at cpmAi= i d0 2Kl 4
di cpmAi = i y - T1we = -(F°x + w7\) - Ql_0 + X QFJp0c0 + J, (5.5.3) at ¡ = i
^ T2(H - hy) + 7>e = - (F°2 - wT0) + Ql + 0 - J, (5.5.4)
ai where the first two equations refer to the southern and northern atmospheric boxes, respectively, the third and fourth refer to the UML and DOL in the upwelling area, the fifth refers to the area of cold deep water formation, and the two last refer to the UML and DOL in the polar ocean.
We now explain the separate terms in these equations and their meanings. Terms on the left-hand sides of the Equations (5.5.1) and (5.5.2) describe changes in time of the mass-weighted, vertically averaged air potential temperature 9 referred to the mean level (ps/2) in the atmosphere; the first terms on the right-hand sides - the meridional transport of sensible heat, the second terms on the right-hand sides - heat sources and sinks in the atmosphere; the first terms on the left-hand sides of Equations (5.5.3)-(5.5.7) - heat content changes in the ocean area examined (here 7i and T2 are the UML and DOL temperature in the upwelling area, T0 is the temperature in the cold deep water formation area, T_t and T_2 are the temperatures of the UML and DOL in the polar ocean, and /i_ x are UML thickness in the upwelling area and in the polar ocean); the second terms on the left-hand sides describe the effect of the entrainment at the UML-DOL interface; the first terms on the right-hand sides of Equations (5.5.3)—(5.5.5) describe the diffusive and advective heat exchange between the upwelling area and the area of cold deep water formation; the second terms in (5.5.3) and (5.5.4) describe the eddy heat flux at the upper boundary of the UML and the equivalent heat flux at the upper boundary of the DOL in the upwelling area; the third term in (5.5.3) and the second term in (5.5.5) describe the heat exchange at the ocean-atmosphere interface; the penultimate term on the right-hand side of (5.5.5) describes the change of heat content in the area of cold deep water formation due to melting ice and snow exported from the polar ocean; the first two terms on the right-hand side of (5.5.6) describe the heat flux at the water-ice interface and the heat exchange between the UML and DOL in the polar ocean; the last terms in (5.5.5)—(5.5.7) describe the change in the heat content created by trapping of water from a neighbouring ocean area as the result of displacement of the southern boundary of sea ice. Here, in addition to the symbols already specified, H is the ocean depth; M, and Ms are ice and snow masses in the polar ocean; = sj/s! and f2 = s\/s2 is the ratio between the area s^, of land in the southern and northern boxes and the area and s2 of these boxes (fractions of continents),1 fx = s°1/s2 and are the fraction and area of sea ice in the northern box, is the area of the cold deep water formation domain connected to area s? of the upwelling domain and to the common ocean area s° in the Northern Hemisphere by the relation = s° — — s°j; mA = ps/g is the mass of the unit atmospheric column; ps is the surface atmospheric pressure; g is gravity; ^ is the dimensional parameter assumed as proportional to M, with
1 The land in each box is represented in the form of a 'segment' of constant extension in the zonal direction with the northern boundary of the land 'segment' in the northern box coinciding with the latitude circle at 71.6°N.
proportionality factor = R/cp; R and cp are the gas constant and the heat capacity of air; p0 and c0 are density and heat capacity of sea water; subscripts A and 0 indicate belonging to the atmosphere and the ocean; double subscripts LA, IA, OA, LS and OS indicate belonging to the atmosphere over land, to the atmosphere over sea ice and over the ocean, to the land and ocean surfaces respectively; the meaning of the remaining symbols will be further clarified.
To find M, and Ms we picture the ice cover in the polar ocean area as a film of finite thickness whose surface can be covered by snow or melted water, and single out three periods within the year cycle. Winter: there is snow on the ice surface; the temperature Ts of the active surface (in this case the snow surface) is less than the water freezing temperature Tso; the growth and melting of ice occurs at the ice-water interface. Spring: the ice surface is covered by a mixture of snow and melted water; Ts= Tso; melting or growth of ice occurs at its low surface. Summer and autumn. The ice surface is covered by a layer of melted water; the temperature of this layer remains equal to rso; melting and growth of ice occurs on both surfaces. The beginning and the end of this period are identified, respectively, with periods when the thickness of the snow and melted water layers are zero.
In these situations the evolution equations for the ice mass M, = 1? snow Ms = ps/iss°!, and melted water Mw = pw/iws° t are written in the form d M, dT
L, di for Ts < Tso; Ms > 0; Mw = 0, for Ts = Ts0; Ms > 0; Mw > 0, (5.5.8)
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