R r 136

We substitute this equality into the second equation of the system (1.3.5). As a result we obtain drjdx = -(1 - f !)/2. (1.3.7)

Integrating (1.3.7) and assuming that t'2 = at i = 0, we find rf - tanh(r/2) 1 - -r(20) tanh(r/2)

In combination with (1.3.6) this expression yields

At t => oo it follows from Equations (1.3.8) and (1.3.9) that = 0 and t2 = -1, that is, on a phase plane (a plane whose coordinates are the state parameters t'2) the asymptotic behaviour of the system is presented as a point with coordinates (0,-1). But, as can be seen from Equation (1.3.5), this equilibrium state is non-unique: there is one more point, with coordinates (0,1).

Let us clarify which of the above is stable. For this purpose we use the perturbation method, and write ir2 in the vicinity of steady states (O^ Y ') as f\ = + r2 = r{2'j) + V2, where = 0 and = 1. Then we substitute expressions r2 =; j <i.3.8>

for and Y2 into (1.3.5), reject the terms containing the products and squares of perturbations and i'~'2 and subtract the equations for "V(1®) and f (2X). As a result we obtain the following equations:

The solution of the first equation takes the form r\ = r\0) exp(f <2c0)t/2), (1.3.10)

where we have taken into account the fact that 'f i = ff* at t = 0.

From (1.3.10) it follows that deviations from the state (0,1) increase exponentially, and deviations from the state (0,-1) decrease exponentially, with time. In other words, the first of these two steady states is unstable, and the second is stable.

First indications of the possible existence of the non-uniqueness of the ocean thermohaline circulation appeared in the work of Chamberlin (1906). He proposed that a deep water formation in the absence of permanent ice cover may occur not as the result of intensive heat transfer to the atmosphere and decrease in sea water temperature at high latitudes, but, rather, as the result of intensive evaporation and increase in water salinity in the subtropics. The existence of a similar thermohaline circulation in the Mesozoic, with its characteristic decrease in temperature difference between the equator and the poles, damping of upwelling in the subtropics and with wide propagation of deep water stagnation, is verified by paleontological data. For indirect evidence in favour of fast (on the geological time scale) reorganization or, in effect, weakening of the ocean thermohaline circulation, data from chemical and isotopic analysis of deep water sediment deposition in the Pleistocene may be used.

Stommel (1961) was the first to present a theoretical proof of the possible existence of direct (with downwelling in high, and upwelling in low, latitudes) and reverse (with upwelling in high, and downwelling in low, latitudes) cells of the ocean thermohaline circulation. He ascertained that in the approximation of the ocean by the three-box system (polar boxes in the Northern and Southern Hemispheres and an equatorial box), two-cell thermohaline circulation is formed, the direction of rotation depending on the relationship between the equator-to-pole difference in temperature and salinity. The next step was taken by Rooth (1982), who showed, using a three-box ocean model, that, in the presence of inter-hemispheric water exchange and finite initial salinity perturbation in one of the polar boxes, two-cell circulation may degenerate into one-cell circulation even if forcing factors and ocean-land area ratio are symmetric about the equator. Numerical experiments carried out by Bryan (1986) within the framework of a three-dimensional sectorial ocean model have confirmed this conclusion. According to Bryan (1986), for constant external forcing the ocean thermohaline circulation has at least three steady states, i.e. two-cell (symmetric about the equator) circulation and two one-cell circulations, respectively, with opposite directions of rotation.

A fourth steady state was discovered by Manabe and Stoufîer (1988) within the framework of a coupled ocean-atmosphere general circulation model (see the description in Section 5.8). This state features the existence of a sharp halocline at high latitudes of both hemispheres, cessation of deep water formation, and degeneration, or even change of direction, of the ocean thermohaline circulation accompanied by a decrease in the meridional heat and salt transport to the poles, and, as a result, a decrease in the sea surface temperature and salinity of the ocean at high latitudes. The decrease in sea surface temperature and salinity is particularly evident at high latitudes of the North Atlantic. It is not caused by change in precipitation-evaporation difference but, rather, by the deceleration of renewal of surface waters due to downwelling cessation. As a result, surface waters propagating along the European continent evolve into a sub-Arctic gyre where they gradually cool down and desalinate.

