Evolution of alfvén waves in a stochastic magnetic field-application of khasminskii´s theorem

The evolution of Alfvén waves in the presence of a spatially uniform but temporally stochastic magnetic field is considered. Using Stratanovich´s method, it is shown that an Alfvén wave is almost surely amplified, if the power spectral density of the magnetic field at twice the wave frequency exceeds a critical value determined by the conductivity of the plasma. Subsequently, a more rigorous and detailed analysis is made using Khasminskii´s Theorem. The evolution of the first and the second moments is analyzed. It is shown that the mean wave is actually damped and the second moment is the one which is amplified. Further, it is found that the wave suffers a frequency renormalization. This frequency renormalization is a totally new effect, arising from the more rigorous analysis. The Markovian limit is taken in a more reasonable way. The results reported in the paper are much stronger than the earlier analysis.