where hm,max is the annual maximal depth of the mixed layer:

where hm is the annual mean mixed layer depth. Note that both of these definitions treat the subduction in the local sense, so these two subduction rates do not include the average over the trajectory downstream. As a result, the rates calculated from these two equations are smaller than the rates calculated from the above-defined SE and SL, both of which include the contribution resulting from the spatial variance of the Ekman pumping velocity and mixed layer depth (Fig. 5.29). Thus, one should not use a simple annual mean for calculating the so-called annual mean circulation; there are always some nonlinear and non-local effects, which must be investigated carefully.

In this example, detainment rate is controlled by the time-dependent term of the mixed layer depth, - dhm/d t; however, due to the contribution of the vertical pumping, detainment starts before -dhm/dt changes its sign from negative to positive. Although the detainment rate is positive for about half of the cycle, only the first quarter of this positive detainment rate, indicated by the shading in Figure 5.29, contributes to the effective entrainment, so this is the part that really contributes to the annual mean subduction rate.

Potential vorticity in the ventilated thermocline A key parameter for mode water formation is the potential vorticity of the newly formed water mass. For simplicity, the setting of potential vorticity of mode water through subduction can be illustrated in a two-dimensional sketch for an idealized case of steady circulation (see Fig. 5.30).

Using density conservation, we have

PoAz po wtr + Ur • Vhm where the overbars indicate the mean over the 1-year trajectory, itml indicates the horizontal velocity in the mixed layer, and the subscript tr indicates the trajectory. The reason for emphasizing the mean over a 1-year trajectory stems from the definition of annual mean subduction rate. For example, the horizontal velocity near the Gulf Stream can be on the order of 0.1 m/s; thus, over 1 year, the trajectory can cover a distance of 2,000-3,000 km. The along-trajectory mean over such a long distance can be considerably different from the local term. Equation 5.40 states that potential vorticity in the ventilated thermocline is linearly proportional to the meridional density gradient of the mixed layer density, and inversely proportional to the sum of the vertical velocity at the base of the mixed layer plus

the horizontal increment of the mixed layer depth. Therefore, low potential vorticity water is formed when there is:

• low meridional gradient of mixed layer density

• strong Ekman pumping (implying a strong vertical velocity)

• a large horizontal gradient of late-winter mixed layer depth and large horizontal velocity.

Upwelling/entrainment prevails in subpolar basins. When a water parcel moves from the permanent pycnocline to the mixed layer in the upper ocean, it loses its original identity, such as temperature and salinity. Thus, water mass is eliminated through erosion.

Similarly to what happens during the ineffective detrainment period, water entrained into the mixed layer may not actually come from the permanent pycnocline; instead, it may come from the seasonal pycnocline whose water was detrained from the mixed layer previously (Woods, 1985; Cushman-Roisin, 1987).

In order to clarify the physical processes involved in entrainment, we use the term obduction. Obduction has been used in geology for describing the process of upward thrusting of a crustal plate over the margin of an adjacent plate. Here obduction is borrowed to describe the process in which water from the permanent pycnocline upwells into the mixed layer and flows over the adjacent layers of water. Although obduction is basically a continuous process between the permanent pycnocline and the seasonal pycnocline, effective entrainment from the seasonal pycnocline to the mixed layer occurs only during part of the entrain-ment period (Fig. 5.31). During the effective entrainment phase, water from the permanent pycnocline, which has not been exposed to surface processes, is entrained into the mixed layer through the seasonal pycnocline. During the rest of the entrainment period (i.e., the ineffective entrainment period), water that has been exposed to air-sea interactions within the past year enters the mixed layer from the seasonal pycnocline, as indicated by the top five lines in Figure 5.31.

Using the special term "obduction" helps us to clarify the irreversible mass flux from the permanent pycnocline to the mixed layer. For example, although mixed-layer entrainment takes place in subtropical basins during a seasonal cycle, in most places water entrained into the mixed layer actually comes from the seasonal pycnocline, so there is no obduction. In fact, obduction takes place only within the subpolar basins and the subtropical-subpolar boundary regions, as will be shown in the following discussion.

The obduction rate can be defined in a way similar to the subduction rate. Though obduc-tion is an antonym of subduction, obduction cannot be calculated as subduction with the opposite sign. There are two major differences between the two terms.

First, the physical processes involved in subduction and obduction are different. Subduction takes place in the subtropical basins, where water geostrophically flows down into the permanent pycnocline. As a result, water subducted to the permanent pycnocline carries late-winter mixed layer properties. In comparison, obduction takes place in the subpolar basin, where water from the permanent pycnocline below flows geostrophically upward into

Ineffective Ineffective Effective detrainment entrainment entrainment

Ineffective Ineffective Effective detrainment entrainment entrainment

the seasonal pycnocline and eventually enters the mixed layer. As water enters the mixed layer, it quickly loses its identity as a result of strong mixing, and it is impossible to keep track of the trajectory of an individual water parcel afterwards.

The difference in the physics is reflected in the mathematical formulation of the suitable boundary value problems for the pycnocline structure in the subtropical and subpolar basins. The pycnocline equation is a nonlinear hyperbolic equation (Huang, 1988a, 1988b). In the subtropical basin, density is specified as an upper boundary condition because the upper surface is the upstream boundary. In the subpolar basin, the upper surface density cannot be specified. In fact, the mixed layer density is determined by the dynamics of the permanent pycnocline and is part of the solution.

Second, the times when effective detrainment and effective entrainment take place are different. It is well known that effective detrainment takes place after late winter, when the mixed layer reaches its annual maximum depth and density and starts to retreat. Effective entrainment takes place between late fall and early winter when the mixed layer deepens quickly, but before it reaches its annual maximum depth and density.

