On the Stability of Small-Scale Ballooning Modes in Mirror Traps

On the stability of small-scale ballooning modes in mirror traps

Igor Kotelnikov∗ Budker Institute of Nuclear Physics SB RAS, Novosibirsk, 630090, Russia and Novosibirsk State University, Novosibirsk, 630090, Russia

Qiusun Zeng Institute of Nuclear Energy Safety Technology HFIPS CAS, Hefei, China (Dated: December 1, 2020) It is shown that steepening of the radial plasma pressure profile leads to a decrease in the critical value of beta, above which small-scale balloon-type perturbations in a mirror trap become unstable. This means that small-scale ballooning instability leads to a smoothing of the radial plasma profile. This fact seems to have received little attention in the available publications. The critical beta values for the real magnetic field of the gas-dynamic trap was also calculated. In the best configuration the critical beta 0.72 is obtained for a plasma with a parabolic radial pressure profile.

I. INTRODUCTION ion Larmor radius) [23]. Nevertheless, even with the listed limitations, the energy principle allows important conclusions There are a number of publications, in one way or another, to be drawn. In particular, this article will show that small- related to the stability of ballooning MHD perturbations in scale ballooning instability leads to a smoothing of the radial open traps [1–35]. Most of them were published in the 80s of plasma pressure profile. the last century. In the next decades, interest in the problem of First part of the study undertaken in the present paper is ballooning instability in open traps significantly weakened (in based on a simplified derivation of the equation for ballooning contrast to what is happening in tokamaks, see e.g. [36–38]), oscillations from the energy principle performed by Dmitri which was a consequence of the termination of the TMX (Tan- Ryutov and Gennady Stupakov [1,2]. The simplification in- dem Mirror eXperiment) and MFTF-B (Mirror Fusion Test cludes (a) the use of the paraxial approximation, and (b) the Facility B) projects in the USA in 1986 [39]. However, the use of the low beta approximation, β 1, which allows us achievement of a high electron temperature and high beta (β, to assume that the magnetic field both inside and outside the the ratio of the plasma pressure to the magnetic field pressure) plasma column is close to the vacuum one. in the Gas-Dynamic Trap (GDT) at the Budker Institute of A more accurate calculation, which used only the first Nuclear Physics in Novosibirsk [40–48] and the emergence of (paraxial) approximation (a), was made by Ol’ga Bushkova new ideas [49] and new projects [50, 51] makes us rethink old and Vladimir Mirnov [3,4]. These authors used the balloon results. equation that was derived by William Newcomb in [5]. The We are planning to prepare a series of three articles and, same equation with a short derivation is given in [8] for the following the logic of previous research, in the first article case of isotropic plasma. It has been studied in [6], and the we turn to the study of the stability of small-scale ballooning effect of strong anisotropy created by a rigid ring of electrons disturbances, which are characterized by large values of the has been included into consideration in [7]. azimuthal wave number m, m 1. In this first article, we re- In the classical paper [9] by Ira Bernstein et al, where the strict ourselves to the paraxial approximation, assuming that energy principle was formulated in its modern form, in the last the transverse size a of the plasma in the trap is small com- section a relatively simple expression for the potential energy pared to its length L, a L. In the next article, the effects of flute and ballooning perturbations in a non-paraxial axially of non-paraxiality will be taken into account, and in the third symmetric open trap is given in the approximation of infinite article, the stability of the rigid ballooning mode m = 1 will azimuthal number m → ∞, when the disturbance is like a be examined. knife blade. Attempts to calculate the equilibrium and stabil- We will use the canonical method for studying ballooning ity of nonparaxial systems were made by Vladimir Arsenin et instability, historically associated with the energy principle in al. [14–19], however, no significant progress seems to have the approximation of one-fluid ideal magnetohydrodynamics been made. [9]. It should be understood that this method has limited ap- It is commonly believed that FLR effects [20–22] are capa- arXiv:2011.14555v1 [physics.plasm-ph] 30 Nov 2020 plicability. In particular, it assumes that in equilibrium the ble to stabilize all modes with the azimuthal numbers m ≥ 2 plasma is stationary, that is, it does not rotate around its axis and impose on the m = 1 mode the shape of a rigid (“solid- and does not flows out through magnetic mirrors. In addition, state”) displacement without deformation of the plasma den- the energy principle ignores the effects of a finite Larmor ra- sity profile in each section of the plasma column. More ac- dius (FLR effects) [20–22], which should stabilize small-scale curate treatment is given in [23]. It leads the conclusion that 2 2 flute and ballooning perturbations with m > a /(Lρi)(ρi is the ballooning modes with m > a /Lρi are stable for any β. The stability of ballooning perturbations with m = 1 was considered in Refs. [24–35]. These and some other papers on the stability of the m = 1 ballooning mode and wall stabilization ∗ [email protected] will be reviewed in a forthcoming paper. 2

