Statistics – Computation
Scientific paper
Jan 1992
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992phdt........78l&link_type=abstract
Thesis (PH.D.)--CORNELL UNIVERSITY, 1992.Source: Dissertation Abstracts International, Volume: 52-12, Section: B, page: 6460.
Statistics
Computation
12
Convection, Hydrodynamics
Scientific paper
Time-dependent magnetohydrodynamic (MHD) convection is studied by numerical modeling on several levels. An intuitive model, the mixing length theory of convection (MLT), is reviewed; from this, a scaling law for convective structures is derived and compared with observations of the solar convection zone. MLT lays the foundation for the magneto-anelastic equations--a model for MHD convection at low Mach and Alfven Mach numbers--which is rederived with particular attention to its relation to the familiar Boussinesq equations. A variant of the anelastic approximation is offered which obviates the need for a first-order pressure equation. The magneto-anelastic model is used as the basis for SUMO, a computer model that generates numerical solutions on a two-dimensional Cartesian mesh by a finite-difference, predictor-corrector algorithm. Thermodynamic properties of the fluid are held fixed and constant at the rigid, stress-free top and bottom boundaries of the computational box, while lateral boundaries are treated as periodic. A numerical study of the equations is made for a zero-order polytrope, at a Rayleigh number 5.44 times the critical value, Prandtl number 1.8, and aspect ratio unity. The dynamical behavior is highly influenced by a horizontal magnetic field which is imposed at the bottom boundary (the top of the domain is attached to a nonconducting region). As the magnetic parameter is increased, the initially tilted character of convection rolls changes. A single-roll, highly sheared, time-dependent state is replaced by a steady "traveling-wave" tilted state; then an oscillatory or "sloshing" state; then a steady two-roll state with no tilting; and finally, a stationary state. The concentrated nature of magnetic fields at the top of these flows is explained by a simple advection-diffusion balance. The complex dynamical behavior is shown to be well-represented by a nine-mode truncation of the Fourier expansion of the Boussinesq equations in a uniform, horizontal magnetic field. The key mode is a "tilting" one, which breaks the up-down mirror symmetry. Qualitative and quantitative comparisons are made between the simulations and the nine -mode model, which demonstrate that the latter captures all the essential physics of the former.
Lantz Steven Richard
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