## Energetics Of The Thermal Wind Equation

The immediate source of kinetic energy for the eddying circulation observed in our baroclinic instability experiment and in the middle-latitude atmosphere, is the potential energy of the fluid. In the spirit of the energetic discussion of convection developed in Section 4.2.3, we now compute the potential energy available for conversion to motion. However, rather than, as there, considering the energy of isolated fluid parcels, here we focus on the potential energy of the whole fluid. It will become apparent that not all the potential energy of a fluid is available for conversion to kinetic energy. We need to identify that component of the potential energy—known as available potential energy—that can be released by a redistribution of mass of the system.

To keep things as simple as possible, we will first focus on an incompressible fluid, such as the water in our tank experiment. We will then go on to address the compressible atmosphere.

### 8.3.1. Potential energy for a fluid system

We assign to a fluid parcel of volume dV = dx dy dz and density p, a potential energy of gz x (parcel mass) = gzp dV. Then the potential energy of the entire fluid is

where the integral is over the whole system. Clearly, PE is a measure of the vertical distribution of mass.

Energy can be released and converted to kinetic energy only if some rearrangement of the fluid results in a lower total potential energy. Such rearrangement is subject to certain constraints, the most obvious of which is that the total mass

Eric Eady (1918—1967). A brilliant theorist who was a forecaster in the early part of his career, Eady expounded the theory of baroclinic instability in a wonderfully lucid paper—Eady (1949)—which attempted to explain the scale, growth rate, and structure of atmospheric weather systems.

cannot change. In fact, we may use Eq. 8-6 to rewrite Eq. 8-5 as where

P dV

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