Negative specific heat of a magnetically self-confined plasma torus

Abstract

It is shown that the thermodynamic maximum-entropy principle predicts negative specific heat for a stationary, magnetically self-confined current-carrying plasma torus. Implications for the magnetic self-confinement of fusion plasma are considered.

The goal of the controlled thermonuclear fusion program is to make the energy source that powers our sun available to human society. Deep in our sun's interior, favorable conditions for the quasistationary nuclear burning of the solar plasma prevail as a result of the immense gravitational self-forces that keep this huge accumulation of matter together. Because gravitational self-confinement is not operative at the reactor and laboratory scale, alternate means of confinement have to be used to achieve sufficiently high plasma densities and temperatures in a reactor. In the perhaps most prominent stationary fusion-reactor scheme, the tokamak, strong electric ring currents are induced in an electrically neutral plasma to achieve axisymmetric magnetic self-confinement in a rotationally invariant toroidal vessel 𝒯. In a torus with a sufficiently large major axis, such a magnetic self-confinement mimics gravitational self-confinement on account of the Biot–Savart law, according to which in a system of parallel current filaments all filaments attract each other magnetically with the same force law as would be the case gravitationally in a system of parallel mass filaments. What makes the magnetic forces more attractive (in the double sense of this phrase) than gravity for laboratory purposes is their very much bigger coupling constant [≈10⁴⁰v²/c² for two electrons moving on parallel trajectories with speed v as measured in the laboratory; note that the even stronger (v²/c² is replaced by 1) electrostatic repulsive forces between the same two electrons are very effectively screened in a neutral plasma and are traditionally neglected to a good approximation]. Of course, the analogy does not extend to all aspects of plasma self-confinement. In particular, the solenoidal-vectorial character of stationary current densities necessitates the toroidal topology of magnetic self-confinement, whereas gravity not only allows but manifestly prefers spherical confinement over toroidal. Unfortunately, axisymmetric toroidal magnetic self-confinement is not known for its stability either. Although major efforts are devoted to the stabilization of the plasma configuration, a vast reservoir of instabilities capable of destroying the confinement has dramatically slowed down the development of an operating tokamak fusion reactor.

Matters are not exactly helped by the fact that our theoretical understanding of the physics on the various space-time scales that govern magnetic plasma confinement is still quite incomplete. In particular, although the solenoidal character of the magnetic induction together with the axisymmetry and stationarity of the law of momentum balance tell us that the poloidal magnetic flux function Ψ and the toroidal current density j must satisfy some local functional relation ℛ(Ψ, j, r) = 0 (in which r is the cylindrical distance from the axis of symmetry), which turns Ampere's law into some in-general nonlinear elliptic partial-differential equation for Ψ (known in the fusion literature as the Grad–Shafranov equation), the actual relation ℛ is not fixed (except for the explicit r dependence in ℛ). Some information about ℛ should be contained in the law of energy balance between current drive (through an applied toroidal electromotive force and other means), ohmic heating, and, ultimately, radiation losses. Unfortunately only the so-called classical and neoclassical transport coefficients have been computed in some detail (1), whereas the small-scale turbulent dissipation mechanisms in a tokamak plasma remain a largely challenging open problem. In this situation, theoreticians have been forced to rely on fair judgment and good taste when guessing some additional principle(s) that would effectively complete the characterization of the stationary, magnetically self-confined plasma torus in a tokamak.

A sizeable fraction of the literature uses a linear approximation of ℛ to get j ∝ Ψ, rendering the Grad–Shafranov equation linear, and some improved-accuracy modeling uses a third-order polynomial approximation of ℛ (2). Subsequently, the filter of linearized dynamical stability analysis, based mostly on macroscopic magnetofluid theory and mesoscopic kinetic theory, is applied to sort out unstable configurations. Although this approach has met with a certain limited success, one does not learn to what the approximations are approximate. Over the years a number of plasma theorists (3–8) have argued that an equilibrium thermodynamics-inspired maximum-entropy principle with a few global dynamical constraints§ should give answers close to the truth. In essence the various formulations in refs. 3–8 give for ℛ the answer j ∝ exp(Ψ), which leads to a nonlinear Grad–Shafranov equation that may have more than one solution Ψ, depending on the domain 𝒯, the boundary conditions for Ψ, and the values of the physical parameters of the problem. In addition, this approach provides a global stability criterion within the class of axisymmetric states satisfying the same dynamical constraints. Only those solutions that maximize the relevant relative-entropy functional will be globally stable.

