Heat Transport in Reduced Order Convection Models

Abstract

Reduced order models (ROMs) are finite systems of ordinary differential equations (ODEs) that approximate the dynamics of the governing partial differential equations. This work considers ROMs for Rayleigh's 1916 mathematical model of thermal convection. A hierarchy of reduced models is developed that satisfy energy, temperature, and vorticity balance laws to promote agreement with the physics of thermal convection. These balance laws are generalized from idealized versions considered by previous authors, and new criteria are established for the vorticity balance in the general case. Each model in the hierarchy is an extension of the Lorenz equations and includes Fourier modes capable of producing zonal flow---horizontal mean flow that vertically shears the fluid.

Upper bounds on time-averaged heat transport are obtained for several models in the hierarchy. Bounds for the ODE models are derived by constructing auxiliary functions such that certain polynomial expressions are nonnegative. Nonnegativity is enforced by requiring these polynomial expressions to admit sum-of-squares representations. Polynomial auxiliary functions subject to such constraints can be optimized computationally with semidefinite programming, minimizing the resulting bound. Upper bounds are compared to particular solutions to the ODEs obtained using bifurcation analysis and numerical integration. An eight-ODE model in the hierarchy is explored in detail, revealing sharp or nearly sharp bounds on mean heat transport for numerous values of the model parameters, the Rayleigh and Prandtl numbers and the domain aspect ratio. In all cases where the Rayleigh number is small enough for the ODE models to closely approximate the physics of the governing equations, mean heat transport is maximized by the steady states that emerge from the first instability of the static state. These equilibria do not exhibit zonal flow, suggesting that this type of flow does not enhance heat transport. Analytical parameter-dependent bounds are derived for the eight-ODE model with quadratic auxiliary functions, and they are sharp for sufficiently small Rayleigh numbers.