V2

are the operator of the total time derivative and the Laplace operator respectively.

To find the sea water density the linear equation of state in Eckart's form is used:

1.000 027[a + b(p0 + ps)l where a, b and p0 are parameters defined as a = 1779.5 + 11.250 — 0.074 592 — (3.8 + O.O10)S; b = 0.698; p0 = 5890 - 380 - O.37502 + 3S; the density is measured in g/cm3; the temperature 6 in degrees Celsius, the salinity in %o.

As in the atmospheric submodel the appearance of the hydrostatic instability is considered to be possible. It is resolved by means of convective adjustment, that is, by instantaneous equalization of the temperature and salinity profiles with an allowance being made for the integral conservation of the latter.

The vertical structure of the upper ocean layer is determined, apart from the convection, by wind mixing. Its effect is described within the framework of the local upper mixed layer model proposed by Kraus and Turner (1967). At the ocean surface the vertical velocity is assumed to be equal to zero, and components of the wind stress and vertical eddy fluxes of heat and salt determined in the atmospheric submodel are considered to be prescribed. The no-flow condition and the condition of vanishing of the vertical heat and salt fluxes are specified at the bottom. On the coastline the normal component of the velocity and the horizontal fluxes of heat and salt are assumed to be zero. The system of equations and boundary conditions is supplemented by the evolution equation for sea ice thickness, taking into account the possibility of change in sea ice thickness due to local factors (accretion, melting, sublimation, etc.) and the influence of the thickness on the sea ice drift velocity.

When solving the ocean submodel equations the standard staggered grid with finer resolution in the vicinity of the west coast is used. At high latitudes where the grid step decreases, the procedure of spatial Fourier smoothing is applied. The ocean is divided along the vertical into 12 layers whose thickness increases unevenly with distance from the free surface to the bottom.

The first numerical experiments within the framework of the coupled model were carried out for the domain representing the truncated spherical sector. From west to east it was limited by meridians separated at 120°, and from south to north by parallels ±81.7°. The zonal extension of the atmosphere and ocean was assumed to be equal everywhere excluding regions of high latitudes. Such a configuration of the domain in question resembles the Atlantic Ocean adjacent to North and South America. In the atmosphere the periodicity condition was prescribed at the boundary meridians and the no-flow conditions were specified at the boundary parallels. The land and ocean surfaces were considered to be flat, the ocean depth was taken to be 4 km. To overcome the difficulties connected with the different heat inertia of the atmosphere and the ocean, an artificial synchronization, that is, combining the ocean state at each time step with the time-averaged atmospheric state, was realized. This was performed using a procedure equivalent to an exponential filter with the weighted function being equal to X~1 exp(t/X) as t < 0 and to zero as t > 0, where X is the time constant equal to one week.

Analysis of the results of integration of the model equations showed that an allowance for the heat transport in the ocean leads to the cooling of the atmosphere in low latitudes and its heating in mid latitudes. In its turn, a decrease in temperature difference between the equator and temperate latitudes causes a weakening of the zonal circulation, and, as a result of the baroclinic atmospheric instability, a decrease in the kinetic energy of macroturbulence in temperate latitudes and in intensity of the Ferrel meridional cell. The heat advection in the ocean also favours deepening of the subtropical and polar atmospheric pressure maxima and baric depression in temperate latitudes providing the transport of air heated over the ocean to the east coast of a model continent. The existence of the cold equatorial upwelling effects a decrease in the precipitation rate in the tropical zone of the oceans and its increase over the continents. In subtropical and high latitudes the warm water transport by the subtropical gyre increases the sensible and latent transfer from the ocean to the atmosphere, as well as the precipitation along the eastern coast of the continent; in northern latitudes this transport causes the appearance of similar effects along the western coast of the continent. In total, the results of calculations using the coupled ocean-atmosphere model turn out to be more realistic than those using the model where the ocean is represented as a 'swamp' of zero heat capacity.

Subsequently, the main results of the above-mentioned numerical experiment were confirmed by more detailed calculations carried out as applied to the real geometry of continents and oceans. The calculations were performed on a grid containing nine levels in the atmosphere and 12 levels in the ocean. The spatial step was taken as equal to 500 km; synchronization of the atmospheric and ocean states was realized by equating 1.3 atmospheric years to 430 ocean years.

Let us dwell on the results of the calculation of the ocean fields. Though the equilibrium of the thermal regime has not been attained, the ocean temperature distribution turned out to be very similar to the observed temperature distribution. It has also been possible to simulate major features of the global salinity distribution: in the Atlantic the salinity was higher than in the Pacific Ocean, and in the Pacific Ocean it was found to be higher in the south than in the north. Also, the model properly reproduced the meridional circulation in the ocean, forming two cells with upwelling at the equator and downwelling in high latitudes. As it turned out, the meridional heat transport was found to reach a maximum in the subtropics where thermohaline and drift components of the current velocity are oriented in one direction. On the other hand, in temperate latitudes where drift and thermohaline components are oriented in opposite directions the meridional heat transport decreases.

Along with the above the inevitable descrepancies between calculated and observed fields of ocean characteristics can be mentioned. They primarily concern overestimation of temperature in high and polar latitudes, underestimation of salinity, clearly expressed in the North Atlantic, and a weakening of the intensity of the circulation as a whole. The last circumstance, as well as underestimation of the rate of cold deep water formation in high latitudes, is explained, according to Manabe et al. (1979), by the absence of the seasonal change of insolation in the model.

The inclusion of the seasonal variations of insolation with small changes in the model (coefficients of the vertical and horizontal eddy diffusion of momentum, heat and salinity in the ocean were prescribed by functions of depth; a synchronization was performed by equating two atmospheric years to 1200 ocean years) improved the reproduction of a number of climatic characteristics of the ocean-atmosphere system such as the distribution of pack ice in the Arctic and snow cover on the continents of the Northern Hemisphere. Consideration of the seasonal cycle of insolation resulted in the elimination of the unrealistically large thickness of sea ice (now it amounts to about 3 m instead of 35 m as it was before) and of the continuous accumulation of snow.

