The equilibrium and stability of rotating plasmas

Abstract

In a rotating equilibrium state, the velocity and magnetic fields are shown to share the same flux surfaces. A simplified derivation is given of a second-order (not necessarily elliptic) partial differential equation which determines axisymmetric equilibrium states. For general configurations, equations on flux surfaces which determine the Alfvén and cusp continuous spectrum are derived and the stability investigated. These equations are written without the use of any particular coordinate system. Similar equations yield a sufficient condition for global stability of axisymmetric equilibria if the flow is parallel to the magnetic field up to a rigid rotation of the plasma. This condition is also necessary for stability in a mirror configuration with no toroidal field and a pure rigid rotation.

On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with flow

“…Apart from a factor 1/(γ−1) in the last term of the right-hand side ([1/(γ−1)]ρ γ S ′ instead of ρ γ S ′ ) Eq. (22) is identical in form with the corresponding ideal MHD equation obtained by Hameiri [12] (Eq. (7) therein).…”

“…It is pointed out that, unlike to the usual procedure followed in equilibrium studies with flow [10,11,12,13,14,15] in the present work an equation of state is not included in the above set of equations from the outset and therefore the equation of state independent Eqs. (15) and (16) below are first derived.…”

“…(22) is not always elliptic; there are three critical values of the poloidal-flow Mach-number M 2 at which the type of this equation changes, i.e. it becomes alternatively elliptic and hyperbolic [10,12]. The toroidal flow is not involved in these transitions because this is incompressible by axisymmetry and, therefore, does not relate to hyperbolicity (see also the discussion in the beginning of Sec.…”

“…(9) leads to S = S(ψ) (S is the specific entropy). It is noted that the case S = S(ψ) was considered in investigations on ideal equilibria with arbitrary flows [11,12] and purely toroidal flows [17,18], as well as on resistive equilibria with purely toroidal flows [9]. In addition, the plasma is assumed to being a perfect gas whose internal energy density W is simply proportional to the temperature.…”

Magnetic island deformation due to sheared flow and viscosity

“…A second-order partial differential equation for the poloidal magnetic flux function, similar to the Grad-Shafranov equation for the static case, 1 can be found for an axisymmetric and nondissipative ͑zero resistivity and viscosity͒ equilibrium. 2 The resistive effect on the equilibrium has been studied in Ref. 3.…”