INHOMOGENEOUS NEARLY INCOMPRESSIBLE DESCRIPTION OF MAGNETOHYDRODYNAMIC TURBULENCE

Abstract

The nearly incompressible theory of magnetohydrodynamics (MHD) is formulated in the presence of a static large-scale inhomogeneous background. The theory is an inhomogeneous generalization of the homogeneous nearly incompressible MHD description of Zank & Matthaeus and a polytropic equation of state is assumed. The theory is primarily developed to describe solar wind turbulence where the assumption of a composition of two-dimensional (2D) and slab turbulence with the dominance of the 2D component has been used for some time. It was however unclear, if in the presence of a large-scale inhomogeneous background, the dominant component will also be mainly 2D and we consider three distinct MHD regimes for the plasma beta ? 1, ? ~ 1, and? 1. For regimes appropriate to the solar wind (? 1, ? ~ 1), compared to the homogeneous description of Zank & Matthaeus, the reduction of dimensionality for the leading-order description from three dimensional (3D) to 2D is only weak, with the parallel component of the velocity field proportional to the large-scale gradients in density and the magnetic field. Close to the Sun, however, where the large-scale magnetic field can be considered as purely radial, the collapse of dimensionality to 2D is complete. Leading-order density fluctuations are shown to be of the order of the sonic Mach number and evolve as a passive scalar mixed by the turbulent velocity field. It is emphasized that the usual pseudosound relation used to relate density and pressure fluctuations through the sound speed as ?? = c 2 s ?p is not valid for the leading-order density fluctuations, and therefore in observational studies, the density fluctuations should not be analyzed through the pressure fluctuations. The pseudosound relation is valid only for higher order density fluctuations, and then only for short-length scales and fast timescales. The spectrum of the leading-order density fluctuations should be modeled as k ?5/3 in the inertial range, followed by a Bessel function solution K ?(k), where for stationary turbulence ? = 1, in the viscous-convective and diffusion range. Other implications for solar wind turbulence with an emphasis on the evolution of density fluctuations are also discussed.