Similarity theory for global oceanatmosphere interaction

As has been mentioned, the primary cause of atmospheric motions is the flux q = (1 — otp)(na2/4na2)S0 of the absorbed short-wave solar radiation (here S0 is the solar constant, ap is the planetary albedo, see below). Besides q, a set

of the external parameters determining the atmospheric circulation includes the radius a of the planet, the angular velocity Q of its rotation, the mass mA = pjg of the atmospheric column of a unit cross-section, the gravity g related to mA and pressure ps at the low boundary of the atmosphere by the relationship g = pJmA, the specific heat of air at constant pressure cp, the specific heat of air at constant volume cv defined by the difference (cp — cv) = R/p (where R is the universal gas constant), the relative air molecular mass p, and, finally, the Stefan-Boltzmann constant a or its reduced value /a (here / is the emittance) determining the outgoing flux of long-wave radiation for a prescribed temperature.

Golitsyn suggested that these eight parameters, which all, with the exception of p, are dimensional, completely determine the global properties of the atmosphere. But because the dimensional parameters contain four independent dimensions (length, time, mass and temperature), then, on the basis of the re-theorem of the dimensional analysis, it is possible to form only the following three dimensionless combinations:

2 L ct a Kga nm = ^ = (/c7)3/8cp_3/V/8 —, tr ma where L = cr/2Q and H = cf/Kg are the horizontal scale of synoptic disturbances and the height of the homogeneous atmosphere; cr = (kRTJp)1/2 is the sound velocity; k = cp/cv is the ratio of heat capacities; Tr = (q/fa)1'4' is the average temperature of the outgoing emission; rp = a/ct is the relaxation time of pressure disturbances; tr = cpmA TJq is the relaxation time of the local radiative equilibrium.

We agree that the angular velocity Q of planetary rotation is only included in all formulae via nn; the gravity g - only via Il9; and mass mA of the atmospheric column of the unit cross-section - only via Ilm. We also exclude from the examination the dimensionless parameter k, assuming it to be equal to a numerical constant. Then any characteristic F of the atmospheric circulation has to have the form

where F0 is the combination with dimension F from the parameters fa, cp, q and a; ij/F is the dimensionless function of non-dimensional combinations

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