provide stability of the flute modes due to “line-tying” [52? ]. ϰ<0 Thus, we narrow the class of possible perturbations allowed to compete while minimizing the potential energy of perturbations, so the found values of βcrit will actually give an upper ϰ>0 bound for this value. At the same time, the difference between the true values of βcrit from this upper boundary should z be small, since behind the plug there is a deep (and plasma- -L 0 L filled) magnetic well (a region of favorable curvature), as a result of which the magnetic field lines are actually rigidly fixed in end plugs, returning us to the condition of being frozen into Figure 1. Ballooning perturbations distort the magnetic field and are localized in the region of unfavorable curvature, where κ < 0; the ends. the plasma tube, shown by the dotted line, floats up in the region of In accordance with the real situation for devices of the GDT negative curvature away from the plasma axis, but is fixed at the ends type, we will assume (following again Refs. [1,2] and many of the trap. subsequent papers) that the magnetic field lines make a small angle with the magnetic axis of the system (paraxial approximation) and that the distance between the magnetic plugs 2L is In what follows, we will adhere to the following order of greater than the total plugs length 2Lm, which can be separated presentation. The equation for ballooning oscillations in the by a quasi-uniform part. The small parameter characterizing low-pressure plasma limit (β 1) is written in sectionII. Its the accuracy of the paraxial approximation is a/Lm, where a is solutions for some model magnetic field and pressure profiles the transverse plasma size in the homogeneous part of the trap are presented in section III. Here the critical value of beta βcrit and Lm < L is the plug length. In the paraxial approximation, is calculated, above which ballooning disturbances become integration over the field line (over dl ) can be replaced by in- unstable. It turned out that the calculated values of βcrit in tegration over the trap axis (over dz ). Let us agree that the co- some cases exceed one. This means that the assumption β ordinates z = ±L correspond to the magnetic mirrors, and the 1, made when deriving the simplified equation, does not hold. homogeneous part of the trap occupies the region |z| < L− Lm. Nevertheless, the low-pressure plasma approximation gives a Assuming also that β 1 and the magnetic field even inside quantitatively correct result for a plasma with a radial pressure the plasma is close to vacuum field, we will assume that it is profile close to a stepwise one, for which the critical beta is described with sufficient accuracy by the function B = B(z), small. which specifies the axial profile of the magnetic field on the Equations in the paraxial approximation for small-scale trap axis. In this approximation, the reduced (that is, divided ballooning perturbations in an axially symmetric plasma with by 2π) magnetic flux at a distance r from the axis is equal to finite pressure (β . 1) and the results obtained in [3,4] for a Ψ = r2B(z)/2, and the field line equation model magnetic field are extended in SectionIV. In SectionV p critical beta is computed for realistic magnetic field of the gas- r(z) = r0 q(z), q(z) = B0/B(z) (2) dynamic trap. Main results are summarized in SectionVI. is obtained from the condition of constancy of the magnetic 2 flux inside the flux tube Ψ = r0 B0/2 = const, where r0 and B0 II. EQUATION OF BALLOONING MODES IN have the meaning of the radius of the flux lines and values of LOW-PRESSURE PLASMA the magnetic field in the median plane z = 0. For perturbations of the flute type in an axially symmetric In contrast to flute perturbations, ballooning modes are lo- mirror cell, for the projection of the displacement vector ξ calized in the region of unfavorable curvature κ of magnetic normal to the field line, the following relation holds: field lines (where κ < 0), distorting the magnetic field approx- ξnrB = ξ0r0B0 = const . (3) imately as shown in Fig.1, approaching zero amplitude in the region of favorable curvature (where κ > 0). Deformation of Hence, in the paraxial approximation, we find magnetic field lines requires energy expenditure, the source of which is the internal plasma energy p/(γ − 1), where p is ξn(z) = ξ0 q(z) ≡ ξfl(z), (4) the plasma pressure and γ is the specific heat ratio; therefore where the amplitude of the flux tube displacement ξ = ballooning-type disturbances are unstable only at a finite value 0 ξ (Ψ, θ) is assumed to be a function of the magnetic flux Ψ of relative plasma pressure β = 8πp/B2, exceeding some limit 0 and the azimuthal angle θ. In the case of a small-scale per- βcrit: turbation, the ξ0 amplitude is sharply dived (localized) near some values of Ψ and θ , but for the time being this fact is β > βcrit (unstable). (1) ∗ ∗ not essential for subsequent calculations. The goal of exact theory is to calculate the βcrit parameter, The displacement (4) does not satisfy the condition of freez- but we will restrict ourselves to simple reasoning and order of ing ξn(±L) = 0 at the ends, therefore instead of (4) one should magnitude estimates. In particular, for simplicity, we assume consider a displacement of a more general form (as it was done in [1,2] and many other papers) that ideally conducting ends are placed directly in the plugs in order to ξn(z) = α(z) ξfl(z), (5) 3 where α(z) is a still unknown dimensionless function. We im- 1.0 pose two boundary conditions on it: 0 0.8 α(±L) = 0, α(0) = 1. (6) ′′ α , 0.6 q3 q′′ -2 q 3

