• 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.

  • View in gallery

    A sketch of the multilayer model

  • View in gallery

    Potential vorticity anomalies and streamlines tracing the original outcrop lines

  • View in gallery

    Layer thickness perturbation maps, generated by cooling within a small area with dy1 = −0.01° and Δx = 4°. Three zonal dashed lines depict the outcrop lines, and I, II, III, and IV indicate different dynamic zones discussed in Fig. 2

  • View in gallery

    Perturbations of (a) interface depth and (b) layer thickness along 36.5°N, generated by cooling within a small area with dy1 = −0.01° and Δx = 2°. The curves for the second and third layer thickness (or interfacial depth) are shifted upward for a clear presentation

  • View in gallery

    Perturbations of (a) interface depth and (b) layer thickness along 32°N, generated by cooling within a small area with dy1 = −0.01° and Δx = 2°

  • View in gallery

    Pertubations of (a) interface depth and (b) layer thickness along 32°N, generated by heating within a large area with dy1 = 0.5° and Δx = 15°

  • View in gallery

    Layer thickness perturbation maps, generated by heating within a large area with dy1 = 0.5° and Δx = 15°

  • View in gallery

    A sketch illustrating the connection between midlatitude and equatorial athermocline

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Climate Variability Induced by Anomalous Buoyancy Forcing in a Multilayer Model of the Ventilated Thermocline

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  • 1 Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
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Abstract

Climate variability induced by surface cooling in a four-moving-layer model of the ventilated thermocline is studied. The perturbations propagate within the characteristic cones defined by streamlines in individual layers where the potential vorticity anomaly is created. Each time the characteristic cones cross a new outcrop line, the number of characteristic cones doubles. Thus, climate variability in a multilayer model can have very complicated structure in the horizontal plane, with anomalies of alternating sign. The vertical structure of such perturbations defines a class of dynamical thermocline modes.

Corresponding author address: Rui Xin Huang, Department of Physical Oceanography, Wood Hole Oceanographic Institution, Woods Hole, MA 02543.

Email: rhuang@whoi.edu

Abstract

Climate variability induced by surface cooling in a four-moving-layer model of the ventilated thermocline is studied. The perturbations propagate within the characteristic cones defined by streamlines in individual layers where the potential vorticity anomaly is created. Each time the characteristic cones cross a new outcrop line, the number of characteristic cones doubles. Thus, climate variability in a multilayer model can have very complicated structure in the horizontal plane, with anomalies of alternating sign. The vertical structure of such perturbations defines a class of dynamical thermocline modes.

Corresponding author address: Rui Xin Huang, Department of Physical Oceanography, Wood Hole Oceanographic Institution, Woods Hole, MA 02543.

Email: rhuang@whoi.edu

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 ith layer are denoted as hi and Hi, respectively; thus, for the top layer n, Hn = hn. The northernmost outcrop line is labeled as f1, 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

First we derive a relation between pressure gradients in two adjacent layers. Using the hydrostatic relation, the pressure gradient in two adjacent layers satisfies
i1520-0485-30-11-3009-e1
where
i1520-0485-30-11-3009-eq1a
is the reduced gravity across the interface. This relation applies to any pair of layers. In addition, one can assume that the lowest layer with subscript 0 is very deep, so that the pressure gradient there is negligible. Thus, the pressure gradient in the mth layer (mn) is
i1520-0485-30-11-3009-e2

b. The Sverdrup relation

By taking the curl of the momentum equations in each layer the vorticity balance is obtained. Multiplying by the layer depth, summing up over all the moving layers, and integrating over [x, xe], one obtains the Sverdrup relation
i1520-0485-30-11-3009-e3
where
i1520-0485-30-11-3009-e4
Perturbing the Sverdrup relation gives, to the lowest order,
i1520-0485-30-11-3009-e5
Since anomalous buoyancy forcing makes no contribution to the right-hand side of (5), climate variability due to surface temperature (or freshwater flux) perturbations must appear as an internal mode, with a depth-weighted zero mean. We call such modes, which satisfy the homogeneous form of (5), 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

