Heat transport in a hierarchy of reduced-order convection models

Abstract

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.

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

“…The analysis for large ρ carries over to other models related to the Lorenz equations (1). For the general context of extensions, we refer to Curry (1978), Sparrow (1982), Park (2021), Olson and Doering (2022) and the references therein. For illustration purposes, let us consider linear additions to (1) in the form…”

“…Their dependence on parameters is of fundamental theoretical interest and also practical importance. The derivation of bounds for such averages by algebraic optimization has received increasing attention in recent years (Fantuzzi 2016;Goluskin 2018;Olson et al 2021;Olson and Doering 2022). Bounds derived by estimates need not be sharp, but also sharp bounds may be misleading, if they are realized by dynamically unstable states (Wen et al 2022a, b).…”

“…The rescaling is chosen such that a bifurcation occurs at ρ = 1 and indeed the Lorenz model accurately captures the onset of steady atmospheric convection rolls for ρ > 1. For larger values of ρ, the Lorenz equations do not accurately capture the full PDE model, but the relation between the hierarchy of higher-order mode truncations with the convection PDE model continue to be of interest, e.g., Felicio and Rech (2018), Park (2021), Olson et al (2021), Olson and Doering (2022).…”