The ﬁrst of them ensures the fulﬁllment of the condition of max

being frozen into the ends, and the second has the meaning 0.4 α q 10 B / -4 of the normalization condition, it fixes the displacement at the B center of the trap. Taking into account the symmetry of the 0.2 problem, we assume that the sought function α(z) is even, that 0.0 -6 is, α(−z) = α(z), so it suffices to find a solution for only one 0 L L L half of the trap, say, for 0 < z < L. An odd solution α(−z) = - m −α(z) also exists. It is obtained for the boundary conditions z α(±L) = α(0) = 0, but it corresponds to a larger value of βcrit. As shown in [53, §29] (see also [1,2]), the energy of the Figure 2. Magnetic field (12), effective potential q3q00 and eigen- ballooning-type perturbation is function α(z) in the Sturm-Liouville problem for the case K = 15, L = 6 , ∆L = 1; the dotted line shows the graph of the function (13) ! Z ! 1 n o for Lm = 2.2∆L. W = dθ dΨ ξ2B dz α02 + 2βq3q00α2 , F 8π 0 0 (7) " with the above boundary conditions (6), is reduced to the where quantum mechanical problem of determining the conditions for the occurrence of a zero energy level in the potential . β = dθ dΨ ξ2 β(Ψ) dθ dΨ ξ2, (8)  0 0 ∞, if z < 0;  8πΨ dp V(z) = 2βq3q00, if 0 < z < L; (11) " β(Ψ) = − " , (9)  2 ∞ B0 dΨ , if z > L. and the prime denotes the derivative with respect to z. We can also say that βcrit is the eigenvalue of the classical According to the energy principle [9], the system is stable Sturm-Liouville problem for Eq. (10) with boundary condi- if the minimum value of the integral (7) is greater than zero. tions (6), and the solution to this problem α(z) is an eigenfunc- The factor inside the first pair of parentheses in (7) is certainly tion. It is easy to prove that the substitution of the eigenfunc- greater than zero, so the sign of WF is determined by the sign tion and the eigenvalue in the integral (7) makes the energy of the integral inside the second pair of parentheses. To find of the ballooning disturbance to zero, which, in accordance the minimum of WF , we take the first variation of the integral with the energy principle, corresponds to the marginal state (7), between stability and instability. The value of βcrit is calculated in Section III, and the actual Z L n o δ dz α02 + 2βα2q3q00 = value of the β parameter to be compared with the critical value 0 depends on the spacial profile of function ξ0 = ξ0(Ψ, θ) and Z L n o pressure profile p(Ψ). = 2 dz α0 δα0 + 2βq3q00α δα , 0 and set it to zero. In this way we obtain an equation that the III. CRITICAL BETA IN LOW PRESSURE PLASMA extremal must satisfy, that is, the function α(z), which min- imizes the perturbation energy WF . Since the values of the For the first example, let us choose the dependence of the function α(z) at the ends of the integration interval are given magnetic field on the coordinate z on the trap axis in the form by the boundary conditions (6), calculating the variation, we −(z−L)2/∆L2 −(L+z)2/∆L2 must assume that δα(0) = δα(L) = 0. Taking this fact into B(z)/B0 = 1 + (K − 1) e + e , (12) account and integrating by parts, we get the integral where K has the meaning of the mirror ratio, and 2L is the dis- Z L n 00 3 00 o tance between the magnetic mirrors. For such a field, the ef- 2 dz −α + 2βq q α δα, fective potential function q3q00 is approximately equal to zero 0 outside the plug (for 0 < |z| < L − Lm) and rapidly decreases which vanishes for any variation of δα if deep into the plug along with the growth of the mirror ratio 2 B/B0 = 1/q , as shown in Figure2. There is a hole at the 00 3 00 α − 2βq q α = 0. (10) entrance to the magnetic mirror on the q3q00 graph. Its width is approximately equal to Lm, and the well itself is located in As one can see, the calculation of the critical value βcrit of the the region of the plug, adjacent to the uniform magnetic field, parameter β, for which there is a nonzero solution to Eq. (10) where B/B0 ∼ 1 and z ≈ L − Lm. Outside the region of the 4

1.0 the parameter beta in the traditional sense 0 8πp 0.8 β 0 , 0 = 2 (16)

′′ B

α , 0 0.6 q -0.5 3 max where p has the meaning of the plasma pressure on its axis at 0.4 3 ′′ 0 q q q 10 B / α Ψ = 0. In the particular case when the pressure has a power- B -1.0 0.2 law proﬁle

