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Fig. 5.165 A sketch of the oscillatory circulation in the meridional plane under the mixed boundary conditions.

Decadal/interdecadal variability of the oceanic circulation can be found through numerical modeling as follows. First, a numerical model for a closed basin is spun up, with relaxation conditions for both temperature and salinity. When the model's solution reaches a quasi-steady state, the equivalent virtual salt flux can be diagnosed from this quasi-steady solution. Next, the model is restarted from this quasi-steady state and run under mixed boundary conditions, i.e., the same relaxation condition for temperature, but salinity is now under a flux condition, specified according to the diagnosed virtual salt flux. In many cases, a wealth of decadal variability can be found through the numerical simulations. Of course, the exact characteristics of such decadal/interdecadal variability depend on the model's geometry and, above all, on the choice of forcing field and parameters.

As an example, the decadal variability was identified in a single-hemisphere ocean model forced by the mixed boundary condition (Weaver et al, 1991). Similarly, the interdecadal variability in an idealized model of the North Atlantic Ocean was discussed by Weaver et al. (1994). Under the mixed boundary condition, the model possesses a limit cycle with a period of approximately 22 years. As shown in Figure 5.166, both the mean kinetic energy and the air-sea heat flux oscillate during each cycle. This limit cycle is associated with a great change of the thermohaline circulation in the model basin. For example, the poleward heat flux varied from a maximum of 0.8 PW at 39° N to a minimum of 0.5 PW. In addition, deepwater formation at the northern boundary of the Labrador Sea was shut off and turned on during each cycle. Therefore, the decadal/interdecadal cycle may play a crucial role in the global oceanic circulation and climate system.

Variability on a diffusive time scale: flushing

Owing to the extremely different natures of the boundary conditions applied to temperature and salinity, there exists one more type of thermohaline variability called flushing (Marotzke, 1989), which is on rather a long diffusive time scale, i.e., on the centennial and millennial time scales. Flushing can be studied using a numerical model as follows. A model is first run to quasi-equilibrium under the relaxation condition for both salinity and temperature with no wind stress. After diagnosing the virtual salt flux, the model is continued, running under the mixed boundary conditions. By adding a small (1%o) fresh (negative salinity) perturbation at high latitudes, the polar halocline catastrophe sets in. The system slowly evolves into a state with very weak equatorial downwelling and upwelling elsewhere, i.e., it is in a haline mode, in reverse of the thermal mode. During a period of several thousand years the horizontal diffusive process produces warm and salty abyssal water in the whole basin.

At high latitudes, cold and fresh water overlays the warm and salty water there, with a relatively weak stratification which is potentially unstable. Imagine a small water parcel moving in a vertical direction, bringing warm and salty deep water to the surface. Near the surface the water parcel quickly loses its heat owing to the strong thermal relaxation. Under the flux condition, surface salinity evolves on a much longer time scale, so it remains virtually unchanged over a relatively short time scale.

As a result of surface cooling, the water parcel becomes heavier than the ambient density and sinks all the way down to the bottom. The GPE released in this process feeds energy to the convective overturning. Consequently, the system goes through a very violent phase, called flushing, until the potentially stored energy is consumed completely. The system is in a quasi-thermal mode during the flushing period, but it returns to the quasi-haline mode after flushing.

Gravitational potential energy balance is the key element in flushing. To demonstrate the connection between GPE and flushing, we analyze the evolution of GPE during the flushing. Assume a model ocean of two layers, with equal thickness of h and unit area (Fig. 5.167). The upper layer is subject to a relaxation temperature T* = T1. At the initial time, temperature and salinity are Ti and Si in the upper layer, and T2 and S2 in the lower layer. Water density is a linear function of T and S: p = po(1 - aT + jS). We also assume, for simplicity, that there is no stratification at the initial time, i.e., p1 = p2, or a(T2 — T1) = j(S2 — S1).

If these two layers are flipped upside down (the second stage in Fig. 5.167), the total GPE remains unchanged. Cooling takes place afterward. After a time period, which is much longer than the relaxation time for surface temperature but much shorter than the time scale for surface salinity change, temperature in the upper layer is reduced to T1, but the salinity there remains unchanged. Changes in GPE can be calculated according to the model used as follows.

If a Boussinesq model is used, the layer thickness remains unchanged after cooling. Due to the artificial source of mass in the model, the total mass of the upper layer increases 8m = hp0a(T2 — T1). Using the bottom as the reference level for GPE, the total GPE

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