## 1. Introduction

Climate variability on decadal timescales can be examined by perturbing the steady solution of the thermocline in a multilayer model. Huang and Pedlosky (1999) studied a simple two-layer model for the ventilated thermocline and revealed the horizontal and vertical structure of the perturbations. In response to a surface cooling anomaly, the perturbations are horizontally confined to a characteristic cone defined by the streamlines stemming from the western and eastern edges of the cooling source.

The vertical structure of the response to the cooling anomaly depends on the structure of the thermocline circulation in which it is embedded. Therefore, we term such structures “dynamical thermocline modes” and they are distinct from, though reminiscent of, the standard geostrophic vertical normal modes of a resting ocean.

An attempt was made by Huang and Pedlosky (1999) to explore the structure of the perturbations for a model with three moving layers. The addition of one more moving layer makes the dynamics much more complicated because a new characteristic cone is created when the primary characteristic cone crosses a new outcrop line. A careful examination reveals that the structure of the perturbations due to a surface forcing anomaly in a multilayer model can be extremely complicated when the number of moving layers increases. In fact, we are chagrined to note that some part of the discussion about the case of three-moving layers presented by Huang and Pedlosky (1999) is in error.

In particular, we will show that the anomalies produced by a localized cooling excite a fan of characteristics that broadens as the perturbations move southward. The disturbance produces a chain reaction of potential vorticity anomalies that because of the beta spiral, constantly expand the zone of influence of a single disturbance. In this paper we will examine the climate variability for a model with three and four moving layers. It is this cascade, or chain reaction, that we overlooked in our previous paper.

## 2. Model formulation

In this paper, we will use a notation slightly different from that used in the classic ventilated thermocline model of Luyten et al. (1983). We denote the bottom and motionless layer as layer 0 and the lowest moving layer as layer 1, and the layers above have number increases upward, Fig. 1. The thickness and the depth of the *i*th layer are denoted as *h*_{i} and *H*_{i}, respectively; thus, for the top layer *n, H*_{n} = *h*_{n}. The northernmost outcrop line is labeled as *f*_{1}, and the outcrop lines southward have number indices that increase accordingly. This notation is more convenient when one has to deal with an arbitrary number of layers.

### a. Pressure gradient in multilayer model

*m*th layer (

*m*⩽

*n*) is

### b. The Sverdrup relation

*x, x*

_{e}], one obtains the Sverdrup relation

*dynamical thermocline modes*(DTM hereafter) since the modal structure will be a function of the background thermocline structure that varies laterally. On the other hand, anomalous wind stress forcing can lead to the first baroclinic DTM.

### c. Region II

*f*

_{2}<

*f*<

*f*

_{1}, denoted as region II in Fig. 2. South of the first outcrop line

*y*

_{1}, potential vorticity in layer 1 is conserved along the streamline,

*h*

_{1}=

*H*

_{1}= const:

*f*=

*f*

_{1},

*h*

_{1}=

*H*

_{1}; thus, the functional form of

*Q*

_{1}is

*Q*

_{1}(

*x*) =

*f*

_{1}/

*x,*and the first layer thickness is

*F,*which is defined as the layer thickness divided by the total depth of the ventilated layers; thus, for region II we have

It is important to note that *f*_{1} in these relations is not necessarily a constant. In fact, we can write it in the form of *f*_{1}(*H*_{1}), which indicates that *f*_{1} is dependent on the latitude of the outcrop line by tracing backward along the streamline *H*_{1} = const. Therefore, in our discussion hereafter, we can write alternatively *f*_{1} ⇒ *f*_{1} + *δf*_{1}, where we imagine that *δf*_{1} represents the outcrop line perturbations induced by heating or cooling anomaly. If *δf*_{1} < 0, this implies that outcrop line moves equatorward from its constant value of *f*_{1} representing cooling.

*H*

^{2}

_{1}

*x*

*H*

^{2}

_{10}

*x*

*O*

*δf*

_{1}

*H*

_{10}(

*x*) is the total layer thickness along the unperturbed outcrop line. Thus, to the lowest-order approximation, the functional relation between

*H*

_{1}and

*x*remains unchanged. Using this relation we can calculate

*x*and

*δf*

_{1}(

*x*) for given

*H*

_{1}in order to complete the necessary inversion to determine

*f*

_{1}(

*H*

_{1}) to lowest order. Accordingly, the solution in region II can be calculated either by solving the nonlinear equations (8) and (9) or using their linearized versions.

