Fig. 5.97 A finite difference grid box in the x — z plane.

(Fig. 5.97), can be written as dJL + ± [(us)+ — (us)—] + [(wS)+ — (wS)—] = (S+ — ^)

where (uS)+ and (uS)— are the salt fluxes advected across the right and left boundaries, and S+ and S— are the horizontal salinity gradients at the right and left boundaries; similarly, (wS)+ and (wS)— are the salt fluxes advected through the upper and lower boundaries, and Sf and (wS)— are the vertical salinity fluxes at the upper and lower boundaries. In addition, vertical velocity is prescribed on the upper surface w = w+, at z = 0 (5.135)

where w+ is the vertical velocity and its value depends on the choice of model.

Relaxation condition

For this case, the rigid lid is interpreted as a solid boundary with no mass flux across it, so that the velocity boundary condition is

As discussed above, however, the rigid-lid approximation does not necessarily require the vertical velocity to be zero at the upper surface. The salinity boundary condition is

where T is the relaxation constant, and S* is specified according to the climatological mean surface salinity. This boundary condition can be traced back to the relaxation boundary condition for the sea surface temperature (Haney, 1971). His original equation is based on a detailed analysis of the heat flux through the air-sea interface, including solar insolation, latent heat flux, sensible heat flux, and the turbulent heat flux:

where T* is the reference temperature, or the equivalent atmospheric temperature, which should be calculated by taking into consideration all heat flux terms; Ts is the sea surface temperature. In many studies the climatological mean SST is used as T*. It is clear that such an approach may introduce certain errors because of the difference between T* and the climatological SST.

Virtual salt flux condition For this case, the rigid lid is also interpreted as a solid boundary with no mass flux across it, so that the velocity boundary condition is

however, the salinity boundary condition is a flux condition

where e - p is evaporation minus precipitation, and S is the salinity in the box. This is the virtual salt flux required for the salinity balance. However, using (e - p)S to force a model may cause the total salt in a basin to increase infinitely. This can be explained as follows. For a steady climate, the global precipitation and evaporation should be nearly balanced:

However, the global integration of the virtual salt flux is larger than zero:

This is due to a positive correlation between evaporation minus precipitation and salinity, as shown in Figure 1.2.3. For example, evaporation is very strong in the subtropical North Atlantic Ocean, giving rise to a salinity of more than 37; meanwhile, excess precipitation in the subpolar Pacific Ocean gives rise to a surface salinity lower than 32. Thus, to avoid the salinity explosion one must use the virtual salt flux defined by

where Ss is the mean sea surface salinity averaged over the whole model domain. This formulation can introduce a systematic bias which we will address later.

Natural boundary condition As discussed above, for a climatic time scale the velocity boundary condition on the upper surface is

and this is the freshwater flux controlling the salinity balance; the corresponding boundary condition for the salinity balance is

Note that within the water column there is always turbulent salt flux kvSz and advective salt flux wS. However, in the air there is no salt, so both these terms are identically zero, although w is not zero. At the air-sea interface kvSz = (e - p)S, i.e., the turbulent flux exactly cancels the advective flux so that there is no salt flux across the air-sea interface, as required by the physics. Thus, we may call this turbulent flux Sf = kvSz an anti-advective salt flux, which is the same as the virtual salt flux discussed above, Sf = (e -p)S. From our discussion, we can see that there is no need for any salt flux across the air-sea interface if we use the natural boundary condition, because these two fluxes exactly cancel each other; however, if we interpret the rigid-lid approximation as a zero vertical velocity condition, the virtual salt flux is needed in order to simulate the effect of freshwater flux through the air-sea interface. Accordingly, under the natural boundary condition, the salinity balance for a surface box is reduced to d S 1 r , — 1 KH /I x KV

- + - [(uS)+ - (uSy]- -(wS)- = ^^ (s+ - s-) - XiSz" (5.146)

The natural boundary condition, in combination with the continuity equation, is a statement of salt conservation

This equation means that the total salt in the world's oceans is conserved. The local salinity change is due to freshwater dilution/concentration, and there is no need for the virtual salt flux. Although our discussion above and Figure 5.97 are based on the case with a rigid lid, the same argument can be applied for the case with free surface, as discussed later.

