Spatial transport and spectral transfer of solar wind turbulence composed of Alfvén waves and convective structures I: The theoretical model

In this paper we give a survey of detailed algebraic developments of a solar wind turbulence model. The numerical solution of the coupled system of spectral transfer equations for turbulence composed of Alfvén waves and convective structures or two-dimensional turbulence is prepared. The underlying theory of spectral transfer equations was established by several authors in the early 1990s. The related numerical turbulence model which is elaborated in detail in this paper is based on a rotationally symmetric solar wind model for the background magnetic and flow velocity fields with the full geometry of Parker's spiral which has to be inserted into the transfer equations. Various sources and sinks for turbulent energy are included and appropriately modelled analytically. Spherical expansion terms related to radial gradients of the background velocity fields are considered as far as possible within a rotational symmetric solar wind model, which excludes vorticity effects. Furthermore, nonlinear interaction terms are considered, justified by phenomenological arguments and evaluated by dimensional analysis. Moreover, parametric conversion terms for Alfvén waves and wave-structure interactions are modelled and a generalized spectral flux function for the residual energy eR is introduced. In addition, we compensate the spectra for WKB trends and f -5/3-slopes in order to prepare a convenient form of the equations for numerical treatment. The modelling of source and sink terms includes a special analytical treatment for correlation tensors. This first part presents a summary of the main ideas and the special approximations used for all these terms, together with details on the basic steps of the algebraic calculations. The description of the numerical scheme and a survey of the numerical results of our model, as well as a discussion of the main physical results are contained in a companion paper.