Upper bounds on time-averaged heat transport are obtained for an eight-mode Galerkin truncation of Rayleigh’s 1916 model of natural thermal convection. Bounds for the ODE model—an extension of Lorenz’s three-ODE system—are derived by constructing auxiliary functions that satisfy sufficient conditions wherein certain polynomial expressions must be nonnegative. Such conditions are enforced by requiring the polynomial expressions to admit sum-of-squares representations, allowing the resulting bounds to be minimized using semidefinite programming. Sharp or nearly sharp bounds on mean heat transport are computed numerically 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 model to be quantitatively close to the PDE model, mean heat transport is maximized by steady states. In some cases at larger Rayleigh number, time-periodic states maximize heat transport in the truncated model. Analytical parameter-dependent bounds are derived using quadratic auxiliary functions, and they are sharp for sufficiently small Rayleigh numbers.

瑞利（Rayleigh）1916年自然对流模型的八模式Galerkin截断法获得了平均时间传热的上限。ODE模型的界线（洛伦兹三ODE系统的扩展）是通过构造满足一定条件（其中某些多项式表达式必须为非负）的辅助函数来得出的。通过要求多项式表达式允许平方和表示来强制执行此类条件，从而允许使用半定编程将结果边界最小化。对于模型参数的许多值：瑞利和普朗特数以及域长宽比，通过数值计算了平均热传递的尖锐或近尖锐边界。在所有瑞利数足够小以使ODE模型在数量上接近PDE模型的情况下，稳态使平均热传递最大化。在某些情况下，在瑞利数较大的情况下，时间周期状态会在截断模型中最大化传热。使用二次辅助函数推导与分析相关的参数范围，对于足够小的瑞利数，范围很清晰。