The time-space distribution of the zonal average surface air temperature over the ocean and land turned out to be very satisfactory. Specifically, it has been possible to simulate a phase shift (lag) and decrease in amplitudes of the seasonal oscillations of the surface air temperature over the ocean compared with their values over land in the temperate and low latitudes and an increasingly earlier occurrence of maximum nearer the pole at high latitudes. The last feature is caused by the influence of sea ice preventing heat exchange between the ocean and the atmosphere, and thereby contributing to the transformation of a maritime climate into a continental one.

A similar conclusion, in terms of accuracy of simulation of the time-space distribution of climatic characteristics, can be made with respect to the average (in the Northern and Southern Hemispheres) values of the air temperature at various heights in the atmosphere, and the water temperature at different depths in the upper ocean layer. It can be seen from Figure 5.8 that in the Northern Hemisphere where the land occupies about 40% of the hemisphere area the amplitude of the seasonal air temperature oscillations increases when approaching the underlying surface, basically because of the high variability of the albedo that relates to the appearance and disappearance of snow on continents. This feature is not found in the Southern Hemisphere. The diagram also demonstrates the asymmetry of the seasonal cycle of the water temperature in the upper ocean layer: maximum positive departures of temperature from its annual mean value in summer are higher than maximum negative departures in winter by almost 0.5 °C. The cause of this is the existence of winter convection limiting the drop in sea surface temperature, and of the seasonal thermocline shielding the warm upper layer from the cold deep ocean. This feature is well reproduced by the model.

The same can be said with reference to seasonal variations of the heat content and the meridional heat transport in the atmosphere and ocean. It was established by Manabe et al. (1975) that in temperate latitudes the meridional heat transport in the atmosphere takes maximum values at the beginning of winter and minimum values at the beginning of summer, for the most part due to appropriate changes in intensity of the baroclinic instability. At low latitudes the meridional heat transport even becomes negative, that is, directed from north to south. In the equatorial area this is performed by the Hadley cell transporting heat from the summer hemisphere to the winter hemisphere.

Data obtained indicate a close correlation between the meridional heat transport in the ocean and the intensity of the zonal atmospheric circulation: the heat transport to the equator is due to the action of westerly winds, and poleward due to the action of easterly winds. In winter when the westerly

Figure 5.8 Seasonal variability of the hemispheric average air temperature (°C) at different heights in the atmosphere, and of the water temperature at different depths in the ocean, according to Manabe et al. (1979): (a) the Northern Hemisphere; (,b) the Southern Hemisphere.

and easterly winds become simultaneously stronger, the heat transport to the pole increases in tropical latitudes of the ocean and decreases in mid latitudes. This feature reproduced by the model is in good agreement with actual data. The amplitudes of seasonal oscillations of the meridional heat transport in the ocean are in worse agreement (they turn out to be several times underestimated). But the worst of this is that the underestimation cannot be explained.

The amplitudes of seasonal oscillations of the sea surface temperature in mid latitudes of the Northern Hemisphere, and of the surface air temperature in high latitudes of the Southern Hemisphere also turned out to be underestimated. On the other hand, the amplitudes of seasonal oscillations of the atmospheric heat content at high latitudes and also the amplitudes of seasonal oscillations of latent heat in snow and sea ice were overestimated. The discrepancies listed are explained by Manabe et al. (1979) to be the result of inaccuracy in prescribing cloudiness (particularly since observational data are unavailable, the field of cloudiness in the Southern Hemisphere is assumed to be the same as in the Northern Hemisphere), and of too low values of the mass transport in the major ocean currents.

IOAS model

The main distinction of this model (see Zilitinkevich et al, 1978) from the GFDL model is the rejection of the artificial synchronization of the oceanic and atmospheric states, and the subdivision of the ocean into the upper active layer naturally synchronized with the atmosphere, and the deep ocean, the state of which is calculated separately. Such an approach was designed for a description of the relatively short-range processes (i.e., seasonal oscillations and interannual variability) in which the state of the deep ocean can be considered as prescribed to a first approximation.

The model consists of four submodels: free atmosphere, the atmospheric planetary boundary, the active ocean layer, and the deep ocean. To simulate the climatic state of the free atmosphere the hydrothermodynamic equations are used in exactly the same form as in the GFDL model; they are distinguished only by the methods of parametrization of the physical processes. Parametrization of small-scale interaction between the atmosphere and the underlying surface accepted in the IOAS model is based on the similarity theory for the Ekman boundary layer, with the latter being considered to be submerged into the low grid layer containing 15% of the total atmospheric mass. The space-time variability of this layer is calculated with the help of equations, and the vertical structure is assumed to be universal, that is, as the similarity theory for the Ekman boundary layer predicts.

The calculation of the characteristics of interaction between the atmosphere and the underlying surface is realized within the framework of laws of resistance, heat and moisture exchange considering the influence of the density stratification and change in the underlying surface roughness on land. The underlying surface temperature is determined from the heat budget equation; in the ocean this equation serves to determine the resulting heat flux at the ocean-atmosphere interface. In the presence of ice cover at the ocean surface the temperature of its upper surface is found in the same fashion as that on land, that is, it is assumed that the ice completely isolates the ocean from the atmosphere. For the permanent ice cover, including pack ice, a restriction is introduced which does not allow the surface temperature to be higher than 0 °C. It is assumed that in this case the resulting heat flux is expended for ice melting.

The IOAS model uses coarse vertical resolution of the atmosphere: the latter is divided into layers with interfaces at heights 1.5, 4.5 and 11 km. But, unlike the GFDL model, cloudiness is not prescribed from climatic data but is determined from the empirical formulae at every time step. A scheme of calculation of radiative fluxes is also simplified in accordance with the coarse vertical resolution. The effects of three-level cloudiness and of absorption by water vapour are taken into account. The mesoscale convection and water vapour phase transitions are calculated in a similar way as in the GFDL model but instead of equalization of the relative humidity with height the condition of similarity for the convective transfer of specific humidity and equivalent potential temperature is used.

