Cylindrical Viscous Boundary Layer For Transonic Equilibrium

Transonic equilibria have been studied in in a number of papers, see e.g. [1-4]. They occur in tokamak geometry when the plasma exhibits a poloidal velocity of the order of the poloidal sound speed (Csp = CsBp/B). It was shown that transonic equilibria develop a radial contact discontinuity with the core plasma region rotating at subsonic poloidal velocities (vp <Csp) and the edge region rotating at supersonic velocities (vp > Csp). In condition of H-mode plasmas with an X-point, Csp can be well below the sound speed Cs and the magnitude of the transonic flow can be modest. In the single-fluid MHD model, transonic equilibria are characterized by a radial discontinuity in all MHD properties:density, pressure and velocity. Physical intuition dictates that including fluid viscosity in the problem will relax the discontinuity to a sharp-gradient layer between the slow- and fast-flowing regions. We use a high aspect ratio expansion of the equilibrium to build a cylindrical model of the viscous, compressible transonic boundary layer. The formulation requires a series expansion in the poloidal angle θ, but only a few terms are needed for good accuracy. We present numerical solutions of there boundary layer in different conditions and highlight challenges and implications for the physics of the transonic discontinuity.

[1] L. Guazzotto, R. Betti, J. Manickam, S. Kaye, Physics of Plasmas 11 2,

604-614

[2] L. Guazzotto, R. Betti, Physics of plasmas 12 5, 056107

[3] L. Guazzotto, E. Hameiri, Physics of Plasmas 21 2, 022512

[4] L. Guazzotto, R. Betti, S.C. Jardin, Physics of Plasmas 20 4, 042502