Lecture 21: Climate variability inferred from models of the thermocline 5/3/2006 4:23 PM

Transcription

1 Lecture : Climate variability inferred from models of the thermocline 5//006 4: PM Climate variability on decadal time scales can be examined by perturbing the steady solution of the thermocline in a multi-layer model. Huang and Pedlosky (999) 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, through reminiscent of, the standard geostrophic vertical normal modes of a resting ocean. In particular, we will show that anomalies produced by a localized cooling excite a fan of characteristics which 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 section, we will examine the climate variability for a model with three and four moving layers. In addition, we will also study the time-dependent problem. Based on the quasi-geostrophic theory and the WKB method, perturbations propagate can be roughly separated into two categories, the non-doppler mode and the advective mode, which will be discussed at the end of this lecture.. Multilayer Model formulation In this section, the notation is slightly different from that used in the classic ventilated thermocline model of Luyten et al. (98). We denote the bottom and motionless layer as layer 0, and the lowest moving layer as layer, and the layers above have number increases upward, Fig.. Fig.. A sketch of the multilayer model. 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, Hn = hn. The northern-most outcrop line is labeled as f, and the outcrop lines southward have number indexes that increase accordingly. This notation is more convenient when one has to deal with an arbitrary number of layers.

2 A) Pressure gradient in multi-layer model First we derive a relation between pressure gradient in two adjacent layers. Using the hydrostatic relation, the pressure gradient in two adjacent layers satisfies pi oi = + γ i Hi () ρi ρi where γ i = g ( ρi ρi) / ρi 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 pressure gradient there is negligible. Thus, the pressure gradient in the m th layer ( m n) is m pm = γ H () ρ 0 i= i i 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, x e ], one obtains the Sverdrup relation n γ ihi = D0 + H0 () i= where f x e D0 = we ( x', y) dx' βγ (4) x n γ i γ ; H γ H = = i 0 i ie γ i= Perturbing the Sverdrup relation gives, to the lowest order, n γ ihiδhi = δd0 (5) i= 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 mode (or the "barotropic" mode because in this study all perturbations discussed are confined to the winddriven circulation in the upper ocean). C. Region II We start our discussion by describing the anomalies produced in layer one and two in the region f < f < f, denoted as region II in Fig.. South of the first outcrop line y, potential vorticity in layer one is conserved along the streamline, h = H = const. Q( H) = f / h = const. (6) At f = f, h = H; thus, the functional form of ( ) f Q x =, and the first layer thickness is x h = fh / f (7)

3 Fig.. Potential vorticity anomalies and streamlines tracing the original outcrop lines. The solution in region II is / D0 + H 0 H = (8) G where f G = + γ (9) f 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 II h f F = H = f (0) II h f F = H = f () 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 f in these relations is not necessary a constant. In fact, we can write it in the form of f( H ), which indicates that f dependents on the latitude of the outcrop line by tracing backward along the streamline H = const.. Therefore, in our discussion hereafter, we can write alternatively f f+ δ f, where we imagine that δ f represents the outcrop line perturbations induced by heating or cooling anomaly. If δ f < 0, this implies that outcrop line moves equatorward from its constant value of f, representing a cooling anomaly. Since along the first outcrop line