Due to limitations in available computer resources the effects of salinity in the active ocean layer are not taken into account. The influence of vertical motions is not taken into account either, and the depth-averaged horizontal components of the current velocity in the active layer are decomposed into climatic constituents borrowed from the deep ocean submodel and drift constituents determined by the wind stress. It is also supposed that the thermal structure of the ocean active layer possesses universality: there is a distinctive upper mixed layer and its underlying thermocline with self-similar temperature distribution. The temperature at the low boundary of the active layer is taken from the deep layer submodel, and the temperature and thickness of the upper mixed layer are determined from the heat and turbulent energy budget equations integrated over the depth of the active layer.

These equations are complemented by the following algorithm, roughly describing the influence of ice cover: if the temperature found from the heat budget equation for the ocean surface becomes less than the sea water freezing point ( — 1.8 °C), it is assumed that ice appears, completely isolating the active layer from the atmosphere. Accordingly, the resulting heat flux at the water-air interface and the kinetic energy flux from the atmosphere into the ocean are assumed to be equal to zero, and the temperature of the subice upper mixed layer is prescribed as equal to the sea water freezing temperature. In so doing, the temperature of the upper ice surface is determined from the heat budget equation for the ocean surface on the assumption that the resulting heat flux is equal to zero. As soon as the temperature of the upper ice surface becomes higher than the sea water freezing point the ice is believed to disappear.

The two-dimensional hydrothermodynamic model of the ocean global circulation, proposed by Kagan et al. (1974), is used as a submodel of the deep ocean. It is based on the following assumptions: the sea water density depends only on temperature; variations in the latter take place only in the baroclinic layer with 2 km thickness below which the temperature is fixed, and in the layer itself it is presented in the form of the product of some standard function of the vertical coordinates and the required function of horizontal coordinates and time, which is determined from the heat transport equation integrated within the limits of the whole ocean thickness. The barotropic components of velocity appearing in this equation are found from the equation for the integral stream function; the vertically averaged velocity components of the drift current are determined from the Ekman equations by the prescribed wind stress at the ocean surface. Finally, the vertically averaged baroclinic velocity components are estimated with the help of quasi-geostrophic relations. The boundary conditions are specified in such a way as to provide realization of heat and mass conservation laws in the World Ocean.

The equations of the deep ocean submodel are integrated for a latitude-longitude grid with 5° angular resolution from the initial state complying with the state of the rest for the horizontally homogeneous (but stratified along the vertical) ocean, up to establishing an equilibrium regime. The wind stress and the resulting heat flux at the ocean surface are calculated by the annual mean fields of atmospheric pressure, air temperature and radiation budget at the ocean surface with fixed values of the Bowen ratio equal to 0.5. Note that only fields of the current velocity and temperature at the low boundary of the active layer need to be provided for the coupled ocean-atmosphere model.

The equations for the atmospheric submodel and the active ocean layer are integrated on Kurihara's spherical grid with a horizontal step of about 1000 km. The annual mean meridional distributions of temperature in the atmosphere and the upper mixed layer of the ocean, the adiabatic vertical distribution of air temperature, the absence of wind as well as constant (in a horizontal plane) values of the surface atmospheric pressure, the relative air humidity and the upper mixed layer thickness are used as initial conditions.

A calculation using the natural synchronization of atmospheric and ocean states was carried out over a period of 1000 days, with an allowance for the seasonal cycle of insolation. An equilibrium quasi-periodical regime of the atmosphere-active ocean layer system was reached after about a year. By that time the mass-averaged wind velocity amounted to 17 m/s (up to 40 m/s in the upper layer); the mass-averaged temperature of the atmosphere amounted to 244 K; the underlying surface temperature amounted to 282 K (minimum monthly averaged values in the Antarctic and maximum in northern Africa turned out to be equal to 234 K and 308 K respectively); the zonal average humidity amounted to 1.6 g/kg; the total fraction of cloudiness amounted to 0.47; evaporation and precipitation amounted to about 3.1 mm/day; the net radiative flux at the underlying surface amounted to 470 W/m2 (80% of this flux is spent as long-wave emission of the Earth's surface, the remaining 20% is used for evaporation and sensible heat exchange). The atmospheric pressure over the ocean turned out to be less than that over land (994 as against 1040 hPa), and, on the other hand, the air temperature, specific humidity, cloudiness, evaporation and precipitation turned out to be more (by 5 °C, 0.9 g/kg, 0.4 and 2.9 mm/day respectively).

The model has been shown to simulate the main features of time-space variability of the atmospheric temperature, precipitation, evaporation, cloudiness, components of the heat budget and the vertical mass flux at the ocean surface, as well as characteristics of the upper mixed ocean layer: its temperature and thickness. In order to confirm the above we mention, in particular, the model detected asymmetry of seasonal changes in air temperature with respect to the equator connected with the different relationship between land and ocean areas in both hemispheres; winter maximum of precipitation in the equatorial region over oceans and summer maximum in tropical latitudes on land; enhancement of evaporation in temperate latitudes of the ocean in winter and on land in summer caused by an increase in temperature difference between the underlying surface and the atmosphere; predominance of cloudiness over the ocean rather than over the land; a six-month phase shift between seasonal oscillations of the resulting heat flux at the ocean surface in the Northern and Southern Hemispheres; and, finally, an increase in the amplitudes of seasonal oscillations of the mass flux in temperate latitudes of the ocean and the retention of its minimum values at the equator throughout a year. The last feature is the result of the small water-air temperature difference and the shielding influence of cloudiness.

The global distribution of the upper mixed layer thickness obtained using the IOAS model demonstrates that from the beginning of summer heating it decreases to several tens of metres in temperate latitudes and to 120-150 m in the subtropics. In summer its distribution is characterized by strong spatial variability. In the winter period the upper mixed layer extends practically over the whole active ocean layer.

CC SBAS model

This model (its detailed description can be found in Marchuk et al, 1984) differs from the models discussed above by its more economic numerical algorithm. Its basis is the use of a combination of two types of splitting (for the time derivative operator and the advective transport and diffusive operators) leading to the fulfilment of the integral laws of conservation of mass, angular momentum, energy (on the adiabatic approximation), moisture, etc. It is appropriate to emphasize here that this algorithm allows us to perform integrations with much larger time steps than is accepted in traditional difference schemes.

