Solving the Grad–Shafranov equation using spectral elements for tokamak equilibrium with toroidal rotation

Abstract

The Grad-Shafranov equation is solved using spectral elements for tokamak equilibrium with toroidal rotation. The Grad-Shafranov solver builds upon and extends the NIMEQ code [Howell and Sovinec, Comput. Phys. Commun. 185 (2014) 1415] previously developed for static tokamak equilibria. Both geometric and algebraic convergence are achieved as the polynomial degree of the spectralelement basis increases. A new analytical solution to the Grad-Shafranov equation is obtained for Solov'ev equilibrium in presence of rigid toroidal rotation, in addition to a previously obtained analytical solution for a defferent set of equilibrium and rotation profiles. The numerical solutions from the extended NIMEQ are benchmarked with the analytical solutions, with good agreements. Besides, the extended NIMEQ code is benchmarked with the FLOW code [L. Guazzotto, R. Betti, et al., Phys. Plasma 11(2004)604].

Formation of edge pressure pedestal and reversed magnetic shear due to toroidal rotation in a tokamak equilibrium

“…Thus, plasma temperature can be considered as a flux function, namely T = T (ψ). Due to P (ψ, R) = n(ψ, R)T (ψ), there will be a shift of number density profile in the direction of major radius, similar to pressure profile, as shown in both analytic and numerical solutions [33]. Physically, this shift arises from the centrifugal force induced by toroidal flow, namely ρ( u • ∇) u = −ρRΩ 2 êR , where êR denotes the direction along the major radius R. Thus, toroidal flow modifies equilibrium through both the pressure profile and the poloidal magnetic flux, as governed by Eq.…”

“…For the static Solov'ev equilibrium, the pressure profile is µ 0 P 0 = p 0 +p 1 ψ and the F profile satisfies F dF dψ = F 0 . In presence of a rigid toroidal flow, the analytical solution for Solov'ev equilibrium becomes [33]…”

“…To further elaborate on this rotational effect on equilibrium, we consider a static L-mode equilibrium with pressure profile P 0 (ψ) = P axis (1 − ψ 2 ) 2 and safety profile q = q axis (1 + (q edge /q axis − 1)ψ 4 ). The equilibrium solver NIMEQ is used for the numerical solution, which is recently extended to self-consistently take into account of the effects of toroidal flow with arbitrary profile [33]. The computational domain is a circular grid with major radius R 0 = 5.0m and minor radius a = 0.5m.…”

MHD analysis on the physical designs of CFETR and HFRC

“…The HFRC is designed to achieve fusion reaction condition through the means of magnetic compression, whose feasibility and efficiency are subject to the constraints imposed from the MHD stability. As a first step, the MHD equilibrium for the initial FRC state prior to magnetic compression is generated using the NIMEQ code [17,18] based on the engineering designs (Fig. 3).…”

A parallel cut-cell algorithm for the free-boundary Grad-Shafranov problem

“…In a fixed-boundary Grad-Shafranov solver, both the location of the computational boundary and the boundary condition for the poloidal magnetic flux are known. Many numerical approaches, including spectral elements [33,21], hybridizable discontinuous Galerkin methods [41,42], boundary integral approaches [39,30], Hermite finite element [37,22] and discontinuous Petrov Galerkin methods [40], have been extended to solve the fixed-boundary problem. A common use of the fixed-boundary Grad-Shafranov solver is to set the computational boundary to be the targeted last closed flux surface, so the plasma shaping is enforced by the computational boundary on which the poloidal flux is a constant.…”