Nearly incompressible fluids. II: Magnetohydrodynamics, turbulence, and waves
- 1 January 1993
- journal article
- Published by AIP Publishing in Physics of Fluids A: Fluid Dynamics
- Vol. 5 (1) , 257-273
- https://doi.org/10.1063/1.858780
Abstract
The theory of nearly incompressible (NI) fluid dynamics developed previously for hydrodynamics is extended to magnetohydrodynamics (MHD). On the basis of a singular expansion technique, modified systems of fluid equations are derived for which the effects of compressibility are admitted only weakly in terms of the different possible incompressible solutions (thus ‘‘nearly incompressible MHD’’). NI MHD represents the interface between the compressible and incompressible magnetofluid descriptions in the subsonic regime. The theory developed here does not hold in the presence of very large thermal, gravitational, or field gradients. It is found that there exist three distinct NI descriptions corresponding to each of the three possible plasma beta (β ≡ the ratio of thermal to magnetic pressure) regimes (β≪1, β∼1, β≫1). In the β≫1 regime, the compressible MHD description converges in the low Mach number limit to the equations of classical incompressible three‐dimensional (3‐D) MHD. However, for the remaining plasma beta regimes, the imposition of a large dc magnetic field forces the equations of fully compressible 3‐D MHD to converge to the equations of 2‐D incompressible MHD in the low Mach number limit. The ‘‘collapse in dimensionality’’ corresponding to the different plasma beta regimes clarifies the distinction between the 3‐D and 2‐D incompressible MHD descriptions (and also that of 21/2‐D incompressible MHD). The collapse in dimensionality that occurs as a result of a decreased plasma beta can carry over to the weakly compressible corrections. For a β∼1 plasma, Alfvén waves propagate parallel to the applied magnetic field (reminiscent of reduced MHD), while for a β≪1 magnetofluid, quasi‐1‐D long‐wavelength acoustic modes propagate parallel to the applied magnetic field. The detailed theory of weakly compressible corrections to the various incompressible MHD descriptions is presented and the implications for the solar wind emphasized.Keywords
This publication has 41 references indexed in Scilit:
- The equations of reduced magnetohydrodynamicsJournal of Plasma Physics, 1992
- Density variations in weakly compressible flowsPhysics of Fluids A: Fluid Dynamics, 1992
- Low Mach number two-dimensional hydrodynamic turbulence: Energy budgets and density fluctuations in a polytropic fluidPhysics of Fluids A: Fluid Dynamics, 1992
- Magnetohydrodynamic simulation of the radial evolution and stream structure of solar-wind turbulencePhysical Review Letters, 1991
- Observations of field-aligned density fluctuations in the inner solar windThe Astrophysical Journal, 1990
- Turbulent polar heliospheric fieldsGeophysical Research Letters, 1990
- Nearly incompressible hydrodynamics and heat conductionPhysical Review Letters, 1990
- Propagation observations of the solar wind near the sunThe Astrophysical Journal, 1989
- Compressible and incompressible fluidsCommunications on Pure and Applied Mathematics, 1982
- Problems with different time scales for partial differential equationsCommunications on Pure and Applied Mathematics, 1980