Plasma models and equilibrium theory

Read 00-model-contracts.md before using these specifications. References refer to 09-literature.md. All proposed Lean declarations and solver bridges remain to be implemented.

Contents

• State, scales, and units
• Ideal MHD
• Static equilibria and boundary conditions
• Weak and stepped-pressure equilibria
• Beyond ideal MHD
• Kinetic reference and reduction contracts
• Theorem backlog

State, scales, and units

Keep mass density ρ, number density nₛ, pressure p, temperature Tₛ, magnetic field B, current J, and fluid velocity u distinct. For a Maxwellian isotropic mixture use a specified equation of state

If temperature is supplied as energy in keV, write pₛ = nₛ Θₛ with Θₛ = k_B Tₛ and perform the energy-to-joule conversion. Do not silently insert keV values into a kelvin formula.

Introduce reference scales L₀, B₀, ρ₀ and derive the nondimensional equations. Track at least

and collision, resistive, thermal, orbit, and operating timescales. The thermal-speed convention is explicitly v_th = √(2Θ/m). A model-validity interface should record the ordering parameters rather than treating “small” as a theorem.

For realistic case studies, import a documented experimental or reactor operating profile together with its geometry; use W7-X transport data or the HELIAS engineering study as entry points. Neither provides a universal minimum field, coil spacing, or material lifetime.

Ideal MHD

For a smooth isotropic adiabatic fluid without gravity, heating, or resistivity:

Equivalently, D_t(pρ^{-Γ})=0 where ρ>0. Define solutions with a declared time interval, regularity, initial data, and boundary conditions. The source model Freidberg, Ideal MHD supplies the physical assumptions and variants.

First prove:

1. div B remains zero because div curl=0, in the chosen classical solution class.
2. Mass is conserved on a fixed impermeable boundary u·n=0.
3. For the appropriate closed conducting boundary, total energy

is conserved; retain boundary fluxes in the general identity.

1. Smooth material magnetic flux is conserved under the ideal induction equation, with a proved flow map.
2. Positive initial density stays positive along a sufficiently regular flow for the existence interval.

Conservation identities alone do not establish global smooth existence or nonlinear stability in a strong norm.

Static equilibria and boundary conditions

For u=0, isotropic static equilibrium means

Specify whether profiles, total currents, fluxes, helicity, or boundary shape are prescribed. Density is not determined by force balance alone; it enters stability through inertia and the operating model.

Fixed boundary. Require an appropriate magnetic boundary, often B·n=0. Initially analyze perturbations with zero boundary trace; later extend to the chosen conducting-wall condition.

Free plasma–vacuum interface. Include a vacuum region, its magnetic field, and coil/external wall boundary data. For an ideal magnetic interface B·n=0 on both sides with no additional imposed traction, require continuity of total pressure:

Use normal-field continuity, tangential jump/current conditions, and the full stress formulation for more general interfaces. An equilibrium computed with a fixed plasma boundary does not supply a free-boundary stability certificate.

Vacuum. J=0 gives curl B=0, in addition to div B=0. On a simply connected region with sufficient regularity, a scalar-potential theorem gives B=∇Φ and ΔΦ=0. On a toroidal region retain harmonic/circulation components.

Force-free. Constant pressure permits J∥B where B≠0. A useful exact counterexample to “zero pressure implies vacuum” uses B₀,L₀>0 on a Cartesian domain periodic in z with period 2πL₀:

This model has nonzero current and zero Lorentz force, with explicit field and length scales preserving SI dimensions. It is a kernel sanity witness, not a stellarator reactor.

Axisymmetry. With cylindrical coordinates and a selected poloidal flux convention,

Prove this reduction on R>0 and track the sign/flux convention. Use it for verified manufactured equilibria and a simpler PDE existence milestone. Axisymmetric examples provide infrastructure and sanity checks, while the practical target remains a three-dimensional stellarator.

Weak and stepped-pressure equilibria

Define the static stress

For smooth divergence-free fields, div S=0 is equivalent to force balance. For piecewise smooth fields formulate div S=0 distributionally, div B=0 distributionally, and the resulting interface jumps. This prevents multiplication of undefined distributional currents by discontinuous fields without a specified interpretation.

For a multi-region relaxed-MHD model:

• decompose the toroidal domain into regions separated by magnetic interfaces;
• use constant pressure within each relaxed region;
• impose curl B = λᵢ B within each region;
• impose interface force balance, flux and helicity/current constraints;
• permit islands and chaotic field lines within the regions.

The SPEC/MRxMHD paper supplies a candidate-generating formulation. Prove the appropriate existence and stability statements for that model; do not silently transfer them to smooth continuously nested ideal-MHD equilibrium.

Enciso–Luque–Peralta-Salas prove piecewise smooth nonconstant-pressure equilibria under nondegenerate toroidal-domain hypotheses. Extract their actual theorem, including the Beltrami/KAM assumptions. This is a substantial existing theorem to formalize, rather than an assertion that all nonaxisymmetric equilibria are unknown.

Smooth nonaxisymmetric equilibrium with a nonconstant pressure profile and a complete prescribed surface foliation raises small-divisor and resonance questions. VMEC/DESC representations impose nested surfaces; a converged finite representation alone is not an existence proof for a classical continuum solution. Research should target a specified class with constructive constants, allow weaker/stepped-pressure alternatives where appropriate, and account for obstructions. Constantin–Drivas–Ginsberg construct equilibria with a small additional force; that force must remain in their conclusion.

Beyond ideal MHD

Keep separate interfaces for:

• resistive Ohm law E+u×B=η_e J, where η_e is electrical resistivity, and induction with magnetic diffusivity η_e/μ₀;
• generalized Ohm law with Hall, electron-pressure, and inertia terms;
• anisotropic stress P=p_⊥ I+(p_∥-p_⊥)bb, b=B/|B|;
• viscosity, heat conduction, sources, and losses;
• equilibrium flows and resistive walls.

In an isothermal/constant-resistivity incompressible toy model with closed boundaries, magnetic plus kinetic energy decreases through resistive and viscous dissipation. In a thermodynamically closed resistive model, Joule heating transfers electromagnetic energy into internal energy rather than destroying total energy. State which balance is being proved.

Static no-flow self-adjoint MHD stability machinery does not automatically apply to a flowing equilibrium or a dissipative nonself-adjoint operator.

Kinetic reference and reduction contracts

A multispecies nonrelativistic kinetic model is

with Maxwell equations and explicitly chosen collision/source operators. Specify decay or velocity boundaries so that moments and integration by parts are legitimate. Define ρ_c=Σqₛ∫fₛdv and J=Σqₛ∫vfₛdv.

Prove moment identities under collision conservation hypotheses. Closure of the pressure tensor and heat flux is an additional assumption or reduction theorem, not a consequence of taking moments alone.

A useful reduction contract has the form

for specified initial data, observables, parameters, and times. Plasma stability in a reduced model transfers only through such a contract and a sufficient robustness margin. Proving useful kinetic-to-fluid and orbit-reduction error bounds is part of the ambitious roadmap.

Theorem backlog