### d. Climate variability induced by surface cooling

### e. Region III

South of *f*_{2} there are two subsurface layers in which the streamlines are *H*_{1} = const and *H*_{1} + *γ*_{21}*H*_{2} = const. Along these streamlines potential vorticity is conserved. Note that, when the potential vorticity anomaly carried by the streamlines in layer 1 (labeled as *δq*_{1} in Fig. 2) crosses the second outcrop line *f*_{2}, a new characteristic is created, and south of *f*_{2} the potential vorticity anomaly in layer 2 propagates along streamlines in layer 2, labeled as *δq*_{2} in Fig. 2. Thus the zone influenced by the cooling broadens in a fanlike manner as the fluid moves southward.

*f*

_{1}obtained by tracing back along

*ψ*

_{1}. We introduce the notation

*f*

_{n,m}

*f*

_{n}

*ψ*

_{m}

*n*th outcrop line influencing a field point by tracing upstream from the point to the outcrop line starting along a streamline of layer

*m.*For an interior point like A in Fig. 2 the potential vorticity in layer 1 is set up by tracing backward to the first outcrop line

*f*

_{1}along

*ψ*

_{1}(the streamline in layer one) in both regions III and II. Thus, the potential vorticity functional form and the thickness of layer 1 are

*ψ*

_{2}in region III. At point B, the lowest layer thickness is

*f*

_{1,2}indicates that the corresponding position on the first outcrop line

*f*

_{1}is now determined by tracing

*ψ*

_{2}backward in region III, and then along

*ψ*

_{1}in region II. At the second outcrop line

*f*

_{2}, thus, we have

*x*

*H*

_{1}

*γ*

_{21}

*H*

_{2}

*f*

_{2}, we have

*f*

_{2},

*Q*

_{2}leads to the fractional layer thickness in region III:

*G*

_{3}= 1 +

*γ*

_{21}(1 −

*F*

^{III}

_{1})

^{2}+

*γ*

_{31}(1 −

*F*

^{III}

_{1}−

*F*

^{III}

_{2})

^{2}.

*f*

_{1,1}and

*f*

_{1,2}are complicated nonlinear functions of the local total layer depth

*H*

_{1}, and these functions are defined by tracing back along streamlines

*ψ*

_{1}and

*ψ*

_{2}, as shown in the upper left part of Fig. 2. For example,

*f*

_{1,2}

*f*

_{1,2}

*H*

_{1}

*γ*

_{21}

*H*

_{2}

*f*

_{1}

*H*

_{1,0}

*H*

_{1}, and this function can be obtained by tracing backward along

*ψ*

_{2}in region III to point B and then following

*ψ*

_{1}in region II, as shown in the upper left part of Fig. 2:

*H*

_{1}

*γ*

_{21}

*f*

*f*

_{1,1}

*H*

_{1,0}

*γ*

_{21}

*f*

_{2}

*f*

_{1,2}

*in region III climate variability induced by anomalous buoyancy forcing comes from two sources*: First, the primary potential vorticity anomaly propagating along

*ψ*

_{1}, as denoted by

*δ*

**f**

_{1,1}; second, the second potential vorticity anomaly propagating along

*ψ*

_{2}, as denoted by

*δ*

**f**

_{1,2}. The corresponding layer thickness perturbations are

### f. Region IV

The solution in region IV is much more complicated because two new characteristics, each carrying a potential vorticity anomaly in layer 3, are created when the trajectories carrying the primary and second potential vorticity anomaly across *f*_{3}, Fig. 2. Thus, *there are four potential vorticity anomaly trajectories, including the primary, the secondary, and the tertiary potential vorticity anomaly.* The solution for this region is included in the appendix.

