Horizontal convection refers to horizontal motions in a fluid slab caused by spatial gradients of fluid density. Rather surprisingly, theoretical arguments show that nonuniform heating or cooling at the top surface of a fluid layer is ineffective in driving the circulation. More specifically, if the only force acting on the fluid is nonuniform surface heating, and if the kinematic viscosity and thermal diffusivity are allowed to approach zero with their ratio fixed, then it can be shown that the energy dissipation per unit mass, determined by integrating over the whole volume of the fluid, goes to zero. Because no energy dissipation implies no turbulence, this is sometimes called the "anti-turbulence theorem." This result is counterintuitive. In fact, it has been thought that the main features of the global-scale ocean circulation might be accounted for by the pole-to-equator temperature gradient, but the anti-turbulence theorem seems to imply that differential heating of the surface cannot by itself drive the circulation. This proposal is motivated by the need to understand the physical processes that account for the anti-turbulence theorem and other purely mathematical constraints on horizontal convection. Implications of these constraints are that fluid motions on the smallest scales must be closely coupled to larger-scale motions as part of a linked, multiscale system. The goal is to relate the improved understanding of horizontal convection to the problem of ocean circulation. The approach consists of a combination of analytical and numerical techniques such as upper bound theory, numerical computation of equilibrium solutions using continuation methods, linear stability analysis of these equilibria, multiscale asymptotic methods, and direct numerical simulation. The work will contribute to improved modeling of the ocean circulation and hence climate. It is supported jointly by the Division of Ocean Sciences and the Division of Mathematical Sciences through the NSF Program, Research Collaborations between the Mathematical Sciences and the Geosciences (CMG).
