2.3 Kinetic energy equation, buoyancy work, and gas pressure work

Taking the scalar product of the velocity with the equation of motion yields an equation that describes the time evolution of the kinetic energy,
∂Ekin- ∂t = − ∇ ⋅ (Ekinu) − u ⋅ ∇P − ρu ⋅ ∇ Φ + viscous terms, (13 )
which shows that local changes of kinetic energy are caused by – in the order of appearance of terms on the right hand side – transport of kinetic energy (divergence of the kinetic energy flux), work by the gas pressure gradient force, work by gravity, and terms related to viscous dissipation. Subtracting off a mean hydrostatic balance ∂P-≈ − ρ∇ Φ = ρgz ∂z (where for simplicity we ignore the turbulent pressure and assume a constant vertical downward acceleration of gravity gz) we can identify a net force ′ ρgz, which is usually referred to as the buoyancy force (cf. the classical principle of Archimedes!).

The work corresponding to the buoyancy force, ρ ′u′zgz, where u ′z 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, 1980Jump To The Next Citation PointHurlburt et al., 1984Jump To The Next Citation PointCattaneo et al., 1991Rieutord 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,

∂ ⟨Ekin⟩ --------= ...+ ⟨ρ′u′zgz⟩, (14 ) ∂t
where u′z is the horizontal fluctuation of the vertical velocity, and the ellipses represent all other terms in the kinetic energy equation. The physical interpretation is apparently clear; fluid that is heavier than average is accelerated downwards while fluid that is lighter than average is accelerated upwards – both give positive contributions to the balance of kinetic energy, so one is led to believe that there is a net positive work done by gravity. However, mass conservation requires that
′ ′ 0 = ⟨ρuz⟩ = ⟨ρ⟩⟨uz⟩ + ⟨ρu z⟩, (15 )
and hence that
′ ′ ⟨ρ⟩⟨uz⟩ = − ⟨ρu z⟩. (16 )
Hence the total work done by gravity vanishes. If we integrate over a volume that entirely encloses the region of convection the integrals of divergence terms vanish, and the only remaining term that can balance the viscous dissipation is
⟨− u ⋅ ∇P ⟩ = ⟨− ∇ ⋅ (uP )⟩ + ⟨P∇ ⋅ u⟩ = ⟨P ∇ ⋅ u ⟩. (17 )
This term is positive on the average because the pressure is higher when the gas expands (on the way up) than when it is compressed (on the way down). So, averaging over the kinetic energy equation tells us that viscous dissipation is balanced by gas pressure work, not by buoyancy work!

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 17View Equation, 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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