotropy is higher for smaller $R$. In the case of oblique NBI (3.2) this connection does not seem obvious. Kesner (Reference Kesner1985) defines the degree of anisotropy as the ratio of the transverse pressure to the longitudinal pressure at the magnetic field minimum

(3.29)\begin{equation} A_{{K}} = {p_{\mathrel\perp}}/{p_{\|}}, \quad \text{at}\ b=b_{\min}=\sqrt{1-\beta }/R. \end{equation}

Examining the quantity $p_{\mathrel \perp }/p_{\|}=nb/(1-b)$ at an arbitrary $b$ reveals it monotonically increases up to infinity as it approaches the turning point $b=1$. The point $b=b_{\min }$ is distinguished only by the fact that the local beta maximum is located there. However, in the model (3.2) the transverse pressure peak is in the field $b=2/(n+1)$, where the ratio ${p_{\mathrel \perp } }/{p_{\|}}=2n/(n-1)$ does not depend on $R$. This and other similar facts indicate that model (3.2) with any $n$ actually describes the pressure distribution with an approximately fixed degree of anisotropy.

In § 3.4 it is shown that the parameter $R$ is related to the angle of injection of neutral atoms. Note, however, that it is possible to invent such a definition of the degree of anisotropy that yields the same result as (3.16) for the A1 model, namely

(3.30)\begin{equation} A_{n} = \frac{p_{\mathrel\perp}}{p_{\mathrel\perp}+n p_{\|}} = \frac{\sqrt{1-\beta }}{R} . \end{equation}

In the A3 model, the true magnetic field in the plasma is expressed in terms of the real root of a fourth-degree polynomial equation

(3.31)\begin{equation} b = \operatorname{Root}\left[ 2 {\#}^4 p - 4 {\#}^3 p + {\#}^2 (2 p+1) -b_{v}^2 \ \& , \ 2 \right] . \end{equation}

Parameter $\beta$ is still defined by (3.8), but (3.21) should be replaced by

(3.32)\begin{equation} p_{0} = \frac{\beta}{2 (1-\beta) \left(1-\sqrt{1-\beta }/R\right)^2} . \end{equation}

The inverse expression is noticeably more complex than (3.22)

(3.33)\begin{align} \beta & = \operatorname{Root}\left[ 4 \#^4 p_{0}^2 + \#^3 \left(8 p_{0}^2 R^2-16 p_{0}^2-4 p_{0} R^2\right) \right.\nonumber\\ & \quad \left. +\, \#^2\left(4 p_{0}^2 R^4-24 p_{0}^2 R^2+24 p_{0}^2+4 p_{0} R^4+8p_{0} R^2+R^4\right) \right.\nonumber\\ & \quad \left. +\, \# \left({-}8 p_{0}^2 R^4+24 p_{0}^2 R^2-16 p_{0}^2-4 p_{0} R^4-4 p_{0} R^2\right) \right.\nonumber\\ & \quad \left. +\, 4 p_{0}^2 R^4-8 p_{0}^2 R^2+4 p_{0}^2\ \& ,\ 1\right]. \end{align}

The threshold value of $p_{0}$, above which the mirror instability is excited, is now greater than prescribed by (3.24)

(3.34)\begin{equation} p_{\text{mm}}(R) = 4 . \end{equation}

The beta threshold is computed as a root of (3.33) at $p=4$

(3.35)\begin{align} \beta_{\text{mm}}(R) & = \operatorname{Root}\left[ 64 {\#}^4 + {\#}^3 \left(112 R^2-256\right) + {\#}^2 \left(81 R^4-352 R^2+384\right) \right.\nonumber\\ & \quad \left. +\, {\#} \left({-}144 R^4+368 R^2-256\right) + 64 R^4-128 R^2+64 \ \& , \ 1 \right] . \end{align}

The firehose instability threshold is described by equally cumbersome formulas

(3.36)\begin{align} p_{\text{fh}}(R) & = \operatorname{Root}\left[ {\#}^3 \left(9R^6+18R^4+9R^2\right) \right.\nonumber\\ & \quad \left.+\, {\#}^2 \left(2R^6-150R^4-264R^2-128\right) + {\#} \left(96R^2-117R^4\right) - 18R^4 \ \& , \ j \right] , \end{align}

where $j=1$ if $R < \sqrt {\tfrac {1}{2} (59+11\sqrt {33})}$, and $j=3$ if $R> \sqrt {\tfrac {1}{2} (59+11\sqrt {33})}$. But the firehose threshold over beta is written almost as simply as (3.27)

(3.37)\begin{equation} \beta_{\text{fh}}(R) = \operatorname{Root}\left[ 4 {\#}^3+{\#}^2 \left(R^2-28\right) +60 {\#}-36 \ \& , \ j \right] , \end{equation}

where $j=1$ if $1 R < R_{\ast }=\tfrac {1}{2}\operatorname {Root}[{\#}^4-236 {\#}^2-2048\ \& , \ 2]$, and $j=3$ if $R> R_{\ast }$.