4 f ( x) x e H = we( x', y) dx' H x e βγ + () it is readily shown that H( x) = H0( x) + O( δ f) where H ( ) 0 x is the total layer thickness along the unperturbed outcrop line. Thus, to the lowestorder approximation, the functional relation between H and x remains unchanged. Using this relation we can calculate x and δ f ( x) for given h in order to complete the necessary inversion to determine f( H ) to the 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. ) Climate variability induced by surface cooling. Cooling can be represented by a southward perturbation of the first outcrop line, δ f < 0 () II f δf = δ f > 0 (4) f II II δf 0 δf = < (5) II f δg = γ( F ) δ f < 0 (6) f Therefore, the cooling induced changes in the layer depth are H δh = δg > 0 (7) G II II H II II δh = F δh+ HδF = γ ( F ) δf 0 G + > (8) H II δh = δh δh = δf < 0 (9) G These results are exactly the same as the two-moving layer case discussed by Huang and Pedlosky (999). As one example, assuming a localized cooling, so the outcrop line moves slightly southward, as a result of the cooling the upper layer becomes thinner, and the lower layer thicker. Furthermore, the increase of the lower layer thickness is more than compensating the thinning of the upper layer; thus, the based of the wind-driven circulation moves downward, Fig.. The most interesting phenomenon is that the perturbations appear in forms of the so-called the second baroclinic mode, i.e., the upper interface moves upward (indicating cooling of the upper layer), but the lower interface moves downward (indicating warming of the lower layer). Surface cooling can induce warming of the lower part of the thermocline -- this is a little anti-intuitive. However, a simple analysis based on the reduced gravity model can explain such a phenomenon. The layer depth in a simple reduced gravity model obeys x f τ h = he + xe x g' ρβ 0 f y ( ) (0) 4

5 Fig.. Perturbations in layer thickness (m) in response to a regional cooling ( Δ θ =.5 o ), represented by a southward shift of the outcrop line, indicated by the thin line in the lowest panel. Assuming the wind stress does not change, taking the variations of this equation leads to x f τ hδ h= ( x ) ' e x δ g () g' ρβ 0 f y Cooling the upper ocean is equivalent to reduce the reduced gravity δ g ' < 0; therefore, δ h > 0, i.e., the depth of the wind-driven gyre, the thermocline, increases in response. We will call this mode as the second dynamic thermocline mode -- it is different from the classic second baroclinic mode inferred from the quasi-geostrophic theory. Cooling creates a potential thickness perturbation signal in lowest layer that propagates along the streamline in this layer -- streamline is characteristics in ideal-fluid thermocline. Cooling also creates thickness anomaly in the layer above; however, this upper layer is exposed to the Ekman pumping directly, so potential thickness of the upper layer is not conserved within this regime. South of f, layer subducts, so the potential thickness anomaly of this layer is now conserved along the streamlines of this layer. Thus, south of f there are two characteristics, the primary characteristics and the secondary characteristics, both of them carry their own signal that propagates in different directions, as shown in Figs. and 4. More interestingly, when the third 5

6 outcrop line f is crossed, these two characteristics will bifurcate again, and resulting in total of four characteristics. Thus, if there are n outcrop lines, there will be n characteristics. Fig. 4. Sketch of the perturbations induced by a localized cooling. Left Panel: Layer thickness perturbation; Right panel: the characteristics carrying signals of the perturbation. ) Climate variability due to changes in the mixed layer depth. If the mixed layer depth has a small perturbation, δ d < 0, there will be perturbations of the thermocline, with the upper layer becomes thinner and the lower layer thick, i.e., the perturbation due to the decline of mixed layer depth is similar to the case of cooling. ) Climate variability due to changes in the Ekman pumping rate. Fig. 5. Perturbations in layer thickness (m) in response to an increase of Ekman pumping, the outcrop line has a negative slope, as depicted by the thin line in the lowest panel. 6

