The work corresponding to the buoyancy force, , where is the horizontal fluctuation of the vertical velocity, is traditionally called the buoyancy work (although buoyancy power would perhaps be more correct). In places where the density of ascending gas is higher than average (e.g., because of the overpressure necessary to accelerate the horizontal flow in large cells) the buoyancy work can be locally negative in upflows – this is referred to as buoyancy braking (Massaguer and Zahn, 1980; Hurlburt et al., 1984; Cattaneo et al., 1991; Rieutord and Zahn, 1995). Averaging the kinetic energy equation over horizontal planes (using to denote horizontal averages) one finds that there is a contribution from the rate of work done by the buoyancy force,
But buoyancy forcing of convection is a fundamental and easily understood mechanism; warm gas is lighter and rises, cold gas is denser and sinks. So, how can the total buoyancy work vanish? The solution to this riddle lies in the word “total”. Buoyancy indeed performs positive work on convection cells, maintaining their kinetic energy. It is only when including the ultimate, global scale, the horizontally averaged velocity, that the energy balance score of the buoyancy work is evened out to zero. Because density and velocity are correlated, and because the average mass flux must vanish, the average vertical velocity is upwards, corresponding to negative buoyancy work of a magnitude that exactly cancels the fluctuating buoyancy work!
The root of the somewhat perplexing situation lies in the particular choice of variables that are to be expanded into ‘averages’ and ‘fluctuations’. As a result of choosing density and velocity, rather than for example density and mass flux (which has a vanishing horizontal average) one must include also the (non-vanishing) product of the averages, and not only the average product of the fluctuations.
Perhaps the clearest way of stating the result is to say that there is on the one hand a (generally positive) work done by the buoyancy force (as defined) on the convective motions, and there is on the other hand an equally large but negative work done by the average force of gravity acting on the mean flow. As a result, the total work done by gravity vanishes.
Broadly speaking the positive buoyancy work goes towards maintaining the fluctuating (convective) velocity field against dissipation, but, as illustrated by Equation 17, the most straightforward analysis of work and dissipation is obtained by considering the gas pressure work rather than the buoyancy work.
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