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Fig. 4.102 Layer thickness perturbation maps, generated by cooling within a small area with dyi = -0.01° and Ax = 4°. Layer thickness anomalies dh (in 10-4m) are presented as follows: for positive anomaly, x = max [log(dh), 0], for negative anomaly, x = min [— log(-dh), 0] (Huang and Pedlosky, 2000b).

|HI|MI|III|III|III|III|III|III|III|III|I

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Fig. 4.102 Layer thickness perturbation maps, generated by cooling within a small area with dyi = -0.01° and Ax = 4°. Layer thickness anomalies dh (in 10-4m) are presented as follows: for positive anomaly, x = max [log(dh), 0], for negative anomaly, x = min [— log(-dh), 0] (Huang and Pedlosky, 2000b).

As the number of moving layers increases, the vertical structure of the perturbations becomes increasingly complicated. As shown in Eqn. (4.527), variability induced by buoyancy forcing must appear in the form of internal modes, the DTM. In region II, it appears in a second baroclinic DTM, M^, where the subscript 2 indicates the number of moving layers and the superscript 1 indicates the layer where the driving potential vorticity source is located.

In region III, the vertical structure of the solution in the two branches appears in a different form. Along the P-branch (the primary potential vorticity anomaly), the perturbation is in the form of M31 mode. This mode is induced by a large positive dh\. Along the S-branch a dZ (cm) b dh (cm)

Fig. 4.i03 Perturbations of interface depth (a) and layer thickness (b) along 36.5° N, generated by cooling within a small area with dyi = -0.0i° and Ax = 2° (Huang and Pedlosky, 2000b).

(the secondary potential vorticity anomaly), the perturbation is in the form of M32 mode, which is induced by a negative dh2 (Fig. 4.i03). Note that |dh2| < |dhi| because the former is a secondary perturbation.

Note that within the S-branch there are small potential vorticity (or layer thickness) perturbations in layer i. Such perturbations are not directly related to the surface forcing on layer i at the outcrop line; instead, they are induced by the secondary potential vor-ticity anomaly in layer 2 in the following way. A potential vorticity anomaly in layer 2 induces a slight shift of streamlines in layer i, and thus induces a change in the potential vorticity there. Similarly, there are small potential vorticity perturbations in layer 2 within branch P.

The vertical structures of M3i and M32 are quite different, because they represent the thermocline's response to potential vorticity perturbations imposed on the first layer and the second layer, respectively. In addition, we note that the perturbation of the depth of the second interface has a very small negative value, due to the almost perfect compensation of the thickness perturbation of layers 2 and 3.

If there were many outcrop lines, all the perturbations would be confined within the so-called characteristic cone. The western edge of the characteristic cone is defined by the streamline on the isopycnal surface of the original cooling source. The eastern edge of the characteristic cone is more complicated. Each time that a new outcrop line is crossed, a new characteristic is created. Owing to the j-spiral, the velocity vector in the upper layer is always to the right of the velocity vector in the subsurface layer. Therefore, the eastern edge of the perturbations is represented by the streamline in the uppermost layer. In the limit, therefore, the eastern edge of the characteristic cone is defined by the envelope of the

streamlines on the uppermost layer. Thus, in a continuous model, the eastern edge of the characteristic cone should be defined as the streamline on the sea surface, stemming from the point source of cooling (Fig. 4.1o4).

4.9.2 Continuously stratified model

The model is set up such that the background stratification is provided by a one-dimensional advection-diffusion balance wpz = Kpzz, where w = 1o-7 m/s is the constant upwelling velocity, and k = 1o-6 - 6o x 1o 6 m2/s is the vertical diffusivity. Using density boundary conditions of p = 1,o23 kg/m3 at z = o and p = 1,o28 kg/m3 at the seafloor z =-5 km, the density profile can be calculated. The sea surface density is a linear function of latitude a = 25 + 2(0 - 0s)/(0n - 0s) kg/m3. Assuming that density is vertically homogenized within the mixed layer, this horizontal density distribution also gives the depth of the mixed layer at each location. The model is under a simple sinusoidal Ekman pumping force we = -1o-6 sin [n (0 - 0s) / (0„ - 0s)] m/s.

Dependence on the diffusivity k

Examples in Figure 4.1o5 show the structure of the thermocline as the vertical mixing rate changes. It is clear that the ideal-fluid thermocline model with k & 1o-5 - 3 x 1o-5 m2/s produces a thermocline structure resembling the main thermocline in the ocean. Most importantly, these examples demonstrate that the ideal-fluid thermocline model can simulate the case with a step-function-like sharp density front associated with the main thermocline, although such a limit case is not realizable in the oceanic circulation under the current climate conditions.

k = 0.6 cm2/s b k = 0.3 cm2/s k = 0.6 cm2/s b k = 0.3 cm2/s a

Thermocline Simple Diagram
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