An atmospheric submodel of the CC SBAS model contains the evolution equations for the zonal and meridional components of the wind velocity, specific humidity and surface pressure. With a view to designing an absolutely stable difference scheme the first four equations are 'symmetrized' by transition from u, v, T and q to the variables pll2u, pl/2v, plJ2T and pl/2q. Parametrization of the horizontal eddy viscosity is carried out with an allowance for two conditions. That is, the term describing the horizontal momentum diffusion is required to be dissipative, and the angular momentum is to be retained. These conditions are met by the following expressions:

where, as before, a = p/ps, kH is the horizontal eddy viscosity coefficient. Indeed, we multiply FA by u, and F(p by v and integrate over the whole area of the atmosphere. Then, for example, for uFx we obtain uFxa2 cos cp dXd(p =

cos3 cp dXdcp < 0, that is, there is dissipation. Further, we multiply Fx by cos (p and integrate over the whole area of the atmosphere. As a result we obtain

Fxa2 cos2 (p dXd<p = 0, whence the conservation of angular momentum follows. The horizontal eddy diffusion of heat and moisture is parametrized in the traditional way:

The parametrization scheme for the atmospheric planetary boundary layer reduces to singling out the logarithmic layer and the overlying well mixed layers within the limits of which wind velocity, temperature and humidity are considered to be constant with height. At the same time, it is assumed that the position of the upper boundary of the planetary boundary layer coincides with the grid level closest to the underlying surface, and the angle between the wind velocity vector in the mixed layer and wind stress at the underlying surface remains constant and equal to 30° in an extra-tropical area over land, 20° over ocean, and 10° over ice. In the tropics this angle is assumed to be equal to zero. To estimate the eddy fluxes of momentum, heat and moisture at the ocean surface the well-known bulk formulae with resistance and heat exchange coefficients depending on wind velocity and stratification are applied.

The parametrization of the processes of convective adjustment and large-scale condensation is performed similarly to that in the GFDL model. The fraction of cloudiness is calculated using empirical relations. The thickness of clouds, their albedo and absorption capacity, as well as the albedo of the ocean surface and ice, ozone and carbon dioxide concentrations, are assumed to be fixed. The albedo of the land surface is determined as a function of the snow cover thickness in water equivalent.

The ocean submodel of the CC SBAS model includes the complete system of the ocean hydrothermodynamic equations, which does not differ much from that adopted in the GFDL model. But the method of solution differs. For this purpose a splitting technique is used where the extended spatial operator is decomposed into a number of simpler ones. As in the GFDL model, artificial synchronization of the atmosphere and ocean states is introduced, whereby one atmospheric year is equated to about 100 ocean years. The time sampling of the information exchange between the atmospheric and ocean submodels is chosen as appropriate to the characteristic time x of mixing in the ocean upper mixed layer with 100 m thickness. For x = 14 days and time step equal to 40 minutes in the atmosphere and two days in the ocean the information exchange is performed every six atmospheric and every seven ocean steps. All atmospheric data transmitted to the ocean submodel are smoothed in time with the difference analogue of the exponential filter.

The model described was tested on January average conditions. The calculation was carried out in two stages. First, the equilibrium regime of the a2 cos2 cp IdA

a2 cos2 cp IdA

+ cos <p — ( kHa cos cp — T, q dcp\ d(p atmospheric circulation with fixed sea surface temperature and the prescribed distribution of sea and continental ice were calculated. Then the ocean submodel with four levels in the vertical was run. These levels were located at 100, 500, 1500 and 3000 m. The total duration of the calculation amounted to 11 ocean years, that is, equivalent to two atmospheric months.

An analysis of the results obtained detects a general decrease in temperature in the ocean of the Northern Hemisphere and its increase in temperate and high latitudes of the Southern Hemisphere. This, naturally, affects the atmospheric circulation. Specifically, there is a shift in the tropical belt of precipitation by about 10° to the south, a diminution (especially in the Southern Hemisphere) of the meridional temperature gradient, a decrease (of approximately 25%) of the available potential energy and its transformation into kinetic energy, an enhancement of the direct Hadley cell in the Northern Hemisphere and a weakening of the opposite Ferrel cells in both hemispheres, a decrease of wind stress, and a deepening of the centres of low pressure over the continents of the Southern Hemisphere.

Consideration of the effects of ocean-atmosphere interaction implies the following sequence of events. At first, on the western coasts of the Pacific Ocean and in the north-west part of the Indian Ocean, negative anomalies of water temperature appear which cause the appearance of strong western boundary currents and the enhancement of upwelling. This, in turn, leads to intensification of the meridional circulation, smoothing of horizontal temperature gradients in the ocean and weakening of the interaction between the ocean and atmosphere.

NCAR model

This model differs from the GFDL model mainly by finite difference approximation of equations of the atmospheric submodel in the vertical direction; by parametrization of the physical processes of the subgrid scale; and by the method of synchronization of the atmosphere and ocean states. In the NCAR model the atmosphere is divided into eight layers, each three kilometres thick. Accordingly, the methods calculation of radiative fluxes, cloudiness and eddy fluxes of momentum, heat and moisture at the underlying surface, as well as the temperature of the latter, are modified. Specifically, when calculating radiative influxes, more accurate functions of absorption of short-wave radiation and of transmission of long-wave radiation are used, and the absorptive properties of various types of underlying surface are taken into account. The diurnal change in insolation is described explicitly. The fractional cover of low and middle level clouds is determined from empirical relations. The base of low-level clouds is located at a height of 1.5 km and the top of the clouds is fixed. The middle-level clouds are assumed to be infinitely thin. The fraction of upper-level clouds is prescribed from the climatological data and their base is placed at a height of 10.5 km over the equator and 7.5 km over the poles; the thickness is taken as 1.5 km.