0.0 k k -1.5 p(Ψ) = p0 (1 − Ψ /Ψa) (17) 0 L-L m L for Ψ < Ψa, and p(Ψ) = 0 for Ψ > Ψa, the maximum value z of β(Ψ) is reached on the boundary field line Ψ = Ψa, where β(Ψ ) = kβ . If k = 1 (pressure profile parabolic in r), the Figure 3. Mirror cell is a system of two coaxial round coils. Mag- a 0 equality β(Ψ ) = β holds, but for a steeper profile with k > 1 netic field (15), effective potential q3q00 and eigenfunction α(z) in the a 0 Sturm-Liouville problem for the case K = 15, L = 6 , b = 0.6, the value β(Ψa) is greater than β0. In the same way, for a steep pressure profile close to a step, the value of β can be much Lm ≈ 3b. larger than β0, so a situation is possible where β ∼ 1 whereas β0 1 and the vacuum magnetic field approximation can be potential well, the equation (14) reduces to the trivial equality completely valid. α00 = 0, so there α(z) will be approximately a linear function The stepped pressure profile can be considered as the limit of the coordinate z. On both sides of the potential well, the of a trapezoidal distribution close to a step with a small but derivative α0 is a constant, but the constant changes in the re- finite width of the boundary layer ∆Ψ. For definiteness, let gion of the well. Let us construct a test function from linear us assume that the plasma pressure is equal to p0 at Ψ < Ψa functions and decreases linearly to zero in the boundary layer with the  width ∆Ψ. The result of calculating β depends on the ratio 1 |z| ≤ L − Lm, of ∆Ψ and the size δΨ of the perturbation with respect to the α1(z) =  L−|z| (13)  L − Lm < |z| ≤ L, variable Ψ.  Lm If the size of the flute tube is small compared to the width of possessing the indicated properties, we substitute it into the the border, δΨ ∆Ψ, the local parameter (9) in the integral integral (7) and equate the result to zero. From the resulting (8) can be considered constant across the size of the tube, so equation, we find the approximate value of the critical beta: that β ≈ β(Ψ∗), where Ψ∗ is the value of Ψ on the field line, near which the disturbance is localized. Local value 1Z L . Z L − 0 2 2 3 00 βcrit = dz α1 dz α1q q . (14) Ψ 2 0 0 β(Ψ ) ≈ β a (18) ∗ 0 ∆Ψ For the magnetic field (12), the calculated value β = 0.580 crit in the boundary layer on the field line, where the disturbance only slightly differs from the exact eigenvalue βcrit = 0.597. The graph of the eigenfunction α(z) is shown in Fig.2, where is localized, is large for a small boundary width and can easily exceed the critical value. Consequently, small-scale pertur- the dashed line shows the test function α1(z). Calculations show that the critical beta is very sensitive to bations will be unstable even if the plasma pressure is much β the magnetic field profile. This fact is confirmed by the second lower than the magnetic field pressure, ie, 0 1. example with a magnetic field However, small-scale oscillations are unlikely to cause great damage to the equilibrium plasma configuration. Large-   B(z)  b3 b3  scale