fluxes) on temperature allows the use of a simple, linear, Newtonian-damping boundary condition. The upper layer of the ocean (or the reservoir representing it in a box model or a more complicated model), is restored to an appropriate reference temperature on a fast time scale, I to 2 months (Haney, 1971). The boundary condition therefore takes the form of a variable flux (in watts per square meter; positive QT means heat out of the ocean),

where is the upper-ocean box (with thickness Δz1) temperature at longitude λ and latitude . is the atmospheric reference temperature, Cp is the specific heat at constant pressure (approximately 4000 J kg−1 °C−1), ρ0 is a reference density (approximately 1000 kg m−3), and τR is a restoring time scale (Haney, 1971).

In ocean models it is appropriate to represent fresh-water fluxes at the ocean surface (due to evaporation, precipitation, river runoff, or ice formation) as a surface boundary condition on salinity. However, evaporation is mainly a function of the air-sea temperature difference, while the distribution of precipitation depends on complicated small-and large-scale atmospheric processes. A Newtonian boundary condition on salinity (units are g salt m−2 s−1) as shown in (2),

then cannot be justified physically; it implies a definite time scale R) for the removal of salinity anomalies, which is not observed. Furthermore, (2) implies that the amount of precipitation or evaporation at any given place depends on the local sea surface salinity , which is clearly incorrect. To resolve this problem in uncoupled ocean models, the imposition of either specified salinity fluxes QS or a salinity flux that depends weakly on the atmosphere-ocean temperature difference is preferred. The salinity fluxes QS may then be converted to implied fresh-water fluxes (P − E, in m yr−1) by

where S0 is a constant reference salinity (about 34.7 psu) and c = 3.16 × 107 is the number of seconds in a year. A constant reference salinity is used in (3) instead of the local salinity so that when (3) is integrated over the surface of the ocean, zero net P − E corresponds to zero net QS.

Surface boundary conditions that involve a Newtonian restoring condition on temperature and a specified flux on salinity are termed mixed boundary conditions. While these boundary conditions are admittedly crude, they do reflect the different nature of the observed sea surface salinity (SSS) and SST coupling between the ocean and the atmosphere. A discussion of some further enhancements to these boundary conditions, which might be employed in future OGCM studies, is presented later. In the uncoupled models discussed below these boundary conditions will usually be used.

Due to the lack of open-ocean observations of surface wind speed, mixing ratios of air above the sea surface (needed to determine evaporation through bulk formulae), and precipitation, it is common to obtain a surface freshwater flux for use in uncoupled ocean models by spinning up a model to equilibrium under restoring boundary conditions on both T and S and then diagnosing the salt flux at the steady state. That is, (1) and (2) are used in the initial spin-up, and then at steady state the right-hand side of (2) is diagnosed at each grid box to yield a two-dimensional salt-flux field. (This field can then be converted to an implied fresh-water flux using (3).) The rationale for this approach is that by spinning up the model using some specified climatological surface restoring fields, one obtains an equilibrium in which the surface fields of T and S are climatologically correct. The diagnosed P − E field is then that field which, in theory, should yield the climatological SSS field. Furthermore, the equilibrium under restoring boundary conditions is also an equilibrium under the diagnosed mixed boundary conditions. Paradoxically, however, if the model simulates the SSS field exactly under restoring boundary conditions, P − E goes to zero.

Diffusive Time-Scale Variability

Marotzke (1989) spun up a single-hemisphere OGCM under restoring boundary conditions on temperature and salinity with no wind forcing. Switching to the diagnosed mixed boundary conditions and adding a small fresh perturbation to the high-latitude salinity budget at equilibrium, precipitated a polar halocline catastrophe. Several thousand years later the system evolved into a quasi-steady state with weak equatorial downwelling (a weak inverse circulation). This state was not stable, since low-latitude diffusion acted to make the deep waters warm and saline while horizontal diffusion acted to homogenize these waters laterally. Eventually, at high latitudes the deep waters became sufficiently warm that the water column became statically unstable and rapid convection set in. As in his zonally averaged model (Marotzke et al., 1988), the result was a flush in which a violent overturning (up to 200 Sv) occurred, whereby the ocean lost in a few decades all the heat it had taken thousands of years to store. At the end of the flush the system continued to oscillate for a few decades until the circulation once more collapsed. In the presence of wind forcing, Marotzke (1990) found that no flush existed. The inevitability of the occurrence of a flush under a purely buoyancy-forced, diffusive regime was illustrated in an analytic model developed by Wright and Stocker (1991).

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