Eddy fluxes of momentum, heat and moisture are calculated, taking into account the dependence of resistance and heat exchange coefficients on stratification of the surface atmospheric layer, and the temperature of the underlying surface - with an allowance for the type of surface. Three types of underlying surface are singled out: the ocean surface, the sea ice surface and the land surface both covered and not covered with vegetation. In turn, the sea ice surface and the land surface not covered with ice are differentiated depending on whether they are covered with snow or not, and depending on the soil moisture. To each type or subtype of the underlying surface is assigned a certain value of albedo, assumed to be a function of snow thickness and of the solar zenith angle. In the presence of vegetation, the temperature distinctions of vegetation cover and the underlying surface, transpiration and evaporation, as well as the transformation of the vertical structure of wind in vegetation cover, are taken into account. A detailed description of the atmospheric submodel of the NCAR model is presented by Washington and Williamson (1977) and Washington et al. (1980).

The ocean submodel of the NCAR model does not differ much from that used in the GFDL model. Compared with the latter, the NCAR model excludes the effects of the horizontal transport and friction in the equation describing the evolution of sea ice thickness (thus, the sea ice model becomes a purely thermodynamic model), and employs other values of coefficients of horizontal and vertical eddy diffusion of heat and momentum. Discretization in the vertical direction is produced differently: the ocean is divided into four layers with thickness (from above) of 50, 450, 1500 and 2000 m.

The method of synchronization of atmospheric and ocean states reduces to the following. The equations of the ocean submodel are integrated with fixed (for January, April, July and October) values of atmospheric parameters for a period of five years. The temperature of the underlying surface and sea ice area obtained in the last year of this period are used as initial information when integrating the equations of the atmospheric submodel separately for each of the four months mentioned above. The values of the atmospheric parameters obtained are assumed to be the initial values when integrating the equations of the ocean submodel during the second five-year period. Again, the atmospheric parameters corresponding to each of the four months of the annual cycle are found and this is continued until a quasi-equilibrium regime is established in all links of the ocean-atmosphere system. According to estimates by Washington et al. (1980), this method of synchronization, from a computing viewpoint, is ten times more effective than that approved for the GFDL model.

The zonal air temperature, longitudinally averaged zonal component of wind velocity, ocean surface temperature, current velocity in the upper 50 metre layer and its underlying 450 metre ocean layer, and, finally, the vertical velocity at the interface between these layers obtained at the end of the fourth five-year period turned out to be close to those observed in terms of quality but not quantity. For example, in January the zero isotherm is at latitude 38 °N according to calculated results, and at latitude 50°N according to observational data. This points to an underestimate of the calculated values of air temperature in the Northern Hemisphere in winter. Next, according to calculated results the summer air temperature at all levels in the troposphere has a maximum in the region 40-50 °N and this is not confirmed by observational data. We note also an overestimation of the meridional air temperature gradient in the Northern and Southern Hemispheres, a shift of the equatorial zone of westerly winds into the Southern Hemisphere in winter and into the Northern Hemisphere in summer and large discrepancies (both in height and in direction) between calculated and observed wind velocities in high latitudes of the Southern Hemisphere.

The model simulated the main features of seasonal variability of the ocean surface temperature and location of the thermal equator correctly. But the horizontal temperature gradients in the regions of the Gulf Stream and Kurosio turned out to be underestimated, and temperature values in the subtropics and in midlatitudes turned out to be overestimated compared with those observed. The first circumstance is caused by the coarse (5°) grid resolution, the second by underestimation of the wind velocity and its associated decrease in the heat exchange between the ocean and atmosphere, and in the heat transport by drift currents. It is emphasized that overestimation of the surface temperature in the Southern Ocean has led to a decrease in the sea ice around the Antarctic. According to the authors of the model this is due to the same causes and to the prescription of a too large value of the horizontal eddy heat diffusion coefficient.

The qualitative agreement of calculated and observed fields is satisfactory. Suffice it to say that in the first grid layer (Figure 5.9) the Antarctic Circumpolar Current, Gulf Stream, Kurosio, West-Australian, Californian and Bengwale currents are apparent. The westward equatorial currents in the Atlantic and Pacific Oceans are found to be in the second grid layer, and the narrow Pacific equatorial countercurrent is found to be in the first layer. Some currents (such as Agulhas and Labrador) are not presented at all

Longitude

Figure 5.9 Field of currents in the upper 50-metre layer of the World Ocean in January, according to Washington et al. (1980).

Longitude

Figure 5.9 Field of currents in the upper 50-metre layer of the World Ocean in January, according to Washington et al. (1980).

because of the coarse grid resolution. The velocities of all currents turned out to be about one-third of those observed.

Literally everything mentioned above can be applied to the vertical motion velocities. The model simulated the intensive equatorial upwelling and regions of downwelling located 15-20° to the north and south of the equator. In both cases extreme velocities of vertical motions (according to calculated results and observational data) take place in the eastern part of the Pacific Ocean. But calculated values of the vertical velocity in coastal upwelling regions turn out to be less than those observed. This cause is the same: the coarse spatial resolution of the grid.

OS U model

The OSU model (see Gates et al., 1985, and Han et al., 1985) differs from the GFDL model mainly by the coarser horizontal and vertical resolution, by the inclusion of cloudiness in a number of unknowns to be determined, and by the rejection of artificial synchronization of ocean and atmospheric states.

The model has the following structure. It consists of two atmospheric and six ocean layers. The upper boundary (a — 0) of the atmosphere is coincident with the isobaric surface pT = 200 hPa; the lower boundary is coincident with the isobaric surface where the pressure p is equal to the surface pressure ps; the interface (a = 1/2) between layers is coincident with the surface p = (pT + ps)/2. Each chosen atmospheric layer is divided in turn into two sublayers with interfaces a = 1/4 (p « 369 hPa) and a = 3/4 (p « 788 hPa).

The horizontal components of wind velocity, air temperature and specific humidity are determined at these levels. The isobaric vertical velocity is calculated at the level a = 1/2 and is assumed to be equal to zero at levels (7 = 0 and (7=1. The interfaces between ocean layers are located at depths of 50, 250, 750, 1550, 2750 and 4350 metres. All layers are divided in half by intermediate depths where horizontal components of the current velocity, sea water temperature and salinity are calculated. The vertical velocity is determined at the interfaces between layers. At the ocean surface it is taken equal to zero, at the bottom it is assumed equal to the orographic vertical velocity defined by the kinematic relation. The upper 50-metre layer is assumed to be well mixed, and the temperature at the intermediate depth z = 25 m is assumed to coincide with the sea surface temperature. The horizontal resolution (equal in the ocean and the atmosphere) is preset equal to 4° in latitude and 5° in longitude.

