Modeling results on spatial transport and spectral transfer of solar wind Alfvénic turbulence
Authors: Marsch E. , Tu C. ‐Y. 
Published: February 2008

In this paper a set of time stationary transport equations for incompressible MHD fluctuations in the solar wind is derived from previous general transport equations (Marsch and Tu, 1989; Zhou and Matthaeus, 1990a), which have been found to give solutions with fast time variations. The present derivation is based on the assumption that the fluctuations are composed of small‐scale convected structures and short‐wavelength Alfvén waves. The different contributions of these two types of fluctuations to the total correlation functions can be evaluated by means of temporal and spatial averaging of the correlations over the small scales. Two linearly decoupled sets of transport equations then result, which separately describe the spatial evolution of the turbulent energies and cross correlations of the structures and waves. For the propagating Alfvén waves a multiple‐scale analysis yields two WKB‐type transfer equations for the autocorrelation functions expressed in terms of Elsässer velocity fields. For the structures a third additional equation is derived, which determines the evolution of the residual energy, that is, the difference between the kinetic and magnetic energy of the convected fluctuations. The final set of equations is slowly varying in time and thus satisfactory from the point of view of conventional statistical turbulence theory. The nonlinearities are modeled by cascading flux functions, which are determined by dimensional analysis following the Kolmogorov phenomenology and based on the time stationarity assumption. The new equations are consistent with this assumption and equivalent to the equations obtained by Tu and Marsch (1993). The present derivation aims at clarifying the relations between the general and the time stationary set of transport equations. Consequently, stationary equations governing the spatial and spectral evolution of the power frequency spectra e± for Alfvénic fluctuations, described in terms of the two Elsässer variables, are established and integrated numerically. As a first step to study the effects of the nonlinear terms, we neglect the coupling terms related to convected structures. This approximation may apply to the fluctuations observed in fast streams near 0.3 AU. We integrate the resulting two coupled transport equations in frequency‐distance‐space by employing a new technique based on the method of characteristics. Interplanetary parametric decay instabilities are also included in the model. The numerical results obtained show that (1) The cascade process which is based on local nonlinear interactions in frequency space cannot transport any initial value of the normalized cross‐helicity from the low‐frequency boundary to the higher‐frequency range. Cascade processes alone invariably result in dynamic alignment and cause the spectra of e+ as well as e− to steepen at higher frequencies. (2) However, a parametric‐decay‐like source term can enforce the normalized cross‐helicity to decrease with increasing heliocentric distance and can also produce and sustain a flatter part in the spectrum of e− in the high‐frequency range. These results are in qualitative agreement with the observations. Research topics which should be dealt with in the future to complete the present preliminary numerical work are also pointed out.