H 2 h

H-h 2 H-h where T is the average (over depth and longitude) temperature; v and w are the meridional and vertical velocity components defined in a similar fashion; u(X0) is the zonal velocity component at the western boundary layer/open ocean interface; qs is the resulting heat flux at the ocean surface normalized to sea water heat capacity; kH is the coefficient of the horizontal eddy diffusion; the first subscript indicates belonging to the open ocean (1) and the western boundary layer (2); the second subscript indicates belonging to the upper (1) and deep (2) layers.

System (5.6.1)-(5.6.4) contains unknown values of horizontal (u, v) and vertical (w) velocity components and heat fluxes qh at the interface between the upper and deep ocean layers. To find the velocity components in the upper layer of the open ocean the following relationships can be used:


^(l-AY^ + n^-fi-A^, (5.6.6) a cos <p \ HJ\ dX dX ) \ HJfh h fdu,, d \

which are obtained by integration of the initial linearized equations of motion and continuity over depth within the limits of the upper layer with an allowance for assumptions of standard vertical temperature distribution of the type lTl2 + (Tli-Tl2)m at z<h, [r12 at z>h, and also those about the absence of momentum fluxes at the interface between upper and deep layers and at the ocean bottom. Here, uu are depth-averaged zonal and meridional velocity components defined by the equalities

"i = (A - A0)— (0! cos (p), curl2t/p0, (5.6.9)

the first one of which is a result of the continuity equation integrated over the ocean depth (note that at the eastern boundary of the ocean, that is, at X = A, the zonal velocity component is considered to be zero); the second equality represents the Sverdrup relation; r/p0 is the zonal component of the surface wind stress normalized to the mean density p0 of sea water; P is the change in the Coriolis parameter / with latitude; &(z) is a prescribed empirical function of the vertical coordinate; m and n are numerical factors depending on the choice of h, H and &(z).

The zonal and meridional velocity components in the deep layer are defined by the condition of vanishing integral (within the limits of the open ocean depth) velocity deviations from their depth-averaged values. This condition, combined with (5.6.5) and (5.6.6), yields ga-rh h f dTx, dT,2\ ,

In the vicinity of the equator where the quasi-geostrophic relations (5.6.5), (5.6.6), (5.6.10) and (5.6.11) are not fulfilled, the baroclinic component of the meridional velocity i'u caused by horizontal inhomogeneity of the temperature field (the second term on the right-hand side of (5.6.6)) and the vertical velocity wt at the interface between the upper and deep layers are assumed to be equal to zero. In this case to estimate un(l), [pn], u12(X0)

and [v12], instead of (5.6.5), (5.6.6), (5.6.10) and (5.6.11), Equations (5.6.9) are used.

The velocity components in the western boundary layer are found from the condition of water mass conservation in the zonal ocean section. In other words, it is assumed that the meridional mass transport in the open ocean is compensated by the opposite transport in the western boundary layer, and that this compensation occurs locally within the limits of every selected layer, that is,

D>2i] = -D>n](A - Ao)Mo, [»22] = ~[»i2](A - A0)/A0,"

[w2] = -[Wi](A - A0)Mo-The heat fluxes at the interface of two layers are parametrized in the form qhl=H/2iTll~ Tll)' q* = HÎ2 {Tl1 ~ Tzz)' (5'6'13)

where kv is the coefficient of the vertical eddy diffusion, assumed to be equal to 5 x 10_4m2/s for Ttl > Tl2, T2l > T22 and 5 x 10~2m2/s for ^11 < Tl2, T21 < T22.

Finally, it is assumed that the water temperature in the northern and southern polar oceans remains constant, and to estimate it the integral condition of the heat budget is used. This condition is written in the form

HvTa cos (p dA

where integration on the left-hand side of the equation is performed along the length of the liquid boundary of the polar ocean; integration on the right-hand side is extended over the area of the polar ocean; the symbol A means averaging over the vertical within the whole depth of the ocean.

Equation (5.6.14) does not take into account the possibility of water phase transitions in the process of ice formation and, hence, requires correction in the case where the temperature in the polar oceans falls below freezing point. In this case it is assumed that the water temperature is maintained at freezing point and that the resulting heat flux qs at the ocean surface is equal to zero. The last condition means that when water temperature reaches freezing point the ocean is instantly covered with ice. This condition, together with the condition of vanishing total (advective + diffusive) heat transport across the coastal boundary, completes the formulation of the ocean submodel.

