SUB-ALFV\'ENIC REDUCED EQUATIONS IN A TOKAMAK

Abstract

Magnetized fusion experiments generally perform under conditions where ideal Alfv'enic modes are stable. It is therefore desirable to develop a reduced formalism which would order out Alfv'enic frequencies. This is challenging because sub-Alfv'enic phenomena are sensitive to magnetic geometries. In this work an attempt has been made to develop a formalism to study plasma phenomena on time scales much longer than the Alfv'enic time scales. In Part I, a reduced set of MHD equations is derived, applicable to large aspect ratio tokamaks and relevant for dynamics sub-Alfv'enic with respect to ideal ballooning modes. Our ordering optimally allows sound waves, Mercier modes, drift modes, geodesic-acoustic modes, zonal flows, and shear Alfv'en waves. Long to intermediate wavelengths are considered. With the inclusion of resistivity, tearing modes, resistive ballooning modes, Pfirsch-Schluter cells, and the Stringer spin-up are also included. A major advantage is that the resulting system is 2D in space, and the system incorporates self-consistent dynamic Shafranov shifts. A limitation is that the system is valid only in radial domains where the tokamak safety factor, $q$, is close to a rational. Various limits of our equations, including axisymmetric and subsonic limits, are considered. In the tokamak core, the system is well suited as a model to study the sawtooth discharge in the presence of Mercier modes. In Part II, we show that the methods of Part I can be directly applied to derive $\sa$ but supersonic reduced fluid equations, for collisionless plasmas, starting from a drift-kinetic description in MHD ordering. In Part III, we begin a reduced description of sub-Alfv'enic phenomena for collisionless kinetic MHD in the subsonic limit. We limit ourselves to discuss axisymmetric modes, including geodesic acoustic modes (GAMs), sound waves and zonal flows. In axisymmetric geometry it is well known that trapped particles precess toroidally at speeds much exceeding the $\E\times \B$ speed. This large flow is expected to contribute a large effective inertia. We show that the kinetic analog of the ``Pfirsch-Schluter" effective inertia $(1+2 q^2)$ is augmented by the well-known Rosenbluth-Hinton factor $\approx 1.6 q^2/\sqrt{\epsilon}$.