disturbances are more dangerous. If δΨ ∆Ψ, per- = 1 + (K − 1)  +  ,  3/2 3/2  forming integration over Ψ in the formula (8), we can assume B0 b2 + (z − L)2 b2 + (L + z)2 (15) that which simulates a mirror cell composed of two coils. In this β(Ψ) = β0 Ψaδ(Ψ − Ψa). (19) example, the critical value βcrit = 1.18 is formally greater than one. Substitution of (19) into the formula (8) results in the estima- Let us discuss the results obtained. The calculated value tion of βcrit turned out to be close to one or even more than one, whereas, starting to derive the equation of ballooning oscil- Ψ β ≈ β a . (20) lations in SectionII, we used the approximation β 1 and 0 δΨ assumed the magnetic field to be vacuum. Therefore, this hy- pothesis of ours was not entirely correct. Nevertheless, our It can be seen from it that β decreases with an increase in work was not entirely useless. the scale of the disturbance; therefore, the largest-scale per- First, note that the β parameter, like the β(Ψ) function, turbations for a given value of β0 can be stable, while small- which are defined by the formulas (8) and (9), are not equal to scale disturbances will be unstable. Thus, the development 5 of ballooning instability will result in the establishment of a where ψ = Ψ/Ψa, “smooth” radial pressure profile. β = 8πp /B2 , f (ψ) = p(ψ)/p , (25a) There is also a second conclusion, which can be drawn from 0 0 vac0 0 the fact established in the above, that in the approximation of b = B/Bvac0, bvac = Bvac/Bvac0, (25b) q a vacuum magnetic field it is formally βcrit & 1. Now we can 2 b(ψ, z) = bvac(z) − β0 f (ψ), (25c) assert that the critical β0 in an axially symmetric mirror cell Z ψ 0 with a smooth pressure profile is not small in comparison with 2 dψ unity. To correctly calculate the critical β it is necessary to r (ψ, z) = 2 p . (25d) 0 0 b2 (z) − β f (ψ0) abandon the vacuum field approximation, which is also called vac 0 the low-pressure approximation. The necessary calculations Varying the integral (24) leads to the equation are relatively easy to perform for paraxial mirror traps. These ! d 1 dX r00 d f are described in the next section. + β X = 0. (26) dz r2b dz 0 rb2 dψ The same equation can be deduced from Eq. (18) in Ref. [8] if we put zero frequency, ω = 0. Calculations were performed IV. PARAXIAL MIRROR TRAP for the pressure profile 2 In the paraxial approximation, the potential energy of bal- f (ψ) = 1 − ψ , (27) looning perturbations is found in the article by William New- which allows one to calculate the integral (25d) and write the comb [5]. For an axially symmetric open trap with anisotropic equation of the field line: plasma, Newcomb gives the following expression (his formula √ p 2 2 177) 2 bvac(z) − β0 1 − ψ + β0 ψ r2(ψ, z) = √ log . (28) p 2 β0 b (z) − β0 " 02 ! # vac 1 Q X 2 02 2κ ∂p 2 WF = dθ dΨ dz + r Y − X . The critical beta was calculated for the boundary field line at 2 B r2B2 rB2 ∂Ψ ψ = 0.99 for a family of axial profiles of the vacuum magnetic (21) $ field