The stability zone bounded by the conditions $\beta <\beta _{\textrm {mm}}(R)$ and $\beta <\beta_{\text{fh}}(R)$ is shown in figure 3(b). Its width shrinks to zero both in the $R\to 1$ limit and in the $R\to \infty$ limit. The maximum width corresponds to the point of intersection of the curves $\beta =\beta _{\textrm {mm}}(R)$ and $\beta =\beta_{\text{fh}}(R)$, where $\beta =0.8386$, $R=2.071$. At the same time, it is worth noting that the maxima of these functions are respectively $\beta _{\textrm {mm}}(\infty )=8/9$ and $\beta_{\text{fh}}(1)=0.9468$. Comparing (a) and (b) in figure 3 one can see that, in model A3, this zone is noticeably larger than in model A2. As can be assumed by looking at the graphs in figure 5 in the next section, this fact is associated with the observation that the distribution function in model A3 is slightly closer to isotropic than in model A2.

Figure 5. Angular distribution of the fast ions in the A1, A2, A3 and A8 models for a set of parameters $R$ indicated in the graphs.

3.4. Angle distribution of fast ions

For an arbitrary index $n$, the oblique injection model reads

(3.38)\begin{gather} p_{\mathrel\perp} = p_{0} f_k(\psi )\, b^2\left(1-b\right)^{n-1}, \end{gather}

(3.39)\begin{gather}p_{\|} = p_{0} f_k(\psi )\, \frac{b}{n}\left(1-b\right)^{n}, \end{gather}

(3.40)\begin{gather}p_{0} = \frac{\beta }{ 2\left(1-\beta\right) \left(1-\sqrt{1-\beta}/R \right)^{n-1}} . \end{gather}

The threshold value of parameter $p_{0}$ with respect to the excitation of the mirror instability is given by

(3.41)\begin{equation} p_{\text{mm}}(R) = \left( \frac{n-2}{n+1} \right)^{2-n} . \end{equation}

It is not possible to write formulas for $\beta _{\textrm {mm}}(R)$, $p_{\text{fh}}(R)$, $\beta_{\text{fh}}(R)$ in compact form.

Let us try to find an answer to the question: How realistic are the analytical models of anisotropic pressure? Is it possible to specify the distribution function of fast ions, which forms the pressure profiles that are given by (3.1) and (3.2)?

It is known that if the distribution of fast ions is given by a function $F(\varepsilon,\mu )$ of two variables, the energy $\varepsilon$ and the magnetic moment $\mu$, then the dependence of the density $N$, transverse $p_{\mathrel \perp }$ and longitudinal pressure $p_{\|}$ on the magnetic field can be calculated through double integrals of the distribution function over the variables $\varepsilon$ and $\mu$ with different weight factors

(3.42)\begin{gather} N(B) = \frac{2 \sqrt{2} {\rm \pi}B}{m^{3/2}} \int_{0}^{\infty} \left(\int_{\epsilon/B_{R}}^{\epsilon/B} \frac{F}{\sqrt{\epsilon -\mu B}} \, \mathrm{d}\mu \right) \, \mathrm{d}\epsilon, \end{gather}

(3.43)\begin{gather}p_{\mathrel\perp}(B) = \frac{2 \sqrt{2} {\rm \pi}B^{2}}{m^{3/2}} \int_{0}^{\infty} \left(\int_{\epsilon/B_{R}}^{\epsilon/B} \frac{\mu F}{\sqrt{\epsilon -\mu B}} \, \mathrm{d}\mu \right) \, \mathrm{d}\epsilon, \end{gather}

(3.44)\begin{gather}p_{\|}(B) = \frac{4 \sqrt{2} {\rm \pi}B}{m^{3/2}} \int_{0}^{\infty} \left(\int_{\epsilon/B_{R}}^{\epsilon/B} \sqrt{\epsilon -\mu B} \, F\, \mathrm{d}\mu \right) \, \mathrm{d}\epsilon . \end{gather}

The inverse problem of recovering the distribution function $F$ depending on two variables from a given dependence of $p_{\mathrel \perp }(B)$ on one variable obviously does not have a unique solution. The situation will not change if $p_{\|}(B)$ is added to $p_{\mathrel \perp }(B)$, since these two functions are related by the longitudinal equilibrium equation. The function $N(B)$ is not yet known to us.

