where Q = S0/4 is the flux of the short-wave solar radiation at the upper atmospheric boundary; S0 is the solar constant; Ts and TA are values of the absolute temperature at the underlying surface (in the case of the ocean surface 7¡ is equal to 7\ or T0) and at the mean level in the atmosphere; y is the short-wave radiation absorption coefficient; v^ and v* are upward and downward long-wave radiation emissivity; v is the coefficient of long-wave radiation absorption; aA is the atmospheric albedo; o is the Stefan-Boltzmann constant; as before, P is the amount of precipitation falling per unit area of the underlying surface per unit of time.
The unknown function P is obtained from the condition of instantaneous precipitation of the moisture excess determined by the difference between evaporation and the meridional water vapour transport by means of synoptic disturbances. This condition is written in the form m»
where C9 is a dimensional constant; the composition of precipitation is identified by the temperature of the underlying surface: if it is less than the water freezing temperature Tso then it is snow that falls, otherwise it is rain.
To find q we use the standard power dependence of specific humidity on pressure p. As a result, we arrive at the relationship
where <?al = J\q'aX + (1 - f\)q°u qa2 = f2q\2 + fxq\2 + (1 - f2 - f\)q°2 are the average (within the limits of southern and northern boxes) values of specific humidity at some fixed level (say, at p/ps = 0.9985) in the surface atmospheric layer; superscripts L, I and O correspond, respectively, to the air surface layer over land, sea ice and ocean; k21 and k22 are numerical constants. Over any type of underlying surface qa can be found using the expression qa = raqm(Ta), where qm(Ta) is the maximum specific humidity defined by the Clausius-Clapeyron equation; Ta and ra are the temperature and specific humidity at the level p/ps = 0.9985.
The relationship (5.5.26) includes one more unknown, Ta. We will assume that the potential temperature is a linear function of pressure. Then the absolute temperature Ta at the level p/ps = 0.9985 is defined by the expression
Ta = (p/Ps)K1[2 (1 - P/Ps)Ta + (1 - 2p/ps)7¡], representing the local dependence between the surface air temperature and the underlying surface temperature. This dependence is valid over land, ice and the upwelling area but is violated over the relatively small area of cold deep water formation, an effect associated with cold air transport from a neighbouring continent and with deep ocean mixing. This peculiarity can be considered in the framework of a box model if the surface air temperature over the area of cold deep water formation is assumed to be described by the interpolation formula:
Ta = kUzT\ + ATÏ + (1 - f2 - /,)r°] + (1 - &)T°a, (5.5.27)
where fi.d is the numerical factor equal to zero as T\ > 273 K, and 1 as T\ < 273 K.
We now consider the heat fluxes Qf appearing in (5.5.3), (5.5.5), (5.5.9), (5.5.10), (5.5.19) and (5.5.24). We discuss the flux Qj of the short-wave solar radiation absorbed by the underlying layer, as well as the net flux Q\ of the long-wave radiation at the underlying surface, and eddy fluxes of sensible Q% and latent Q\ heat. Let us represent them in the form where c1 = dqm/d|i|r=7-a, c2 = (1 — ra/rs)qm(Ta); as is the albedo of the underlying surface; C„ is the resistance coefficient; pa is the air density.
We define the relative humidity (rs) at the underlying surface and the relative humidity in the surface atmospheric layer as rs = W/Wc and ra = r*W/Wc, where Wis the moisture content and Wc is its critical value for which evaporation is equal to potential évapotranspiration (evaporability); r* is a numerical constant, and then we recollect that the climatic system contains four types of underlying surface: the sea surface, the sea ice surface, the snow-covered ice and the snow-free land surfaces. We assume that W/Wc = 1 for the first three types of underlying surface and that the critical moisture content Wc and soil moisture capacity W0 are connected with each other by the relation WQ = 0.75 W0 (see Manabe and Bryan, 1969). Next, using data presented by Korzun (1974) we assume that in the case where the temperature Ts of the land surface is higher than the water freezing temperature Tso and the land moisture content W is less than the moisture capacity W0, a certain part of precipitation (say, yR(W/W0)(P/p0), here P/p0 is the thickness of the precipitation layer in water equivalent; p0 is the fresh water density, yR is a numerical factor) is consumed for run-off formation. Meanwhile the remaining part is used for land moisture content change. But if W> W0,
Q! = ( 1 - X)(l - «s)(l - oca)Q, Qi = o&n - Tf), ef = PaC„1/X(7;-7;),
then the whole moisture excess resulting from the difference between precipitation and evaporation is transformed into run-off. Hence, for the southern box, where the temperature of the land surface is always higher than Tso, we have
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