amplitude by a factor of 2. As a result, the negative amplitude of |T| is smaller relative to |S|, but the phase relationships are qualitatively the same.
The aim of this paper is to develop a very simple model of Atlantic climate variability on decadal and longer time scales by combining elements of existing models of Stommel (1961) and Hasselmann (1976). In this model random forcing at the ocean surface drives the water mass distribution of the subarctic gyre of the North Atlantic away from its climatologically balanced and stable steady state. A random forcing by unstable cyclones and anticyclones passing over the ocean would produce a uniform forcing of both high and low frequencies. Feedback through air-sea interaction is assumed to be purely negative, restoring temperature to its equilibrium state, but with no corresponding feedback for salinity.
As shown by Hasselmann (1976), the storage capacity of the ocean greatly amplifies the response of the ocean to forcing at very low frequencies. Extremely large responses at very low frequencies are tempered by thermal damping and by the effects of the thermohaline circulation that responds to density gradients that build up between the subarctic gyre and the ocean at lower latitudes. The combined effect of these two mechanisms is to flatten the response spectra for salinity and temperature at low frequencies. Details of the forcing and thermal damping determine the exact amplitude and phase of the temperature and salinity fluctuations.
Although the response of the simple toy model of this study is devoid of the sharp spectral peaks that have been found in more complex models, we can reproduce the gross features of the results of those complex models. The response of the model to white-noise forcing is simply red noise at frequencies less than the basic time scale of the thermohaline circulation, and an equilibrium response at frequencies much less than that of the thermohaline circulation. The response of the model to small perturbations is always stable in the thermally dominated regime, corresponding to present-day climate. Since there is no possibility of resolving the upper thermocline, instabilities that turn on or shut off convection are excluded. The thermohaline circulation always acts to restore the ocean climate to its equilibrium state.
The simplicity of the model allows some interesting insights on the phase relationships between the thermohaline circulation and temperature and salinity fluctuations. If fluctuations of heating and E - P are negatively correlated, as suggested by the coupled-model results of Delworth et al. (1995), the toy model predicts that density perturbations due to temperature in the subarctic box will lead the thermohaline circulation, while density perturbations corresponding to salinity will lag behind. This result corresponds to the phase relationships found in the more detailed three-dimensional model of Delworth et al. (1995). On the other hand, very different phase relationships are found in the toy model when heating and E - P are taken to be positively correlated.
The realistic phase behavior of the model may be described in terms of counterclockwise orbits in the T - S plane, caused by the fact that temperature is damped much more than salinity due to the air-sea interaction. Counterclockwise orbits are required by a negative correlation between stochastic heating and stochastic evaporation-minus-precipitation.
It is assumed that forcing is sufficiently small that the system oscillates in an essentially linear fashion about its equilibrium state. This would seem to be appropriate for the climate that has existed in the North Atlantic since the last ice age, and is consistent with the simulations of Delworth et al. (1995). At the close of the last ice age the hydrological cycle of the North Atlantic was strongly perturbed by the melting of large ice sheets. In that case much larger excursions from equilibrium would be expected, and it is not clear whether useful insights could be obtained with such a simple model as we have described in this study.
The authors would like to thank Tom Delworth, Syukuro Manabe, and Edward Sarachik for generously sharing their ideas and results with us. This research was supported in part by funding from the Atlantic Climate Change Program of the NOAA Office of Climate and Global Change.