Nearly incompressible fluids: hydrodynamics and large scale inhomogeneity

A system of hydrodynamic equations in the presence of large-scale inhomogeneities for a high plasma beta solar wind is derived. The theory is derived under the assumption of low turbulent Mach number and is developed for the flows where the usual incompressible description is not satisfactory and a full compressible treatment is too complex for any analytical studies. When the effects of compressibility are incorporated only weakly, a new description, referred to as "nearly incompressible hydrodynamics," is obtained. The nearly incompressible theory, was originally applied to homogeneous flows. However, large-scale gradients in density, pressure, temperature, etc., are typical in the solar wind and it was unclear how inhomogeneities would affect the usual incompressible and nearly incompressible descriptions. In the homogeneous case, the lowest order expansion of the fully compressible equations leads to the usual incompressible equations, followed at higher orders by the nearly incompressible equations, as introduced by Zank and Matthaeus. With this work we show that the inclusion of large-scale inhomogeneities (in this case time-independent and radially symmetric background solar wind) modifies the leading-order incompressible description of solar wind flow. We find, for example, that the divergence of velocity fluctuations is nonsolenoidal and that density fluctuations can be described to leading order as a passive scalar. Locally (for small lengthscales), this system of equations converges to the usual incompressible equations and we therefore use the term "locally incompressible" to describe the equations. This term should be distinguished from the term "nearly incompressible," which is reserved for higher-order corrections. Furthermore, we find that density fluctuations scale with Mach number linearly, in contrast to the original homogeneous nearly incompressible theory, in which density fluctuations scale with the square of Mach number. Inhomogeneous nearly incompressible equations for higher order fluctuation components are derived and it is shown that they converge to the usual homogeneous nearly incompressible equations in the limit of no large-scale background. We use a time and length scale separation procedure to obtain wave equations for the acoustic pressure and velocity perturbations propagating on fast-time-short-wavelength scales. On these scales, the pseudosound relation, used to relate density and pressure fluctuations, is also obtained. In both cases, the speed of propagation (sound speed) depends on background variables and therefore varies spatially. For slow-time scales, a simple pseudosound relation cannot be obtained and density and pressure fluctuations are implicitly related through a relation which can be solved only numerically. Subject to some simplifications, a generalized inhomogeneous pseudosound relation is derived. With this paper, we extend the theory of nearly incompressible hydrodynamics to flows, including the solar wind, which include large-scale inhomogeneities (in this case radially symmetric and in equilibrium).