Dynamical accessibility and physical realizability of the invariant-constrained axisymmetric equilibria of incompressible MHD (Chandrasekhar, 1958)

Chandrasekhar (1958) characterized the equilibria of an incompressible, perfectly conducting fluid whose motions and magnetic fields are axisymmetric, using a variational principle constrained by the seven conserved integrals of that formulation — $I_1, \ldots, I_6$ together with the energy $E$. Three of the six $I_j$ — $I_1$ (in fact an infinite family, $I_1 = \iint f(\varpi^2 P)\, \varpi\, d\varpi\, dz$ for arbitrary $f$), $I_4$, and $I_6$ — exist only under axisymmetry. The equilibrium family that results contains an arbitrary function $\Phi(\varpi^2 P)$ conjugate to $I_1$, and the paper concluded, heuristically, that axisymmetric force-free states ($\Phi \equiv 0$) are 'in general, unattainable' via axisymmetric evolution, and conjectured that equilibria depending on $I_1$, $I_4$, $I_6$ 'are not likely to have any practically realizable counterparts.' The subsequent literature — Woltjer's completion of the invariant set and equivalence proof, the statistical mechanics of the 'Chandrasekhar model,' energy-Casimir and selective-decay frameworks, and stellar fossil-field simulations — has settled the mathematical structure of the problem but left both the accessibility criteria and the realizability conjecture undecided. The problem is to make these 1958 assertions precise and prove or refute them: determine which of the invariant-constrained equilibria are dynamically accessible from given initial data (including the determination of $\Phi$), establish their stability to symmetry-breaking perturbations and weak dissipation, and characterize the exceptional initial data for which force-free states are attainable.

Background

Magnetohydrodynamics (MHD) is the fluid theory of electrically conducting media in which the magnetic field $H$ influences the flow and the conductivity is high enough that the field is effectively frozen into the fluid ('ideal' MHD). In 1958, motivated by the theory of magnetized stars (Chandrasekhar & Prendergast 1956), S. Chandrasekhar posed and partially solved the following problem: which equilibrium states can an incompressible ideal fluid reach if both the motion and the magnetic field are constrained to be axisymmetric, i.e. invariant under rotations about a fixed axis? Working in cylindrical coordinates $(\varpi, \varphi, z)$, any divergence-free axisymmetric field splits into a poloidal part (lying in meridional planes) and a toroidal part (purely azimuthal), so the magnetic field in Alfvén-speed units $h = H/\sqrt{4\pi\rho}$ (the Alfvén speed being the speed at which magnetic disturbances propagate along the field lines in a conducting fluid) and the velocity $v$ are represented by four scalars: $P$ (the poloidal magnetic flux function — $\varpi^2 P$ is proportional to the magnetic flux through a horizontal disk of radius $\varpi$), $T$ (toroidal magnetic field), and the analogous velocity scalars $U$ and $V$.

The ideal evolution equations for $(P, T, U, V)$ (Chandrasekhar 1958, eqs. (4)–(7); the bracket $[A, B]$ appearing there is the Jacobian of $A$ and $B$ with respect to $\varpi$ and $z$; the boundary conditions are $P = T = U = 0$ at the rigid wall) conserve seven integrals (eqs. (11)–(17)): $I_1, \ldots, I_6$ and the energy $E$. Among the $I_j$ are analogues of the magnetic helicity $\int A \cdot h\, dV$ (where $A$ is a vector potential for $h$) and the cross helicity $\int v \cdot h\, dV$ — the two invariants of general (non-axisymmetric) ideal MHD — as well as angular-momentum-type invariants. The paper emphasizes that three of the six, $I_1$, $I_4$ and $I_6$, 'have no analogues in the non-axisymmetric case': they are peculiar to axisymmetry. The foremost is $I_1$: since $f$ is arbitrary,

$$I_1 = \iint f(\varpi^2 P)\, \varpi\, d\varpi\, dz$$

is an infinite family of invariants — in modern language, a Casimir family (functionals conserved for any choice of $f$ because they Poisson-commute with all admissible variations). Conservation of $I_1$ for all $f$ is equivalent to conservation of the volume distribution of the values of $\varpi^2 P$: no reshuffling of flux-function values across volume is possible (Woltjer 1959b). This is also, as Woltjer noted, the integral that makes an axisymmetric dynamo impossible.

Equilibria were sought by the variational principle of Woltjer (1958b): extremize $E$ at fixed invariants, $\delta E + \sum_j \alpha_j \delta I_j = 0$, where the $\alpha_j$ are constants (Lagrange multipliers). The resulting Euler–Lagrange equations (Chandrasekhar 1958, eqs. (31)–(39)) give $U = \alpha_3 P$, eliminate $T$ and $V$ algebraically (except in the degenerate case $\alpha_3^2 = 1$, where the elimination divides by zero and the paper proceeds no further), and leave a single partial differential equation for $P$ of Grad–Shafranov type (the standard equilibrium equation of axisymmetric plasma confinement, in which the flux function satisfies a nonlinear Poisson equation with free functions of the flux) containing an arbitrary function $\Phi(\varpi^2 P)$ — a free profile that the paper traces directly to $I_1$. A force-free field (one in equilibrium with zero Lorentz force, $\nabla \times h = \alpha h$; Woltjer (1958a) had proved that a field of given helicity with minimum energy must be force-free) corresponds to $\Phi \equiv 0$; the case of purely rotational motion recovers Ferraro's isorotation law (the classical consequence of ideal axisymmetric evolution that the fluid's angular velocity is constant on each poloidal magnetic field line). From this structure Chandrasekhar drew two physical conclusions: (i) an axisymmetric force-free state 'is, in general, unattainable as an equilibrium state via axially symmetric non-equilibrium states,' the reason being the conservation of $I_1$; and (ii) equilibria whose structure depends on $I_1$, $I_4$ and $I_6$ 'are not likely to have any practically realizable counterparts.' Both conclusions are heuristics: the paper contains no stability analysis to non-axisymmetric perturbations, no derivation of $\Phi$ from initial data, and no proof that the dynamics relaxes to the constrained energy minimum rather than merely visiting extremals of the variational problem.