As has already been noted the model provides the determination of cloudiness. Depending on the mechanism of its formation four types of cloud are distinguished, corresponding to penetrative and cumulus convection at the middle level a = 1/2 (type 1), large-scale condensation at levels <r = 3/4 (type 2) and a = 1/2 (type 4) and to cumulus convection at the level cr = 3/4 (type 3). Cumulonimbus clouds with base at a = 1/4 and top at cr = 0 are clouds of the first type; stratus and cirrus clouds of the lower level with base at a = 3/4; and top at the level a = 1/2 are clouds of the second type; thin cumulus and cirrus clouds of the middle level at (7 = 3/4 are clouds of the third type; and stratus and cirrus clouds of the upper level with base at c7 = 1/2 and top at (7=1/4 are clouds of the fourth type. It is considered that cumulus clouds are formed in the case where the vertical gradient of the equivalent potential temperature becomes less than zero, and the local relative humidity becomes more than 80%. Meanwhile, stratus and cirrus clouds are formed if the local relative humidity is more than 80% and the vertical gradient of equivalent potential temperature takes either positive or zero values (the atmosphere is hydrostatically stable). It is also assumed that clouds of the second, third and fourth types cannot coexist simultaneously with clouds of the first type, and that clouds of the third type cannot coexist simultaneously with clouds of the second type. However, the coexistence of clouds of the second and third types with clouds of the fourth type is possible. In the last case all clouds are ascribed to the lower level.

Calculation of atmospheric characteristics is carried out in two stages: first (every ten minutes) changes are found that are determined by horizontal and vertical advection, then their values are corrected at the expense of internal and external sources and sinks of heat and momentum. Non-adiabatic factors are addressed every hour. The time step for integrating the ocean submodel equations is one hour, which makes it possible to synchronize the heat and momentum change between the ocean and atmosphere without the need for any additional procedures of time smoothing or averaging.

Preliminary results concerning the evolution of the ocean-atmosphere system during the first 16 years of the simulation are discussed by Han et al. (1985). We dwell only on some global characteristics here. The model correctly simulates maximum positive values of the net radiative flux at the upper atmospheric boundary in January and underestimates its minimum negative values in June. This last circumstance is connected with underestimation of the outward long-wave radiation flux, and the latter in turn with errors when determining cloudiness in winter in the Southern Hemisphere, and in summer in the Northern Hemisphere. The annual mean global net radiation flux at the upper atmospheric boundary amounts to 4-5 W/m2 instead of 0, as it needs to be in the steady state. The authors explain this feature by the influence of numerical viscosity and by disregarding the transformation of mechanical energy into heat due to viscous forces.

The model underestimates the extremal values of the resulting heat flux at the underlying surface in June and February caused by overestimation of sensible and latent heat fluxes in temperate and high latitudes of the ocean in winter, and as a result does not reproduce observed variations of the sea surface temperature: after ten years from the start of integration the calculated and observed seasonal oscillations of the sea surface temperature turn out to be out of phase with each other. The cause is the progressive decrease of the sea ice area in the Southern Ocean leading to an increase in the contribution of the oceans of the Southern Hemisphere to the formation of seasonal oscillations of the global average sea surface temperature (in reality these oscillations are mainly controlled by the oceans of the Northern Hemisphere).

We note a systematic underestimation (by approximately 1.5 °C) of the average temperature of the troposphere and a gradual increase in seasonal oscillations of the surface air temperature. Underestimation of the average tropospheric temperature relates to the coarse vertical resolution, and an increase in seasonal oscillations of the surface temperature relates to a decrease in the sea ice area in the Southern Ocean. By the way, the disadvantages detected (underestimation of the sea ice area and, accordingly, overestimation of the surface temperature in the Southern Ocean) are inherent not only in this model but also in other coupled ocean-atmosphere general circulation models. But it is interpreted differently in other models: the authors of the GFDL model explain it by overestimation of the absorbed solar radiation flux at the ocean surface and by underestimation of the heat transport to the equator by drift currents; the authors of the NCAR model explain it by overestimation of the horizontal eddy heat diffusion coefficient; and the authors of the OSU model explain it by overestimation of the absorbed solar radiation flux and by the inadequate reproduction of the Antarctic upwelling when using models with a coarse horizontal resolution. It is obvious that the first, second and third explanations have some grounds but it is not clear whether these are sufficient or not.

UKMO model

This model consists of atmospheric, ocean and sea ice submodels. The atmospheric submodel is an 11-level version of the climatic model of the United Kingdom Meteorological Office which is more advanced, as compared with that presented by Corby et al. (1977) and that presented by Slingo and Pearson (1987), in the following respect. First, it contains a parametrization of the orographic gravity waves drag determined by the interaction between gravity waves and elements of the Earth's surface orography. Second, it is assumed that the planetary boundary layer of the atmosphere includes the surface layer and the overlying mixed layer with the inversion above it. The height of the planetary boundary layer identified with the inversion base is assumed to be variable. This is found with the help of the evolution equation incorporating air entrainment in a regime of convective instability. Provision is made for the fact that under stable stratification the thickness of the planetary boundary layer can decrease down to its minimum value determined by mechanical mixing. Third, the model describes the vertical distribution of temperature in the active soil layer. For this purpose the soil temperature is calculated on four levels. Finally, the cloudiness is assumed to be interactive, that is, generated by the model, and radiative properties of clouds are assumed to be constant and equal to their characteristic values for the Earth as a whole.

The ocean submodel is represented by the 17-level version of the model taking account of isopycnic diffusion and the dependence of the vertical diffusion coefficients on stratification. Sea ice is described within the framework of the thermodynamic model where the prognostic variables are the thickness of ice and snow, surface temperature and ice concentration, the latter being included in the number of unknowns allows for the presence of leads.