The situation changes radically if the distribution function can be rewritten in the form of $F=F(\varepsilon,\mu /\varepsilon )$, for example, $F=g(\varepsilon )\,F_{n}(\mu B_{R}/\varepsilon )$. Then the integrals are separated. By replacing $\mu =\varepsilon \sin ^{2}\theta /B_{\min }$, $B=B_{R}b$, $B_{R}=B_{\min }R$, $\sin ^{2}\theta =X/R$ in (3.43), one obtains

(3.45)\begin{equation} p_{\mathrel\perp}(b) = \frac{2\sqrt{2}{\rm \pi} b^{2}}{m^{3/2}} \left( \int_{0}^{\infty} \varepsilon^{3/2}g(\epsilon ) \, {\rm d}\epsilon \right) \left( \int_{1}^{1/b} \frac{X F_{n}(X)}{\sqrt{1 -b X}} \, {\rm d}X \right) . \end{equation}

The integral in the first pair of brackets gives a constant independent of $b$. Omitting this and other constants leads to the Volterra integral equation of the first kind (see, e.g. Polyanin & Manzhirov Reference Polyanin and Manzhirov2008, chap. 10) for determining the function $F_{n}(X)$

(3.46)\begin{equation} (1-b)^{n-1} = \int_{1}^{1/b} \frac{X F_{n}(X)}{\sqrt{1-b X}} \, \mathrm{d}{X} .\end{equation}

It is solved using the Laplace transform and the convolution theorem. To bring the last equation to the canonical formulation of the convolution theorem, one needs to make a few more changes of variables: $b=1/t$; $t=1+\tau$, $X=1+x$. This yields the equation

(3.47)\begin{equation} h_{n}(\tau) = \frac{\tau^{n-1}}{\left(1+\tau\right)^{n-1/2}} = \int_{0}^{\tau} K(\tau-x)\,Q_{n}(x) \mathrm{d}{x} , \end{equation}

where $K(\tau )=1/\sqrt {\tau }$ is the kernel of the integral equation, and $Q_{n}(x)=(x+1)F_{n}(x+1)$. By the convolution theorem, the Laplace image of the desired function $Q_{n}(x)$ is equal to the Laplace image of the function $h_{n}(\tau )$ on the left-hand side of (3.47) divided by the Laplace image of $K (\tau )$. After performing the inverse Laplace transform, it is possible to restore the function $F_{n}(X)$. The result takes an unexpectedly simple form

(3.48)\begin{equation} F_{n}(X) = \frac{\varGamma(n)}{\sqrt{\rm \pi}\varGamma(n-1/2)} \frac{(X-1)^{n-3/2}}{X^{n+1}} . \end{equation}

Recall that $X=R\sin ^{2}\theta$, where $\theta$ is the pitch angle in the median plane. Function (3.48) passes through a maximum at $X=2(n+1)/5$. In terms of the pitch angle, the last relation means that

(3.49)\begin{equation} \sin^{2}\theta = 2(n+1)/5R . \end{equation}

Thus, parameter $R$ can be interpreted as a measure of the slope of the NBI.

Examples of the angular distribution of charged particles are shown in figure 5 for the A1, A2, A3 and A8 pressure models. Comparing these figures, it can be concluded that the width of the angular distribution increases, and the degree of anisotropy in its intuitive interpretation decreases as the index $n$ gets larger.

Knowing the function $F_{n}(X)$, one can find the plasma density distribution

(3.50)\begin{gather} N = n_0 f_k(\psi) \frac{b}{2n}\left(1-b\right)^{n-1} \left(1+(2n-1)\,b\right) , \end{gather}

(3.51)\begin{gather}p_{0}/n_{0} = \left.\int_{0}^{\infty }\varepsilon^{3/2}g(\varepsilon )\,\mathrm{d}\varepsilon \right/ \int_{0}^{\infty }\varepsilon^{1/2}g(\varepsilon )\,\mathrm{d}\varepsilon . \end{gather}

In a similar way, it is possible to restore the angular part of the distribution function in model A1

(3.52)\begin{equation} F_{1}(X) = \frac{ X^2+3 \sqrt{X-1}\, X^2 \tan ^{{-}1} \left(\sqrt{X-1}\right) -X-2 }{2{\rm \pi} \sqrt{X-1}\, X^2} , \end{equation}

and then the dependence of density on the magnetic field

(3.53)\begin{equation} N = n_0 f_k(\psi)\, \frac{1}{2}\, (1-b) (b+3) . \end{equation}

4.1. Minimal vacuum gap

There is hardly any need to prove that the stabilizing effect of the lateral conducting wall is maximum when it is located closest to the surface of the plasma column. Therefore, as a first study of a new configuration, it is reasonable to carry out calculations of the critical beta with the conducting wall located as close as possible to the surface of the plasma column. For the pressure model A1, corresponding to the transverse NBI, such calculations are given in Kotelnikov et al. (Reference Kotelnikov, Prikhodko and Yakovlev2023) and Zeng & Kotelnikov (Reference Zeng and Kotelnikov2024). Figures 6 and 7 are compiled to compare the stabilizing effect of the lateral wall for the three pressure models: A1, A2, A3. They show a series of graphs for the case $\varLambda _{0}=500$ and illustrate the dependence of $\beta _{\textrm {cr}2}$ on $R$ and the lateral wall shape. Graphs in odd and even rows are drawn for proportional and straightened conducting walls, respectively.