7 It can be shown readily that if the outcrop line is zonal, change in Ekman pumping rate at a confine location can create perturbations in forms of the first baroclinic mode that propagate westward. There is no perturbation south of the latitudinal band of Ekman pumping anomaly. On the other hand, if the outcrop lines are non-zonal, small perturbation of Ekman pumping rate at a given location can create perturbations in forms of the second dynamical thermocline mode that propagate southwest of the Ekman pumping anomaly. The analysis of such a phenomenon can be found in Huang and Pedlosky (999). D. Region III. South of f there are two subsurface layers in which the streamlines are H = const. and H+ γ H = const.. Along these streamlines potential vorticity is conserved. Note that when the potential vorticity anomaly carried by the streamlines in layer one (labeled as δ q in Fig. ) crosses the second outcrop line f, a new characteristic is created, and south of f the potential vorticity anomaly in layer two propagates along streamlines in layer two, labeled as δ q in Fig.. Thus the zone influenced by the cooling broadens in a fan-like manner as the fluid moves southward. E. Region IV. The solution in Region IV is much more complicated because two new characteristics, each carrying a potential vorticity anomaly in layer three, are created when the trajectories carrying the primary and second potential vorticity anomaly across f, Fig.. Thus, there are four potential vorticity anomaly trajectories, including the primary, the secondary, and the tertiary potential vorticity anomaly. The solution for this region was discussed by Huang and Pedlosky (000). F. Climate variability 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 o wide and covers the region 0 o N to 50 o N, with three outcrop lines at y = 45.5 o N, y = 4 o N, and y = 5 o N. The first layer thickness along the eastern boundary is set to 00 m, and so the complete solution includes a shadow zone. We will carefully choose our forcing to limit its region of influence to the 4 ventilated zone for simplicity. The Ekman pumping rate is we = 0 sin { ( y ys) / Δy π} (cm/s) where y 0 o s = and Δ y = 0 o. The discussion presented here is for the case with cooling imposed on f ; 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 x0, 0 0 δ y = if x Δx x x +Δx; () Δx = 0, otherwise, where x 0 = 0 o. We chose a small perturbation to illustrate the fundamental structure of the variability induced by a point source of buoyancy forcing, so dy = 0.0 o and Δ x = 4 o for the first experiment. Of course, since the perturbation solution is linear, the case of anomalous heating simply corresponds to a change of sign of all perturbation variables. 7

8 The climate variability is defined as the difference between the unperturbed solution and the perturbed solution. As discussed by Huang and Pedlosky (999a), surface cooling along f leads to an increase of h in the region of surface cooling. South of the first outcrop line, this primary potential vorticity anomaly propagates along ψ and induces a decline in h in region II, as shown in the upper panels in Fig. 6. Thus, potential vorticity anomaly is created in layer. South of f, the second layer is subducted, so the potential vorticity anomaly in the second layer is preserved and propagates along ψ. Thus, in region III we have two characteristic cones consisting of potential vorticity anomaly trajectories -- the primary potential vorticity anomaly δ q q ψ, that propagates along ψ, and the second potential vorticity anomaly δ that propagates along as shown in Fig. 6a,.6b, and 6c. Note that in the previous study by Huang and Pedlosky (999a) 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, 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. Fig. 6. Layer thickness perturbation maps, generated by cooling within a small area with dy = 0.0 o and Δ x = 4 o. From the discussion above, we argue that the number of the characteristic cones carrying the perturbation information doubles each time they cross a new outcrop line. The exponential 8

9 growth of the number of characteristic cones is a major difficulty in dealing with the multi-layer model. In this section we will focus on the case with four moving layers. 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.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. 6. 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. As the number of moving layer increases, the vertical structure of the perturbations becomes increasingly complicated. As shown in equation (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, where the subscript indicates the number of moving layers and the superscript indicates the layer where the driving potential vorticity source is located. The solution has been discussed by Huang and Pedlosky (999). In region III, the vertical structure of the solution in the two branches appears in different form. Along the P-branch (the primarily potential vorticity anomaly), the perturbation is in forms of M mode. This mode is induced by a large positive dh. Along the S-branch (the secondary potential vorticity anomaly), the perturbation is in the form of M mode, which is induced by a negative dh, Fig. 7. Note that dh < dh because the former is secondary perturbation. Fig. 7. Perturbations of interface depth (a) and layer thickness (b) along 6.5 o N, generated by cooling within a small area with dy = 0.0 o and Δ x = o. Note that within the S-branch there are small potential vorticity (or layer thickness) perturbations in layer one. Such perturbations are not directly related to the surface forcing on layer one at the outcrop line; instead, they are induced by the secondary potential vorticity anomaly in layer two in the following way. A potential vorticity anomaly in layer two induces a slight shift of 9

