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Next: III. Computational Model Up: MHD Effects on Previous: I. Introduction

II. Physical Model

When a pellet is injected into a plasma, the pellet rapidly heats, and ablates. The ablated gas then ionizes and becomes a cloud of high density plasma, which is cooler than its surroundings. The plasma cloud moves together with the background plasma, according to the dissipative MHD equations,




where tex2html_wrap_inline819 is the resistivity and tex2html_wrap_inline821 a scalar viscosity. The density satisfies the continuity equation:


where D is a small diffusion coefficient, kept to improve numerical stability. A single temperature is assumed for simplicity, whose evolution equation includes a large thermal transport parallel to the magnetic field, to be described below in (12) and (13). This tends to make the temperature approximately constant on magnetic field lines,


An initial non equilibrium state, will move towards equilibrium, because it has lower potential energy. An MHD equilibrium satisfies


which together with (5) requires


The initial conditions for the simulations are obtained assuming that the ablation, ionization and parallel thermal conduction time scales are short compared to the typical Alfvén time for MHD motion. The details of the ablation process [5] are not considered, but may be taken into account in in future simulations. The most important fact is that when the pellet material ionizes, it cannot be distributed uniformly on magnetic flux surfaces, because the non ionized material is not constrained by the magnetic field.

In principle, the injection process is adiabatic, imparting no energy to the plasma. The temperature is constant on flux surfaces before and after the pellet injection. After injection, the temperature is lower on flux surfaces that intersect the pellet cloud. To calculate the new temperature, adiabaticity implies that the pressure contained between flux surfaces is the same, before and after the pellet injection. Equivalently, the flux surface average ;SPMlt;p;SPMgt; of the pressure p remains the same, where the flux surface average is


and tex2html_wrap_inline829 is the toroidal angle. Let tex2html_wrap_inline831 and tex2html_wrap_inline833 refer to the unperturbed state before the pellet, and tex2html_wrap_inline835 and tex2html_wrap_inline837 refer to the perturbed state containing the pellet. The temperature is a flux function, tex2html_wrap_inline839 tex2html_wrap_inline841 , as well as the initial density, tex2html_wrap_inline843 . Adiabaticity tex2html_wrap_inline845 implies that


which shows how the temperature is reduced by the density perturbation, such that T is a flux function. The pellet - perturbed density tex2html_wrap_inline849 is not constant on a flux surface. The ratio of peak to average density is large, and the density ablation cloud is localized in three dimensions, so (7) is far from being satisfied. The pellet perturbed pressure is given by


The flux surface averaged pressure perturbation (10) vanishes, but the unaveraged perturbed pressure is not constant on magnetic field lines. Hence (6) is not satisfied, and the plasma is out of MHD equilibrium. The subsequent motion of the plasma will be studied in numerical simulations.

The density blob spreads out because of plasma compression effects. Eqs. (1) and (4) lead to (neglecting diffusion)


If tex2html_wrap_inline851 the density gradient along the magnetic field can change. The MH3D code treats both fast and slow magnetosonic waves, which both contribute to the evolution of the density. The fast magnetosonic waves cause the relatively rapid expansion and equilibration of the pellet cloud pressure with the magnetic pressure. On a slow time scale, sound waves with tex2html_wrap_inline853 cause the density to equalize on magnetic field lines.

next up previous
Next: III. Computational Model Up: MHD Effects on Previous: I. Introduction

Hank Strauss
Wed Jan 7 14:23:34 EST 1998