Figure 6. Second critical beta vs $R$ in the limit $\varLambda \to \infty$ for three pressure models: A1 (a,d,g,j), A2 (b,e,h,k) and A3 (c,f,i,l). Stability zone of the ballooning rigid mode for a radial pressure profile with a given index $k$ is located above the margin curve, coloured as indicated in the legend under (j–l).

Figure 7. Second critical beta vs $R$ in the limit $\varLambda \to \infty$ for three pressure models: A1 (a,d,g,j), A2 (b,e,h,k) and A3 (c,f,i,l). Stability zone of the ballooning rigid mode for a radial pressure profile with a given index $k$ is located above the margin curve, coloured as indicated in the legend under (j–l).

Both figures confirm the previously discovered strong dependence of the critical beta $\beta _{\textrm {cr}2}$ on the shape of the radial pressure profile, represented by the index $k$. The region of stability with respect to the flute and ballooning disturbances in these figures is located above the marginal beta curve $\beta _{\textrm {cr}2}(R)$, the colour of which is associated with the index $k$ according to the legend under the bottom row of graphs in each figure. The areas of mirror and firehose instabilities are hatched by the scheme presented in figure 3. Minimal area of stability in respect to ballooning modes is shaded with the colour of the most upper curve. Most often, this is the blue colour corresponding to the most smooth radial profile $k=1$, but in figure 7(b,h) this blue zone is invisible since the corresponding blue curve is too short because it crosses the lower margin of the mirror mode. The PEK code quits above the mirror threshold where paraxial equilibrium is not possible.

Previously, it was found for model A1 (Kotelnikov et al. Reference Kotelnikov, Prikhodko and Yakovlev2023; Zeng & Kotelnikov Reference Zeng and Kotelnikov2024) that, when the end MHD stabilizers are switched off, the critical value $\beta _{\textrm {cr}2}$ depends only very weakly on both the mirror ratio $M$ and index $q$, which controls the axial profile of the magnetic field. For this reason, in figure 6, illustrating the dependence on the parameter $M$, only graphs with index $q=4$ are kept, and the set of values of $M$ is reduced to two, $M=4$ and $M=16$. Similarly, in figure 7, illustrating the dependence on $q$, only graphs with the mirror ratio $M=8$ and a pair of index values $q$ are kept, namely: $q=2$ and $q=8$. In general, all these graphs only confirm the conclusion about the weak influence of the magnetic field on the stabilizing properties of the lateral conducting wall within the same model of anisotropic pressure.

However, the differences between the three studied anisotropic pressure models A1, A2 and A3 are visible to the naked eye. These conclusions partially coincide with those of Kesner (Kesner Reference Kesner1985). He stated that the critical beta was insensitive to the axial profile of the magnetic field, but did not specify the value of the index $n$ in (3.2) used in his calculations.

In model A2, the stabilization boundary of ballooning instability is closely adjacent to the threshold of mirror instability, so that the region of joint stability $\beta _{\textrm {cr}2} < \beta < \beta _{\textrm {mm}}$ turns out to be very narrow. In model A3 this region is noticeably wider, especially in the region $R\lesssim 2$. In both models, the zone of stabilization of ballooning modes partially overlaps the region of firehose instability at $R\gtrsim 4$.

Thus, it can be concluded that the stability properties of an anisotropic plasma are very sensitive to details of the axial pressure profile.

In the most general terms, the last statement can hardly be disputed. At the same time, it should be noted that model A2 apparently does not adequately describe a real experiment. As follows from the discussion of (3.28), the point of origin of mirror instability in this model coincides with the turning point $b=1$ of fast ions, since the derivative ${\partial p_{\mathrel \perp }}/{\partial b}$ at this point experiences a jump as a result of mathematical idealization. A quick look at figure 5 shows that both the increase in anisotropy and the decrease in NBI tilt angle make it more difficult to stabilize an anisotropic plasma.