The atmospheric submodel includes quasi-stationary equations for the heat and moisture budget (of the type given by Equation (5.5.1)), written with an allowance for the meridional resolution, and with rather simplified (compared with 5.5.24) parametrization of heat sources and sinks. In particular, the coefficients of short-wave and long-wave radiation absorption and the upward and downward long-wave radiation emissivities are assumed to be constant. An analogous simplification is made with regard to the relative humidity at the land surface that excludes the necessity of examining the continental part of the hydrological cycle. The meridional heat and moisture transport are parametrized in terms of a non-linear diffusive approximation.

When constructing the third submodel (unit of continental ice) it is assumed that ice sheets behave as a viscous liquid spread under its own weight, and because of this their height is related to the area, and the latter depends in turn on continental size and ice sheet mass budget originating from solid precipitation, ablation and iceberg discharge when the ice sheet reaches the edge of the continent. The resulting relations obtained for the ice sheet height and area, together with the integral equation for the heat budget, as well as an approximation of the vertical temperature distribution in the ice sheet by a polynomial of second degree and with appropriate boundary conditions, provide all the information required from this submodel: the height and area of the ice sheet and the temperature of its upper surface.

Numerical integration of the model equations is performed using artificial synchronization of separate subsystem states, namely, the equations for the evolution of the ice sheet are integrated with a time step equal to 1000 years. Then at every such step a stationary solution of the atmospheric equations and a time-dependent solution of the ocean equations are found, with the time step for equations of the ocean submodel assumed to be equal to three days and the duration of the integration period assumed to be 10 years. The integration is completed when the climatic system as a whole has achieved an approximate statistical equilibrium.

Tests of the model as applied to modern conditions reveal that it simulates the meridional distribution of the mass-weighted temperature in the atmosphere, as well as ocean and land surface temperatures, the meridional heat transport and characteristics of the Antarctic ice sheet. But perhaps the most remarkable peculiarity is that the model demonstrates indications of intran-sitivity: the climatic system has four steady states with fixed external parameters and different initial conditions for the ice sheet area in the Northern and Southern Hemisphere and for the ocean temperature. The first state corresponds to the present-day climate characterized by the absence of the ice sheet in the Northern Hemisphere (the Greenland ice sheet is not reproduced on a grid with five-degree angular resolution) and by the presence of the Antarctic ice sheet; the second and third states are characterized by maximum development and the total disappearance of ice sheets in both hemispheres;

and the fourth state is characterized by maximum development of the ice sheet in the Northern Hemisphere and its absence in the Southern Hemisphere. A series of numerical experiments performed by Verbitskii and Chalikov (1986) shows that the distinctions of steady states become apparent only in the temperate and high latitudes of both hemispheres and do not extend to equatorial latitudes. The scales of climatic changes associated with them can be judged from the following figures. The appearance of the continental ice sheet is accompanied by a decrease in the mass-averaged temperature of the atmosphere and the temperature of the underlying surface in temperate and high latitudes of the Atlantic and Pacific Oceans of almost 5 °C. The temperature of the land surface changes most of all in the following areas: where land is covered by ice its surface temperature falls by 20 °C. But at small distances from the edge of the ice sheet the decrease in surface temperature does not exceed 5 °C.

One more important result of numerical experiments is the high sensitivity of climatic characteristics and of the number of stationary solutions to mutual locations of land and ocean. This result is easier to understand when recollecting that the area of the southern ice sheet is controlled by Antarctica, and the northern ice sheet is controlled by ocean temperature. Thus, according to Verbitskii and Chalikov (1986) in the case where at the initial time that the southern boundary of the northern ice sheet was located to the south from parallel 40 °N, and the temperature in northern parts of the Atlantic and Pacific Oceans was higher than at present, the ice sheet receded immediately. If the water temperature was lower than at present then at first heat transfer from the Southern Hemisphere to the Northern Hemisphere, together with an increase in water temperature, occurred and only after the ice sheet receded.

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