Topic/Type: 1.2 Fusion Plasmas (magnetic & inertial confinement), Oral

### Numerical Stability Analysis of Analytical Tokamak Plasma Equilibria with Toroidal Flow

J. W. Haverkort1, 2, J. W. S. Blokland1, B. Koren2, 3

1 FOM Institute for Plasma Physics Rijnhuizen, Assocation EURATOM-FOM, Nieu
2 Center for Mathematics and Computer Science (CWI), Amsterdam, The Netherlands
3 Leiden University, Leiden, The Netherlands

Axisymmetric plasma equilibria with purely toroidal flow can be described by the extended Grad-Shafranov equation for the poloidal magnetic flux $\small \psi$. Unlike in the static case, the pressure $\small p$ is no longer a function depending on $\small \psi$ only (a flux function), but is subject to $\small \nabla p |_{\psi} = R\Omega^2 \nabla R$ with $\small \Omega$ the angular frequency and $\small R$ the radial cylindrical coordinate. This equation can be solved analytically under the assumption that either the temperature, the density, or the entropy is a flux function, yielding an extended Grad-Shafranov equation which in general has to be solved numerically. Assuming linear profiles for the static pressure and the stream function for the poloidal current density and assuming a constant ratio $\small \Lambda = \Omega^2 / 2T_0$ between the angular frequency squared and the static temperature, Maschke and Perrin found exact analytical solutions [3] for the case that either the temperature or the entropy is a flux function. We extend these solutions to the case in which the density is a flux function. We use these three separate solutions to validate the implementation and test the fourth order accuracy behavior of the equilibrium code FINESSE (FINite Element Solver for Steady Equilibria [1]). The analytical solutions are also extended to enable specification of the triangularity of the poloidal plasma cross-section.

The studied exact analytical solutions, although limited in their applicability, provide a good standardized test case for equilibria and stability codes including toroidal flow. We investigate these equilibria using the linear stability code PHOENIX [2], which allows for the inclusion of a small resistivity. The linearized MHD equations are projected on straight field line coordinates and discretized, in weak form, using both finite element and spectral methods. The resulting generalized eigenvalue problem is solved with the Jacobi-Davidson algorithm. We report on the oscillation frequency and growth rate of various waves and instabilities arising in rigidly rotating and sheared plasma flows. In the analytical solutions the shape of the toroidal velocity profile is completely free, only its ratio $\small \Lambda$ with the static temperature has to be constant. This makes it possible to study the influence of the specific flow and shear profiles, while maintaining the same equilibrium solution $\small \psi$.

As a special instance of the considered analytical solutions we investigate an isothermal tokamak. Lacking the temperature gradients driving some of the most damaging instabilities in present-day tokamaks, isothermal tokamak plasmas are assumed to have very favorable stability properties [4]. Using an analytical equilibrium solution, we numerically investigate the stability of such an isothermal confinement device.

[1] A. J. C. Beli?n et al., J. Comp. Phys. 182, 91-117 (2002)

[2] J. W. S. Blokland et al., J. Comp. Phys. 226, 509-533 (2007)

[3] E. K. Maschke and H. Perrin, Plasma Phys. 22, 579-594 (1980)

[4] P. J. Catto and R. D. Hazeltine, Phys. Plasmas 13, 122508 (2006)