h −γ/2 µi−2/γ It is assumed here that the prime denotes the partial derivative bvac(z) = 1 − 1 − K z (29) with respect to z, and all functions except for Bvac = Bvac(z), X = X(θ, Ψ, z), and Y = Y(θ, Ψ, z), depend on Ψ and z, with mirror ratio K = 100 and a set of parameters µ and γ. In order to check our code, we recalculate their results assuming Z Ψ the boundary conditions 2 dΨ 2 2 r = 2 , B = Bvac(z) − 8πp⊥, (22a) 0 B(Ψ, z) X(0) = 1, X(1) = 0 (30) 2 p⊥ + pk Q = B + 4π p⊥ − pk , p = , (22b) and obtained similar results which are listed in TableI. 2 The performed calculations show that the profile (29), ξθ which is optimal for stabilizing flute modes at µ = 1, γ = 2, X = ξ · ∇Ψ = rBξn, Y = ξ · ∇θ = , (22c) r becomes unstable with respect to ballooning vibrations at 00 κ = r . (22d) β0 > 0.359. To increase the critical values of β0, Bushkova and Mirnov proposed to go to steeper axial profiles of the Newcomb pointed out that the function Q is positive under the magnetic field described by Eq. (29) with µ, γ ≈ 5 ÷ 6. In condition of stabilization of the hose instability this way, one can raise βcrit to the values above 0.7. To study the effect of the radial plasma pressure profile, we

βk > 2 + β⊥, (23) calculated the critical beta values for a smoother pressure profile, described by the function f (ψ) = 1 − ψ, and a steeper pressure profile, described by f (ψ) = 1 − ψ4, with the same which was always assumed to be fulfilled. Otherwise, the first set of values for the parameters µ and γ as in TableI. The cal- term in square brackets in the integrand (21) becomes negative culation results are presented respectively in TablesII and III. (destabilizing), like the second term. 00 As would be expected from the discussion in Section III, the Minimizing WF over Y gives Y = 0, which allows us to 02 smooth plasma pressure profile is more robust against small- consider the term with Y in (21) as zero. The calculations scale ballooning disturbances. The calculation for a parabolic for this case were performed by Bushkova and Mirnov [3,4]. radial pressure profile (TableII) gave very high values of beta They investigated the limit of isotropic plasma, when p⊥ = approaching unity. These values are obtained for large param- pk = p(ψ) and rewrote Eq. (21) in the dimensionless form eters µ and γ. They correspond to the axial profile of the mag- convenient for numerical calculations: netic field that grows very steeply near the magnetic mirror, Z " 02 00 # which can hardly be formed in a real mirror trap. We made ∆ψ X r d f 2 WF = dz − β0 X , (24) the calculation for such values of µ and γ just to compare our 8π r2b rb2 dψ results with the results of [3,4]. 6