Because in nonequilibrium statistical mechanics the maximum-entropy principle has not acquired a status anywhere near as fundamental as in equilibrium statistical mechanics, it is mandatory to register some arguments in its favor for the case at hand. Thus, the relation j ∝ exp(Ψ) has been shown to be almost universally singled out also by a truly dissipative Fokker–Planck approach to stationary, magnetically confined plasma (9, 10). Perhaps the most compelling reason to give it serious consideration, however, is the successful application of the maximum-entropy approach to the physically distinct but mathematically quite similar problems of stationary planar incompressible flows,¶ where the vorticity plays a role closely analogous to the current density in the plasma torus, and the strongly magnetized pure electron plasma in a circular cylinder,‖ where the charge density plays that role. In this spirit, we conducted a thorough investigation of the thermodynamic-type maximum-entropy approach to the magnetically self-confined stationary plasma torus (10).

A most curious finding of the study (10) is that the gravity-inspired toroidal magnetic plasma self-confinement scheme inherits from the stars their gravothermal negative specific heat.** This result is a little surprising, because it follows from what we said earlier that the plasma torus should actually more closely mimic a cylindrical caricature of a star, and the specific heat of a “cylindrical maximum-entropy star” (29) and its plasma physical clone, Bennett's cylindrical plasma beam (30, 31), is nonnegative! The existence of a maximum-entropy plasma torus with negative specific heat therefore is a truly nontrivial fact. The purpose of this article is to point out some potentially important consequences of our finding for plasma physical applications.‡‡ To pave the way for the discussion we first describe the model and our results.

The Model

Because the whole problem is rotationally invariant, we work with conventional cylindrical coordinates r, θ, z. The magnetic induction field decomposes accordingly as B = B_T + B_P, where B_T ∥ e_θ and B_P ⊥ e_θ. In an actual tokamak, the toroidal component B_T and a part, B₀, of the poloidal component are externally generated harmonic fields that serve the purpose of azimuthal and radial stabilization. The total poloidal induction B_P = ∇Ψ × ∇θ is the sum of B₀ and a component that is generated by the electric plasma current density vector, je_θ, via the toroidal Ampere's law

1

In a similar manner one decomposes the electric field, E = E_T + E_P, where E_T is driving the plasma current, and the poloidal part is determined by Coulomb's law, ∇⋅E = 4πρ, where ρ is the electric charge density. Usually the so-called quasineutrality approximation is invoked, which determines the poloidal electric field in leading order through a singular perturbative approach to Coulomb's law. To keep matters as simple as possible, we consider an “electron-positron” plasma, which is totally charge-symmetric with regard to the particle species so that ρ vanishes exactly. In that case the poloidal electric field vanishes identically, too, whereas the toroidal one is implicitly contained in the electric plasma current I = Nqω/2π. Here N is total number of plasma particles, q is the elementary charge, and ω is the mean absolute angular frequency of a species. This settles the electromagnetic part of the model, and we now turn to the statistical mechanics part.

Although most works in refs. 3–8 are formulated at the macroscopic magnetofluid level, they can be recovered, in essence, from the statistical mechanics approach of Kiessling et al. in ref. 4, which begins with the Hamiltonian N particle formulation and takes the kinetic limit. At this kinetic level, the plasma particles are described by distribution functions on single-particle phase space. We seek those distribution functions that maximize the familiar Boltzmann entropy functional under the constraints that the two separating integrals of motion, particle number and energy, take prescribed values N and E, and given that the plasma carries a prescribed electric current I. Because the current is not a separating integral, one has to resort to a ruse and prescribe the canonical angular momentum of each species, which in the axisymmetric kinetic limit is a separating integral, and subsequently pass from this microcanonical angular momenta ensemble to its canonical convex dual, characterized by prescribed ω, namely I. The solutions of the corresponding Euler–Lagrange equations for this variational principle are also stationary solutions to the axisymmetric kinetic equations of Vlasov. Over velocity space the resulting distribution functions are simply rigidly rotating Maxwellians with temperature T = (k_Bβ)⁻¹ > 0, tied to the energy constraint, and angular frequencies ±ω(∝ I), microcanonically tied to the angular momentum constraints but canonically prescribed. This allows one to explicitly integrate over the velocity space to retain only the effective macroscopic entropy principle for the total number density of plasma particles, n(x) = n̄(r, z), which in our charge-symmetric plasma is just twice the value of the common space-dependent Boltzmann factor of each species' distribution function. In effect this renders the entropy a functional of n,

2

where λ_dB = h/ is the thermal de Broglie wave length. This entropy functional has to be maximized under the constraints that n ≥ 0 is axisymmetric, ∫n d𝒯 = N, and that the effective energy functional††

