The simplest choice is a two-box model for a single-hemisphere ocean, in which temperature is relaxed toward the specified reference temperature of T= To, T| = 0 and an air-sea freshwater flux [p, —p]. To simulate the effect of the wind-driven gyre, a volume transport m between the two boxes is added (Huang and Stommel, 1992). Following Guan and Huang (2008), this volume transport is prescribed a priori, independent of the energy for mixing and temperature/salinity (Fig. 5.131).

Assuming that the flow is poleward in the upper layer and equatorward in the abyssal ocean (the so-called thermal mode), the temperature balance in Box 1 obeys

HL2 — = —HLuT1 + L2wT2 + L2r(T0 — T1) — HLm(Ti — T2) (5.214)

dt where L and H are the width and depth of each box, T1 and T2 are the temperature in each box, u and w are the horizontal and vertical velocity, m is the strength of gyration, and r is the surface relaxation constant. Introducing the nondimensional variables through u = TlW, w = rw', m = H t = H t', and dropping the primes, the temperature balance for Box 1 is reduced to dT1 = — uTx + wT2 + T0 — T1 — m(T 1 — T2) (5.215)

Similarly, the temperature balance for Box 2 is

The salinity balances for Boxes 1 and 2 are dSl = -uSi + wS2 - m(Si - S2) (5.217)

Subtracting Eqn. (5.216) from Eqn. (5.215) and using Eqn. (5.219), we obtain d

-(T1 - T2) = -2M(T1 - T2) + 2pT1 + To - T1 + T2 - 2rn(T1 - T2). (5.220)

Integrating the sum of Eqns. (5.215) and (5.216) leads to a constraint: for any initial conditions, the sum of temperature in the two boxes converges exponentially, T1 + T2 ^ T0. Thus, the 2pT 1 term in Eqn. (5.220) can be replaced by p(T0 + T1 - T2). Adding Eqns. (5.217) and (5.218) leads to the salt conservation

where S0 is the mean reference salinity of the model ocean. We now introduce new temperature and salinity variables: AT = T1 - T2, AS = S1 - S2. Using Eqns. (5.219) and (5.220), we obtain a pair of equations:

dt d AS dt

When the circulation is in the haline mode, the direction of mass transport between these two boxes is reversed, but the contribution due to wind-driven circulation stays unchanged. Thus, equations corresponding to Eqns. (5.222) and (5.223) are d AT

dt d AS dt

These sets of ordinary differential equations, (5.224) and (5.225), govern the thermohaline circulation, including the effect of wind-driven circulation. In order to calculate the circulation, we need one more constraint that determines the vertical velocity w.

The constraint regulating the meridional overturning rate is the remaining critical part of the model formulation. Stommel (1961) postulated a constraint which can be classified as a buoyancy constraint. Using Eqn. (5.175), the corresponding vertical velocity scale is as follows:

where c is a constant. This formulation implies that the circulation rate is regulated by the surface buoyancy difference. In other words, surface thermohaline forcing drives thermohaline circulation.

Stommel's assumption has been widely adopted in most existing box models. A careful examination of Stommel's classical assumption, namely Eqn. (5.226), suggests that the reasoning behind Stommel's choice of such a constraint is not obvious; in addition, it is not the only possible constraint and may not even be the best constraint for modeling thermohaline circulation under different climate conditions.

More suspiciously, this constant c has been treated as a constant intrinsic to each model, and it is assumed to be invariant under different climate conditions. From the most basic consideration of balance of mechanical energy, however, to maintain a steady circulation in the ocean, dense deep water must be brought upward through the thermocline, and heat absorbed in the upper ocean must be mixed downward against the upwelling of the cold and dense water through diapycnal mixing. Therefore, the balance of the external sources and the dissipation of mechanical energy are what really control the thermohaline circulation. As an alternative, a constraint based on mechanical energy sustaining diapycnal mixing can be formulated as follows. Assuming a one-dimensional balance of density in the vertical direction wepz = Kpzz (5.227)

where k is the vertical (diapycnal) diffusivity, we obtain the following scale for the vertical velocity:

where D is the depth scale of the main thermocline. The differences in temperature, salinity, and density between the surface and abyssal ocean are AT, AS, and Ap. Note that Ap = p0 (-a AT + jAS) = p1 - p2 < 0. In a two-layer model, the rate of GPE increase (per unit area) due to vertical mixing k is -gKAp. Using the above notations, the rate of GPE created by mixing in the box model is