We start our discussion by describing the anomalies produced in layer 1 and 2 in the region f2 < f < f1, denoted as region II in Fig. 2. South of the first outcrop line y1, potential vorticity in layer 1 is conserved along the streamline, h1 = H1 = const:
i1520-0485-30-11-3009-e6
At f = f1, h1 = H1; thus, the functional form of Q1 is Q1(x) = f1/x, and the first layer thickness is
i1520-0485-30-11-3009-e7
The solution in region II is
i1520-0485-30-11-3009-e8
where
i1520-0485-30-11-3009-e9
We introduce the fractional layer thicknesses, F, which is defined as the layer thickness divided by the total depth of the ventilated layers; thus, for region II we have
i1520-0485-30-11-3009-e10
where the superscript II indicates that the definition applies to region II with two moving layers, and the subscripts indicate the individual layers.

It is important to note that f1 in these relations is not necessarily a constant. In fact, we can write it in the form of f1(H1), which indicates that f1 is dependent on the latitude of the outcrop line by tracing backward along the streamline H1 = const. Therefore, in our discussion hereafter, we can write alternatively f1f1 + δf1, where we imagine that δf1 represents the outcrop line perturbations induced by heating or cooling anomaly. If δf1 < 0, this implies that outcrop line moves equatorward from its constant value of f1 representing cooling.

Since along the first outcrop line
i1520-0485-30-11-3009-e12
it is readily shown that
H21xH210xOδf1
where H10(x) is the total layer thickness along the unperturbed outcrop line. Thus, to the lowest-order approximation, the functional relation between H1 and x remains unchanged. Using this relation we can calculate x and δf1(x) for given H1 in order to complete the necessary inversion to determine f1(H1) 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

Cooling can be represented by a southward perturbation of the first outcrop line,
i1520-0485-30-11-3009-e13
Therefore, the cooling-induced changes in the layer depth are
i1520-0485-30-11-3009-e17
These results are exactly the same as the two-moving-layer case discussed by Huang and Pedlosky (1999).

e. Region III

South of f2 there are two subsurface layers in which the streamlines are H1 = const and H1 + γ21H2 = const. Along these streamlines potential vorticity is conserved. Note that, when the potential vorticity anomaly carried by the streamlines in layer 1 (labeled as δq1 in Fig. 2) crosses the second outcrop line f2, a new characteristic is created, and south of f2 the potential vorticity anomaly in layer 2 propagates along streamlines in layer 2, labeled as δq2 in Fig. 2. Thus the zone influenced by the cooling broadens in a fanlike manner as the fluid moves southward.

Similar to the discussion in the previous section, at an interior point A in region III potential vorticity anomalies in layer 1 are set up by perturbations at positions along the first outcrop line f1 obtained by tracing back along ψ1. We introduce the notation
fn,mfnψm
to be the position of the nth 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 f1 along ψ1 (the streamline in layer one) in both regions III and II. Thus, the potential vorticity functional form and the thickness of layer 1 are
i1520-0485-30-11-3009-e21
The potential vorticity in layer 2 at the same point A is, however, set up at point B, which is defined by tracing backward along ψ2 in region III. At point B, the lowest layer thickness is
i1520-0485-30-11-3009-e22
where the subscript in f1,2 indicates that the corresponding position on the first outcrop line f1 is now determined by tracing ψ2 backward in region III, and then along ψ1 in region II. At the second outcrop line f2, thus, we have
i1520-0485-30-11-3009-e23
Let
xH1γ21H2
Along f2, we have
i1520-0485-30-11-3009-e25
Therefore the functional form of potential vorticity in the second layer is
i1520-0485-30-11-3009-e26
South of f2,
i1520-0485-30-11-3009-e27
Thus, from (24) we obtain
i1520-0485-30-11-3009-e28
Substituting (28) into the relation of Q2 leads to the fractional layer thickness in region III:
i1520-0485-30-11-3009-e29
The Sverdrup relation for region III is
i1520-0485-30-11-3009-e32
where
G3 = 1 + γ21(1 − FIII1)2 + γ31(1 − FIII1 − FIII2)2.
The solution in region III can be obtained by solving the nonlinear equations (29)–(33). Note that both f1,1 and f1,2 are complicated nonlinear functions of the local total layer depth H1, and these functions are defined by tracing back along streamlines ψ1 and ψ2, as shown in the upper left part of Fig. 2. For example,
f1,2f1,2H1γ21H2f1H1,0
is a nonlinear function of the local total layer depth H1, and this function can be obtained by tracing backward along ψ2 in region III to point B and then following ψ1 in region II, as shown in the upper left part of Fig. 2:
H1γ21ff1,1H1,0γ21f2f1,2
Climate variability induced by anomalous buoyancy forcing can be calculated by taking the perturbations of these equations
i1520-0485-30-11-3009-e36
These relations clearly indicate that in region III climate variability induced by anomalous buoyancy forcing comes from two sources: First, the primary potential vorticity anomaly propagating along ψ1, as denoted by δf1,1; second, the second potential vorticity anomaly propagating along ψ2, as denoted by δf1,2. The corresponding layer thickness perturbations are
i1520-0485-30-11-3009-e38