Comparison of graphs in odd and even rows of figures 6 and 7 reveals that the shape of the conducting chamber significantly deforms the shape of the $\beta _{\textrm {cr}2}(R)$ curves in the case of oblique NBI simulated by the models A2 and A3. For the model A1, Zeng & Kotelnikov (Reference Zeng and Kotelnikov2024) came to the opposite conclusion that the effect of the shape of the conducting chamber is insignificant. The apparent contradiction can be explained by the fact that the width of the gap between the conducting wall and the plasma at the location of the pressure peak plays a decisive role. In model A1 (transverse NBI), the pressure peak is located in the midplane of the trap. For the same value of the parameter $\varLambda _{0}$, the vacuum gap in the midplane is the same for the proportional and straightened chambers, so the difference in the gap width in other sections of the trap is not very significant. In the case of the A2 and A3 models (oblique NBI), the peak is located outside the midplane. The size of the gap at the location of the peak in the strengthened chamber will be larger than in the proportional chamber; accordingly, the stabilizing effect of the conducting chamber will be less.

Comparison of graphs in successive columns of figures 6 and 7 shows that the critical beta $\beta _{\textrm {cr}2}$ is definitely smaller in the case of oblique injection (which is good), but the stability zone disappears at large values of parameter $R$, since the plot of $\beta _{\textrm {cr}2}(R)$ crosses the mirror instability threshold $\beta _{\textrm {mm}}(R)$ at approximately $R\approx 3.5\unicode{x2013}4$. Considering that (3.49) relates the parameter $R$ to the injection angle $\theta _{\textrm {inj}}$, one obtains $\theta _{\textrm {inj}}\approx 40^{\circ }$ in model A2 and $\theta _{\textrm {inj}}\approx 47^{\circ }$ in model A3. In a straightened chamber, the $\beta _{\textrm {cr}2}$ curve intersects the $\beta _{\textrm {mm}}$ curve at a lower value of the parameter $R\approx 2$, which corresponds to $\theta _{\textrm {inj }}\approx 56^{\circ }$ in model A2 and $\theta _{\textrm {inj}}\approx 70^{\circ }$ in model A3.

The second observation is that the straightened chamber stabilizes the smoothest radial pressure profile with index $k=1$ noticeably worse in the sense that the range of values of the $R$ parameter at which this profile can be stabilized is noticeably narrower compared with the proportional chamber. In § 6, it is shown that, in the presence of the end MHD stabilizers, on the contrary, the smoothest radial profile is the easiest to stabilize because the unstable zones on the stability maps in figures 11–18 are the smallest for the $k=1$ radial profile.

It is useful to compare the above depicted calculations with the results of predecessors, in particular, with the works of Kesner and his co-authors (Kesner Reference Kesner1985; Li et al. Reference Li, Kesner and Lane1985, Reference Li, Kesner and Lane1987a, Reference Li, Kesner and LoDestrob). In all these papers, the authors analyse the plasma model with a sharp boundary, which corresponds to the infinite index $k=\infty$. Direct comparison is possible only with the first of the listed publications, and then only with certain reservations. In other works, the differences in the formulation of the problem are too great.

In his first work (Kesner Reference Kesner1985), Kesner approximated the vacuum magnetic field with a parabola. In the model of magnetic field (2.3), a parabola can approximate the field near the midplane in the case of $q=2$. Therefore, results can be compared in cases where the pressure peak is concentrated near the median plane, that is, at small values of the parameter $R\in \{1.1,1.2,1.3\}$. The necessary data can be extracted fromFootnote ² figure 2 of Kesner (Reference Kesner1985), which shows the calculation results (in our terms) for the model A1-LwPr and zero vacuum gap. Comparing these data with figure 2(a,d,g,j) of Kotelnikov et al. (Reference Kotelnikov, Zeng, Prikhodko, Yakovlev, Zhang, Chen and Yu2022b), one can see quite satisfactory agreement.

It is more difficult to compare the results of calculations in the case of oblique injection. Figure 3 of Kesner (Reference Kesner1985) shows the graph of $\beta _{\textrm {cr}2}$ depending on Kesner's anisotropy degree (3.29) in the range from $A_{{K}}=1$ to $A_{{K}} =6$. The explanation of that figure indicates that the plasma occupies the entire length of the trap up to the magnetic mirrors, i.e. $R=M$, but the magnetic field was still approximated by a parabola. The values of the parameters $M$, $R$, $n$ are not specified in the article, however, taking into account the definition (3.29) of the anisotropy parameter in that article, one can assume that there should be $n\gg M$ in order to have such values of $A_{{K}}$. Formally, this means that comparison with my calculations for $n=2$ (i.e. model A2) and $n=3$ (model A3) is impossible.

