## Thermodynamic Equation

The equation governing the evolution of temperature can be derived from the first law of thermodynamics applied to a moving parcel of fluid. Dividing Eq. 4-12 by St and letting St —> 0 we find:

DQ Dt

DQ/Dt is known as the diabatic heating rate per unit mass. In the atmosphere, this is mostly due to latent heating and cooling (from condensation and evaporation of H2O) and radiative heating and cooling (due to absorption and emission of radiation). If the heating rate is zero then DT/Dt = P- Dp/Dt, and, as discussed in Section 4.3.1, the temperature of a parcel will decrease in ascent (as it moves to lower pressure) and increase in descent (as it moves to higher pressure). Of course this is why we introduced potential temperature in Section 4.3.2; in adiabatic motion, 9

The three equations in 6-7, together with 6-11 or 6-12, and 6-14 are our five equations in five unknowns. Together with initial conditions and boundary conditions, they are sufficient to determine the evolution of the flow.

Before going on, we make some remarks about restrictions in the application of our governing equations. The equations themselves apply very accurately to the detailed motion. In practice, however, variables are always averages over large volumes. We can only tentatively suppose that the equations are applicable to the average motion, such as the wind integrated over a 100-km square box. Indeed, the assumption that the equations do apply to average motion is often incorrect. This fact is associated with the representation of turbulent scales, both small scale and large scale. The treatment of turbulent motions remains one of the major challenges in dynamical meteorology and oceanography. Finally, our governing equations have been derived relative to a 'fixed' coordinate system. As we now go on to discuss, this is not really a restriction, but is usually an inconvenience.

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