MHD Stability and Energy Principle for Two-Dimensional Equilibria without Assumption of Nested Magnetic Surfaces

Abstract

Abandoning the assumption of nested magnetic surfaces in tokamak plasma expands the field of research and opens up new approaches for both theoretical and experimental plasma physics. The computer code KINX for calculations of the ideal MHD stability was developed for studies of doublet plasmas with two magnetic axes and using block-structured grids in each subdomain with nested magnetic surfaces. Then, the MHD_NX code on unstructured grids was developed to calculate the stability of two-dimensional equilibria with an arbitrary topology of magnetic surfaces. The study of equilibrium and stability of equilibrium configurations with toroidal current density reversal and axisymmetric n = 0 islands, which are associated with internal transport barrier and low current density at the magnetic axis, as well as with the operation of tokamaks in the alternating current regime, leads to more general issues of MHD stability of two-dimensional solutions of the Grad−Shafranov equations with islands under other types of symmetry—chain of islands in helical symmetry and cylindrically symmetric m = 0 islands in configurations with the longitudinal field reversal. New ideal MHD unstable modes have been discovered for various types of two-dimensional island configurations. The energy principle with MHD-compatible boundary conditions at open magnetic field lines is necessary for the self-consistent stability analysis of divertor configurations in tokamaks with a finite current density at the separatrix, taking into account the plasma outside the separatrix. Several codes have been developed for calculations of plasma equilibrium and stability, taking into account the influence of currents outside the separatrix, which are ready for integration with other codes for edge plasma modeling.

This work was supported by the Russian Science Foundation (grant no. 16-11-10278).

Authors and Affiliations

1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

   S. Yu. Medvedev, A. A. Martynov, V. V. Drozdov, A. A. Ivanov & Yu. Yu. Poshekhonov

2. National Research Center “Kurchatov Institute”, 123182, Moscow, Russia

   S. Yu. Medvedev, A. A. Martynov & S. V. Konovalov

3. National Research Nuclear University “MEPhI”, 115409, Moscow, Russia

   S. Yu. Medvedev

4. Swiss Plasma Center, EPFL-SB-SPC, CH-1015, Lausanne, Switzerland

   L. Villard

Corresponding author

Correspondence to S. Yu. Medvedev.

The article was translated by the authors.

APPENDIX

Following the classical work [11], where the energy principle for the problem of ideal MHD stability is formulated, we show how the parts of the potential energy functional are transformed in the case when the boundary \({{S}_{p}}\) between the plasma and the vacuum includes the divertor plates \({{S}_{d}}\). To do this, we establish a relation between the displacement \({\mathbf{\xi }}\) and the vector of the perturbed electric field \({\mathbf{e}}\). From the general condition at the plasma–vacuum interface \({\mathbf{n}} \times \left\langle {\mathbf{E}} \right\rangle = {\mathbf{n}} \cdot {\mathbf{v}}\left\langle {\mathbf{B}} \right\rangle \) (Eq. (2.10) in [11]) for a perfectly conducting plasma, it follows that

where the subscript \(v\) indicates the value from the vacuum side. On the part of the boundary coinciding with the equilibrium magnetic surface, \({\mathbf{n}} \times {{{\mathbf{e}}}_{v}} = ({\mathbf{n}} \cdot {\mathbf{\xi }}){{{\mathbf{B}}}_{v}}\) follows from Eq. (A.1). On the divertor plates, we have boundary condition (3) (\({\mathbf{n}} \cdot {\mathbf{\xi }} = 0\)) and equality (A.1). Taking into account these relations and the linearized condition of the total pressure continuous across plasma vacuum interface (Eq. (2.32) in [11]) on \({{S}_{p}}{\backslash }{{S}_{d}}\), the surface term (see Eq. (3.14) in [11]) can be written as

where \({{W}_{S}} = \frac{1}{2}\int_{{{S}_{p}}} {{{{({\mathbf{n}} \cdot {\mathbf{\xi }})}}^{2}}{\mathbf{n}} \cdot \left\langle {\nabla (p + {{B}^{2}}{\text{/}}2)} \right\rangle dS} \) and the angular brackets denote the jump of the corresponding value across the plasma boundary, for example, \(\left\langle {\mathbf{B}} \right\rangle = {{{\mathbf{B}}}_{v}} - {{{\mathbf{B}}}_{p}}\). The last term in Eq. (A.2) is zeroed due to the assumption of continuity of the normal component of the magnetic field through thin divertor plates. The next two terms from the end of Eq. (A.2), after using the Gauss formula and the relations and \(\nabla \times \delta {{{\mathbf{B}}}_{v}} = 0\) in vacuum, give

coinciding with expressions (2) and (4). In the absence of an equilibrium surface current, the integral \({{W}_{S}}\) is zeroed and relation (A.1) is reduced to the continuity of the tangential component of the perturbed electric field, \(\left\langle {{\mathbf{e}} \times {\mathbf{n}}} \right\rangle = 0\).

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