An alternative possibility to ‘organize’ a large parameter $A_{{K}}$ involves the simultaneous limits $R\to 1$ and $\beta \to 0$. Parameter $A_{{K}}$ may well fall into the range $1\unicode{x2013} 6$ for $n=2\unicode{x2013} 3$ if $\beta \ll 1$ and $R=1.1\unicode{x2013} 2$. In order to draw figure 8 in $\{A_{{K}}, \beta \}$ coordinates, suitable for comparison with figure 3 in Kesner (Reference Kesner1985), $\beta _{\textrm {cr}2}$ was calculated for a set of discrete values of $R$ with four values of the mirror ratio $M\in \{2, 4, 8, 16\}$. Comparing figure 8 and figure 3 of Kesner (Reference Kesner1985), one can see some qualitative discrepancies. The most evident of them is that, in figure 8(b,c), the width of the stable zone between the $\beta _{\textrm {cr}2}$ and $\beta _{\textrm {mm}}$ curves decreases as $A_{{K}}$ becomes smaller whereas figure 8 demonstrates an opposite tendency.

Figure 8. Second critical beta $\beta _{\textrm {cr}2}$ vs Kesner's degree of anisotropy $A_{{K}}$ defined by (3.29) as in Kesner (Reference Kesner1985) for three models of anisotropic pressure and four mirror ratios, $M\in \{4,8,16\}$. Dashed curve shows the mirror mode thresholds. Compare with figure 3 in Kesner (Reference Kesner1985).

It does not follow from this fact that Kesner's results are wrong. In fact, we do not quite understand his prerequisites because some important details are missed. In addition, the choice of parameters $A_{{K}}$ and $\beta$ for figure 3 of Kesner (Reference Kesner1985) axes is unsuccessful, since parameter $A_{{K}}$ itself depends on $\beta$.

4.2. Effect of the vacuum gap

The width of the vacuum gap between the lateral surface of the plasma column and the inner surface of the conducting chamber of a selected type is determined by the parameter

(4.2)\begin{equation} r_{w0}/a_{0}=\sqrt{(\varLambda_{0}+1)/(\varLambda_{0}-1)} . \end{equation}

In § 4.1 it was shown that, in the limit $r_{w0}/a_{0}\to 1$, the critical beta remains essentially unchanged when both the mirror ratio $M$ and the index $q$ change. This conclusion holds for any value of $r_{w0}/a_{0}$, so it is sufficient to show only one (say, the average) value of the index and one value of the mirror ratio. In addition, the range of changes in the parameter $R$ is limited below by the interval $R\in [1.1\ldots 4]$, since at $R\gtrsim 4$ the margin of stability with respect to ballooning oscillations lies inside the firehose instability zone.

Figures 9 and 10 contain series of graphs of $\beta _{\textrm {cr}2}$ vs ratio $r_{w0}/a_{0}$ at fixed mirror ratio $M = 8$ and fixed axial profile index $q=4$. Figure 9 is compiled for a proportional chamber LwPr, and figure 10 repeats the same graphs for the LwSt configuration.

Figure 9. Stability maps for the LwPr configuration and three pressure models: A1 (a,d,g,j,m), A2 (b,e,h,k,n) and A3 (c,f,i,l,o). The second critical beta is drawn as a function of $r_{w}/a$ for the model magnetic field (2.3) with index $q=4$, set of mirror ratios $R\in \{1.1,1.2,1.5,2,4,8\}$ at the turning point and fixed mirror ratio $M=8$. The stable zone for a radial pressure profile with an index $k$ is located above the curve $\beta _{\textrm {cr}2}$ coloured according to the legend under (m–o). Compare with figure 10.

Figure 10. Stability maps for the LwSt configuration and three pressure models: A1 (a,d,g,j,m), A2 (b,e,h,k,n) and A3 (c,f,i,l,o). The second critical beta is drawn as a function of $r_{w}/a$ for the model magnetic field (2.3) with index $q=4$, set of mirror ratios $R\in \{1.1,1.2,1.5,2,4,8\}$ at the turning point and fixed mirror ratio $M=8$. The stable zone for a radial pressure profile with an index $k$ is located above the curve $\beta _{\textrm {cr}2}$ coloured according to the legend under (m–o). Compare with figure 9.

Comparison of graphs within any row demonstrates a strong dependence of $\beta _{\textrm {cr}2}$ on the pressure model. It can be seen that, for $r_{w}/a\approx 2$, the lowest critical beta is much smaller in the case of the oblique NBI imitated by model A2 (middle column) compared with the transverse NBI described by model A1 (left column). In other words, with an oblique NBI it is easier to achieve a stable plasma confinement mode. The effect is even more significant in model A3 (right column). However, it is very difficult to stabilize a plasma with a maximally smooth radial pressure profile $k=1$ during oblique injection. In model A2, the blue curve corresponding to the index $k=1$ is completely absent from most graphs. In addition, in model A2, the boundary of the stability zone $\beta =\beta _{\textrm {cr}2}$ rests on the threshold of mirror instability $\beta =\beta _{\textrm {mm}}$ even for a not very large ratio $r_{w}/a$. As is noted above, model A2 is not realistic enough. Model A3 predicts a wider range of $r_{w}/a$ values at which stabilization by a properly designed conducting chamber is possible.

