7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Comparison between the generalized Grad-Shafranov equation for MHDequilibrium with flow and its reduced models
Atsushi ItoNational Institute for Fusion Sciencee-mail (speaker): ito.atsushi2@nifs.ac.jp

Axisymmetric plasma equilibrium with flow inmagnetohydrodynamics (MHD) is described by thegeneralized Grad-Shafranov (GGS) equation [1]. It is thegeneralized equation of the Grad-Shafranov (GS)equation to include effects of flow and coupled with theBernoulli law for density in the direction parallel to themagnetic field. In this presentation, the GGS is comparedwith the reduced MHD equilibrium equations in thepresence of flow for tokamaks, theoretically andnumerically.
The GGS equation coupled with the Bernoulli law is a
non-linear second-order partial differential equation(PDE) with five free functions for the magnetic flux
function. It can be elliptic, hyperbolic or singulardepending on the poloidal flow velocity in toroidal
geometry. The pressure is not constant on magnetic flux
surfaces. Compared with the GS equation for static
equilibrium that is an elliptic PDE with two freefunctions of the magnetic flux function includingpressure, the GGS equations is complicated and difficult
to solve. To solve numerically, an iteration method withan initial guess of the magnetic flux function is exploited.
The Bernoulli law is an algebraic equation for density
and solved with a root-finding method with a given
magnetic flux function. The GGS with solved with aPoisson solver by substituting the initial guess and thesolution of the density. These procedures are iterated
until the solution converges. Since the GGS equation is
nonlinear, the elliptic PDE for the poloidal flow depends
on the solution. To find a solution, the five free functions
should be carefully chosen.
By using asymptotic expansions with respect to theinverse aspect ratio of a torus of magnetically confinedplasma, reduced equilibrium equations are derived. InRef. [2], reduced equilibrium equations for high-betatokamaks are shown. The derivation depends on theorder of poloidal flow velocity. Two cases with flow
velocity comparable to the poloidal sound velocity andthe poloidal Alfven velocity are studied. These are easy
to solve compared to the GGS equation and theiranalytical and numerical solutions are obtained [3-5].
The solutions show qualitative difference in the shift ofmagnetic flux surfaces and pressure isosurfaces for thecases below and beyond the characteristic velocities.
In this presentation, we will show results of thecomparison between the GGS equation and reduced
equilibrium equation. To compare the solutions, theequivalent free functions must be used. In GGS equation,
free functions are in the form of entropy and enthalpy
while, in reduced equations, they are pressure andpoloidal Alfven Mach number. The correspondence of
these free functions is derived. By using these freefunctions, benchmark results of numerical calculations
will be shown. We have developed a code for thecoupled equations of the GGS equation and the Bernoulli
law. The GGS equation is solved with the finite element
method. The Bernoulli law is solved with the Newton
method. Numerical solutions of these equations will becompared with those of the reduced equilibrium
equations.
References[1] E. Hameiri, Phys. Fluids 26, 230 (1983).
[2] A. Ito et al., Plasma Fusion Res. 3, 034 (2008).
[3] A. Ito et al., AIP Conf. Proc. 1069, 121 (2008).
[4] A. Ito et al., Plasma Phys. Control. Fusion 51,035007 (2009).
[5] A. Ito et al., Phys. Scr. 96, 035602 (2021).