Geophysical Turbulence and Buoyant Flows

Abstract

The behavior of turbulence can be greatly affected by fluctuating body forces if the latter are correlated with the velocity fluctuations. The most common example is the large effect of gravity on flows with density fluctuations. If the density fluctuations arise because there is a mean density gradient in the same direction as the mean velocity gradient (as in a boundary layer on a heated or cooled horizontal surface) or if the flow is actually driven by the mean density differences (as in buoyant plumes in still air) then the density and velocity fluctuations are highly correlated and buoyancy can have a large effect. If the density increases upward (heavy fluid on top of light fluid) the flow is “unstable” and the density-velocity correlation can convert potential energy into turbulent kinetic energy. Conversely if the density decreases upward at a faster rate than is expected for fluid in hydrostatic equilibrium, existing turbulent energy can be converted into potential energy (because turbulent mixing tends to reduce the density gradient and thus raise the center of gravity of the fluid). We shall see below that a convenient parameter for correlating the effects of a density difference difference Δϱ across a fluid layer of thickness h with a typical velocity U is

$$ \frac{{\Delta \varrho gh}}{{\varrho {U^2}}} $$

which is the ratio of the hydrostatic pressure difference across the layer to (twice) a typical dynamic pressure.