Again the last term on the left-hand side describes an amount of heat lost from a given volume by small-scale vertical eddy motion. The external heat sources are radiation and, according to the implicit way of treating latent heat, the heat of condensation of water vapour (plus possibly other phase change contributions), p Q = R + C, where R is the amount of heat added to a unit volume by radiation and C is the contribution from condensation processes.

For the approximations considered, a complete set of equations for the atmosphere is constituted by (2.45), (2.47), (2.48), (2.53) and a number of equations (2.49), the most important of which may be that of water vapour. Auxiliary relations include the equation of state, (2.50), and the equations defining the potential temperature (2.51), the heat sources and the source functions for those other constituents, which have been included in (2.49). The water vapour equation is important for calculating the condensation contribution to the heat source function Q. The calculation of the radiative contribution to Q involves, in addition to knowing the external radiation input from space and from continents and oceans as a function of time, a fairly detailed modelling of those constituents of the atmosphere which influence the absorption of radiation, e.g. ozone, water in all forms (distribution of clouds, etc.) and particulate matter.

The atmospheric heat source function

Alternatively, but with a loss of predictive power, one may take Q as an empirical function, in order to obtain a closed system of equations without (2.49), which - if the turbulent eddy term is also left out - allows a determination of the averaged quantities V*, w*, p, P and T* or B*. Any calculation including the dissipation by turbulent eddies must supplement the equations listed above with either the equations of motion for the fluctuations V' and w', or (as is more common) by some parametrisation of or empirical approximation to the time averages pw'V', pw'9' and eventually pw} A' .

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