Thus, the meridional velocity scale is proportional to the square root of the mean salinity and the amplitude of freshwater flux, and is independent of wind stress and diffusivity. The corresponding meridional volumetric flux is

The volumetric balance equation (5.315) leads to

are the halocline depth scales associated with wind stress and diffusion. The solution of Eqn. (5.338) is

There are two limit cases:

The case of strong wind stress and weak diapycnal mixing Introduce a small parameter:

so from Eqn. (5.340), the halocline scale depth is

D = hw (1 + £2 - 2e4 + ...) = ^VL1 (1 + £2 + ■■■) (5.342)

and the meridional overturning rate is

therefore, both the halocline depth and the meridional overturning rate are linearly proportional to the Ekman pumping rate, and the diffusivity makes a small linear correction.

The case of weak wind stress and strong diapycnal mixing Introduce a small parameter:

so from Eqn. (5.340), the halocline scale depth is

and the meridional overturning rate is

M = (K VLy)l/2 Lx(l 1 + À + («Ly4/2( f) ^ (l + (5.346)

thus, both the halocline depth and the meridional overturning rate are proportional to the 2 power of the diffusivity, as discussed by Huang and Chou (1994), while the Ekman pumping rate makes a small linear correction. This displays a much stronger dependence on the diapycnal mixing than the |-power law for the case of the relaxation conditions discussed above.

In addition, the strength of the meridional circulation depends on the 1 power of the precipitation amplitude and the mean salinity in the ocean. For example, if there were no salt in the ocean, the barotropic gyres described by Goldsbrough (1933) would be the only circulation driven by evaporation and precipitation, with no baroclinic return flow at all.

Circulation under mixed boundary conditions The general case is circulation under mixed boundary conditions, i.e., the thermal balance is forced by a temperature relaxation toward a specified reference temperature, but the salt balance is driven by the specified air-sea freshwater flux. The meridional density difference corresponding to Eqn. (5.335) is now

The case of fixed diapycnal diffusivity Assuming that the zonal velocity and the meridional velocity obey the same empirical relation, V = cU, the thermal wind relation, Eqn. (5.300), is reduced to

This equation includes the two limit cases corresponding to the cases discussed above, i.e., the case with fixed reference temperature and the case with a fixed rate of freshwater flux specified at the upper surface. The solution to this equation is cgaAT

2fLy J f

Substituting this relation into the meridional volumetric flux balance equation leads to a cubic equation, which includes a square root term in the cubic term. This equation can be solved using the same perturbation method (Zhang et al, 1999).

The case of fixed rate of energy sustaining mixing Following discussion in the subsection entitled "Extension of the two-box model: The energy constraint and wind-driven gyration" in Section 5.4.2, the total amount of external mechanical energy supporting diapycnal mixing Em is fixed, which is used to support the diapycnal mixing, i.e.,

where H and A are the mean depth and the total horizontal area of the ocean. The salinity balance associated with the meridional overturning cell gives rise to a simple relation

where E is the freshwater flux. Thus, Eqn. (5.350) is reduced to e + 6sS E

where e = Em/gp0HA represents the strength of the external source of mechanical energy and, under the temperature relaxation condition AT ~ AT*, can be used as a good approximation ( AT* is the meridional difference of reference temperature). The corresponding meridional density difference can be rewritten as

Using the scaling relation between the meridional and vertical velocity V = Why/D, Eqn. (5.301) is reduced to the pycnocline depth scale

The corresponding poleward heat flux is defined as

The scaling analysis implies that strong freshwater forcing enhances the upwelling rate, and thus the meridional overturning rate and poleward heat flux. On the other hand, a large meridional temperature difference implies strong stratification, and thus it can suppress the overturning rate and poleward heat flux. This scaling law has been verified by some idealized model experiments (Huang, 1999; Nilsson etal., 2003); however, model behavior under energy constraint may not exactly follow the scaling law discussed above, and the suitable scaling law remains unclear for the general cases.

A scaling law for the meridional circulation in the Atlantic Ocean The pycnocline depth and the meridional overturning cell in the Atlantic Ocean are closely related. In order to consider a scaling law for these two aspects of the circulation, it is very important to include the crucial role played by the ACC and the associated wind stress forcing due to the southern westerly (Gnanadesikan, 1999). The basic methodology is a volumetric transport balance of all the relevant components of the circulation system, as depicted in Figure 5.175.

According to Eqn. (5.308), meridional volume flux associated with the northern sinking is

N PL"

where L^ is the meridional length scale of the North Atlantic Basin, and D is the scale depth of the main pycnocline.

A major physical process unique to the Southern Ocean is the strong eddy activity associated with the ACC. The eddy-induced flow gives rise to a reduction of the transport, indicated by the TE term on the left-hand edge of Figure 5.175.

Physically, eddy activity reduces the horizontal density gradient, leading to a decline of the isopycnal slope. Imagining the isopycnal surface as a rubber bow, then baroclinic instability tends to bend the southern edge of the rubber bow and makes it flatter, thus causing the water to leak out. This is a sink of water, which can be written as

where Lx is the circumference of the Earth at the latitude of the Drake Passage, veddy = Aj dSl/dz = Aj /Ly, where Aj is an eddy diffusion coefficient, Si = D/Lsy is the slope of the isopycnals, and Lsy is the meridional width of the pycnocline associated with the ACC. The contribution due to wind stress is

and the contribution due to upwelling is

where Ly is the meridional scale of the wind-driven circulation. The volume flux balance for the whole basin is

which leads to a cubic equation for D. Once again, dimensional homogeneity of this equation suggests that we write it in the following form:

where three pycnocline depth scales are introduced, i.e., the eddy-controlled pycnocline depth scale de, the wind stress-controlled pycnocline depth scale dw, and the diffusion controlled pycnocline depth scale dK

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