DECEMBER 2002 NOTES AND CORRESPONDENCE 3657

NOTES AND CORRESPONDENCE

Buoyancy and Mixed Layer Effects on the Sea Surface Height Response in an Isopycnal Model of the North Paci®c

LUANNE THOMPSON School of , University of Washington, Seattle, Washington

KATHRYN A. KELLY Applied Physics Laboratory, University of Washington, Seattle, Washington

DAVID DARR School of Oceanography, University of Washington, Seattle, Washington

ROBERT HALLBERG NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

16 August 1999 and 25 July 2002

ABSTRACT An isopycnal model of the North Paci®c is used to demonstrate that the seasonal cycle of heating and cooling and the resulting mixed layer depth entrainment and detrainment cycle play a role in the propagation of wind- driven Rossby waves. The model is forced by realistic winds and seasonal heat ¯ux to examine the interaction of nearly annual wind-driven Rossby waves with the seasonal mixed layer cycle. Comparison among four model runs, one adiabatic (without diapycnal mixing or explicit mixed layer dynamics), one diabatic (with diapycnal mixing and explicit mixed layer dynamics), one with the seasonal cycle of heating only, and one with only variable winds suggests that mixed layer entrainment changes the structure of the response substantially, par- ticularly at midlatitudes. Speci®cally, the mixed layer seasonal cycle works against Ekman pumping in the forcing of ®rst-mode Rossby waves between 17Њ and 28ЊN. South of there the mixed layer seasonal cycle has little in¯uence on the Rossby waves, while in the north, seasonal Rossby waves do not propagate. To examine the ®rst baroclinic mode response in detail, a modal decomposition of the numerical model output is done. In addition, a comparison of the forcing by diapycnal pumping and Ekman pumping is done by a projection of Ekman pumping and diapycnal velocities on to the quasigeostrophic potential vorticity equation for each vertical mode. The ®rst baroclinic mode's forcing is split between Ekman pumping and diapycnal velocity at midlatitudes, providing an explanation for the changes in the response when a seasonal mixed layer response is included. This is con®rmed by doing a comparison of the modal decomposition in the four runs described above, and by calculation of the ®rst baroclinic mode response using the one-dimensional Rossby wave equation.

1. Introduction (Chelton and Schlax 1996; Vivier et al. 1999, VKT hereafter; Stammer 1997; Stammer et al. 1996, Plate 3). The TOPEX/Poseidon altimeter observations of SSH Studies in the North Paci®c show that the steric response (sea surface height) show a variety of responses to at- to heating is large at high latitudes (VKT; Stammer mospheric forcing and give information about how the 1997). A quasi-steady topographic Sverdrup balance is ocean adjusts to this forcing. The SSH observations can also found over the North Paci®c to latitudes as low as be explained primarily by the adjustment of the ocean 25ЊN. Equatorward of that, the ¯at-bottomed Sverdrup to seasonal variations of wind and heat-¯ux forcing balance is important (VKT). Forced ®rst-mode baro- clinic Rossby waves are seen to propagate south of about 40ЊN. Corresponding author address: Dr. LuAnne Thompson, School of Oceanography, University of Washington, Box 355351, Seattle, WA A feature of the observed Rossby waves not explained 98195-5351. by the simple models is apparent damping of Rossby E-mail: [email protected] waves as they propagate from east to west across the

