Generation of large-scale winds in horizontally anisotropic convection

We simulate three-dimensional, horizontally periodic Rayleigh-B\'enard convection between free-slip horizontal plates, rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the vertical turbulent heat flux is much lower on average than when there is no wind

Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming

We present methods for bounding infinite-time averages in dynamical systems governed by nonlinear PDEs. The methods rely on auxiliary functionals, which are similar to Lyapunov functionals but satisfy different inequalities. The inequalities are enforced by requiring certain expressions to be sums of squares of polynomials, and the optimal choice of auxiliary functional is posed as a semidefinite program (SDP) that can be solved computationally. To formulate these SDPs we approximate the PDE by truncated systems of ODEs and proceed in one of two ways. The first approach is to compute bounds for the ODE systems, increasing the truncation order until bounds converge numerically. The second approach incorporates the ODE systems with analytical estimates on their deviation from the PDE, thereby using finite truncations to produce bounds for the full PDE. We apply both methods to the Kuramoto--Sivashinsky equation. In particular, we compute upper bounds on the spatiotemporal average of energy by employing polynomial auxiliary functionals up to degree six. The first approach is used for most computations, but a subset of results are checked using the second approach, and the results agree to high precision. These bounds apply to all odd solutions of period 2πL, where L is varied. Sharp bounds are obtained for L≤10, and trends suggest that more expensive computations would yield sharp bounds at larger L also. The bounds are known to be sharp (to within 0.1% numerical error) because they are saturated by the simplest nonzero steady states, which apparently have the largest mean energy among all odd solutions. Prior authors have conjectured that mean energy remains O(1) for L>>1 since no particular solutions with larger energy have been found. Our bounds constitute the first positive evidence for this conjecture, albeit up to finite L, and they offer some guidance for analytical proofs

Penetrative internally heated convection in two and three dimensions

Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows proportionally to the heating rate (H) without convection, grows proportionally to H4=5 when convection is strong in both two and three dimensions. The ratio of the heating rate to the fluid temperature is likened to the Nusselt number of Rayleigh–Bénard convection. Simulations are reported for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in terms of the heating rate) up to 5 1010

Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization

We present a method for finding lower bounds on the global infima of integral variational problems, wherein $\int_\Omega f(x,u(x),\nabla u(x)){\rm d}x$ is minimized over functions $u\colon\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$ satisfying given equality or inequality constraints. Each constraint may be imposed over $\Omega$ or its boundary, either pointwise or in an integral sense. These global minimizations are generally non-convex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the original problem. The PDR can be derived by dualizing and relaxing the original problem; its constraints are pointwise equalities or inequalities over finite-dimensional sets, rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of $(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions $m,n$ and number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat. Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincar\'e inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several examples