A virtual natural boundary condition Evaporation and precipitation data are difficult to collect, and there are no reliable historical data. As a compromise, one can use the sea surface salinity as a forcing field to reconstruct the salinity balance in the past, using the virtual natural boundary condition as follows.

First, the equivalent evaporation and precipitation field can be inferred from the historical sea surface salinity data from the model as e - p = V(S* - S)/Ss (5.148)

where S * and S are the climatological mean salinity and the observed salinity at a given time in the past, and Ss is the global mean sea surface salinity. Second, this equivalent evaporation and precipitation field can be used as the freshwater flux through the air-sea interface in the model.

The natural boundary condition discussed above is based on the rigid-lid approximation, which was used extensively in the past. However, in the new generation of numerical models, the free surface is the preferred choice. The suitable boundary condition for salinity balance is a simple mathematical statement thatp — e is a source of freshwater at the upper surface of the ocean, with no salinity flux through the air-sea interface. We note that the upper surface of the ocean is neither a Eulerian surface nor a Lagrangian surface, because freshwater moves across this surface, as discussed in Chapter 3.

As an example, we discuss the suitable upper boundary condition for salinity used in mass-conserving models. A convenient choice of the vertical coordinate for a mass-conserving numerical model is the pressure coordinate discussed in Section 2.8. Since both the surface and bottom pressure change with time, the pressure-^ coordinates can be used. The concept of the ^-coordinate system was introduced by Mesinger and Janjic (1985). In a pressure-^ coordinate system (Huang and Jin, 2007), the vertical coordinate is defined as wherepb = pb(x,y, t) is the bottom pressure, pt = pt(x, y, t) is the hydrostatic pressure at the upper surface of the water column, and pB = pB (x, y) is the time-invariant reference bottom pressure, which is calculated from the basin-averaged stratification prescribed in the initial state.

Sincept is the specified pressure at the upper boundary, owing to evaporation and precipitation the increment in hydrostatic pressure is S (p — pt) = —p/gSQE—P, where pf is the density of freshwater and QE—P is the freshwater flux across the air-sea interface associated with evaporation and precipitation. Thus, the upper boundary condition is where n = d n/dt is the virtual vertical velocity, which has a dimension different from the vertical velocity used in the traditional z-coordinates.

The bottom pressure is prognostically calculated from the bottom pressure tendency equation (Huang et al, 2001). We begin with the continuity equation in pressure-n coordinates

Salinity condition for models with free surface n = (p - Pt) /rp, rp = Pbt/PB, Pbt = Pb - pt

Integrating Eqn. (5.151) from n = 0 (sea surface) to n = pB (bottom) and applying the corresponding boundary condition at the sea surface, we obtain the bottom pressure tendency

where Vbaro is the vertically integrated horizontal velocity and Vh is the horizontal divergence operator. Thus, precipitation minus evaporation can directly affect the bottom pressure due to the adding of mass. The contribution of the air-sea freshwater flux regulates the salinity distribution through the dilution of seawater by the mean of mass continuity.

