N 30n 45n 60n

Fig. 5.199 a Poleward heat flux in the 3-D ocean model (in 1015 W); b meridionally accumulated energy required for sustaining subsurface diapycnal mixing in the 3-D ocean model (in 109 W) (Huang et al., 2007).

a the ocean interior, which may enhance the MOC and poleward heat flux. On the other hand, more mechanical energy input can increase the mixed layer depth. This process also plays an important role in regulating the MOC and poleward heat flux. For example, tropical cyclones can input a large amount of mechanical energy into the ocean, which not only enhances diapycnal mixing in the ocean interior but also deepens the mixed layer greatly (Price, 1981), so tropical cyclones play an essential role in driving the MOC and poleward heat flux (Emanuel, 2001). However, there is no simple linear relationship between the amount of mechanical energy input and the strength of the MOC and the related poleward heat flux. The complicated nature of such connections remains a critically important issue for further study.

Another interesting result is that the energy required for sustaining subsurface diapyc-nal mixing is actually reduced for the cases with a deeper mixed layer at lower latitudes (Fig. 5.199b). This is due to the fact that a deeper mixed layer implies a smaller density difference between the bottom water and the base of the mixed layer. Since the amount of energy required for sustaining subsurface diapycnal mixing is proportional to this density difference, it is reduced with this smaller density difference.

where p is the in situ temperature, cp is the specific heat under constant pressure, v is the meridional velocity, and © is the potential temperature. It is well known that if the system has a net mass flux through the section, the heat flux calculated from this formula may depend on the choice of temperature scale. In general, heat flux itself has no direct physical meaning, and the most essential quantity associated with the heat flux is its divergence. There are two criteria in defining heat flux: first, the divergence of heat flux should satisfy the local heat balance equation; second, the heat flux should depend on the local properties only.

The best-known example is the circulation in the South Pacific and Indian Oceans. Due to the Indonesian Throughflow, the poleward heat flux in the South Pacific and Indian Oceans is not uniquely defined. To overcome the complication due to the net mass flux through the sections, many different approaches have been used. For example, Zhang and Marotzke (1999) proposed a procedure in which the heat flux associated with the westward flow in the strait is used to evaluate the poleward heat flux. Although the heat flux definition they proposed satisfies the first criterion, it does not satisfy the second criterion. In fact, their definition includes a term that depends on the thermal condition in the cross-section of the Throughflow. If the condition within the strait changes, the heat flux evaluated will also change, even if the circulation and water properties at the given section south of the strait do not change.

To overcome such a problem, a much simpler definition is recommended:

where © and ©0 are potential temperature and the reference potential temperature, which can be chosen arbitrarily; the most obvious choice is the global-mean potential temperature, about 2° C. This definition satisfies the two criteria listed above. We analyze the meaning of the heat flux defined in this way as follows.

Appendix: Definition of the oceanic sensible heat flux

Traditionally, the poleward oceanic heat flux is defined as

In the oceans, there is indeed a net mass flux through any given section due to either evaporation/precipitation or throughflow:

where ~pv = ff pva cos 0dXdz/ ff a cos 0dXdz is the mass flux averaged over the section. The first component, Ho,0, is the oceanic sensible heat flux through this section, carried by the circulation without the net mass flux. By definition, it is independent of the choice of reference temperature. On the other hand, the second component, Ho,i, clearly depends on the choice of reference temperature, so it should be isolated from the first component. In the oceans, the net mass flux through a given section is due either to evaporation/precipitation or to an inter-basin loop current, such as the Indonesian Throughflow. We will separate the net mass flux through each section into two parts:

where pv¡oop is the zonal-mean meridional mass flux associated with the loop current going through the whole basin, and the total amount of mass flux associated with this component should be constant at any zonal section, such as the Indonesian Throughflow; and pvemp is the zonal-mean meridional mass flux associated with evaporation and precipitation. This component of the net mass flux through the section is associated with the water vapor cycle in the atmosphere, so it should not be counted as part of the oceanic sensible heat flux.

Accordingly, poleward sensible heat flux in the Atlantic Ocean can be defined as

HAr„= I I p Cnv fI pCpv (© — ©0) a cos 0dXdz — Hiemp (5.A7)

Hiemp = Cp ff pvemp(0) ©a(0) — ©0]a cos 0dXdz = CpMempA(0) ©a(0) — ©0]

where ©A (0) is the mean potential temperature at a given zonal section in the Atlantic Ocean, and Memp,A is the zonally integrated meridional mass flux due to evaporation/precipitation. Similarly, the poleward heat flux in the Pacific and Indian Oceans are defined as

Hpsen = ff pCpv (© — ©0) a cos 0dXdz — Hpemp, Hpemp = CpMempp (0) © (0) — ©0] (5.A9)

H0,sen = jj P°PV © - ©°) a C0S °dXdZ - Ho,emp>

The sum of water flux due to evaporation and precipitation, including the river run-off, in all oceans should equal the water vapor flux in the atmosphere. Thus, the sum of the evaporation and precipitation in three basins is

The heat flux associated with this mass flux is

Ho ,emp cp [Memp,A®A(0) + MempP®P (0) + Memp,I®I (0)] - CpMemp©0 (5.A12)

This heat flux is independent of the choice of temperature scale.

This example demonstrates that a working definition of heat flux should be independent of the specific choice of temperature scale. In order to achieve this goal, one has to separate the net mass flux through a given section, and link this net mass flux with the corresponding return flow in the climate system, i.e., the water vapor flux in the atmosphere,

HZ-*™ = -Memp {Lh + Cp [©atmos(0) - ©°]} + Ho,emp (5.A13)

where the first term indicates the heat content flux in the atmospheric branch of the water vapor loop, and the second term is the heat content flux in the oceanic branch of the water cycle. This definition is, of course, independent of the choice of temperature scale.

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