We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh-Bénard convection, focusing on its ability to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers (P r) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number (Ra) sufficiently, and we explore the resulting convection for Ra up to 10 10 . When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as Ra → ∞. The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with Ra. When the large-scale shear is present with P r 2, the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with Ra for P r = 1. When the shear is present with P r 3, the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of Ra, but the growth rates are slower than any previously reported for Rayleigh-Bénard convection without large-scale shear. We find the Nusselt numbers grow proportionally to Ra 0.077 when P r = 3 and to Ra 0.19 when P r = 10. Analogies with tokamak plasmas are described.

Abstract. We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.

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 $H^{4/5}$ 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-B\'enard 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 $5\times10^{10}$.Comment: 12 pages, 5 figures, 3 tables, 1 movie; v2: text updated, figures and tables added, figure 4 correcte

For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxiliary function provides phase space volumes in which all nearly maximal trajectories are guaranteed to lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz system.