Standard GDT γ \ µ 1 2 3 4 5 6 f (ψ) \ zend 329.5 cm 350 cm 400 cm 0.5 0.189 0.332 0.383 0.409 0.424 0.434 1 − ψ2 0.411 0.389 0.343 2 0.359 0.486 0.526 0.545 0.556 0.564 1 − ψ 0.719 0.690 0.621 6 0.595 0.682 0.706 0.718 0.724 0.729 1 − ψ4 0.229 0.215 0.186 Table I. critical beta for local perturbations on the extreme field line GDT with Short Pit 2 ψ = 1 for f (ψ) = 1 − ψ , mirror ratio K = 100, and different values f (ψ) \ zend 329.5 cm 350 cm 400 cm of the parameters µ and γ. 1 − ψ2 0.316 0.299 0.262 1 − ψ 0.579 0.552 0.491 γ \ µ 1 2 3 4 5 6 1 − ψ4 0.170 0.160 0.140 0.5 0.365 0.618 0.701 0.741 0.764 0.780 2 0.646 0.832 0.882 0.905 0.918 0.927 Table IV. Critical beta for local perturbations on the extreme field 6 0.911 0.982 0.995 0.999 1.00 1.00 line ψ = 1 for three radial pressure profiles f (ψ) and three positions zend of plasma limiter or receiver. Table II. Same as in TableI for more smooth radial plasma profile, f (ψ) = 1 − ψ. of neutral atoms at the angle of 45◦ to the axis of the trap. γ \ µ 1 2 3 4 5 6 0.5 0.0971 0.175 0.204 0.218 0.227 0.233 Due to the presence of fast ions, the plasma pressure in the 2 0.193 0.269 0.294 0.306 0.313 0.318 GDT is anisotropic, i.e. p⊥ , pk, and the function p⊥ depends 6 0.356 0.418 0.437 0.445 0.450 0.454 not only on the magnetic flux ψ, but also on the magnitude of the magnetic field B. In the central section of the GDT Table III. Same as in TableI for more sharp radial plasma profile, (i.e., between the magnetic mirrors), function p⊥(ψ, B) has a f (ψ) = 1 − ψ4. maximum near the stopping point of fast ions, and in the expander (i.e., outside the central section) it decreases roughly B(T) proportional to B. As a result of all this, it turns out that Eq. (22a) cannot be solved analytically for B and r, as in the 10 case of an isotropic plasma. Thus, calculating the critical beta 5 in anisotropic plasma requires much more complex calculations than the method described in SectionIV allows. To sim- plify our task, we ignore the fact of anisotropy and still use the 1 method from SectionIV, taking into account that the plasma 0.50 pressure varies only slightly along the magnetic field lines in the region |z| ≤ 400 cm, where the magnetic field is not less than in the middle of the central section at z = 0 cm. 0.10 Since the axial profile of the magnetic field along the GDT axis is rigidly set by the configuration of the magnetic sys- z(cm) 100 200 300 400 500 tem, the only free parameter that can affect the critical beta is the coordinate zend of an imaginary or real conducting plasma GDT with Short Pit Standard GDT diaphragm, on which the perturbation X vanishes. In other words, we calculated the limit for beta by solving Eq. (26) Figure 4. Axial profile of vacuum magnetic field in gas-dynamic with the boundary conditions trap. X(0) = 1, X(zend) = 0. (31) The results of calculations are presented in TableIV for two V. GAS-DYNAMIC TRAP magnetic field configurations and three values of zend. The coordinate zend = 400 cm corresponds to the placement of the The calculations performed in SectionIV for the model plasma receiver in the expander, and zend = 350 cm in the field (29) can be repeated for more realistic profiles of the magnetic plug. Finally, zend = 329.5 cm marks the coordinate vacuum magnetic field along the trap axis. As the last exam- of an actually existing conducting diaphragm, which is actu- ple, let us calculate the critical beta for the Gas-Dynamic Trap ally capable of playing the role of a stabilizer of small-scale [54, 55]. flute disturbances on the lateral surface of the plasma column. The vacuum magnetic field in the GDT is symmetric about Analysis of the data in the table shows that there is a very the equatorial plane z = 0. Fig.4 shows the axial profile of strong dependence of the critical value of beta on the steep- the magnetic field in one half of the GDT for two options for ness of the radial plasma pressure profile, but the dependence switching magnetic coils: the so-called standard version of the on the coordinate zend does not seem to be very significant. GDT and the GDT with a short pit (short mirror cell). In the The second point to pay attention to is a noticeable decrease second variant, a shallow local minimum of the magnetic field in critical beta in a GDT configuration with a short pit com- is formed near the stopping point of fast ions (z ≈ 140 cm). A pared to the standard version of GDT. This is the expected population of fast ions arises upon oblique injection of beams result. 7