10 streamlines in layer one, and thus induces a change in the potential vorticity there. Similarly, there are small potential vorticity perturbations in layer two within branch P. The vertical structure of the M and M is 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 layer two and three. In region IV, the vertical structure of the solution in the four branches are different, Fig. 8. The structure of the P-branch is in forms of the M 4 mode, and the S-branch is in forms of the M 4 ode. In comparison, the M 4 has a structure similar to the M, and the M 4 has a structure similar to the M. There are two new characteristic cones, T and T, that are generated by the tertiary potential vorticity in layer three, induced by the primary (secondary) potential vorticity in layer one (two). The structure of these two branches is in forms of the M 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. 4 Fig. 8. Perturbations of interface depth (a) and layer thickness (b) along o N, generated by cooling within a small area with dy = 0.0 o and Δ x = o. If there were many outcrop lines, all the perturbations will be confined with 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 when a new outcrop line is crossed, a new characteristic is created. Due to 0

11 the β -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, thus, 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, stemmed from the point source of cooling, Fig. 9. Fig. 9. The definition of the edge of the characteristic cone.. Continuously stratified model Fig. 0. Sketch of a model of the ideal-fluid thermocline, including a mixed layer of variable depth, the ventilated thermocline where the potential vorticity is to be determined as part of the solution, the unventilated thermocline where potential vorticity is specified a priori, and the stagnant abyssal water there the stratification is also specified.

12 The model (Huang, 000a,b) is set up, in which the background stratification is provided by a onedimensional advection-diffusion balance wρz = κρzz 5 where w= 0 cm/ s is the constant upwelling velocity, and κ = cm / s is the vertical diffusivity. Using density boundary conditions of ρ = at z=0 and ρ = 8 at the sea floor z=- 5km, the density profile can be calculated. The sea surface density is a linear function of latitude σ = 5 + ( y ys) /( yn ys). 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 structure of the model ocean is shown schematically in Fig. 0 A. Dependence on the diffusivity κ Examples in Fig. show the structure of the thermocline as the vertical mixing rate changes. It is clear that oceans correspond to the ideal-fluid thermocline solutions with κ > cm / s. Most importantly these examples demonstrate that the ideal-fluid thermocline model can simulate the case with 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. One of the most outstanding features in these solutions is the low potential vorticity water formed near the northern edge of the subtropical gyre. This is a fundamental issue about the mode water formation -- why the potential vorticity of mode water is so low? Many people have been working on this problem; however, we do not have a satisfactory answer so far. Fig.. Isopycnal sections taken along the outer edge of the western boundary (5 o away from the wall), with the solid line depicting ventilated thermocline and the dashed lines depicting the unventilated thermocline, contour interval Δ σ = 0.. The blank portion in the upper ocean indicates the mixed layer where density is vertically uniform.

13 B. Shifting of atmospheric conditions Using this model, we explored four different cases of climate variability. On decadal time scale or longer, the position of the Gulf Stream can move meridionally, or the sea surface density can increase or decline. Under such different forcing boundary conditions, the thermocline variability is shown in Fig.. One of the most outstanding features is the very low potential vorticity mode water created for the case of anomalous surface cooling, Figs. d and b. The dynamic meaning of this experiment remains to be explored in further study. Fig.. Isopycnal sections taken along the outer edge of the western boundary (5 o away from the wall): a) The intergyre boundary moved to 5 o N, with the maximum Ekman pumping rate unchanged; b) the intergyre boundary moved to 5 o N, with the maximum Ekman pumping rate increased 50% in order to have the same total amount of Ekman pumping; c) surface warming; d) surface cooling. To quantify the dynamical difference in these cases, we plotted the stratification and potential vorticity for a water column taken at a station ( 6 o N, 5 o E ), which is near the western boundary and at a location corresponding to the Bermuda station in the Atlantic. Most interestingly, all perturbations appear in forms of the second dynamic thermocline mode like features, Fig. a.

