-AM. + s?!(P2 + Q^JLJ for Ts < Tso; M > 0; M -j?,Ms + dM'Jdt for TS = TS

dMw di

= < -j?,Mw + t(P2 + Q'lJLJ - dM'Jdt for Ts = Tso; Ms > 0; Mw > 0, + s°_,(P2 + e'_s14/Z,J - dM'Jdt for Js = Tso; Ms = 0; Mw > 0.

where the first terms on the right-hand sides describe the ice, snow, and melted water export from the polar basin; the second term in (5.5.8) describes the growth or melting of ice at its low surface; the second terms in the first equation of (5.5.9) and in the two last equations of (5.5.10) describe the increase or decrease in snow and melted water masses due to precipitation and evaporation; the remaining terms describe the change in ice, snow and melted water masses owing to water phase transitions at the upper surface of the ice-snow cover. Here hlt hs, and hw are the ice, snow and melted water layer thicknesses; p,, ps, and pw are their densities, P2 and QIS14 are the precipitation and latent heat flux; L+ is the heat of sublimation equal to L + Lx when Ts < Tso, and to L when Ts > Tso; L and L, are latent heat of condensation and melting; the double subscript IS indicates belonging to the upper surface of the ice-snow cover.

The required values of Ts, dM's/dt and dM'Jdt are obtained from the heat budget equation for the upper surface:

Ll dM' for Ts = Tso; Ms = 0; Mw > 0, dt where, as in (5.5.3) and (5.5.5), the term containing the symbol £ represents the sum of fluxes of the absorbed short-wave (i = 1) and long-wave (i = 2) radiation, as well as of sensible (i = 3) and latent (i = 4) heat; the second term on the left-hand side of (5.5.11) describes the vertical heat flux in the ice-snow cover determined by the condition of the quasi-stationarity of the temperature in the form

'(Ts - MXy + hM~1 for Ts < Tso; Ms > 0; Mw = 0,

<2, = (rs - T_1)(hl/Xl + hjxs + KIK)'X for Ts = Tso; Ms > 0; Mw > 0, (Ts - T- MKT1 for Ts = Tso; Ms = 0; Mw > 0.

Here A,, 2.s and Aw are ice, snow and liquid water thermal conductivity.

We determine the vertical heat flux Q-® at the low ice surface. For this purpose we assume that the UML thickness in the polar ocean remains constant in time, and temperature is equal to the sea water freezing temperature. Then on the basis of (5.5.6) we have

This equation is supplemented by the expressions

yoh-iiTo - T_ J ds° Jdt for ds2 x/di > 0, .0 for ds°x/di < 0,

(.H - h.jXTo - T_2) ds? i/dt for ds° t/di > 0, 0 for ds°i/di < 0,

at ds° Jdt < 0, for the heat flux , at the UML-DOL interface and the rate of heat content change in the polar ocean (Dl^ £>I2) and in the area of cold deep water formation (Dj) due to trapping of water from the neighbouring areas, and also by the relationship connecting the sea ice thickness /i, and area .s° j and complying with the conditions /i, => 0 as si x => 0 and /i, => hlm as t => oo, where hlm and lm are limiting values of the sea ice thickness and area (the latter in this case is equal to the ocean area s° in the Northern Hemisphere); k0 is a dimensional factor; y0 is the non-dimensional constant characterizing the ratio between the amount of heat trapped by the polar ocean from the area of cold deep water formation and the amount of heat consumed for sea ice melting (it is assumed that the remaining trapped heat remains in the area of cold deep water formation and is expended immediately in an increase in the heat exchange between the ocean and the atmosphere in the northern box as this takes place in reality when leads are available.

We direct our attention to the determination of the thickness of the UML, as well as of the eddy heat flux 0 at the low boundary of the UML, and the equivalent heat flux Qj1 + 0 at the upper boundary of the DOL in the upwelling area. When it is assumed, as was mentioned above, that the flux Q^ _ o determined by entrainment of water from the DOL into the UML differs from zero only in periods of UML deepening, and that cutting of part of the UML accompanying the formation of a new thermocline and its integration with the DOL are tantamount to an assignment of equivalent flux Ql + 0 at the upper boundary of the DOL defined by the condition of heat conservation in the UML-DOL system, then

for we > 0, for we < 0, for we > 0, for we < 0,

-(Tx-T2)we for we < 0, where we = (d/i 1,/'di + w) is the entrainment velocity, w is the velocity of vertical motions in the upwelling area.

To estimate we use the turbulent energy budget equation integrated within UML limits. At the same time we suppose that the integral dissipation and generation of turbulent energy of mechanical and convective origin are proportional to each other, and that the turbulent energy of mechanical origin does not propagate beyond the limits of the Ekman boundary layer. Then d h,

0 0

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