ϰ(1/cm) the m = 1 rigid ballooning mode [33].

0.0002 VI. DISCUSSION 0.0001 We have shown that steepening of the radial plasma pres- z(cm) sure profile leads to a decrease in the critical value of beta, 100 200 300 400 500 above which small-scale balloon-type disturbances in a mirror -0.0001 trap become unstable. It means that small-scale ballooning in- -0.0002 stability leads to a smoothing of the radial plasma profile. This fact does not seem to have received due attention in the available publications. It is known that the study of the stability GDT with Short Pit Standard GDT of the global ballooning mode leads to the opposite conclusion. For example, Ref. [31] states that wall stabilization of Figure 5. Axial profile of the vacuum magnetic field curvature in the global ballooning mode is more effective for plasma with gas-dynamic trap along a field line that has unit radius at z = 0. a hollow pressure profile. We also calculated the critical beta values for the real magnetic field of the GDT in two configurations: a standard GDT and the GDT with a short well. For the version of the standard GDT, the calculation result turned out to be close to the On the other hand, comparison of the calculation results in results of calculations in the model used earlier by Bushkova tableIV for zend = 350 cm (they are in bold) with the calcu- and Mirnov [3,4]. In the best case, the critical beta is 0 .719. It lation results for the model field (29) leads to an unexpected is obtained for a plasma with a parabolic radial pressure pro- conclusion. When designing the GDT, the magnetic field was file, which was not analyzed in Refs. [3,4]. The maximum tuned so as to minimize the destabilizing contribution of the beta, which has been achieved in experiments at the GDT to central section to the Rosenbluth-Longmire criterion of flute date, is 0.6 [41]. stability [56]. Such a minimum is provided by a magnetic field with the profile that is described by Eq. (29) for µ = 1 and γ = 2. The corresponding critical beta values in tables ACKNOWLEDGMENTS I,II, III are also shown in bold. Due to the discrete structure of the magnetic system, the real magnetic field in the GDT is The work was financially supported by the Ministry of Ed- slightly rippled and only approximately follows the formula ucation and Science of the Russian Federation. This study (29). This is clearly seen in Fig.5, where it is shown that the was also supported by Chinese Academy of Sciences Pres- curvature of the field line oscillates, whereas according to the ident’s International Fellowship Initiative (PIFI) under the formula (29) it should be a smooth function. We expected that Grant No. 2019VMA0024 and Chinese Academy of Sci- curvature ripples would result in a decrease in critical beta, ences International Partnership Program under the Grant but according to tableIV, the critical beta turned out to be No. 116134KYSB20200001. The authors are grateful to slightly higher than in the tablesI,II and III. We currently Vladimir Mirnov for valuable explanations of the method he have no explanation for this fact, although it worth noting that used to solve Eq. (26), and to Dmitry Yakovlev for providing the magnetic field ripples are known to improve stability of the results of calculating the magnetic field in the GDT.

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