3

takes a prescribed value, E. Here, K(x, x′) = (2 πI/cN)²G(x, x′), and G is the Green's function for −∇⋅(r⁻²∇) in 𝒯 for boundary conditions detailed below. With the help of a variant of Moser's corollary (33) of the Trudinger–Moser inequality, it can be shown that given I, the entropy functional S(n) takes its finite maximum on the set of nonnegative axisymmetric densities n(x) = n̄(r, z) satisfying ∫n(x)d𝒯 = N > 0 and W(n) = E, and the maximizer is a regular solution of the Euler–Lagrange equation.§§ We remark that more than one maximizing density function n might exist, and in addition the nonlinear Euler–Lagrange equation may have other types of solutions. We call a solution S-stable if it is a global maximizer of the entropy (for the given constraints), S-metastable if it is merely a local maximizer, and unstable otherwise. Of course, only those S-stable solutions that exhibit magnetic self-confinement are of interest.

Results

Explicitly carrying out the variations and converting the Euler–Lagrange equation for n into an equation for Ψ, using Ψ(x) = c⁻¹∫ G(x, x′)j(x^′)/r′d𝒯′ and j(x) = qn(x)ωr, we obtain Pfirsch's (34) nonlinear Grad–Shafranov equation,

4

which is to be solved in the torus 𝒯 for the boundary conditions encoded in G. Solving Eq. 4 is in general only possible numerically on a computer. However, some explicit analytical control is available if one simplifies the actual laboratory geometry somewhat and considers a torus 𝒯 with rectangular cross section {r, θ, z|r_i < r < r_o; θ fixed; 0 < z < H}. The poloidal flux function Ψ(x) = Ψ̄(r, z) is assumed to satisfy periodic conditions at the z boundary and to be constant at r_i and r_o so that the radial component of B_P vanishes at the inner and outer boundaries of 𝒯. In this setting the harmonic poloidal part is simply a homogeneous B₀ ∥ e_z, which we choose so that Ψ̄(r_i, z) = Ψ̄(r_o, z). By gauge freedom we can now even set Ψ̄(r_i, z) = 0. Beside the desired self-confined configurations, these boundary conditions also allow unconfined ones, namely Pfirsch's toroidal plasma sheet (34), given by the e_z-invariant solution of Eq. 4,

5

with κ ∈ (0, ∞) a parameter and β(κ²) given by

6

Although these solutions do not describe a magnetically self-confined plasma torus, they serve as our jumping-off point for the numerical computations of the confined configurations. Our strategy, which was also contemplated by K. Schindler (personal communication), is to look for ring-like bifurcations from the toroidal sheet solution (Eq. 5). At an infinite sequence of discrete values E₁ > E₂ > … , with E_k ↘ E_∞ = −π²I²(r − r)/2Hc², other solutions bifurcate off of the plasma sheet sequence, breaking its z invariance. The bifurcation points are determined by setting Ψ(x) = Ψ_Pf(r) + ɛψ(r, z) + O(ɛ²), with ψ(r_i,o, z) = 0 and ψ(r, z + H) = ψ(r, z), and expanding Eq. 4 to first order in ɛ, giving the linearized problem

7

where 〈ψ〉 = ∫ψ(r, z)V(r)d𝒯/∫V(r)d𝒯, and

8

By Fredholm's alternative (35), the solution of Eq. 7 is trivial except for certain discrete values of κ at which the bifurcations occur. We have proved (10) that for our 𝒯 all bifurcations off of the plasma sheet are due to modes ψ_k, k = 1, 2, … , which satisfy 〈ψ_k〉 = 0. The first mode is of the form ψ₁(r, z) = R(r)cos(2π[z − z₀]/H), with z₀ arbitrary and R(r) satisfying

9

for R(r_i) = R(r_o) = 0. With realistic domain dimensions r_i = 1, r_o = , and H = 2, a standard Runge–Kutta solver finds the unique nontrivial solution at κ = κ₁ = 1.62, giving E₁ = 2.72 W_•, with W_• = 2π²r_iI²/25c². Numerical solutions of Eq. 4 with r_i = 1, r_o = , and H = 2 then were computed with a well tested bifurcation code (36–42) based on a continuation method (43). Our code reproduced the analytical solution for the toroidal plasma sheet and its first bifurcation point in excellent agreement with our independently obtained semianalytical results. We then numerically followed the first bifurcating branch that emerges from ψ₁ to nonlinear amplitudes, where it develops into a toroidal plasma ring with a double-X magnetic structure (see Fig. 1) similar to the double-X structure in the PDX–PBX tokamak experiment at the Princeton Plasma Physics Laboratory. Retrospectively, this vindicates our choice of boundary conditions.

Figure 1.