6.1. Wide pressure peak, $M=R$

The series figures 14–16 visualizes the calculation results for the special case $R=M$ when the hot plasma component occupies the entire mirror trap from plug to plug. Initially, this case was highlighted among many others in Kotelnikov et al. (Reference Kotelnikov, Chen, Bagryansky, Sudnikov, Zeng, Yakovlev, Wang, Ivanov and Wu2020) in order to compare the stability of anisotropic plasma, which is formed under transverse NBI, with the stability of isotropic plasma. In the A1 model, the $R=M$ variant corresponds to the minimum anisotropy. However, in models A2 and A3, parameter $R$ cannot serve as a measure of anisotropy, as shown in § 3.4, but it is related to the injection angle $\theta _{\textrm {inj}}$ by (3.49). Too large values of $R$, $R\gg 2(n+1)/5$ correspond to too small angles of injection $\theta _{\textrm {inj}}$, which can hardly be of interest from a practical point of view. Taking into account this explanation, the range of the parameter $R$ is further reduced compared with the range that was adopted in previously published analogues for the transverse NBI model, in particular in figure 9 of Kotelnikov et al. (Reference Kotelnikov, Prikhodko and Yakovlev2023).

Figure 14. Stability maps for model magnetic field (2.3) and anisotropic plasma pressure models (3.10) (a–f), (3.18) (g–l) and (3.28) (m–r) at combined MHD stabilization by lateral wall and end MHD anchors, $q\in \{2, 4, 8\}$, $M=R=16$. The unstable zone is located between the lower $\beta _{\textrm {cr}1}(r_{w}/a_{0})$ and upper branches $\beta _{\textrm {cr}2}(r_{w}/a_{0})$ of every curve. Correspondence of the index $k$ to the colour of the curves is shown at the bottom of the figure. Shaded common zone of stability lies to the left of the curve for the most steep radial pressure profile ($k = \infty$). (a,d,g,j,m,p) Show the maps for a ‘parabolic’ magnetic field with the index $q=2$, (c,f,i,l,o,r) show the maps for the ‘quasi-flat hole’ magnetic field with $q=8$. Panels (a–c,g–i,m–o) and (d–f,j–l,p–r) show maps for the CwPr and CwSt configurations, respectively.

Figure 15. Same as in figure 14 but for $M=R=8$.

Figure 16. Same as in figure 14 but for $M=R=4$. Rows for the A2 and A3 models are dropped since the rigid ballooning mode is stable in the entire region below the mirror instability threshold.

In the first two rows, figures 14–16 present the stability maps for the A1 model, corresponding to the transverse NBI. In figures 14 and 15, composed respectively for $M=16$ and $M=8$, they are followed by two rows of graphs for the oblique NBI model A2 and the last two rows contain maps for oblique NBI model A3. Figure 16 has only two tows since the plasma produced by oblique NBI is stable against ballooning perturbation in the entire range of $\beta$ below the mirror instability threshold.

The difference between the first two rows and the four following ones is so obvious that it almost could not be commented on. With oblique injection, there is practically no upper stability zone $\beta _{\textrm {cr}2}$, which was present both during transverse NBI and in isotropic plasma, as shown in figure 10 in Kotelnikov et al. (Reference Kotelnikov, Prikhodko and Yakovlev2023). It can be said that, with oblique NBI, the upper stability zone is absorbed into the region of mirror mode instability.

One can also notice a significant decrease in the value of $\beta _{\textrm {cr}1}$ for all radial pressure profiles. This effect is accompanied by the appearance of an instability zone for smooth radial profiles (with indices $k=1$, $k=2$) in the range of small values of parameter $r_{w}/a$ at which such zones were not present in the A1 model. The brown hatching from the third to sixth rows means that, in the A2 and A3 models, the margin of the stability zone of the rigid ballooning mode passes entirely inside the zone of firehose MHD instability. It is important to emphasize that the threshold of firehose instability is determined by the criterion (3.6), which is obtained for a homogeneous magnetic field. In an inhomogeneous magnetic field, the threshold should be slightly lower, as shown in V. Mirnov's dissertation (Mirnov Reference Mirnov1986). We are not aware of experiments with plasma in open traps in which manifestations of hose instability would be reliably recorded. Therefore, it would be premature to assert that a transition beyond the formal boundary of firehose instability is not possible.