## 3. Climate variability

In this section we will primarily focus on the climate variability induced by anomalous buoyancy forcing imposed along the outcrop lines. The model basin is a rectangular basin 60° wide and covers the region 20° to 50°N, with three outcrop lines at *y*_{1} = 45.5°N, *y*_{2} = 41°N, and *y*_{3} = 35°N. The first layer thickness along the eastern boundary is set to 300 m, and so the complete solution includes a shadow zone. We will carefully choose our forcing to limit its region of influence to the ventilated zone for simplicity. The Ekman pumping rate [cm s^{−1}] is *w*_{e} = 1.0 × 10^{−4} sin{[(*y* − *y*_{s})/Δ*y*]*π*}, where *y*_{s} = 20° and Δ*y* = 30°.

*f*

_{1}; cooling/heating imposed on other outcrop lines can be treated in a similar way. A cooling anomaly is imposed in terms of a southward migration of the outcrop line with a small patch

*x*

_{0}= 20°. We chose a small perturbation to illustrate the fundamental structure of the variability induced by a point source of buoyancy forcing, so

*dy*

_{1}= −0.01° and Δ

*x*= 4° for the first experiment. Such an equatorial shift of the outcrop line represents a localized increase of the surface density due either to cooling of the surface or the production of a localized enhanced patch of salinity. The precise relation between such changes in surface temperature or salinity and the anomalous heat flux or precipitation/evaporation anomaly required to produce them is beyond the scope of our simple model. Since the perturbation solution is linear, the case of anomalous heating simply corresponds to a change of sign of all perturbation variables.

The climate variability is defined as the difference between the unperturbed solution and the perturbed solution. As discussed by Huang and Pedlosky (1999), surface cooling along *f*_{1} leads to an increase of *h*_{1} in the region of surface cooling. South of the first outcrop line, this primary potential vorticity anomaly propagates along *ψ*_{1} and induces a decrease in *h*_{2} in region II, as shown in the upper panels in Fig. 3. Thus, potential vorticity anomaly is created in layer 2.

Note that within these characteristic cones, the amplitude of the layer thickness perturbations varies over two orders of magnitude. In order to show the pattern of the perturbations over a large range a specially designed contour map is produced. First, the perturbation is converted into units of 0.1 mm. Second, all positive values are converted into *x* = log(*dh*), and all negative values of *x* are set to zero. Third, all negative values are converted into *x* = −log(−*dh*), and all positive values of *x* are set to zero. The horizontal pattern of the layer thickness perturbations is shown in Fig. 3. Even with just three outcrop lines, the pattern of the perturbations is rather complicated, and it is clear that the sign of layer thickness perturbations alternate in the horizontal plane.

South of *f*_{2}, the second layer is subducted, so the potential vorticity anomaly in the second layer is preserved and propagates along *ψ*_{2}. Thus, in region III we have two characteristic cones consisting of potential vorticity anomaly trajectories—the primary potential vorticity anomaly *δq*_{1} that propagates along *ψ*_{1} and the second potential vorticity anomaly *δq*_{2} that propagates along *ψ*_{2}, as shown in Figs. 3a–c. Note that in the previous study by Huang and Pedlosky (1999) the discussion about a model with three moving layers is in error because the dynamic role of the secondary potential vorticity anomaly was overlooked.

South of *f*_{3}, the third layer is subducted, so the tertiary potential vorticity anomalies are created in the third layer, induced by the primary potential vorticity anomaly in the first layer and the secondary potential vorticity anomaly in the second layer. Thus, there are two new characteristics cones in region IV.

From the discussion above, we argue that the number of characteristic cones carrying the perturbation information doubles each time they cross a new outcrop line. The exponential growth of the number of characteristic cones is a major difficulty in dealing with the multilayer model. In this study we will focus on the case with four moving layers.

As the number of moving layer increases, the vertical structure of the perturbations becomes increasingly complicated. As shown in Eq. (5), 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*^{1}_{2}

In region III, the vertical structure of the solution in the two branches appears in different forms. Along the P branch (the primarily potential vorticity anomaly), the perturbation is in forms of *M*^{1}_{3}*dh*_{1}. Along the S branch (the secondary potential vorticity anomaly), the perturbation is in the form of *M*^{2}_{3}*dh*_{2} (Fig. 4). Note that |*dh*_{2}| < |*dh*_{1}| because the former is secondary perturbation.

