where, as before, u = (u, v, w) and v = (u, v) are the three-dimensional and two-dimensional (in the <p, ¿-plane) velocity vectors with components u — a cos (p d/L/dt, v = a d(p/dt, w = dz/di; F, FT and Fq are sources and sinks of momentum, heat and moisture referred to the unit mass; p and p are the air pressure and density; Q is the angular velocity of the Earth's rotation; k is the unit vector directed along the z-axis (vertically upward); other designations are the same.

Five equations ((2.5.1)—(2.5.5)) involve six unknown functions: v, w, p, p, T and q. To close the system we take advantage of the equation of state of unsaturated moist air p = pRT, (2.5.6)

where R is the gas constant, and insert definitions of potential energy <h = gz, internal energy I = cvT, latent heat Lq, kinetic energy K = \u\2/2, and total energy S = K + <£> + I + Lq refer to the unit mass. The budget equation for the potential energy,

ot results from the definition of 4), if it is affected by the operator (d/dt + uV) and the identity w = dz/di is taken into consideration.

The budget equation for the internal energy is found by a combination of Equations (2.5.3), (2.5.4) and (2.5.6). Namely, combining (2.5.3) and (2.5.6) and substituting in (2.5.4) the resulting expression

0 0

Post a comment