FIGURE 3 

Steady-state solutions of the model are given by the  intersection points of the solid lines (bold-faced numbers), which  correspond to the different values of the damping parameter, and  the dashed lines (italic numbers), which correspond to different  values of E, the nondimensional evaporation-minus-precipitation  forcing. The diagonal line separates the thermal regime, in which  temperature differences dominate the poleward density gradients,  from the haline, in which salinity dominates.

equation (15) gives a family of curves that penetrate further below T = 1 for weaker haline forcing.

STOCHASTIC FORCING OF THE THERMALLY DOMINATED REGIME

In this section we will consider the results obtained by stochastic forcing of a version of the Stommel model with no restoring of the salinity field. The stability of the Stommel two-box model has been discussed by previous authors (Huang et al., 1992; Marotzke, 1990; Stommel, 1961; Walin, 1985). The results of the linear analysis of our model are shown in Figure 4. Damped harmonic solutions exist for T < S < T/(2T − 1)2. Unstable real roots exist in the region bounded by S = T(2 − T)/2 and T = S. The remaining areas have simple damped solutions. In the North Atlantic, geologic evidence suggests that a thermally dominated thermohaline regime has existed since the close of the last ice age, 10,000 years ago. Therefore, it appears that the stable regime in which S < T(2 − T)/2 in our model corresponds to the present climate of the North Atlantic. The remainder of this paper is concerned with the analysis of the response to forced oscillation about the stable equilibrium in the thermally dominated regime of the model.

To check our calculations we calculated the linear response analytically and compared it to the results of direct

FIGURE 4 

A map of the T-S plane showing the results of a linear-stability analysis of steady-state solutions of the model. Unstable  solutions are only found in the region T(2 − T)/2 < S < T. In the  remaining regions, solutions are either simply damped or harmonically damped.

numerical integration of the full nonlinear model with very small stochastic forcing. It is appealing to think of the effect of air-sea fluxes associated with cyclones and anticyclones passing over the ocean as causing a random walk in vertically integrated water-mass properties. Results from the GFDL coupled model show that heating and net evaporation-minus-precipitation at the ocean surface are negatively correlated. The results of Delworth et al. (1995) show that surface fluxes tend to heat and simultaneously freshen the ocean surface or, conversely, cool the ocean surface and make it more saline. Rather than making heating and evaporation-minus-precipitation independent random variables, we made them proportional to one another, with opposite sign, in our stochastic model.

Analytic spectra for the three cases are shown in Figure 5. Figure 5a corresponds to the case in which η = 2 and the steady component of E = 0.1. Q' is 10 times larger than E' and of opposite sign, causing the temperature fluctuations to be larger than the salinity fluctuations at all frequencies. At high frequencies, both spectra have a slope of ω–2. Damping becomes important for salinity only at frequencies less than 0.1 cycles/unit time, which corresponds to a period of 240 years. Figure 5b shows a case that is the same as that of Figure 5a, except that ηQ' is now twice −E'. |T2| is exactly four times longer than |S2| at high frequencies, but a crossover point is reached at a period between 50 and 100 years. The effect of removing the thermohaline coupling term is shown in Figure 5c. This case corresponds to the



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