Thus, it turns out that the ocean thermohaline circulation may have four stable states: two two-cell and two single-cell states with opposite directions of rotation. In practice, as was established by Welander (1986), within the framework of a three-box ocean model there are nine steady states, but only four of them are stable, the others being unstable. However, if we reject this three-box model and replace it by a more realistic model representing the ocean as a system of three ventilated (participating in heat and moisture exchange with the atmosphere) layers - the surface layer, the intermediate layer and the bottom layer - then, according to Kagan and Maslova (1991), the number of steady states will be equal to 81, including 16 stable states. These steady states differ from each other in the number of cells (one-cell, two-cell and four-cell circulations), direction of rotation (clockwise and anticlockwise), properties of symmetry (symmetric and asymmetric circulations about the equator) and the vertical extension of cells (extension over the whole thickness of the ocean or only over a layer of finite depth). Such a variety of ocean thermohaline circulation forms has far-reaching consequences in respect of ocean climate predictability: even small changes in little-known initial fields of salinity and/or salt flux at the ocean surface may lead to qualitatively different distributions of climatic characteristics.

A similar situation takes place in the atmospheric circulation. According to Lorenz (1967), when using a two-layer quasi-geostrophic model of the general atmospheric circulation and retention for six modes, the increase in the equator-to-pole temperature difference is accompanied by the following change in regimes of the large-scale circulation. Firstly, when the temperature difference does not exceed 58 K there is only one steady solution representing the direct meridional circulation cell (so-called Hadley cell) where warm air ascends at the equator and cold air descends at the pole. A distinctive feature of this circulation regime is the absence of zonal flow in the lower layer and, hence, of its interaction with undulations of the underlying surface. When the temperature difference is equal to 58 K a birfurcation (replacement of one circulation regime by the other) occurs. The new regime is characterized by the existence of three steady state solutions; one of them (the Hadley cell) is unstable, and the two others, describing wave motions and, respectively, westerly and easterly flows, are stable. The second bifurcation occurs at the equator-to-pole temperature difference of 72 K. The subsequent increase in temperature difference results in the simultaneous existence of five steady solutions, two of them conforming to low and high zonality indices (large and small amplitudes of wave oscillations or, in other words, the presence and absence of blocking) respectively. Of these five steady state solutions only one is stable with regard to disturbances of the first meridional mode. It is a solution describing relatively weak zonal flow with superposition of high-amplitude wave constituents.

It should be emphasized that the results mentioned above concerning the change in atmospheric circulation regimes were obtained within the framework of a model based on an indirect assumption of the absence of all wave components with the exception of large-scale forced oscillations, and, hence, of the absence of all forms of non-linear interaction apart from the interaction between large-scale forced waves and zonal flow. In practice, because of a strong baroclinic instability generating free wave motions of synoptic scale, there are always other forms of non-linear interaction in the atmosphere, such as interaction between free waves and stationary flow, between forced and free waves and between free waves. Taking these forms of interaction into account within the framework of a multicomponent spectral model results in only one stable solution and the disappearance of all others. Moreover, it turns out that the unique possible stable solution corresponds to the wave regime of the atmospheric circulation with low zonality index, that is, it coincides with the only steady solution in a small-component model. The main distinction of these stable solutions is the fact that for different initial conditions the trajectories of the solution corresponding to the six-mode model are attracted to one and the same point of phase space (space of parameter), while in the multicomponent model they contract to a certain restricted domain inside which the behaviour of the solution becomes so complicated that the solution appears to describe stochastic oscillations of large-scale circulation. From the viewpoint of predictability it means that detailed long-range prediction of large-scale atmospheric circulation is impossible. But since the 'stochastic' oscillations of a large-scale circulation are relatively small (the attractor basin is restricted) then, as was mentioned above, the low-frequency variability of large-scale circulation can be considered as predictable on the average, that is, under conditions of preliminary filtration of high-frequency oscillations.

0 0

Post a comment