The Transport of Low-frequency Turbulence in Astrophysical Flows. I. Governing Equations

Numerous problems in space physics and astrophysics require a detailed understanding of the transport and dissipation of low-frequency turbulence in an expanding magnetized flow. We employ a scale-separated decomposition of the incompressible MHD equations (based on an Elssasser description) and develop a moment hierarchy to describe the transport of the total energy density in fluctuations, the cross-helicity, the energy difference, and correlation lengths corresponding to forward- and backward-propagating modes and to the energy difference. The dissipation terms for the various transport equations are derived. One-point closure schemes are utilized. The technical elements of this work that distinguish it from previous studies are (1) the inclusion of the large-scale background inhomogeneous Alfvenic velocity VA at a level of detail greater than before, (2) the introduction of a tractable slow timescale closure to eliminate high-frequency interference terms that is likely to prove a useful approximation for practical problems related to the transport of turbulence in an inhomogeneous flow such as the solar wind or solar corona, and finally, (3) we develop a simplified phenomenology for the energy difference or equivalently residual energy that may be useful for practical applications. This yields a coupled system of six equations that describes the transport of turbulence in inhomogeneous sub-Alfvenic and super-Alfvenic flows. The turbulence transport equations are quasi-linear in their spatial evolution operators and nonlinear in the dissipation terms, making the model equations relatively tractable to analysis.