Generalized Solovev equilibrium with sheared flow of arbitrary direction and stability consideration

Abstract

A Solovev-like solution describing equilibria with field aligned incompressible flows [G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 19, 014504 (2012)] is extended to non parallel flows. The solution expressed as a superposition of Bessel functions contains an arbitrary number of free parameters which are exploited to construct a variety of configurations including ITER shaped ones. For parallel flows, application of a sufficient condition for linear stability shows that this condition is satisfied in an appreciable part of the plasma region on the high-field side mostly due to the variation of the magnetic field perpendicular to the magnetic surfaces. Also, the results indicate that depending on the shape of the Mach-function profile and the values of the free parameters the flow and flow shear may have either stabilizing or destabilizing effects.

Generalized Conditional Symmetries, Related Solutions and Conservation Laws of the Grad-Shafranov Equation with Arbitrary Flow

“…Therefore, by exploiting the respective arbitrary number of free parameters, a variety of equilibria with desirable shaping and useful confinement figures have been reported. Further extensions [12,32] to confined plasmas with incompressible flows parallel to the magnetic field and respectively to plasmas with incompressible flows of an arbitrary direction have been achieved on the basis of generalized Grad-Shafranov equation (GGSE). And furthermore, a possible extension that involves various stochastic magnetic field configurations, with or without shear, in turbulent plasmas may be analyzed [22-24, 26, 30].…”

A tokamak pertinent analytic equilibrium with plasma flow of arbitrary direction

“…Turning now to a particular solution of (10) we assume that f n (ρ) := 0 for n ≥ N , implying that (10) becomes homogeneous for n = N − 1 and n = N − 2. Therefore, the inhomogeneous "source" term in (10) for n = N − 3 is known in terms of the generic solution of (11).…”

“…Crucial for the success of this methodology is the number of points to be exploited, that is, usually the boundary shape is decently reproduced if this number is sufficiently large. However, upon employing a shaping method introduced in [8] and utilized also in [9,10] we can reduce the number of points that have to be used, in principle to three for up-down symmetric configurations and even solutions in ξ and to four for asymmetric configurations or/and non-even solutions in ξ. This set of shaping conditions incorporates equations concerning the boundary values of the flux function u(ρ, ξ) and its first and second order derivatives at the top, lower, inner and outer points of the configuration given by the coordinate pairs…”

“…= a/R 0 is the inverse aspect ratio of the tokamak, δ, is the triangularity and κ is the elongation along the z axis. The justification for the utilization of these conditions is given in [8] and [9,10]; therefore we omit here further discussion regarding their origin. The shaping conditions are summarized as follows…”

Μελέτη Ισορροπίας Και Ευστάθειας Ελικοειδώς Συμμετρικού Μαγνητισμένου Πλάσματος

“…The partial differential equation (4.24), consisting of a reduction of the respective one found in reference [221], admits a generalized Solovev analytical solution of the form…”