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 f3, 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 y1 = 45.5°N, y2 = 41°N, and y3 = 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 we = 1.0 × 10−4 sin{[(yys)/Δy]π}, where ys = 20° and Δy = 30°.

The discussion presented here is for the case with cooling imposed on f1; 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
i1520-0485-30-11-3009-e43
where x0 = 20°. We chose a small perturbation to illustrate the fundamental structure of the variability induced by a point source of buoyancy forcing, so dy1 = −0.01° and Δx = 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 f1 leads to an increase of h1 in the region of surface cooling. South of the first outcrop line, this primary potential vorticity anomaly propagates along ψ1 and induces a decrease in h2 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 f2, the second layer is subducted, so the potential vorticity anomaly in the second layer is preserved and propagates along ψ2. Thus, in region III we have two characteristic cones consisting of potential vorticity anomaly trajectories—the primary potential vorticity anomaly δq1 that propagates along ψ1 and the second potential vorticity anomaly δq2 that propagates along ψ2, 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 f3, 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, M12, 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. The solution has been discussed by Huang and Pedlosky (1999).

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 M13 mode. This mode is induced by a large positive dh1. Along the S branch (the secondary potential vorticity anomaly), the perturbation is in the form of M23 mode, which is induced by a negative dh2 (Fig. 4). Note that |dh2| < |dh1| 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 M13 and M23 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 notice 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 (see Table 1).

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 M14 mode, and the S branch is in forms of the M24 mode. In comparison, the M14 has a structure similar to the M13, and the M24 has a structure similar to the M23.

There are two new characteristic cones, T1 and T2, 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 M34 mode, with opposite signs.

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, dy1 = −0.01°. For a realistic amplitude, say dy1 = −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 dy1 = 0.5° and Δx = 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 Δx = 15° an outcrop line displacement larger than 0.5° will lead to a nonmonotonic function H1(x) defined in (12). When H1(x) is nonmonotonic, one cannot invert this function to find the suitable outcrop line position. Physically, the nonmonotonic function H1(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 f1, as these two characteristics across outcrop line f3, each of them bifurcate, so there are four characteristics south of f3; 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 f2, 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:

  1. The first layer. The fractional thickness for the first layer is
    i1520-0485-30-11-3009-ea1
    where f1,11 indicates that the position of the first outcrop line used here is obtained by tracing backward along streamline ψ1 in regions IV, III, and II, as shown in Fig. 2.
  2. The second layer. In order to determine the potential vorticity function Q2, we notice that this function is set at point D along f2. Repeating the same argument as in the previous section, we obtain the functional relation
    i1520-0485-30-11-3009-ea2
    There are two differences, compared with the expression in region III: First, the second interfacial depth here is H2 because there is one more moving layer here; second, f1,2 is replaced by f1,22 because now the corresponding outcrop position f1 should be obtained by tracing ψ2 in both regions IV and III.
    Since in region IV
    i1520-0485-30-11-3009-ea3
    the corresponding fractional layer thickness is
    i1520-0485-30-11-3009-ea4
  3. The third layer. The potential vorticity in layer three is set up at point E along f3 (Fig. 2). Along f3, the fractional thickness for the first layer is
    i1520-0485-30-11-3009-ea5
    where f1,31 indicates that the position of the first outcrop line is obtained by tracing back along ψ3 in region IV, and then ψ1 in region III.