Note that within the S branch there are small potential vorticity (or layer thickness) perturbations in layer 1. Such perturbations are not directly related to the surface forcing on layer 1 at the outcrop line; instead, they are induced by the secondary potential vorticity anomaly in layer 2 in the following way. A potential vorticity anomaly in layer 2 induces a slight shift of streamlines in layer 1, 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 structure of the *M*^{1}_{3}*M*^{2}_{3}

In region IV, the vertical structure of the solution in the four branches are different, Fig. 5. The structure of the P branch is in forms of the *M*^{1}_{4}*M*^{2}_{4}*M*^{1}_{4}*M*^{1}_{3}*M*^{2}_{4}*M*^{2}_{3}

There are two new characteristic cones, *T*_{1} and *T*_{2}, that are generated by the tertiary potential vorticity in layer 3, induced by the primary (secondary) potential vorticity in layer 1 (2). The structure of these two branches is in forms of the *M*^{3}_{4}

Since there are four moving layers, the vertical structure becomes more complicated, with the possibility of higher modes. However, we do not have a clear way of defining the higher modes in the DTM. Although, one can certainly define a set of eigenmodes and project the perturbation solution onto these eigenmodes, the meaning of such eigenmodes is not clear. The fundamental difficulty is that the perturbation solution to the thermocline equation involves not only the local stratification, but also other dynamic variables, such as the horizontal gradient of potential vorticity. Thus, any eigenfunction defined only by the local stratification does not carry the complete dynamic message implied by the thermocline structure of the unperturbed flow field.

Note that we have chosen a rather small amplitude perturbation of the outcrop line, *dy*_{1} = −0.01°. For a realistic amplitude, say *dy*_{1} = −1°, the perturbations shown in Tables 1 and 2 should increase 100 times.

Climate anomalies identified from climate records generally consist of a subsurface anomaly with a horizontal area large enough to survive for decades. As an example, we present a solution with *dy*_{1} = 0.5° and Δ*x* = 15°, Fig. 6. Since the northward displacement corresponding to warming, the induced perturbations have opposite signs compared with the case of cooling. Furthermore, the anomaly has a finite amplitude, so all four branches overlap, leading to a complicated nonlinear solution within the characteristic cones. The interface depth and layer thickness perturbations are on the order of 5 m. The primary branch stands out clearly, while the secondary and tertiary branches merge. In this case we have chosen a small outcrop line displacement because for Δ*x* = 15° an outcrop line displacement larger than 0.5° will lead to a nonmonotonic function *H*_{1}(*x*) defined in (12). When *H*_{1}(*x*) is nonmonotonic, one cannot invert this function to find the suitable outcrop line position. Physically, the nonmonotonic function *H*_{1}(*x*) indicates that the slope of the outcrop line is too large, there is no consistent solution for the ventilated thermocline (e.g., Huang and Pedlosky 1999).

The structure of the perturbations on each isopycnal surface also look quite different compared with the previous cases. Now in the southern basin the four branches of the perturbations overlap, so there is no clearly defined boundary between them, Fig. 7. This is in contrast to the solution shown in Fig. 3, where each branch is well separated from the others. The overlapping of different branches of the perturbations may make the analysis of decadal climate variability more difficult.

Since the perturbation near the upper surface may be lost to influences from other anomalous forcing coming into play later, only the subsurface perturbations will be likely to survive and can be identified from observations. From this four-layer model, we speculate that patterns shown in Fig. 6 may be a representative pattern for decadal climate variability in the oceans. Of course, this will either be confirmed or denied by further observations.

## 4. Discussion

As discussed by Huang and Pedlosky (1999), perturbations are horizontally confined within the characteristic cones defined by streamlines that carry the potential vorticity anomaly. The thickness of the lowest layer increases due to surface cooling, and the subduction of this layer creates the primary potential vorticity anomaly that drives the climate variability in the ventilated thermocline downstream.

Each time the characteristic cones carrying a potential vorticity anomaly cross a new outcrop line, the number of the characteristic cones involved in the perturbation doubles. In this paper, we present the climate variability of a four-moving-layer model of the ventilated thermocline. As the number of moving layers increases, the structure of the perturbation becomes exponentially complicated. The structure of perturbations to a model with a large number of moving layers is not clear at this time.