14 Fig.. Density and potential vorticity profiles at a station near the western boundary. The solid line is for the standard case; the crosses are for the case with surface warming; the thin solid line is for the case with surface cooling; the thin dotted line is for the case with intergyre boundary moved to 5 o N, and same Ekman pumping rate as the standard case; the long dashed line depicts the case with the intergyre boundary moved to 5 o N, and the Ekman pumping rate increased 50%. C. Localized cooling In one numerical experiment, the model ocean is cooled within a patch near 40 o E and y0 = 0 o N. This cooling induces a depression of the sea surface elevation, Fig. 4a, a shoaling of the isopycnal surface σ = 6. (about 0m); however, both the deep isopycnal σ = 7 moves down 0m and the base of the wind-driven gyre moves down more than 70 m. Here again, the perturbations appear in forms of the second dynamic thermocline mode. The vertical structure of the perturbation can be seen better through a vertical profile taken at the center of the perturbations, Fig. 5. Fig. 4. Perturbation of the mean fields under a localized cooling: a) Sea surface elevation (in cm); b) depth of isopycnal surface σ = 6. (in m); c) depth of isopycnal surface σ = 7 (in m); d) depth of the base of the wind-driven gyre (in m). 4

15 Fig. 5. A density profile at 0 o N, 40 o E ; a) the vertical displacement of isopycnals due to cooling; b) the density profiles, with the solid line for the standard case and the dashed line for the case with cooling. The thin dashed line indicates the background stratification. D. Point source of cooling A most interesting question is what happens when the size of the cooling anomaly gradually shrinks to a δ -function like point source. Within the relative crude horizontal resolution of this model, we carried several numerical experiments with point source of cooling or mixed layer depth anomaly. The perturbations induced by a point of cooling source have a wave train-like feature, Fig. 6. This picture suggests that perturbations are the combination of lower-order modes and higher-order modes, with the lower-order modes dominate. Such complicated spatial structure is not inconsistent with the alternating signs shown in Fig. 6. 5

16 Fig. 6. Perturbation of the mean fields from a model with 8 grid points and under a localized cooling along a single outcrop line, with δ y = 0.05 o, and a width of δ x = 0. o. The western edge of the perturbation zone is defined by the streamline on the isopycnal surface of the primary characteristic due to cooling; the eastern edge of the perturbation zone is defined by the envelope of the streamline on the sea surface. E. Changes in subpolar mode water formation Subpolar mode water formation rate changes on time scale from interannual to millennia. As a result, the background stratification, or the potential vorticity for the unventilated water changes in response. We explored this issue, using our model under the same forcing, except alternating the potential thickness for the unventilated thermocline, as indicated by the solid line in the right part of Fig. 7. With this change in the potential thickness of the unventilated thermocline, all ventilated and unventilated isopycnals move down in response. Results from our model are quite consistent with analysis based on historical data, Joyce et al. (999). 6

17 Fig. 7. Depth perturbation at a station ( 8.5 o N, 0 o E ) in the model basin. The solid line indicates the vertical displacements on each isopycnal, and the thin line indicates thickness perturbation in each isopycnal layer ( Δ ρ = 0.σ ).. Theory of planetary wave propagation in the thermocline Decadal climate variability can be explained in terms of planetary waves, our discussion below is adapted from the papers by Liu (999). The thermocline can be examined through a.5 layer model, and the corresponding quasi-geostrophic potential vorticity equations are t + J( ψ, ) q = ( we wd) / d (a) t + J( ψ, ) q = wd / d (b) q y ψ ψ / d q = y+ ψ ψ / d are potential vorticity in each layer, where = + ( ) and ( ) d = D /( D + D ) and d D / ( D D ) = + are the non-dimensional layer thickness. Other nondimensional variables are defined in terms of the corresponding dimensional variables (with * ) as x, y = x*, y* / L, t = t*/ T, ψ = ψ / V L (4) ( ) ( ) * * * we = we / W, wd = wd / W where L is the gyre scale, V ( L g' ( D D ) s n n s = β L is the velocity scale for the planetary waves, D D = +, g' = gδ ρ / ρ are the deformation radius and the reduced gravity), T = L/ Vs is the time scale for the gyre scale advection, W is the scale of vertical velocity that satisfies the Sverdrup relation β Vs ( D+ D) = f0w. Assume the disturbunces φ n are of small amplitude, i.e., ' ψn =Φ n( xy, ) + φn( xy, ), φn Φn ; similarly, we = We + w' e, wd = Wd + wd. The linearized perturbation potential vorticity equations are in forms + U + V q + J φ, Q = w w / d (5a) ( t x y) ( ) ( e d) ( t U x V y) q J( φ, Q) wd / d where ( Un, Vn) ( y n, x n), n, = (5b) = Φ Φ = are the mean currents, Q y ( ) / d = + ( Φ Φ )/ are the mean potential vorticity in each layer, and q ( φ φ ) ( φ φ )/ d Q y d q = + Φ Φ and = / d and = are potential vorticity perturbations. These equations can be solved using the WKB method. Therefore, for waves with scale (~00km) much smaller than the basin scale the potential vorticity and mean current of the basin circulation (~ thousands of km) can be treated as homogeneous medium. Therefore, we are looking for perturbation solutions in forms ' iθ( xyt,, ) ' iθ( xyt,, ) ' iθ( xyt,, ) φn = φne, n=,; we = we e, wd = wde (6) where the amplitude functions are assumed locally constant. After dropping the primes, the WKB solutions at the first order yields Uk+ Vl σ φ φ / d + φ R = i w w / d (7a) ( )( ) ( e d) ( Uk Vl σ)( φ φ )/ d φ R iw / d + + = (7b) d 7

