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### Solution of Fokker-Planck Equation

Given the source rate of beam ions and the local plasma densities and temperatures, SNAP solves the Fokker-Planck equation in each radial shell for the distribution function of the thermalizing beam-ion population as a function of energy and pitch-angle. The source rate in a given shell is computed by the beam neutral deposition routine, and is essentially a set of delta functions in energy (one for each energy component of each beam source) multiplied by narrow functions in . Here represents the pitch-angle of the beam ions as they cross the horizontal midplane. The equation solved is:

The charge-exchange lifetime is in the limit where the beam ion velocity v is much greater than the average neutral velocity. Since in supershots the neutral temperature--which is comparable to the local ion temperature--can approach the third-component beam energy, SNAP integrates the reaction rate over the neutral velocity distribution, , where and are the relative energy velocity between the beam ion and neutral.

Up to this point the beam charge-exchange is regarded as a complete loss, i.e., the possibility of reionization of the beam neutral before it leaves the plasma has not been considered (indeed, this is probably not a bad assumption for a single shell, given its small width--but it will ionize some neutrals born in other shells). As discussed in section 3.4, SNAP includes a crude correction for this reionization term: a fixed fraction of the charge-exchange losses are considered to be reionized locally. The fraction is selected by the user, and is constant in space, energy, and pitch angle. This is done simply by increasing by the factor where R is the reionization fraction.

Note that electric field and trapping effects are neglected. The equation is solved by expanding in a series of Legendre polynomials: .     Next: Details of Beam Power Up: Deposition of Beam Neutrals Previous: Deposition of Beam Neutrals

Marilee Thompson
Fri Jul 11 15:18:44 EDT 1997