studies the fundamental importance of Stommel's 1961 paper has become more widely recognized.
Hasselmann (1976) demonstrated in a simple way how the heat-storage capacity of the upper ocean acts to integrate random heat impulses from the atmosphere. This integrating property of the upper ocean greatly amplifies the response of sea surface temperature to low-frequency heating inputs. In Hasselmann's stochastic model the excursions of sea surface temperature about equilibrium are taken to be the result of a random-walk process, which is limited in amplitude by atmospheric feedback. In Hasselmann's model, air-sea interaction is purely local within the ocean. However, this elegantly simple model runs into difficulties when an attempt is made to generalize it to include salinity as well as temperature. For a random-walk process on the temperature-salinity plane driven by stochastic forcing there is no physically plausible mechanism to limit extreme anomalies of vertically averaged salinity, as there is for temperature. This difficulty is removed only by allowing for a nonlocal process that permits exchange with other parts of the world ocean. A stable regime in Stommel's two-box model of the thermohaline circulation driven by stochastic forcing represents the simplest extension of Hasselmann's ocean climate model, which can include salinity as well as temperature.
The motivation for attempting to construct a simple toy model of this kind is the success of much more complex numerical models that illustrate oscillations of thermohaline circulation. Examples are studies by Weaver and Sarachik (1991b), Mikolajewicz and Maier-Reimer (1990), and Delworth et al. (1995, in this volume). The calculations by Mikolajewicz and Maier-Reimer were carried out for a numerical model of the world ocean. Surface temperature is damped toward observed values, but stochastic forcing is used to simulate observed variations of the net water flux at the surface. It was found that stable oscillations of the Atlantic thermohaline circulation took place with a maximum amplitude at a period of several centuries. Delworth et al. studied a fully coupled ocean-atmosphere model. Their results illustrate climate fluctuations consistent with Bjerknes' (1964) hypothesis for Atlantic climate variability. Variations in the strength of the thermohaline circulation are correlated with decadal-scale sea surface temperature anomalies that seem to be quite realistic when compared with analyses of the Comprehensive Ocean-Atmosphere Data Set (COADS) by Kushnir (1994). The aim of the present study is to construct a simpler framework for understanding the important results of these physically complete but highly complex models.
Since Hasselmann's (1976) stochastic model of climate is an important point of departure for the present study, we will review it briefly. Consider a reservoir of upper ocean water as shown in Figure 2a. Let T' be the departure of the temperature from its climatological average. The governing equation is
where D is the depth of the reservoir, and Q is the average temperature change in the reservoir due to heating, which is associated with the random fluctuations of cyclones and anticyclones at the ocean surface. Temperature fluctuations are damped by the term, − λT'/D, on the right-hand side of (1). This negative feedback term represents the combined effects of long-wave radiation, evaporative cooling, and sensible heating, all of which are closely related to ocean surface temperature. Let
Hasselmann (1976) assumes that the effect of ''weather" over the ocean can be represented as white noise, where |Qw| is thus uniform at all frequencies. The power spectrum of the solution of (1) may be written as
The spectrum given by (3) has two very different regimes. In the high-frequency regime, where ω λ/D, "white noise" forcing gives rise to a "red noise" response. This can be understood physically as the effect of the memory of the ocean. Hasselmann (1976) describes it as a random-walk process in one dimension. Excursions from the origin become longer and longer over greater and greater time scales.