18 where k = θ, l = θσ, = θ are local wave-numbers and frequency. x y t ( ) ( ) / ( ) ( ) R = k yq l xq = k + U U k+ V V l d (8a) R = k yq l xq = k + U U k + V V l / d (8b) represent the contribution due to mean advection. Assume a patch of anomalous forcing in forms of w ' i( mx+ ny ωt) = w e, w ' i( mx+ ny ωt) = w e. (9) e e d d We look for perturbations which has the same frequency and wavenumber. There are two types of solutions to the dispersion relation, the N mode and the A mode. The propagation of these modes can be explored using the ray tracing theory. As shown in Fig. 8, there are two types of waves. The N-Mode (the non-doppler mode) propagates westward, mostly un-affected by the mean current. Of course, this N mode is affected by the mean current in a slightly and complicated way; however, for the lowest order approximation, we can treat them as nearly non-interact with the mean current. These waves move westward, despite the eastward current in the northern part of the subtropical gyre. These waves resemble the first baroclinic Rossby waves discussed in the classical quasi-geostrophic theory, and they are closely related to the sea surface height anomaly and the thermocline depth. The other important mode is the A-mode, or the advective mode, which closely follow the current direction in the subtropical gyre, as shown in Fig. 8b. However, their speed can be noticeably different from the mean flow. These waves resemble the second baroclinic mode in the QG theory, and they are closely related to temperature anomaly in the upper part of the thermocline. Fig. 8. Wave rays in a ventilated thermocline. Six rays starte in the ventilated zone and four rays start within the shadow zone, as marked by the heavy dots. The forcing frequency is ω = (decadal) and the initial meridional wavenumber is n =. a) The non-doppler mode; b) the advection mode (Liu, 999). 4. Decal climate variability diagnosed from data and numerical models A. Climate variability diagnosed from data The early development along this line was due to Deser et al. (996). By analyzing the climate data, they were able to show how the temperature anomaly was able to penetrate to the subsurface layers, 8

19 Fig. 9. It is amazing to see that in decadal time scale how the temperature anomaly moves basically along isopycnal surface, as shown in Fig. 0. From Fig. 9, during the period from 977 to 989 temperature anomaly moved down about 60m, so the vertical velocity is slightly more than 0m/year, which is on the same order as the vertical velocity in this area of ocean, as inferred from the Ekman pumping rate on the order of 0m/y. Since Deser's work, there have been many papers published along this line, and they all show similar structure. Fig. 9. Evolution of the anomalous temperature in the central North Pacific region, after Deser et al Fig. 0. The 0. o C anomaly isotherms for (dashed line), (dotted line), and (dot-dashed line), super-imposed upon the mean late-winter isopycnals ( kg / m ; thin solid lines), after Deser et al B. Re-appearance of temperature anomaly in the upper ocean. 9