Em = -gKApL2 = -gDL2WeAp = gDL2We (p0 a AT - p0jAS) (5.229)

Therefore, the scale of We associated with meridional overturning satisfies e

p0aAT - p0jAS

where e = = §, E = (E is the rate, per unit area, of external mechanical energy supply) has a dimension of density multiplied by velocity and represents the strength of the external source of mechanical energy sustaining mixing. A noteworthy point here is that an external source of mechanical energy can vary with climate; thus, e in the energy-constraint model is treated as an external parameter which can change with climate.

Scaling laws

Some useful insights can be obtained through simple scaling. The continuity relation is scaled as

where U and W are horizontal and vertical velocity scales. The salinity balance associated with the meridional overturning cell obeys a simple scaling relation

where F is the scale of freshwater flux, and Q = wD/L is the scale of gyration. Therefore, the scale of salinity difference is

When Q = 0, the vertical velocity scale under the energy constraint is reduced to

Under the temperature relaxation condition, AT ~ T0 is a good approximation. The corresponding scale for the poleward heat flux is

CpL2

In other words, the scaling analysis implies that strong freshwater forcing enhances the upwelling rate, and thus both the meridional overturning rate and the poleward heat flux are enhanced.

In contrast, the Stommel-like model predicts that strong freshwater forcing reduces the meridional density difference, and thus the meridional overturning rate. Indeed, the assumptions of the fixed buoyancy constant c or the energy constant e represent two extreme conditions, and the real-world condition may be somewhere in between.

If w = 0 (i.e., no wind-driven gyre), it is well known that in a quite wide region of the parameter space there exist three steady states in the Stommel-like model (Fig. 5.132a): a stable thermal state, an unstable thermal state, and a stable haline state; whereas three

Fig. 5.132 Thermohaline circulation bifurcation for a the Stommel-like model, c in 10-7 m4/kg/s; b the energy-constraint model, e in 10-7 kg /m2/s,To = 15°C, So = 35, p = 2 m/yr. Heavy lines indicate the thermal modes, thin lines indicate the haline modes, dotted lines for unstable modes; S is the saddle-node bifurcation point (dashed line) (Guan and Huang, 2008).

steady states existing in the energy-constraint model (Fig. 5.132b) are a stable thermal state, a stable haline state, and an unstable haline state.

It can readily be seen that when m = 0 the bifurcation structure of the Stommel-like model is dramatically different from that of the energy-constraint model. In particular, the stable haline mode from the energy-constraint model is characterized by an extremely large salinity difference, i.e., salinity in the pole box is nearly zero. However, including the gyration dramatically reduces the salinity difference, AS, for the haline mode in the energy-constraint model; and it becomes much closer to the corresponding solution obtained from the Stommel-like model. The fact that the inclusion of wind-driven circulation can change the bifurcation structure of thermohaline circulation clearly demonstrates the importance of coupling the wind-driven gyre with thermohaline circulation.

In this section we closely examine the role of gyration in detail. We adopt the following set of parameters for both the Stommel-like model and the energy-constraint model: the basin has a length scale of 4,000 km, is 4 km deep, has a reference temperature T0 = 15°C, the mean salinity is S0 = 35, and the other parameters are the same as in the above discussion. As shown in Figure 5.133, for a certain parameter range, changes in wind-driven circulation alone can induce a bifurcation of thermohaline circulation in both the Stommel-like model and the energy-constraint model.

For the Stommel-like model, an increasing wind-driven circulation enhances the meridional overturning rate in the thermal mode and reduces the meridional overturning rate

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