The corresponding salinity condition at the upper surface is that the net salt flux due to salt advection and vertical salt diffusion exactly cancel each other, i.e.,

Sf = Sf adv + Sf ,diffu = 0, at the surface (5.153)

Note that, in the pressure coordinate, the sea surface elevation is a diagnostic variable calculated by integrating the hydrostatic relation

In the traditional z-coordinate, the corresponding salinity condition at the upper surface is the same as Eqn. (5.153), i.e., there is no net salt flux across the air-sea interface. The effect of air-sea freshwater flux in the system is reflected in terms of a mass source in the free surface elevation. In the z-coordinate model, the vertical velocity at the sea surface Z satisfies dZ

where uh is the horizontal velocity, and pf and ps are the freshwater density and sea surface density. Vertical integration of this equation leads to a prognostic equation for the free surface dZ ( d CZ d CZ \

it \dx J H udz + dyj H vdz) + (p - e)Pf /Ps + RT (5.156)

where RT means rest terms associated with thermohaline processes in the water column. Thus, precipitation tends to increase the sea level. However, the local sea level is also closely related to the horizontal convergence/divergence of the vertically integrated velocity field (Huang and Jin, 2002b).

The pitfalls of the relaxation and virtual salt flux conditions

The relaxation condition implies a strong negative feedback on the surface salinity. As a result, the solutions obtained under this boundary condition match the observed surface salinity; in addition, the solutions are stable for most cases. These features can be advantageous for simulating the present climate. However, this boundary condition may not be suitable for simulating the oceanic circulation for general situations.

First, although the relaxation condition, Eqn. (5.138), for sea surface temperature is based on sound physical reasoning, the salt relaxation condition, Eqn. (5.137), lacks physical background. In addition, using such a large relaxation time seems unreasonable.

Second, the salinity relaxation condition is not suitable for climate study or forecasting because the reference salinity is unknown for climate conditions different from the present-day climate.

The virtual salt flux condition is also unphysical. First, to balance the salt in the oceans, a huge virtual salt flux through the air-sea interface and the atmosphere is required for taking up the salt from the subpolar basin, transporting it equatorward, and dumping it there. The virtual salt flux required in the North Atlantic Ocean can be estimated as

It is obvious that such a huge virtual salt flux is not real and should be avoided.

Note that the original definition of virtual salt flux, Eqn. (5.140), includes a weakly positive feedback because of the fact that S tends to be high wherever evaporation overpowers precipitation. However, the virtual salt flux is based on the basin-mean salinity Sx; thus, it has nothing to do with the local salinity. Even if the virtual salt flux can be accepted as a parameterization, the physics associated with the virtual salt flux is distorted.

In addition, this constraint can introduce large errors wherever the local salinity is much different from the basin or global mean. When the local salinity is very low, this formulation exaggerates the equivalent haline forcing and could give rise to negative salinity due to the large exaggerated virtual salt flux. For example, near the mouth of the Amazon River, sea surface salinity is very low. As a result, the model's salinity near the mouth of the river may become negative. For the world's oceans the virtual salt flux condition can also introduce non-negligible errors. Surface salinity in the North Pacific Ocean can be lower than 33, and the surface salinity in the North Atlantic Ocean can be higher than 37. Supposing that Ss is defined as 35 for a world ocean circulation model, a systematic bias of 10% will be introduced through the upper boundary condition for the salinity.

The most important differences between thermal and haline boundary conditions at the sea surface are the following. First, thermal forcing in the upper ocean is a flux of internal energy, while the surface haline forcing is a freshwater flux, which is a mass flux plus a small amount of gravitational potential energy. According to Eqn. (3.5.31), the amount of GPE due to precipitation is ff gpZ^dA, where Z is the free surface elevation and rn is the rate of precipitation.

Second, sea surface thermal forcing is associated with a strong negative feedback between the surface temperature and air-sea heat flux. As a result, the e-folding decay time scale for thermal anomalies tends to be short, i.e., a thermal anomaly cannot survive for a long time. On the other hand, there is no direct feedback between the local salinity and

The nonlinear nature of the surface buoyancy boundary conditions Differences in the surface thermohaline forcing conditions evaporation/precipitation. Thus, salinity anomalies tend to last much longer than thermal anomalies.

