KLEIN-GORDON EQUATION AND THE LOCAL CRITICAL FREQUENCY FOR ALFVEN WAVES PROPAGATING IN AN ISOTHERMAL ATMOSPHERE

A Klein-Gordon equation approach developed by Musielak, Fontenla, and Moore for assessing reflection of Alfven waves in a smoothly nonuniform medium is reexamined. In this approach, the local critical frequency for strong reflection is simply found by transforming the wave equations into their Klein-Gordon forms and then choosing the largest positive coefficient of the zeroth-order term to be the square of the local critical frequency. In this paper, we verify this approach for a particular atmosphere and show that the local critical frequency can be alternatively defined by using the turning-point property of Euler's equation. Our results are obtained specifically for steady state, linear Alfven waves in an isothermal atmosphere with constant gravity and uniform vertical magnetic field. The upward Alfven waves (those above the wave source) are standing waves and the downward waves (those below the wave source) are propagating waves. We demonstrate that for any given wave frequency both upward and downward waves have the same turning point or critical height. This height is determined by the condition omega = Omega(A) = V-A/2H, where V-A is the Alfven velocity and H is the scale height; Omega(A) can be taken as the local critical frequency for strong reflection for the upward waves and as the local critical frequency for free propagation for the downward waves. Our turning-point analysis also yields another interesting result: for our particular model atmosphere the magnetic field perturbation wave equation yields the local critical frequency but the velocity-perturbation wave equation does not. Thus, for this model atmosphere, we find that the Klein-Gordon equation approach of Musielak, Fontenla, and Moore is correct in (1) its choice of the magnetic-field-perturbation wave equation for finding the local critical frequency, and (2) its assumption that the upward and downward waves have the same critical frequency.