What does Sandstrom's Theorem tell us about the ocean overturning circulation?

Sandstrom (1908) derived a theorem which can be expressed as follows: In order for heating and cooling to drive a steady circulation, the heating must be located above the cooling. Since in the ocean, the heating and cooling are essentially at the same level, this theorem has been used to imply that the thermohaline circulation cannot be driven by buoyancy fluxes. However, Sandstrom's theorem ignores the effect of diffusion, viscosity, and salinity. It also appears to be disproved by the laboratory experiments of Park and Whitehead (JPO, 1999) in which a rotating thermohaline circulation is generated in a basin in which the heating and cooling are both at the bottom. Does this theorem have any relevance to the real ocean? In this talk, we examine the buoyancy budget in a number of circulation models. We show that Sandstrom's theorem can be reduced to a statement about the buoyancy flux. When the buoyancy flux is upwards (as it is in the atmosphere), and this flux is carried by the mean circulation,  buoyancy work represents an important source of energy for the mean circulation.  

By analyzing the output of a number of circulation models, however, we find that the buoyancy flux in the ocean is actually downwards, and that this downward flux is carried by the mean circulation. The downward flux arises because of the nonlinearity of the equation of state, which results in a sink of energy amounting to 0.4 TW. The fact that this downward buoyancy flux is carried by the mean circulation means that the horizontal pressure work is negative.  Since geostrophic flows are associated with no pressure work and frictional flows in boundary currents with negative work – only Ekman flows, which remove water from regions of low pressure and deposit it in regions high pressure have the right sign to explain this circulation. We argue from this that wind stress and mesoscale eddies, rather than turbulent mixing, play the dominant role in the buoyancy budget of the real ocean.

 

Figure 1: Heat, salt, and buoyancy transport due to heat and salt in two circulation models with realistic temperature, salinity, radiocarbon, and oxygen structures. The terms associated with the mean advection are shown in black, those associated with convective adjustment in red and those associated with all other eddy mixing processes in green. Note that the buoyancy transport is dominated by the heat term. The solid lines and those with the symbols represent the two different models. (from Gnanadesikan et al., in press, J. Climate).  

Reference: Gnanadesikan, A., R.D. Slater, P.S. Swathi, and G.K. Vallis, The energetics of ocean heat transport, in press. J. Climate.