High-beta analytic equilibria in circular, elliptical, and D-shaped large aspect ratio axisymmetric configurations with poloidal and toroidal flows

Abstract

The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows. Using a variational perturbative approach [E. Hameiri, Phys. Plasmas 20, 024504 (2013)], analytic approximations for high-beta equilibria in circular, elliptical, and D-shaped cross sections in the high aspect ratio approximation are found, which include finite toroidal and poloidal flows. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to an inhomogeneous Helmholtz partial differential equation (PDE) subject to homogeneous Dirichlet conditions. An application of the Green's function method leads to a closed form for the circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. To extend the elliptical solution to a D-shaped domain, a boundary perturbation in terms of the triangularity is used. A comparison with the code FLOW [L. Guazzotto et al., Phys. Plasmas 11(2), 604–614 (2004)] is presented for relevant scenarios.

Stability of a tokamak plasma with diffuse toroidal rotation

“…The numerical implementation shoots from a position near the origin ‘pointing towards’ the plasma–vacuum transition. Unfortunately, due to the fact that the equilibrium introduced in López & Guazzotto (2017) possesses a singularity at the origin, the flow shear drive introduces this unphysical singularity at the eigenmode equation level. This issue has been addressed by constructing an alternative representation for the equilibrium that is regular on the complete plasma domain (appendix A).…”

“…For an ideal gas γ = 5/3, the isothermal case is recovered as γ approaches unity. In López & Guazzotto (2017), a self-consistent, high-β tokamak equilibrium solution for a circular plasma with toroidal and poloidal flows is developed as an asymptotic expansion of the complete GSB system in terms of the inverse aspect ratio. In this work, we utilize a reduced version of this equilibrium solution, where poloidal flows are set equal to zero, as our starting point.…”

“…(2.13) For the convenience of readers, the equilibrium solution for the poloidal flux that was originally constructed by López & Guazzotto (2017) is provided in appendix A. In particular, Ψ 0 and Ψ 1 are given in closed form by (A 4) and (A 5), respectively.…”

“…For the purposes of modelling realistic high-β fast rotating plasmas, as in NSTX, it would be important to incorporate cross-sectional shaping, as done in (Chapman et al 2011(Chapman et al , 2012. Despite the fact that an analytic D-shaped equilibrium solution for a flowing plasma has been constructed in López & Guazzotto (2017), the present stability analysis is limited to circular cross-sections. In addition, even if the safety factor is divergent in configurations with a separatrix, we consider scenarios without internal resonances; this restriction could be relaxed in future work.…”

“…Accordingly, our stability calculations have been built upon a high-β analytic equilibrium with diffuse profiles in the safety factor and in the angular toroidal frequency. This equilibrium has been developed for a large aspect ratio plasma (López & Guazzotto 2017). For the purposes of modelling realistic high-β fast rotating plasmas, as in NSTX, it would be important to incorporate cross-sectional shaping, as done in (Chapman et al 2011(Chapman et al , 2012.…”