The subsequent literature settled the mathematical structure but not the physics. Woltjer (1959a), responding to a criticism by Kruskal & Kulsrud (1958), conceded that both his and Chandrasekhar's invariant sets were incomplete, noted that only some variational extrema are stable (the second variation decides), and proved the lowest force-free mode to be the absolute energy minimum. Woltjer (1959b) then obtained the complete solution: the general axisymmetric incompressible equilibrium family with arbitrary functions of $\varpi^2 P$ (Chandrasekhar's family is a subcase), the complete invariant set including the full Casimir family, and a proof that the variational principle with the complete set is equivalent to the direct equilibrium equations; Woltjer (1959c) extended this to compressible media. In modern terminology the problem lives in the energy-Casimir framework (Morrison 1998) — equilibrium and stability analysis by extremizing the energy together with the Casimir invariants, within which 'dynamically accessible' variations are those perturbations that the unmodified ideal dynamics can actually reach: symmetric MHD equilibria and their Casimir invariants were systematized by Andreussi, Morrison & Pegoraro (2010, 2012, 2016), and the selective-decay hypothesis (weak dissipation destroys energy faster than the invariants, so the system settles to the energy minimum at fixed invariants — the laboratory-plasma archetype being Taylor (1974), where only helicity survives) was given a geometric formulation by Gay-Balmaz & Holm (2014). Leprovost, Dubrulle & Chavanis (2005) built the statistical mechanics of exactly this system (naming the linear-Casimir limit the 'Chandrasekhar model'), and the stellar fossil-field program (Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006; Duez & Mathis 2010; Duez, Braithwaite & Mathis 2011) uses invariants of the $I_1$ type — the mass enclosed in poloidal flux surfaces — to select relaxed magnetic equilibria in stellar radiative zones. None of these works decides Chandrasekhar's realizability conjecture or constructs the accessibility map.

Problem Statement

Consider the ideal incompressible MHD system restricted to axisymmetric states in the formulation of Chandrasekhar (1958): divergence-free magnetic and velocity fields represented by the scalars $(P, T, U, V)$ in cylindrical coordinates $(\varpi, \varphi, z)$ — $P$ the poloidal magnetic flux function, $T$ the toroidal magnetic field, and $U$, $V$ the corresponding velocity scalars — with $P = T = U = 0$ on the boundary, evolving under eqs. (4)–(7) with conserved energy $E$ and integrals $I_1$–$I_6$, where $I_1 = \iint f(\varpi^2 P)\, \varpi\, d\varpi\, dz$ for arbitrary $f$ is an infinite Casimir family and $I_1$, $I_4$, $I_6$ are specific to axisymmetry. The variational equilibria $\delta E + \sum_j \alpha_j \delta I_j = 0$ form the family of eqs. (35)–(39) (equivalently, the subcase of the complete family of Woltjer 1959b), parameterized by multipliers $\alpha_j$ and an arbitrary function $\Phi(\varpi^2 P)$ conjugate to the $I_1$ family; Chandrasekhar concluded (i) that an axisymmetric force-free state ($\Phi \equiv 0$) is 'in general, unattainable' from axisymmetric non-equilibrium states because $I_1$ is conserved, and (ii) that equilibria depending on $I_1$, $I_4$, $I_6$ 'are not likely to have any practically realizable counterparts.' The problem is to make (i) and (ii) precise and to decide them. Specifically:

(a) Accessibility criteria. Establish necessary and/or sufficient conditions on the initial axisymmetric state — equivalently, on the initial Casimir profile of $\varpi^2 P$ and the constants $I_2, \ldots, I_6$ — under which the evolution, either strictly ideal or ideal with weak dissipation preserving the axisymmetric invariants (selective decay), converges to a given member of the equilibrium family.

(b) Realizability and robustness to symmetry breaking. Determine whether the equilibria whose structure depends on $I_1$, $I_4$, $I_6$ are physically realizable end states — that is, prove or refute Chandrasekhar's expectation — and how weak dissipation (reconnection, resistivity) erodes the axisymmetry-specific invariants.

(c) The force-free exception set. Characterize exactly the initial data for which an axisymmetric force-free equilibrium ($\Phi \equiv 0$) is dynamically accessible, converting Chandrasekhar's 'in general, unattainable' into a theorem with explicit hypotheses.

Since the 1958 statements are semi-qualitative, deciding (i) and (ii) requires fixing a precise formal notion of dynamical accessibility and physical realizability (for example, convergence of weak solutions of the axisymmetric ideal system; stability in the energy-Casimir sense; robustness under three-dimensional perturbations of specified size). The relevant dynamical framing is attainment as an end state of axisymmetric evolution starting from non-equilibrium initial data and preserving the integrals of the motion (in particular the full $I_1$ family), under the paper's relaxation premise (Section 4) that the attainable final equilibrium is the state of lowest energy compatible with the constancy of $I_1, \ldots, I_6$; the paper's claim concerns attainment 'via axially symmetric non-equilibrium states,' so the initial data are non-equilibrium and the equilibrium is the end state of the evolution. Parts (a), (b) and (c) are related: (a) asks for necessary and/or sufficient conditions for accessibility, while (c) is the special case of (a) in which the criterion is exact and the member is the force-free state $\Phi \equiv 0$.

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