On the other hand, the potential vorticity function Q2 at this point is set up at point F along f2 (Fig. 2). By following a similar analysis, we have the functional form of Q2
i1520-0485-30-11-3009-ea6
where f1,32 indicates that the position of the first outcrop line is obtained by tracing back along ψ3 in region IV, and then ψ2 in region III.
Approaching point E, f1 is defined by f1,31, that is, the position of the first outcrop line obtained by tracing backward along ψ3 in region IV, and ψ2 in region III. The corresponding fractional layer thickness is
i1520-0485-30-11-3009-ea7
Along f3, the fractional layer thickness for the lowest two moving layers are
i1520-0485-30-11-3009-ea8
Note that along f3,
i1520-0485-30-11-3009-ea10
Thus, the potential vorticity function of layer 3 in region IV is
i1520-0485-30-11-3009-ea11
Note that south of f3,
H1γ21H2γ31H3H1γ21FIV1γ31F1FIV2
From these equations, we obtain the fractional layer thickness
i1520-0485-30-11-3009-ea13

4) How to determine the outcrop position. The calculation of the perturbed solution requires us to link the total depth of the ventilated thermocline H1 at a given location to the corresponding total depth of the ventilated thermocline along f1, H1,0. For a given point in region IV there are five possible trajectories:

Determining f1,11 by tracing backward along ψ1: Since H1 is conserved,
i1520-0485-30-11-3009-ea17
Determining f1,21 by tracing backward along ψ2ψ1: Since the streamlines ψ2 are defined by H1 + γ21H2,
i1520-0485-30-11-3009-ea18
Note that f1,11 is the outcrop position for interior point, but f1,21 is the corresponding outcrop position for the streamline that meets f3. As a result, the relation for the total depth is
i1520-0485-30-11-3009-ea19
Determining f1,22 by tracing backward along ψ2ψ2: Similar to the previous case, streamlines ψ2 are defined by H1 + γ21H2, so at f3 the equivalent total depth is
i1520-0485-30-11-3009-ea20
From f3 the trajectory is continued along ψ2 northward; thus, a similar relation holds,
i1520-0485-30-11-3009-ea21
Determining f1,31 by tracing backward along ψ3ψ1: Streamlines ψ3 is defined by H1 + γ21H2 + γ31H3, so
i1520-0485-30-11-3009-eq2
Thus, along the ψ3ψ1 trajectory, the depth ratio is
i1520-0485-30-11-3009-ea22
Determining f1,32 by tracing backward along ψ3ψ2: There is a additional factor due to the contribution from the ψ2 trajectory,
i1520-0485-30-11-3009-ea23
Therefore, the depth ratio is
i1520-0485-30-11-3009-ea24

Fig. 1.
Fig. 1.

A sketch of the multilayer model

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

Fig. 2.
Fig. 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

Fig. 3.
Fig. 3.

Layer thickness perturbation maps, generated by cooling within a small area with dy1 = −0.01° and Δx = 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

Fig. 4.
Fig. 4.

Perturbations of (a) interface depth and (b) layer thickness along 36.5°N, generated by cooling within a small area with dy1 = −0.01° and Δx = 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

Fig. 5.
Fig. 5.

Perturbations of (a) interface depth and (b) layer thickness along 32°N, generated by cooling within a small area with dy1 = −0.01° and Δx = 2°

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

Fig. 6.
Fig. 6.

Pertubations of (a) interface depth and (b) layer thickness along 32°N, generated by heating within a large area with dy1 = 0.5° and Δx = 15°

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

Fig. 7.
Fig. 7.

Layer thickness perturbation maps, generated by heating within a large area with dy1 = 0.5° and Δx = 15°

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

Fig. 8.
Fig. 8.

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

Table 1.

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

Table 1.
Table 2.

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

Table 2.

* Woods Hole Oceanographic Institution Contribution Number 10098.

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