where T is the time period over which the average is taken, assumed to be 1 year because of the seasonal cycle; wtr is the vertical velocity along the 1-year trajectory; and Ahm,L represents the mixed layer depth change accumulated over a 1-year trajectory in Lagrangian coordinates. Thus, this definition includes both the temporal average and the spatial average over a 1-year trajectory. The schematic diagram in Figure 5.27 illustrates this definition for a two-dimensional case. An instrument called a Bobber is released in late winter, when effective detrainment starts at a station. This instrument can be checked continuously by means of acoustic signals. If we were following the instrument, we would see that during the first part of the trajectory effective detrainment takes place, i.e., the mixed layer retreats and leaves stratified water behind. During the second half of the trajectory, mixed layer entrainment takes place and re-takes part of the water which entered the seasonal pycnocline (at an earlier time). Thus, the seasonal cycle can be divided into three phases; (1) the effective detrainment phase, during which water that left the mixed layer flows geostrophically into the seasonal pycnocline and enters the permanent pycnocline irreversibly; (2) the ineffective detrainment phase, during which water that entered the seasonal pycnocline will be re-taken later (at a downstream location); and (3) the entrain-ment phase. Calculation of the subduction rate requires accurate information about the kinematic structure of the mixed layer and the velocity field in the pycnocline. Because such detailed information is very difficult to obtain from oceanic climatology, a simplified formula is

where dtr,0 and dtr,i are the depth of the trajectory at the beginning and end of 1 year, hm,0 and hm,1 are the depth of the mixed layer at the beginning and end of 1 year, and T = 1 year is the duration of the motion. In Figure 5.27 we have assumed a simple case for the northern part of the subtropical basin, where the winter mixed layer depth increases northward. By definition, the subduction rate should be non-negative; therefore, a negative value calculated from this definition should be interpreted as a zero subduction rate.

The annual mean subduction rate can also be defined in Eulerian coordinates. In this case, we stand at a fixed station. In order to calculate the annual mean subduction rate, we monitor the local mixed layer detrainment and entrainment. In addition, we have to follow the trajectories of particles released from this station to see whether these particles will eventually enter the permanent pycnocline (overtaken by the mixed layer entrainment) downstream.

Fig. 5.28 Finding the critical trajectory that defines the end of the effective detrainment period by tracing trajectories released at a fixed station and regular time intervals: a time since ejection of a trajectory; b time since ejection of the first trajectory.

For example, let us assume that the seasonal cycle of mixed layer depth is a simple sinusoidal function of time which is independent of the geographic location. Effective detrainment takes place very slightly before the mixed layer depth reaches the annual maximum, and we will choose this as the zero point of time axis in Figure 5.28. At this station, at regular time intervals, one water parcel is released at the base of the mixed layer and its trajectory is monitored over 1 year. The trajectory of a water parcel released at the beginning of the effective detrainment period is called the first trajectory. By our choice of time axis, the effective detrainment first starts at time to = 0, and continues to to = 0.18 because this is the last trajectory of the water parcel that barely escapes the overtaking of a mixed layer downstream (Fig. 5.28a). Although the trajectories shown in Figure 5.28a started from the same position, they began at different times, as shown in Figure 5.28b. Since the mixed layer depth is assumed to be a function of time only, the local mixed layer depth appears to be the same for all trajectories. If the mixed layer depth is also a function of geographic location, the corresponding mixed layer depth for individual trajectories should be different as well.

Similar to the case in Lagrangian coordinates shown in Figure 5.27, the seasonal cycle in the mixed layer at a given station can be divided into three phases; effective detrainment, ineffective detrainment, and entrainment. The annual mean subduction rate is defined as


where TS and TE indicate the starting and ending times of the effective detrainment.

Subduction rates calculated from these definitions for an idealized model are shown in Figure 5.29. The model is set for the northern part of the subtropical basin, where the mixed layer depth increases northward and the Ekman pumping velocity increases southward. In addition, the seasonal cycle is assumed to be a simple sinusoidal function of time. For comparison, two additional terms are introduced below

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