[Paper Review] MHD Equilibrium Equation in Symmetric Systems

M Y Kucinski, Iberê L. Caldas|arXiv (Cornell University)|Mar 25, 2011

Advanced Thermodynamics and Statistical Mechanics1 references3 citations

TL;DR

This paper derives a general MHD equilibrium equation in curvilinear coordinates for symmetric plasma systems, where physical quantities depend only on two variables. By assuming an ignorable coordinate and using magnetic surface functions with the same symmetry, it formulates a unified equilibrium equation applicable to non-orthogonal generalized coordinates, enabling derivation of known equations like Grad-Shafranov and helical system equations as special cases.

ABSTRACT

In MHD symmetric systems the equilibrium physical quantities are dependent on two variables only. In this cases it is possible to find a magnetic surface function that has the same symmetry. Under the assumption that the metric determinant is also independent of a third, ignorable coordinate, a general MHD equilibrium equation in curvilinear coordinates is deduced. This equation is specially useful when non-orthogonal generalized coordinates are used.

Motivation & Objective

• To develop a general MHD equilibrium equation applicable to symmetric plasma systems with two independent variables.
• To formulate the equilibrium equation in non-orthogonal generalized curvilinear coordinates, enhancing flexibility in coordinate system choice.
• To derive the equilibrium equation using a magnetic surface function Ψ and a current function I that preserve the system's symmetry.
• To demonstrate the applicability of the general equation by deriving known equilibrium equations in toroidal, helical, and natural coordinates.
• To unify the treatment of MHD equilibrium in symmetric systems by expressing B and J fields in terms of Ψ and I via metric tensor components.

Proposed method

• Uses curvilinear coordinates (u₁, u₂, u₃) with u₃ as an ignorable coordinate, assuming periodicity in u₂ and u₃.
• Defines the transverse magnetic flux Ψ(u₁, u₂) as the flux through a surface bounded by u₃, derived from the magnetic field component B².
• Introduces the current function I(u₁, u₂) via the average of B₃ over u₃, analogous to Ψ for the magnetic field.
• Derives the general MHD equilibrium equation by combining ∇×B = μ₀J and ∇P = J×B, expressing P′ in terms of Ψ and I using metric components.
• Applies the general equation to derive specific forms in toroidal (Appendix A), helical (Appendix B), and natural (Appendix C) coordinates.
• Uses vector calculus identities and metric tensor relations (e.g., g₃₃, √g) to express B and J in terms of ∇Ψ and ∇I, ensuring symmetry preservation.

Experimental results

Research questions

• RQ1 How can a unified MHD equilibrium equation be derived for symmetric systems using non-orthogonal generalized coordinates?
• RQ2 What is the role of the magnetic surface function Ψ and current function I in preserving symmetry in curvilinear coordinates?
• RQ3 How does the assumption of an ignorable coordinate (u₃) simplify the derivation of the equilibrium equation?
• RQ4 Can the general equilibrium equation reproduce known results such as the Grad-Shafranov equation and helical system equations?
• RQ5 How are the magnetic and current density fields expressed in terms of Ψ and I using the metric tensor in arbitrary curvilinear systems?

Key findings

• The general MHD equilibrium equation is derived in curvilinear coordinates as ∂²Ψ/∂u₁² + ∂²Ψ/∂u₂² = -μ₀P′ - μ₀²II′, valid for symmetric systems with ignorable coordinate u₃.
• The magnetic field is expressed as B = (e₃/g₃₃)×∇Ψ + B₃e₃/g₃₃, ensuring ∇Ψ·B = 0 and confirming Ψ = constant defines a magnetic surface.
• The current density is given by J = (e₃/g₃₃)×∇I + J₃e₃/g₃₃, analogous to the magnetic field expression, preserving symmetry.
• In toroidal coordinates, the equilibrium equation reduces to a form involving ∂/∂ξ and ∂/∂ω, with a source term depending on pressure and current derivatives.
• For helical systems with straight magnetic axis, the equation becomes a radial and poloidal derivative form involving (1+α²r²)/r terms and pressure/current gradients.
• In natural coordinates, the equilibrium equation simplifies to P′ = -J₃χ′/g₃₃ - μ₀II′/g₃₃, and integrating over a surface yields P′V′ = -χ′J′ - Φ′I′, linking fluxes and currents.