As for the threshold of mirror instability, the PEK code is currently unable to perform calculations in the area above its threshold.

As can be seen from the analysis of figures 14–16, with joint stabilization by the lateral wall and end MHD stabilizers, the instability zone $\beta _{\textrm {cr}1} <\beta < \beta _{\textrm {cr}2}$ decreases as $q$ increases. In other words, open traps with short magnetic plugs are more stable than traps in which the magnetic field increases smoothly as you move from the centre of the trap to the magnetic plugs. This observation is consistent with the long-standing calculations of V. Mirnov and O. Bushkova. They calculated the threshold of ballooning instability with respect to small-scale disturbances (Bushkova & Mirnov Reference Bushkova and Mirnov1986), ignoring the effects of the finite Larmor radius. Later, the results of those calculations were confirmed in the work (Kotelnikov et al. Reference Kotelnikov, Lizunov and Zeng2022a).

With fixed parameters $q$, $M$, $R$, the instability zone is maximal for the steepest radial pressure profile ($k=\infty$) and may be absent altogether for smooth profiles ($k=1$, $k=2$). Without stabilization by the end MHD anchors, the opposite situation occurs: the stability zone is smaller and may be absent altogether for smooth profiles.

6.2. Effect of pressure peak width, $1< R< M$

The case $R=M$, considered in § 6.1, corresponds to the widest pressure profile along the trap axis. Recall that the parameter $R$ in the magnetic field model (2.3) makes sense of a plug ratio in the section of an open trap where the pressure of hot ions decreases to almost zero. Thus, parameter $R$ is a characteristic of the extent of the plasma pressure peak localization region, which is universally suitable for all A1, A2, A3, $\ldots$ anisotropic pressure models studied so far. In addition, in the A1 model, which simulates transverse NBI, parameter $R$ characterizes the degree of plasma anisotropy, whereas in the case of oblique NBI, described by the A2 and A3 models, it is associated with the angle of inclination of the NBI to the direction of the magnetic field.

To study the effect of the extent of the region occupied by fast ions, which are formed during the injection of beams of neutral atoms, in real experiments at the GDT facility at the Budker Institute of Nuclear Physics, the magnetic field is reprofiled in the axial direction, forming a short well (Shmigelsky et al. Reference Shmigelsky, Lizunov, Meyster, Pinzhenin, Solomakhin and Yakovlev2024). A similar operation in the numerical study with the model field (2.3) in use involves increasing parameters $q$ and $M$. As can be seen from figure 1, short magnetic plugs as in the GDT correspond to higher values of $q$ and $M$ than those adopted in the current and earlier works. Corresponding calculations for model magnetic field (2.3) are planned for publication together with calculations for the real magnetic field in GDT and ALIANCE facilities.

Maps of stability for the case $q=4$, $M=16$ and five values of the parameter $R\in {1.2, 1.5, 2, 4, 8}$ are presented in figure 17 for the proportional lateral wall chamber and figure 18 for the straightened chamber. The first thing that catches eye when analysing these figures is the absence of an unstable zone at small values of the parameter $R$ for all three models of anisotropic pressure A1, A2, A3, and the marginal value of $R$ for transverse NBI is noticeably smaller than for oblique NBI. From this point of view, oblique NBI appears to be preferable to transverse NBI.

Figure 17. Stability maps vs ratio $r_{w}/a$ for the A1, A2 and A3 pressure models in the CwPr configuration simulating combined stabilization by the proportional lateral conducting chamber and end MHD anchors; $q=4$, $M=16$, $R\in \{1.2, 1.5, 2, 4, 8\}$. The instability zone is located between $\beta _{\textrm {cr}1}(r_{w}/a)$ (lower branch of the marginal curve) and $\beta _{\textrm {cr}2}(r_{w}/a)$ (upper branch of the same colour); in the case of inclined injection, which corresponds to the A2 and A3 models, the upper branch is completely or partially absorbed by the region of mirror instability; the stability zone is shaded for a plasma with a sharp boundary ($k=\infty$), for which it has the minimum dimensions.

Figure 18. Same as in figure 17 but for CwSt configuration simulating combined stabilization by the straightened lateral conducting wall and the end MHD anchors.

The second conclusion that arises when comparing graphs in the same row  in these figures is that transverse NBI, with other things being equal, allows us to achieve higher values of relative pressure $\beta$ than oblique NBI. This circumstance is due to the fact that the limiting beta in the case of oblique injection is limited by the mirror instability, not the ballooning one. From this point of view, transverse NBI appears to be preferable to oblique NBI.