Survival of temperature and salinity anomalies in the surface layer

Two aspects of physics affect the survival characteristics of surface temperature and salinity anomalies. First, the equation of state is nonlinear. In particular, the thermal expansion coefficient is large at high temperatures, but is very small near the freezing point. As a result, density structure and thus the circulation at high latitudes are primarily controlled by salinity rather than temperature. On the other hand, density and circulation at low latitudes are primarily controlled by temperature rather than salinity, with exceptions near the mouths of rivers where the large salinity difference may play a dominating role.

Second, surface thermal anomaly is subject to a rather strong negative feedback; thus, thermal anomalies can be dissipated rather quickly owing to the strong air-sea heat flux feedback. On the other hand, salinity is not directly linked to the evaporation and precipitation rate across the air-sea interface.

The following two figures (Figs. 5.98 and 5.99) illustrate the difference between temperature and salinity anomalies at low and high latitudes in the North Atlantic Ocean. The heavy solid lines indicate temperature, salinity, and density profiles at two stations, near 29° N and 69° N. For the climatic mean state, a stable stratification is maintained in the following way. At low latitudes, stratification is characterized by warm and salty water lying over cold and fresh water. On the other hand, at high latitudes, relatively cold and fresh water lies over warm and salty water. Assume that there are surface temperature/salinity perturbations in the upper ocean, with linear profiles in the upper 75 m and the maximum

Fig. 5.98 a Temperature, b salinity, and c density profiles at a station in the North Atlantic (20.5° W, 29.5° N). Thin solid lines indicate perturbations due to temperature anomaly in the upper 75 m, thin dashed lines indicate perturbations due to salinity anomaly in the upper 75 m.
Fig. 5.99 a Temperature, b salinity, and c density profiles at a station in the North Atlantic (20.5° W, 69.5° N). Thin solid lines indicate perturbations due to temperature anomaly in the upper 75 m, thin dashed lines indicate perturbations due to salinity anomaly in the upper 75 m.

values of 3°C and 0.5 at the surface. It is clear that, at low latitudes, the density anomaly is primarily due to temperature (Fig. 5.98c); however, at high latitudes, it is primarily due to the salinity anomaly (Fig. 5.99c).

Note that a negative salinity anomaly (or freshwater anomaly) in the upper ocean can survive much longer. This is due to the fact that a freshwater anomaly in the upper ocean lowers the surface density, thus forming a strong halocline which is rather stable to perturbations. The lack of deep convection can lead to a further decline of heat loss to the atmosphere and a smaller evaporation rate. These physical processes work together to promote the longevity of surface freshwater anomalies in the high-latitude oceans. These stable freshwater layers on the top of the ocean can cause the halocline catastrophe, which will be discussed later.

Freshwater transport in a Boussinesq model Most numerical models currently used in simulating ocean circulation and climate are based on the Boussinesq approximations. These models use the volume conservation to replace the physically more accurate mass conservation. As a result, virtual salt fluxes across the air-sea interface and meridional sections appear in these models. Such salt fluxes are artifacts of these models; so the meaning of such salt fluxes is unclear when the model is in a state of transition.

However, when the model reaches a quasi-steady state, the virtual salt fluxes diagnosed from the model may be interpreted as follows. When the model in a closed basin reaches a quasi-steady state, the meridional transport of salt across a latitude circle should vanish

where S is the basin mean salinity; thus, the freshwater transport through this section is jI p vdxdz = J J pv(1 - S /S)dxdz (5.159)

Since seawater density is nearly constant, as a good approximation the freshwater volumetric transport in these models can be defined as

Because the meridional volume transport for a Boussinesq model in a closed basin vanishes in a quasi-steady state, this equation can be rewritten as

i.e., the equivalent freshwater flux through a meridional section is equal to the meridional salt flux diagnosed from the model, multiplied by a negative sign and divided by the basin mean salinity (Bryan, 1969; Huang, 1993b).

