Dimensional box models

In Section 1.1 we showed that the evolution of the surface air temperature is determined not only by one (inherent to the atmosphere) time scale but depends crucially on the time scales of the upper mixed layer (UML) and deep ocean layer (DOL) interacting with each other and with the atmosphere. This consideration is the basis of the box thermodynamic model suggested by Kagan et al. (1990) and is intended to simulate the seasonal variability of the climatic system of the Northern Hemisphere.

The seasonal time scale of variability predetermines the necessity of singling out the UML as an independent subsystem with all consequences arising with respect to the description of the processes at the UML-DOL interface.

We approximate the seasonal thermocline separating the UML and the DOL by a temperature jump, and single out two periods within the annual cycle - periods of falling and rising of the lower boundary of the UML. We will also suppose that an increase in UML thickness occurs exclusively owing to the entrainment of colder water from the DOL. The last condition is equivalent to the assumption that the eddy heat flux at the lower boundary of the UML differs from zero (it is used for heating water entrained from below) when UML thickness increases, and otherwise is equal to zero when entrainment is absent. Vanishing of this heat flux means that in a period of rising of the lower boundary of the UML any heat exchange between the UML and DOL stops and within the UML a new (shallower) thermocline is formed preventing heat penetration into the lower part of the UML. But isolation of part of the UML and its joining with the DOL are tantamount to the assignment of an equivalent heat flux at the upper boundary of the DOL defined by the condition of heat conservation in the UML-DOL system. In other words, within the framework of the adopted parametrization for the heat exchange at the UML-DOL interface, the heat transfer from the UML into the DOL occurs only in the period of rising, and is absent in the period of falling, of the lower boundary of the UML.

Another factor controlling the thermal regime of the DOL when considering this layer as a whole is an ordered vertical transfer of cold deep water created by upwelling. According to Munk (1966) the balance of these two factors -the heat transfer from above and the cold transfer from below - provides the existence of the main thermocline in the ocean. But owing to the law of mass conservation the vertical velocity integrated over the ocean area must vanish. Hence it is necessary to suggest the existence in the ocean of a downwelling area with a degenerate main thermocline. From general considerations confirmed by the results of laboratory experiments using revolving tanks, it is clear that the most intense downwelling has to be timed to an area of cold deep water formation and that this area and the area of upwelling must be connected to each other by the cold water transport from the first area into the second one in the deep layer, and by an opposite transport of warm water in the surface layer. These are the major prerequisites used in the design of the model in question.

The conditional concepts as to the character of the meridional circulation in the ocean form the basis of the model. That is, it is suggested that the area of upwelling is located in temperate and low latitudes and that in this area the upwelling is regulated by the inflow of cold deep waters from high latitudes and by the sinking of warm surface waters. Calculated results of the meridional heat transport in the ocean (see Section 2.3) and observed spatial distributions of the potential temperature, salinity and phosphate concentration obtained within the framework of GEOSECS and TTO programmes testify to such a meridional circulation cell.

Three areas can be singled out in the ocean - an area of temperate and low latitudes (hereinafter the upwelling area), the area of cold deep water formation in high latitudes and the polar ocean area covered with ice. It is assumed that in the upwelling area, where ocean warming occurs, the UML and DOL are separated by the clearly marked seasonal thermocline preventing heat exchange between them (in the polar ocean this function is performed by the halocline), and in the area of cold deep water formation both layers unite as a result of strong cooling accompanied by the development of intensive convection. Similarly, two zonal areas are singled out in the atmosphere: the area of temperate and low latitudes stretching over the upwelling area and part of the land contiguous with it, and the area of high latitudes located over the remaining part of the land, over the area of cold deep water formation and the polar ocean. As a result the atmosphere and the ocean are represented in the form of a system of seven boxes (Figure 5.4): two atmospheric boxes (northern and southern) and five ocean boxes (the UML and DOL in the upwelling area and in the polar ocean, and the area of cold deep water formation).

The evolutionary equations describing the heat budget in each separate box have the form

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