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Heat transport in a hierarchy of reduced-order convection models

Reduced-order models (ROMs) are systems of ordinary differential equations (ODEs) designed to approximate the dynamics of partial differential equations (PDEs). In this work, a distinguished hierarchy of ROMs is constructed for Rayleigh's 1916 model of natural thermal convection. These models are distinguished in the sense that they preserve energy and vorticity balances derived from the governing equations, and each is capable of modeling zonal flow. Various models from the hierarchy are analyzed to determine the maximal heat transport in a given model, measured by the dimensionless Nusselt number, for a given Rayleigh number. Lower bounds on the maximal heat transport are ascertained by computing the Nusselt number among equilibria of the chosen model using numerical continuation. A method known as sum-of-squares optimization is applied to construct upper bounds on the time-averaged Nusselt number. In this case, the sum-of-squares approach involves constructing a polynomial quantity whose global nonnegativity implies the upper bound along all solutions to a chosen ROM. The minimum such bound is determined through a type of convex optimization called semidefinite programming. For the ROMs studied in this work, the Nusselt number is maximized by equilibria whenever the Rayleigh number is sufficiently small. In this range of Rayleigh number, the equilibria maximizing heat transport are those that bifurcate first from the zero state. Analyzing this primary equilibrium branch provides a possible mechanism for the increase in heat transport near the onset of convection.