where [[spi:math]] (19) At the ionospheric end of the field line ([[spi:math]]), the Alfven wave electric field Ey = vx B0/c drives a conduction current [[spi:math]] where [[spi:math]] is the ionospheric Pedersen conductivity. Ampere’s law then requires that [[spi:math]]. (20) From Fig. 9 we note that the Alfven speed [[spi:math]] varies a great deal along z; therefore, the assumption of piecewise constancy of [[spi:math]] and B0 above is valid only if we divide up the field line into many segments in order to perform the numerical analysis. We have divided each field line into 15 segments. At the intersection of adjacent segments the continuity of [[spi:math]] and their first derivatives with respect to z are maintained. The reader 22 Integration of Ampere’s law over the entire ionospheric layer yields the ionospheric boundary condition on [[spi:math]] [[spi:math]] where [[spi:math]] is the height integrated ionospheric Pedersen conductivity. The problem is then solution of (15) with the initial condition (17) and boundary conditions (18) and (19). The solution of this problem is rather difficult, even numerically, primarily because of the complexities of the boundary conditions. On the other hand, it is noted that if [[spi:math]] and B0 are piecewise constant over a segment of z the entire problem can be solved in closed form, albeit with the use of a summation of an infinite set of Laguerre functions. familiar with potential field problems of quantum mechanics will immediately recognize in (15) that with the aid of the transformation
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