The model equations are integrated on a latitude-longitude 2.5° x 3.75° grid. The exchange of information between ocean and atmospheric submodels is performed discretely: the fluxes of heat, fresh water and momentum which are necessary for renewal of the boundary conditions in the ocean submodel, as well as the sea surface temperature, area and concentration of sea ice which are necessary for renewal of the boundary conditions in the atmospheric submodel, are transmitted from one submodel to another every five days.

Data of calculations presented by Foreman et al. (1988) were obtained as the result of a four-year integration of the coupled atmospheric-ocean-sea ice model. Such an integration period is not sufficient to reach an equilibrium state of the system but it is acceptable for detecting tendencies to changes in the solution. In particular, it has been found that the model overestimates (sometimes by several degrees) the sea surface temperature in the tropics. One of the reasons for this is an underestimation of sensible and latent heat fluxes in regions with high sea surface temperature and low wind velocity. Another reason is the existence of local feedback between the temperature and salinity of the ocean surface: overestimation of sea surface temperature contributes to intensification of convection in the atmosphere, and the latter leads to an increase in precipitation, the appearance of a surface layer with relatively fresh water, an increase in static stability, weakening of the vertical mixing, and, finally, to a rise in the sea surface temperature. One more possible reason for the overestimation of sea surface temperature in the tropics can be underestimation of cloudiness, causing overestimation of the absorbed solar radiative flux. It follows from the data presented that the model underestimates mass transports of the western boundary currents. For example, the maximum mass transport of the Gulf Stream turned out to be equal to 35 x 106 m3/s, while from observations it is of the order of 100 x 106 m3/s. The cause of this is clear: the coarse spatial resolution of the grid does not allow simulation of the real western boundary currents with scales less than the grid size.

The inadequate reproduction of the meridional heat transport in the ocean of the Northern Hemisphere is closely connected with underestimation of the mass transport in the western boundary currents. This explains the underestimation of calculated values of sea surface temperature to the north of the 30 °N parallel. The model also underestimates seasonal variations of the sea ice area in the Southern Ocean that, according to the authors, is caused by neglecting to take the dynamics of sea ice into account. But the worst of it is that the annual mean area of the Antarctic sea ice tends to decrease with time. On the other hand, the ice area in the Arctic turns out to be overestimated throughout the year and is almost unaffected by seasonal variations. It is also remarkable that there is an underestimation, by 10-15 °C, of summer surface temperatures over the continental areas to the north of 55°, connected with overestimation of the ice cover area.

The amount of low-level cloud over midlatitudes of land in the Northern Hemisphere turns out to be underestimated, especially in winter and in spring.

This results not only in an overestimation of the active surface temperature but in a delay (almost by a month) of the beginning of snow melting. As a result the summer period becomes shorter, and the summer temperatures become less than those observed by 4-5 °C. This circumstance has one more consequence: a decrease in the advective heat transport from the continents to the Arctic basin. Accordingly, the sea ice area in the Arctic increases, and the temperature of its active surface decreases.

The above effect is intensified further due to the significant decrease in summer values of the absorbed solar radiation flux in the Arctic Ocean. This is determined not only by the existence of feedback between the albedo and temperature of the underlying surface but also by underestimation by more than twice the incoming solar radiation. The last feature is caused by fixing the radiative properties of clouds: the amount of low-level clouds in the Arctic turns out to be realistic but, owing to the fact that their radiative properties are assigned as being uniform everywhere and complying with global average values, that is, more appropriate for clouds in low and temperate latitudes, the absorption of solar radiation by clouds turns out to be too large, and the solar radiation flux to the underlying surface is too small. All this is aggravated by underestimation of the meridional heat transport in the polar zone of the atmosphere and ocean.

MPI model

At first this model was intended not so much for simulation of the present-day climate as for a demonstration of the procedure of elimination of one objectionable feature appearing during the process of solving the problem. We mean the so-called solution drift, that is, the slow transition of the solution from one steady state complying with autonomous (non-interactive) models of the ocean and atmosphere to a new steady state corresponding to interactive models of the two media. This phenomenon is caused by inadequacy in the separate submodels and a mismatch between them, a consequence of which is that the new steady state turns out to be far from the real state despite the fact that both autonomous models have been thoroughly calibrated and tested for their agreement with observational data prior to their coupling.

The essence of the proposed procedure reduces to the following (Sausen et al, 1988). If the vector of the atmospheric state is designated as <!> and the vector of the ocean state as *F, and if it is considered that the evolution of the model climatic system differs from the evolution of the real climatic system by an error equal to EA for the atmosphere and E0 for the ocean, then the equations describing the evolution of the interacting atmosphere and ocean can be presented in the form ddf/dt = Ga(<D, t) + F(<Db, i) + £a(<D, T, i), = G0(<D, t) - F(0>b, ¥b, f) + ^(O, T, t),

where GA, G0 and F are sources and sinks of substances in the atmosphere and ocean and the exchange between them respectively; superscript 'b' designates boundary values of the functions O and (say wind velocity, temperature and moisture in the surface atmospheric layer, the sea surface temperature, etc.) which are necessary to calculate the exchange.

Similarly, the equations describing the evolution of uncoupled atmosphere and ocean take the form

<3<J>Jdt = GA(4>U, i) + F(<Db, Tb, 0 + £a, jdt = Goeru, t) - F(Ob , ¥b, i) + E0..

Here, as is common practice, the boundary values 4>b, Tb of the variables <l>u, are replaced by empirical data C>b , *Fb .

In (5.8.13) we add the components AFA, AF0 and define them in such a way that the solutions obtained within the framework of interactive and autonomous models should coincide with each other. Then instead of (5.8.13) we arrive at the following equations:

d®/dt = Ga(0>, t) + F(4>b, ¥b, i) + £a + AFa, dv/dt = Go(^, t) - F(Ob, ¥b, t) + E0- AF0..

In (5.8.15) we replace ® and ¥ by and *FU and then subtract Equations (5.8.14). As a result we obtain

AFA = F(<Db,¥b,i)-F(<&b,¥b,i), AF0 = F«, t) - F(<Db, i),

from which it follows that AFA, AF0 are differences of the exchange found using empirical information and information from the autonomous models of the ocean and atmosphere, without using empirical data. Thus, the problem reduces to the determination of correcting the components AFA, AF0 and the subsequent integration of Equations (5.8.15).

