On WKB expansions for Alfvén waves in the solar wind

Publisher Website
Google Scholar
Abstract
We reexamine the WKB expansion for “toroidal” Alfvén waves in the solar wind, as described by equations (9) of Heinemann and Olbert (1980). Our principal conclusions are as follows: (1) The WKB expansion used by Belcher (1971) and Hollweg (1973) is nonuniformly convergent. (2) Using the method of multiple scales (Nayfeh, 1981), we obtain an expansion which is uniform. (3) The uniform expansion takes into account the small modification to the Alfvén wave phase speed due to spatial gradients of the background. (4) Both the uniform and nonuniform expansions reveal that each “normal mode” has both Elsässer variables δz+ ≠ 0 and δz− ≠ 0. Thus if δz− corresponds to the outgoing mode in a homogeneous background, an observation of δz+ ≠ 0 does not necessarily imply the presence of the inward propagating mode, as is commonly assumed. (5) Even at the Alfvén critical point (where V = υA) we find that δz+ ≠ 0. Thus incompressible MHD turbulence, which requires both δz+ ≠ 0 and δz− ≠ 0, can proceed at the Alfvén critical point (cf. Roberts, 1989). (6) With very few exceptions, the predictions of these calculations do not agree with recent observations (Marsch and Tu, 1990) of the power spectra of δz+ and δz− in the solar wind. Thus the evolution of Alfvén waves in the solar wind is governed by dynamics not included in the Heinemann and Olbert equations.