Suppose we double the number of layers used to represent the flow south of the outcrop line where the primary potential vorticity perturbation is created. Since the total density difference on the surface is invariant we must correspondingly halve the density differences between the new layers. As a consequence of the beta spiral (see, e.g., Pedlosky 1996) the turning angle between the flows in adjacent layers is proportional to the density difference between the layers and so would be correspondingly reduced as new layers are added. Hence we expect that in the limit of a continuously stratified fluid the eastern edge of the characteristic cone containing the anomalies would be grossly similar to our four-layer model. Although the streamlines in the most recently subducted layer always turn eastward, the overall pattern should tend to resemble the multilayer model although there may be a quantitative alteration in the limit. Note, though, that, as new layers are added, any eastward extension of the cone becomes thinner in the vertical direction. The numerical solutions with 20 ventilated layers provides some very interesting hints as to the structure of the climate variability, including complicated modal structure in both the horizontal and vertical directions (see Huang 2000).

However, we argue that, in principle, the overall domain of the perturbations is well defined, as shown in Fig. 8. First, the western edge of the perturbation (depicted by the solid line) is defined by the streamline in the layer where the primary potential vorticity is generated by surface cooling. Second, the eastern edge of the perturbation (depicted by the dashed line) is defined as an envelope in the following way. Starting from point P in Fig. 8, when the characteristic carrying the potential vorticity anomaly crosses a new outcrop line, a new characteristic carrying the potential vorticity anomaly induced in the layer above is created. Due to the rotation of the horizontal velocity, the so-called beta spiral, trajectories in the shallow layer travel to the right of the trajectories below. Thus, the streamline in the layer above defines the eastern edge of the perturbation zone south of the new outcrop line.

Similarly, each time when the characteristics carrying the potential vorticity anomaly cross a new outcrop line, the signal is passed to the streamlines in the newly subducted layer above. Therefore, the envelope of the streamlines in the series of the most recently subducted layers defines the eastern edge of the perturbation zone in the model ocean, as shown by the dashed line in Fig. 8.

Accordingly, a point source of buoyancy forcing anomaly creates perturbations that propagate downstream within a characteristic cone. A given surface cooling anomaly of finite size can be treated as a superposition of such point sources, although the superposition is nonlinear in nature. Consequently, the variability induced by surface cooling can propagate to a location much closer to the equator than implied by the streamline carrying the primary potential vorticity anomaly. Although the vertical structure of the perturbations along the western edge of this characteristic cone is clearly defined by the nature of the primary potential vorticity, we do not know a priori the structure over the other parts of the characteristic cone at this time. As our calculations indicate, the anomaly at a given location may consist of either a local heating or cooling perturbation.

The perturbations created in the extratropics can propagate to the equatorial region, and a model of this process with a two moving layer has been discussed by Huang and Pedlosky (2000). However, for a multilayer model, such communication really is rather complicated. Most importantly, this communication depends on the position of the characteristic cone, carrying the perturbations created in the extratropics, relative to the latitudinal location where the western boundary current bifurcates, as shown in Fig. 8.

If the southern edge of the characteristic cone (labled as S) is north of the matching latitude, all the perturbations should move northward, not affecting the equatorial thermocline. If the northern edge of the characteristic cone (labled as N) is south of the matching latitude, all perturbations should move into the equatorial thermocline. In particular, the primary potential vorticity anomaly should propagate to the equator. The climate variability at the equator induced by cooling at midlatitudes is discussed by Huang and Pedlosky (2000). On the other hand, if the characteristic cone straddles the matching latitude (as shown in Fig. 8), only the southern part of the perturbation will move to the equatorial region, and its dynamic effect on the equatorial circulation is not clear because it depends on the structure of the perturbation within this part of the characteristic cone that remains a sensitive function of initial position.

## Acknowledgments

RXH was supported by National Science Foundation through Grant OCE-9616950 to the Woods Hole Oceanographic Institution. JP was supported by National Science Foundation through Grant OCE-9301845 to the Woods Hole Oceanographic Institution.

## REFERENCES

Huang, R. X., 2000: Climate variability inferred from a continuously stratified model of the ideal-fluid thermocline.