In calculating the obduction rate, it is important to trace back to the origin of the entrained water. To make the presentation clearer, assume t = 0 at late winter, say March 1. We begin with a simple case where the mixed layer depth has a simple sinusoidal cycle, and its amplitude is spatially uniform. The upwelling rate is 18 m/yr, and it is also uniform along the trajectory. Because the mixed layer is relatively shallow, we assume that the vertical velocity is approximately the same as the Ekman sucking rate everywhere along the trajectory. Below the base of the mixed layer, mixing is negligible, so the identity of water parcels is preserved. As a result, the particle trajectory can be used to trace the origin of the water before it is entrained into the mixed layer. As soon as water parcels enter the mixed layer, they lose their identity because of the strong vertical mixing within the mixed layer.

During spring and early summer the mixed layer retreats and leaves the stratified water behind, so this is the period of detrainment, although it is only a temporary detrainment in the present case. Beginning in early fall, the mixed layer deepens and entrainment takes place. The water entering the mixed layer during the first period does not really come from the permanent pycnocline. In fact, these water parcels were detrained into the seasonal pycnocline at an earlier time and at some upstream locations, as indicated by the top five lines in Figure 5.31. Such water was "contaminated" in the mixed layer during the past year, so this is not genuine effective entrainment. It is only within the second phase of entrainment that water from the permanent pycnocline enters the mixed layer, as indicated by the lower trajectories.

For the situation shown in Figure 5.31 a simple calculation shows that for the case with a uniform upwelling of 18 m/yr, the effective entrainment starts at TS = 0.8762 and ends at Te = 1.0088. In an ordinary year, the effective entrainment starts at January 18 and ends at March 4, yielding a duration of about 44 days.

The contrasts between subduction and obduction are shown in Table 5.3.

Obduction rate defined as an integral quantity The annual mean obduction rate can be defined slightly differently, depending on the coordinates used. First, the obduction rate can be defined with the Lagrangian coordinates (Woods, 1985). Accordingly, the annual mean obduction rate is defined as where T is the time duration over which the average is taken, which is assumed to be 1 year because of the seasonal cycle; the subscript tr indicates the critical trajectory in Figure 5.31 (which marks the end of obduction); and Ahm,L indicates the mixed layer depth change accumulated over a 1-year Lagrangian trajectory. Note that both terms include the temporal average over the past year and the spatial average over the 1-year trajectory. By definition, an obduction rate should be non-negative, and a negative value calculated from this definition should be interpreted as a zero obduction rate.

Second, an instant entrainment rate in Eulerian coordinates can be defined as where the subscript mb indicates the base of the mixed layer. The instantaneous entrainment rate fluctuates greatly during one seasonal cycle. In addition, some of the entrained water during the phase of ineffective entrainment does not contribute to obduction. Thus, the

Table 5.3. Ventilation: subduction vs. obduction

Mass flux

Water mass Time

Atmospheric forcing Mixed layer Trajectory tracking

Subduction

From the mixed layer to the permanent pycnocline

Formation

Spring

Heating

Shoaling

Downstream

Obduction

From the permanent pycnocline to the mixed layer

Erosion

Winter

Cooling

Upstream entrainment cannot be used as an index for the water mass conversion rate. Similar to the definition of the annual-mean Eulerian subduction rate, it is more meaningful to use the annual mean obduction rate defined as

1 jtS

where TS and TE are the times when the effective entrainment started and ended, and E is the instantaneous entrainment rate defined in Eqn. (5.42).

Some incorrect definitions for water mass erosion rate have been used in previous studies. In some cases, people have simply used "upwelling" as the term to describe the process opposite to subduction. As discussed above, upwelling is only part of the obduction term, and the other part is associated with lateral induction, which can be the dominant term in the subpolar basin.

Another potential pitfall in calculating the annual mean obduction rate arises from using a simple Eulerian mean at a given station over the whole year

1 f0

where substituting Eqn. (5.42) would lead to a wrong estimation of O = Wmb + umb ■ Vhm. In general, this substitution tends to underestimate the annual mean obduction rate.

The major differences between these two definitions of obduction rate are as follows. In Eulerian coordinates, water parcels entrained into the mixed layer at one station are monitored, and trajectories of parcels are traced back upstream for 1 year to determine whether the water comes from the permanent pycnocline or not. Accordingly, the beginning and the end of the effective entrainment are determined, and the annual obduction rate can be calculated by using Eqn. (5.43). In Lagrangian coordinates, water parcels entrained into the mixed layer at a station are monitored to determine the critical trajectory that marks the end of obduction. Given this trajectory, one can trace back along the trajectory for 1 year and calculate the obduction rate by using Eqn. (5.41).

Calculating the annual mean obduction rate according to these two definitions requires accurate information on the spatial and temporal evolution of the mixed layer, and such detailed information is almost impossible to obtain from any climatic data. Thus, a simple definition of the annual mean Lagrangian obduction rate can be used where d and h are the depth of the trajectory and the mixed layer base, and subscripts 0 and -1 indicate the fact that these quantities are calculated at the beginning of the year and the previous location by upstream-tracing of the critical trajectory for 1 year.

Although obduction rates calculated according to these definitions are slightly different, their difference is quite small compared with the errors existing in present-day climatology data. Therefore, the definition in Eqn. (5.45) can serve as a convenient and practical tool for calculating the obduction rate from climatological data with errors comparable to those of other processes.

Water mass formation and erosion take place in the oceans through the subduction and obduction processes discussed above. Along with the mixed layer and the Stommel demon, these processes can be schematically viewed in Figure 5.32. The ventilation/obduction rate has two components, the lateral induction and vertical pumping terms: S = Su + Svp, and

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