where [[spi:math]]. The ambient plasma mass density [[spi:math]] is the sum of various constituents (H+, He+, O+) and is given by the model of Chiu et al. (1979). Eq* (13) can be rewritten as [[spi:math]] (14) Using these relations we derive differential equations for the plasma displacement [[spi:math]] and the Alfven wave magnetic field Bx. [[spi:math]] (15) (16) The boundary and initial conditions for [[spi:math]] and Bx are somewhat more complex. The initial condition is obviously that the plasma speed at the equator (z=0) at t=0 is the beam speed v0. To derive the boundary condition of (15) at the equator (z=0), we assume that the beam plasma (speed in the z-direction but concentrated at z=0) moves with the same displacement [[spi:math]] near z= so that we can integrate (15) once over z to obtain 21 [[spi:math]] [[spi:math]] (17) [[spi:math]] (18)
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