It is worth noting that the definition of equivalent freshwater flux includes the basin mean salinity. Thus, the equivalent freshwater fluxes defined for different regions may not be comparable, because they are based on different mean salinity. This inconsistency is, unfortunately, intrinsic to the definition itself. It is worth emphasizing that the formula in Eqn. (5.161) is applicable for a closed basin only. At the latitudes of ACC, it cannot be used to infer the equivalent meridional freshwater transport for individual sectors, such as the Atlantic or Pacific sectors, because the meridional volumetric transport for individual sectors is non-zero.

5.3.3 Haline circulation induced by evaporation and precipitation

Historically, evaporation/precipitation was the first mechanism explored as the driving force for the oceanic general circulation. However, the physics involved in the haline circulation in the ocean has been overlooked for many decades. In fact, in most previous oceanic circulation models, the haline circulation was driven by either the salt relaxation condition or the virtual salt flux condition; thus, the real physics of haline circulation remained obscure for a long time. Before 1990, there were only a very few papers published in which the haline component of the oceanic circulation was forced by evaporation and precipitation.

In Chapter 3 we discussed in great detail why surface thermal forcing alone cannot drive or maintain a meridional overturning circulation. The situation in relation to surface freshwater flux is quite similar. The major difference between surface heat flux and freshwater flux is as follows. Surface freshwater flux is always associated with a mass transport across the air-sea interface. In general, freshwater is taken up from the surface at low latitudes in the form of moisture. Through the meridional circulation in the atmosphere, the moisture is transported to high latitudes; where it is put back into the ocean. If there were no other forcing, such as wind stress, heat flux, and tidal dissipation, the air-sea freshwater flux alone could drive a barotropic circulation in the ocean, which is called the Goldsbrough-Stommel circulation.

As will be discussed below, the barotropic circulation in the world's oceans driven by evaporation and precipitation is quite slow, with the total transport being on the order of 1 Sv. In the ocean, however, the haline component of the thermohaline circulation is one order of magnitude stronger. Surface freshwater flux alone cannot provide the mechanical energy required for sustaining such a strong circulation against friction. Thus, evaporation and precipitation alone cannot drive the baroclinic component of the haline circulation energetically. However, in the following discussion we will keep the use of the word "drive" for the presentation of the classical results, and the question of what really drives the thermal and haline circulation in the ocean will be discussed in detail in Section 5.4.

Classical circulations driven by evaporation and precipitation Hough's solution

Evaporation and precipitation can drive oceanic circulation, and this was first explored by Hough (1897) in his tidal paper. Hough originally assumed the P - E pattern to be a second Legendre polynomial:

i.e., precipitation over high latitudes and evaporation over low latitudes.

Hough did not include any figures in his paper; his solution became much clearer after it was illustrated graphically by Stommel (1957). For simplicity, Stommel assumed a simpler pattern of precipitation over the Northern Hemisphere and evaporation over the Southern Hemisphere (Fig. 5.100).

Hough did not know how to parameterize friction, so there was no friction in his model. In fact, his solution may be valid only for the initial stage of the problem, when the solution can be treated as infinitesimal, so that the nonlinear effect can be neglected. In the beginning, the ocean covers the solid Earth with a uniform depth of water. Water from precipitation enters the sea surface in the Northern Hemisphere and leaves the sea surface in the Southern Hemisphere. Owing to the small difference in the meridional sea level, there is a meridional velocity pointed to the south (Fig. 5.100a).

As time progresses, more water is piled up in the Northern Hemisphere and less water is left in the Southern Hemisphere; this leads to an asymmetric shape of the water sphere. Owing to the Coriolis force, a zonal velocity gradually builds up, which is westward in the Northern Hemisphere and eastward in the Southern Hemisphere (Fig. 5.100b).

Since there is no friction in the model, it is not suitable for describing the long-term evolution of the solution. As time goes on, the free surface elevation and the zonal velocity would be unbounded, so the solution is not valid. An accurate solution can be found through more rigorous theory and modeling.

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