To illustrate the procedure, which is called the procedure of exchange correction, we examine, in accordance with Saussen et al. (1988), a simple box model of the ocean-atmosphere system, the type we discussed in Section 1.1. Let the evolution of the heat budget in the atmosphere and ocean be described by the system of equations cA drA/d( = Ra - âaTa + àao(Tq - rA), c0 dr0/dt = R0 - X0T0 - aA0(Tq - Ta),

where TA and T0 are temperatures of the atmosphere and ocean; RA and R0 are heat sources of radiative origin; /A and A0 are the parameters of the feedback between radiative heat sinks and the temperature of the respective medium; aao is the heat exchange coefficient at the ocean-atmosphere interface; cA and cQ are atmospheric and ocean heat capacities. We take RA as constant and R0 as varying, depending on the ocean temperature, as t _ T*1)

Ro(To) = { K1' + J(°2) _ r°(1) (K2> - W) at TS1» < To < Tg\ (5.8.18) R{o2) at T0 > P02\

where Rq ] > Such a prescription of radiative sources of heat in the ocean allows for the existence of feedback between the albedo of the snow-ice cover and temperature: for T0 < Tq> the ocean surface is assumed to be covered with ice; for T0 > Tq] it is assumed to be free from ice.

In the case where autonomous models of the atmosphere and ocean are used, and the evolution of the heat budget in the atmosphere is controlled, apart from everything else, by a change in the observed ocean temperature T{™\ and the evolution of the heat budget in the ocean is controlled by a change in the observed atmospheric temperature Equations (5.8.17) are rewritten in the form cA dir/df = Ra - Aa 7T + ¿ao(T- 2T), (5.8.19)

c0 dT^/dt = R0-A0T<?> - aA0(T<0»> - nm)). (5.8.20)

Equation (5.8.19) has the unique steady state solution nu) = (Ra + ¿ao TV)/(Xa + kAO), (5.8.21)

while Equation (5.8.20) can have either one or three steady state solutions, with two of these three solutions being stable and one being unstable. Stable steady solutions comply with the top and bottom lines of Equation (5.8.18) and are defined by the formula

where the index i can take values 1 or 2 depending on whether T(q] belongs to one or another range of temperature change in (5.8.18).

Let cA = 107 J/m2 K, cQ = 108 J/m2 K, RA = 130 W/m2, «{}> = 120 W/m2, R(2) = 125 W/m2, Aao = 10 W/m2 K, /A = 0.5242 W/m2 K and A0 = 0.3472 W/m2 K. These model parameters and observed values of temperature in the atmosphere (T(A} = 286 K) and ocean (T^ = 288 K) are met by the unique stable steady solution T[A = 286.12 K, T(q] = 288.20 K which differs only slightly from the empirical estimates. Respective heat flux values for autonomous models of the atmosphere and ocean are equal to Fa = F(T^\ T^) = aao(T^ - T^) = 18.78 W/m2 and FQ = F(Tf \ T^) = 4o(7ou) - = 22.01 W/m2 In the interactive model one or two stable steady state solutions:

are also obtained and for chosen model parameters the desired variables TA, T0 and F(Ta, Tq) = Xaq(T0 - Ta) turn out to be equal to 295.57 K, 297.94 K and 23.7 W/m2. This solution differs markedly from the initial solution found with the help of empirical data. Hence, solution drift in the interactive model of the ocean- atmosphere system is inevitable.

We take advantage of the procedure described above. With this purpose in mind we turn to (5.8.16) and find AFa = F(T%\ - F(T%\ T^) = -XA0(T^ - T{?>) = -2.01 W/m2, AF0 = F(T<F\ rg») - F(T%\ r{J>) = AAO(r^u) - T™) = 1.22 W/m2. Then we rewrite Equations (5.8.17) in the form cA drA/di = RA-1ATA + aA0(T0 - Ta) - aao(7T - rg»),| 8

Co drQ/di = Ro - Ao To - ?.AO(TQ - TA) - Aao(T%> - 7f), J

and integrate them. As a result it turns out that the temperature of the atmosphere and ocean at different instants of time will coincide exactly with the initial values TA] = 286.12 K, Tq> = 288.20 K, and, therefore, the procedure of exchange correction completely excludes the solution drift.

We now turn to testing the procedure of exchange correction as applied to the MPI global model. We first indicate its features. The atmospheric submodel is represented by a low resolving spectral model from the European Centre for Medium-Range Weather Forecasts (ECMWF), where the fields of prognostic variables (relative vorticity, velocity divergence, temperature, specific humidity, geopotential and surface pressure) are approximated in the form of finite series in spherical functions. As a result, the initial equations reduce to the appropriate equations for expansion coefficients depending only on time and the vertical coordinate. The vertical structure of the expansion coefficients is found with the help of discrete presentation. The version of the model adopted uses 21 spherical harmonics and 16 levels in the vertical.

The distinguishing features of the model are the introduction of a so-called hybrid vertical coordinate (a combination of the c-coordinate with the isobaric coordinate) tracking the topography of the underlying surface in the low layer and coinciding with the isobaric coordinate in the upper atmospheric layers, as well as an incorporation of the 'enveloping' orography which approximately takes into account undulations of the underlying surface of the subgrid scale. A number of features which parametrization of the physical processes should include are, first, a detailed description of cloudiness determined as a function of relative humidity and height; second, an allowance for the dependence of radiative fluxes on cloudiness, temperature, specific humidity and concentration of carbon dioxide, ozone and aerosols; third, the inclusion of the influence of stratification on eddy fluxes of momentum, heat and moisture within the framework of the Monin-Obukhov similarity theory in the surface atmospheric layer and K-theory beyond the limits of this layer, and, finally, the determination of the time when the moisture convection appears depending not only on establishing a superadiabatic temperature gradient but on the difference in the divergence of the large-scale moisture transport in the upper layers, and the divergence of the eddy moisture flux in the surface layer of the atmosphere.

To simulate the ocean climate a quasi-geostrophic model is used. Its basis is the principle of decomposition of current velocity into two components: barotropic and baroclinic. It is assumed that the barotropic component is immediately adapted t

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