Magnetoconvection dynamics in a stratified layer. 2: A low-order model of the tilting instability

Gyrostatic extensions of the Howard-Krishnamurti model of thermal convection with shear

“…The resulting LOM has neither the offending invariant manifold nor solutions diverging to infinity; instead, it exhibits better agreement with experimental heat flux data than the original HK model (Thiffeault and Horton, 1996). Meanwhile, Hermiz et al (1995) showed that, in order to satisfy a vorticity balance condition derived by Lantz (1995), the HK model should include an additional mode in the streamfunction expansion.…”

“…If one's goal is to formulate a numerically accurate CFD simulation, this also requires investigating the convergence of the Galerkin approximation as one continues to add modes (or gyrostat modules). Lantz (1995) extended the HK model to the case of 2-D magnetoconvection with an imposed horizontal magnetic field. His aim was to model magnetoconvection with spontaneously generated vertical shear, a phenomenon originally observed in a numerical simulation of the magnetoconvection partial differential equations (Lantz and Sudan, 1995).…”

“…His aim was to model magnetoconvection with spontaneously generated vertical shear, a phenomenon originally observed in a numerical simulation of the magnetoconvection partial differential equations (Lantz and Sudan, 1995). The basic momentum, heat, and magnetic flux equations, in nondimensional form, are as follows (Lantz, 1995):…”

“…Also, Pr is the Prandtl number, Pr m is the magnetic Prandtl number, Ra is the Rayleigh number, and Q is the Chandrasekhar number which characterizes the "initial strength of the mean-squared horizontal magnetic field in the region" (Lantz, 1995). (See Lantz and Sudan (1995) for the precise definition of Q.)…”

“…Subsequently, we added the streamfunction mode of Hermiz et al (1995) to the Thiffeault-Horton LOM, which results in a system of six coupled gyrostats that respects both energy and vorticity balance . In this paper, we expand the scope of the coupled gyrostats approach for designing LOMs by showing how the HK model can be further extended to include (1) a representation of externally forced shear, by extending our previous LOM , and (2) magnetoconvection with shear, by extending the LOM of Lantz (1995). These will be discussed in the next two sections, respectively.…”