The structure of energy conserving low-order models

Abstract

General principles for developing physically sound low-order models in fluid dynamics are discussed. It is shown that low-order models with energy conserving properties of the original equations that arise in various important problems (Rayleigh–Bénard convection, rotating fluid, magnetohydrodynamic convection) may be presented in the form of coupled three-mode systems known in mechanics as Volterra gyrostats (plus terms describing forcing and friction). When these models are expanded by increasing the order of approximation or by adding new physical mechanisms, they still have the structure of coupled gyrostats. Conversely, when a low-order model cannot be transformed into coupled gyrostats, this may indicate that its conservation properties should be questioned. For instance, while the widely used (in convection studies) Howard–Krishnamurti model [J. Fluid Mech. 170, 385 (1986)] is not energy conserving and does not have a gyrostatic form, its simple extension to a system of coupled gyrostats possesses inviscid energy invariants. Integrals of motion in the fluid are shown to have their analogs in systems of coupled gyrostats. Thus, giving low-order models a gyrostatic structure ensures that certain important physics from the original fluid dynamical equations is retained. Finally, this approach is used to develop a coupled gyrostat model of turbulence that exhibits Kolmogorov spectral behavior.