Klein-Gordon Equation and Reflection of Alfv\'en Waves

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Session 121 -- Early Stars, X-Ray Binaries, Theory
Oral presentation, Thursday, 12, 1995, 2:00pm - 3:30pm

[121.06] Klein-Gordon Equation and Reflection of Alfv\'en Waves

Z. E. Musielak (UAH), R. L. Moore (NASA/MSFC)

It is of some interest to know the physical conditions that lead to efficient reflection of Alfv\'en waves in the solar and stellar atmospheres. The problem seems to be important because these waves may play some role in non-radiative heating of the solar and stellar chromosphere and coronae, and may also be responsible for acceleration of the solar and cool massive stellar winds. A significant effort has been made by a number of authors to understand the behavior of these waves in highly inhomogeneous stellar atmospheres. The simplest treatment of the problem seems to be the so-called Klein-Gordon equation approach, which allows obtaining local critical frequencies by transforming the wave equations into their Klein-Gordon forms and then choosing the largest positive coefficient to be the square of the local critical frequency. In this paper, we 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 Alfv\'en waves propagating in an isothermal atmosphere with constant gravity and uniform vertical magnetic field. We demonstrate that Alfv\'en waves in the upper (above the wave source) part of our model always form a standing wave pattern and that the waves in the lower (below the wave source) part of the model are always propagating (but partially reflected) waves. We also show that the turning point for the upward and downward waves is located at the height where the condition $\omega = \Omega_A$ is satisfied and that $\Omega_A = V_A / 2 H$, where $V_A$ is the Alfv\'en velocity and $H$ is the scale height, can be taken as a local critical frequency because the waves undergo strong reflection in this region of the atmosphere where $\omega \leq \Omega_A$. By applying our turning-point analysis to the Alfv\'en wave equations for the velocity and magnetic field perturbation, we obtain an 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. A physical interpretation of the obtained results will be given.

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