*J. Phys. Oceanor.,***30,**1389–1406.——, and J. Pedlosky, 1999: Climate variability inferred from a layered model of the ventilated thermocline.

*J. Phys. Oceanogr.,***29,**779–790.——, and ——, 2000: Climate variability of the equatorial thermocline inferred from a two-moving-layer model of the ventilated thermocline.

*J. Phys. Oceanogr.,***30,**2610–2626.Luyten, J. R., J. Pedlosky, and H. Stommel, 1983: The ventilated thermocline.

*J. Phys. Oceanogr.,***13,**292–309.Pedlosky, J., 1996:

*Ocean Circulation Theory.*Springer-Verlag, 453 pp.

## APPENDIX

### The Four-Layer Region

As discussed above, there are two sets of characteristics in region III, which carry the potential vorticity anomaly. If we imagine a point source of cooling along *f*_{1}, as these two characteristics across outcrop line *f*_{3}, each of them bifurcate, so there are four characteristics south of *f*_{3}; that is, a streamline in layer 1 (heavy solid line in Fig. 2) that carries the potential vorticity perturbation originally created in the cooling region, a streamline in layer 2 (heavy long-dashed line in Fig. 2) that carries a secondary potential vorticity anomaly created along *f*_{2}, and two streamlines in layer 3 that carry the tertiary potential vorticity anomaly, which is created when the first and second potential vorticity anomaly lines across the third outcrop line.

For the interior of region IV, we have the fractional layer thickness:

- The first layer. The fractional thickness for the first layer iswhere
*f*_{1,11}indicates that the position of the first outcrop line used here is obtained by tracing backward along streamline*ψ*_{1}in regions IV, III, and II, as shown in Fig. 2. - The second layer. In order to determine the potential vorticity function
*Q*_{2}, we notice that this function is set at point D along*f*_{2}. Repeating the same argument as in the previous section, we obtain the functional relationThere are two differences, compared with the expression in region III: First, the second interfacial depth here is*H*_{2}because there is one more moving layer here; second,*f*_{1,2}is replaced by*f*_{1,22}because now the corresponding outcrop position*f*_{1}should be obtained by tracing*ψ*_{2}in both regions IV and III.Since in region IVthe corresponding fractional layer thickness is - The third layer. The potential vorticity in layer three is set up at point E along
*f*_{3}(Fig. 2). Along*f*_{3}, the fractional thickness for the first layer iswhere*f*_{1,31}indicates that the position of the first outcrop line is obtained by tracing back along*ψ*_{3}in region IV, and then*ψ*_{1}in region III.

*Q*

_{2}at this point is set up at point F along

*f*

_{2}(Fig. 2). By following a similar analysis, we have the functional form of

*Q*

_{2}

*f*

_{1,32}indicates that the position of the first outcrop line is obtained by tracing back along

*ψ*

_{3}in region IV, and then

*ψ*

_{2}in region III.

*f*

_{1}is defined by

*f*

_{1,31}, that is, the position of the first outcrop line obtained by tracing backward along

*ψ*

_{3}in region IV, and

*ψ*

_{2}in region III. The corresponding fractional layer thickness is

*f*

_{3}, the fractional layer thickness for the lowest two moving layers are

*f*

_{3},

*f*

_{3},

*H*

_{1}

*γ*

_{21}

*H*

_{2}

*γ*

_{31}

*H*

_{3}

*H*

_{1}

*γ*

_{21}

*F*

^{IV}

_{1}

*γ*

_{31}

*F*

_{1}

*F*

^{IV}

_{2}

4) How to determine the outcrop position. The calculation of the perturbed solution requires us to link the total depth of the ventilated thermocline *H*_{1} at a given location to the corresponding total depth of the ventilated thermocline along *f*_{1}, *H*_{1,0}. For a given point in region IV there are five possible trajectories:

*f*

_{1,11}by tracing backward along

*ψ*

_{1}: Since

*H*

_{1}is conserved,

*f*

_{1,21}by tracing backward along

*ψ*

_{2}→

*ψ*

_{1}: Since the streamlines

*ψ*

_{2}are defined by

*H*

_{1}+

*γ*

_{21}

*H*

_{2},

*f*

_{1,11}is the outcrop position for interior point, but

*f*

_{1,21}is the corresponding outcrop position for the streamline that meets

*f*

_{3}. As a result, the relation for the total depth is

*f*

_{1,22}by tracing backward along

*ψ*

_{2}→

*ψ*

_{2}: Similar to the previous case, streamlines

*ψ*

_{2}are defined by

*H*

_{1}+

*γ*

_{21}

*H*

_{2}, so at

*f*

_{3}the equivalent total depth is

*f*

_{3}the trajectory is continued along

*ψ*

_{2}northward; thus, a similar relation holds,

*f*

_{1,31}by tracing backward along

*ψ*

_{3}→

*ψ*

_{1}: Streamlines

*ψ*

_{3}is defined by

*H*

_{1}+

*γ*

_{21}

*H*

_{2}+

*γ*

_{31}

*H*

_{3}, so

*ψ*

_{3}→

*ψ*

_{1}trajectory, the depth ratio is

*f*

_{1,32}by tracing backward along

*ψ*

_{3}→

*ψ*

_{2}: There is a additional factor due to the contribution from the

*ψ*

_{2}trajectory,

Potential vorticity anomalies and streamlines tracing the original outcrop lines

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Potential vorticity anomalies and streamlines tracing the original outcrop lines

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Potential vorticity anomalies and streamlines tracing the original outcrop lines

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Layer thickness perturbation maps, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 4°. Three zonal dashed lines depict the outcrop lines, and I, II, III, and IV indicate different dynamic zones discussed in Fig. 2

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Layer thickness perturbation maps, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 4°. Three zonal dashed lines depict the outcrop lines, and I, II, III, and IV indicate different dynamic zones discussed in Fig. 2

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Layer thickness perturbation maps, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 4°. Three zonal dashed lines depict the outcrop lines, and I, II, III, and IV indicate different dynamic zones discussed in Fig. 2

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of (a) interface depth and (b) layer thickness along 36.5°N, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 2°. The curves for the second and third layer thickness (or interfacial depth) are shifted upward for a clear presentation

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of (a) interface depth and (b) layer thickness along 36.5°N, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 2°. The curves for the second and third layer thickness (or interfacial depth) are shifted upward for a clear presentation

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of (a) interface depth and (b) layer thickness along 36.5°N, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 2°. The curves for the second and third layer thickness (or interfacial depth) are shifted upward for a clear presentation

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of (a) interface depth and (b) layer thickness along 32°N, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 2°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of (a) interface depth and (b) layer thickness along 32°N, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 2°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of (a) interface depth and (b) layer thickness along 32°N, generated by cooling within a small area with *dy*_{1} = −0.01° and Δ*x* = 2°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Pertubations of (a) interface depth and (b) layer thickness along 32°N, generated by heating within a large area with *dy*_{1} = 0.5° and Δ*x* = 15°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Pertubations of (a) interface depth and (b) layer thickness along 32°N, generated by heating within a large area with *dy*_{1} = 0.5° and Δ*x* = 15°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Pertubations of (a) interface depth and (b) layer thickness along 32°N, generated by heating within a large area with *dy*_{1} = 0.5° and Δ*x* = 15°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Layer thickness perturbation maps, generated by heating within a large area with *dy*_{1} = 0.5° and Δ*x* = 15°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Layer thickness perturbation maps, generated by heating within a large area with *dy*_{1} = 0.5° and Δ*x* = 15°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Layer thickness perturbation maps, generated by heating within a large area with *dy*_{1} = 0.5° and Δ*x* = 15°

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

A sketch illustrating the connection between midlatitude and equatorial athermocline

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

A sketch illustrating the connection between midlatitude and equatorial athermocline

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

A sketch illustrating the connection between midlatitude and equatorial athermocline

Citation: Journal of Physical Oceanography 30, 11; 10.1175/1520-0485(2001)031<3009:CVIBAB>2.0.CO;2

Perturbations of interface depth (*dZ* = −*dH*) and layer thickness along 36.5°N (region III), in centimeters

Perturbations of interface depth (*dZ* = −*dH*) and layer thickness along 32°N (region IV), in centimeters

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* Woods Hole Oceanographic Institution Contribution Number 10098.