20 An interesting phenomenon identifiable from Fig. 9 is the reappearance of temperature anomaly in the upper ocean several years after the initial impact of a strong cool anomaly. Based on the ideas of ventilated thermocline theory, we can define an annual subduction depth Ds = SannT Where S ann is the annual mean subduction rate, and T = yr is the time duration in defining the annual mean subduction rate. The subduction depth ratio is defined as Rs = Ds/ hmax The distribution of this depth ratio for the North Pacific, inferred from an ideal-fluid thermocline model with continuous stratification is shown in the next figure. The inverse of this ratio is closely related to the time (in yr) for surface temperature anomaly can survive in the upper ocean, which can be seen by comparing this figure with the corresponding figures by Frankignoul and Reynolds (98). Using a simple one-dimensional model to analyze the thermal balance of the upper ocean heat content, it is readily seen that it takes about / R s yr to renew the water property in the mixed layer. In addition, it is clear that the optimal season to form the anomaly is late winter (March ) when the mixed layer is coldest and densest. In addition, only cold anomaly is able to survive because warm anomaly does not penetrate deep, and it will be wiped out in the following late winter of a normal year when late winter mixed layer is denser and deeper. Fig.. Subduction depth ratio for the North Pacific, inferred from an ideal-fluid thermocline model (Huang and Russell, 994). C) Climate variability inferred from numerical models The time evolution of the thermocline anomaly was demonstrated through a numerical experiment by Lazar et al. (00). The model ocean is forced by a thermal anomaly over the period of month, and allowed to run freely afterward. The temperature anomaly identified from the numerical model 0

21 was traced both on isopycnals (Fig. ) and a vertical section along the core of the anomaly (Fig. ). Note that perturbations appear in both positive and negative signs. Fig.. 0-year evolution of the temperature anomaly projected on σ = 5. of the reference run, for each other year in December, except for March of the first year; contour interval 0. o C. Bernoulli contour in green. Fig.. 0-year evolution of the temperature anomaly projected onto the vertical plane, along the trajectory of the core of the temperature anomaly. Density stratification of the reference (solid) and anomalous runs (dashed) in green; contour interval 0. o C. References Deser, C., M. A. Alexander, and M. S. Timlin: Upper-ocean thermal variations in the North Pacific during J. Climate,

22 Frankignoul, C. and R. W. Reynolds, 98. Testing a dynamical model for midlatitude sea surface temperature anomalies. J.Phys. Oceanogr., 7, -45. Huang, R. X., 000a. Parameter study of a continuously stratified model of the ideal-fluid thermocline. J. Phys. Oceanor, 0, Huang, R. X., 000b. Climate variability inferred from a continuously stratified model of the idealfluid thermocline. J. Phys. Oceanogr, 0, Huang, R. X. and J. Pedlosky, 999: Climate variability inferred from a layered model of the ventilated thermocline. J. Phys. Oceanogr., 9, Huang, R. X. and J. Pedlosky, 000: Climate variability induced by anomalous buoyancy forcing in a multilayer model of the ventilated thermocline. J. Phys. Oceanogr., 0, Joyce, T. M., R. S. Pickart, and R. C. Millard, 999: Long-term hydrographic changes at 5 and 6 o W in the North Atlantic subtropical gyre and Caribbean. Deep-Sea Res. II, 46, Lazar, A., R. Murtugudde, and A. Busalacchi, 00. A model study of oceanic temperature anomaly propagation from subtropics to tropics in the southern Atlantic thermocline. Geophy. Res. Lett., 8, Liu, Z.-Y., 999. Forced planetary wave response in a thermocline gyre. J. Phys. Oceanogr., 9, Luyten, J. R., J. Pedlosky, and H. Stommel, 98: The ventilated thermocline, J. Phys. Oceanogr.,, Pedlosky, J. and P. Robbins, 99: The role of finite mixed-layer thickness in the structure of the ventilated thermocline. J. Phys. Oceanogr.,, 08-9.