Poloidal magnetic lines of force of maximum-entropy solutions near the second-order phase transition at E₁ = 2.72 W_•. (Left) Plasma torus: W(n) = 2.34 W_•, β = 0.29N/W_•, z₀ = 0. (Right) Plasma sheet: W(n) = 3.00 W_•, β = 0.30N/W_•. The toroidal hoop effect is neatly visible.

Our primary interest is in the energy–entropy diagram. Shown in Fig. 2 is ΔS(n) versus W(n), where ΔS(n) = S(n) − Nk_Bln, with e being the Euler number and the density function n running along the computed bifurcation sequences of plasma torus and plasma sheet. At sufficiently high effective energies, Pfirsch's plasma sheet is the unique solution of Eq. 4 for the stipulated boundary conditions, hence maximizing entropy. Numerically it appears to be the case for all W(n) > E₁ = 2.72 W_• (see Fig. 2). For all W(n) < E₁ down to W(n) = −0.5 W_•, where we terminated the computation, the plasma torus has higher entropy than the toroidal plasma sheet at the same effective energy. By asymptotic analysis we found that also for W(n) ↘ −∞, and by continuity for W(n) ≪ −W_•, the maximum-entropy configuration consists of a highly concentrated plasma torus that, in rescaled coordinates centered at the density maximum, converges to Bennett's cylindrical plasma beam (30, 31) as W(n) ↘ −∞. On the basis of this evidence we surmise that the plasma torus has maximum entropy for all W(n) < E₁, implying its S stability in the class of rotationally invariant plasma with effective energy W(n) < E₁ and current I. We remark that the first bifurcation off of the toroidal plasma sheet into the S-stable plasma torus branch is a symmetry-breaking second-order phase transition.

Figure 2.

ΔS(n) versus W(n) for toroidal plasma sheet (concave branch) and plasma torus (convex branch).

Remaining to be determined is the specific heat of the configurations, which we recall is negatively proportional to the second derivative of S with respect to E. Thus we inspect the curvature of the graphs of the entropy as a function of energy for the various solutions (given in Fig. 2). The graph representing the plasma sheet is concave. However, the graph for the plasma torus is manifestly convex over the whole computed range of energies −0.5 W_• < W(n) < E₁. We were also able to prove the convexity analytically to second order in perturbation theory away from the bifurcation point. This confirms what we have announced earlier: the specific heat of the plasma torus is negative!

Discussion

At last, we discuss the potential implications that our finding of gravothermal-type negative specific heat has for the problem of toroidal magnetic self-confinement of plasma. In a gravitationally bound plasma, negative specific heat on one hand aids the ignition of nuclear burning in a protostar by heating it up when it loses energy by radiation, but it is also responsible for some more spectacular instabilities once the nuclear burning expires, similar to the onset of the red-giant structure (23, 24). It would be intriguing enough if the negative specific heat of a magnetically self-confined plasma torus should be confirmed to aid the ignition of nuclear burning in a tokamak. For this to be so, one would have to be able to hold N and I fixed and secure the toroidal invariance (which is what one wants to achieve anyhow), whereas E would have to decrease (the plasma radiation would seem to help in this respect) slow enough such that one would essentially evolve along the plasma torus branch in Fig. 2 to the left, thereby heating up the plasma while it concentrates more strongly. This would not seem unwelcome. For now, however, energy leakage by radiation is a serious problem, and at the same time the electromotive-force current drive leads to yet-uncontrolled ohmic heating of the plasma. In this case where E is allowed to fluctuate too widely, the negative specific heat will have a very unwanted effect on the confinement. This can be illustrated by considering the temperature rather than energy E to be controlled by the competition of ohmic heating and radiation (still assuming N and I fixed and toroidal invariance). In that case the canonical ensemble determines the stability. But microcanonical and canonical ensembles are not equivalent when the microcanonical one exhibits states with negative specific heat (14, 25, 32, 44), and sure enough, none of the computed plasma solutions with negative specific heat minimizes the free energy functional

10

Actually, F is unbounded below for these β, N, and I values, any minimizing sequence concentrating on a singular ring current (cf. ref. 45 for a good discussion of the translation-invariant analog). Of course, a real plasma would not get anywhere near such a singular ring current configuration, because a highly concentrated plasma torus is known to be susceptible to magnetofluid dynamical instabilities that destroy the axisymmetry.

Conclusion

In the covered energy range, a plasma torus is S-stable but has negative specific heat of the gravothermal type. It therefore will be stable if and only if N, E, and I are essentially fixed and the toroidal invariance is secured, in which case the negative specific heat may aid the ignition of thermonuclear burning. The “if” part is good news; the bad news is the “only if” part.