᭧ 2002 American Meteorological Society 3658 JOURNAL OF VOLUME 32 basin. Qiu et al. (1997) show that Laplacian or bihar- appendix and has been used to study the seasonal cycle monic mixing should be added to the simple long-wave and water mass formation (Ladd and Thompson 2001). model for better agreement with SSH observations. This To maintain the model's free surface while increasing form of damping represents lateral mixing by eddies, the time step, the value of gravity at the free surface is and is scale selective. Another possible form of damping 10 times smaller than the actual value. The SSH re- is Newtonian cooling proportional to SSH in a reduced sponse is 10 times larger than what it would be in the gravity model. This formulation could represent simple ocean so that in all plots of SSH from the model, we thermal damping of the SSH, assuming SSH is propor- divide the magnitudes by 10. High-frequency barotropic tional to SST. Newtonian damping is also interpretable motions will not be well represented in the model, owing as potential vorticity mixing, indicative of vertical mix- to both the arti®cial value of g as well as the closed ing processes in the ocean. This suggests that diabatic domain. An experiment with the true value of g was effects may play a role in the dynamics of the response done and compared against the control run described of the ocean to variability in wind forcing. below for year 1992. The response is quantitatively sim- In reduced gravity models of the ocean's response to ilar, with rms SSH locally no more than 10% larger. wind forcing (Qiu et al. 1997), many physical processes The low-frequency (annual period) motions of interest are ignored; these include the coupling of the Rossby are very well represented by the approximation used waves to bottom topography (Killworth and Blundell here. In particular, quasi-steady barotropic topographic 1999), coupling among different vertical modes, and Sverdrup response on the annual period is well repro- vertical mixing processes. More sophisticated numerical duced in the model, as demonstrated by VKT. models include these effects but are dif®cult to interpret. The model domain is the North Paci®c between 13ЊS Here we focus on the in¯uence of diapycnal processes, and 60ЊN. Resolution is 2Њ in each direction. The hor- primarily the seasonal cycle of entrainment and detrain- izontal mixing is biharmonic (with the coef®cient set at ment of the mixed layer, on the large-scale response of 2 ϫ 1015 m 4 sϪ1), which favors mixing at smaller lateral the ocean to both wind and heat-¯ux forcing by doing scales. Thus, the solution is less viscously controlled a series of model runs with different forcing con®gu- than one with the numerically required value of Lapla- rations. The possibility of forcing of Rossby waves by cian mixing. The diapycnal mixing coef®cient is 1 ϫ diapycnal processes has been explored previously in the 10Ϫ5 m 2 sϪ1 unless otherwise noted. There is also hor- context of decadal variability (Liu 1999). With simpli- izontal thickness diffusion of 20 m2 sϪ1, required for ®ed geometry and forcing, Liu ®nds that Ekman pump- numerical stability. Bathymetry is taken from ETOPO ing forces the ®rst baroclinic mode, while heating and 60. The model is forced by National Centers for En- cooling at the surface primarily forces the second bar- vironmental Prediction (NCEP) daily wind ®elds, using oclinic mode. Here we quantify these results in a model perpetual 1992 winds to spin up the model for 20 years that has realistic forcing to determine if vigorous mixing from the September initial conditions based on Levitus associated with the seasonal evolution of the mixed lay- et al. (1994) and Levitus and Boyer (1994). Year 1992 er in¯uences the ®rst baroclinic mode evolution and thus is used for sensitivity studies. The forcing is the SSH in the ocean. a combination of da Silva et al. (1994) heat ¯uxes and The outline of the paper is as follows. The numerical a relaxation to Levitus et al. sea surface buoyancy. The model used in this study is described in section 2. Four timescale of the relaxation is 100 days for a mixed layer model runs are discussed in section 3. The dynamics at 100 m thick (Rothstein et al. 1998). The coef®cient of work are explored in the context of a quasigeostrophic thermal expansion varies with location and season so modal decomposition of the oceanic response in section that the equation of state is nonlinear in temperature. 4. Consequences of the results for the interpretation of Two different density layer con®gurations are used (Ta- the response of the ocean to seasonal forcing are dis- ble 1). In the ®rst case, an adiabatic simulation is done cussed in section 5. with 7 isopycnal layers. In the second, 8 isopycnal layers are used along with a buffer layer and a mixed layer, giving a total of 10 layers. 2. The isopycnal model formulation The model used for this study is the Hallberg Iso- 3. The role of diapycnal and mixed layer processes pycnal Model (Hallberg 2000; Hallberg and Rhines 1996; Ladd and Thompson 2001), an isopycnal model To elucidate the role of diapycnal processes in the similar to MICOM [New et al. (1995) and references response of the ocean to seasonal forcing, we present therein]. The model includes diapycnal mixing and an the results from four model runs, run 1: wind forcing embedded mixed layer. The mixed layer has Kraus± but no diapycnal processes either in the mixed layer or Turner physics with additional shear-dependent mixing the interior and no heat ¯ux forcing; run 2: heat ¯ux, below the mixed layer, and the entrainment algorithm wind forcing, and interior diapycnal mixing; run 3: wind used in MICOM. However, detrainment occurs through forcing but heat ¯ux is given by relaxation to the an- a buffer layer, an additional layer with variable density. nually averaged SST and wind mixing is turned off in The detrainment method is described in detail in the the mixed layer; and run 4: seasonal heat ¯ux forcing DECEMBER 2002 NOTES AND CORRESPONDENCE 3659

TABLE 1. Simulations. horizontal or vertical resolution is needed or the winds Layer Density that forced the model had too low amplitude. In addition, the relaxation part of the heat ¯ux could cause damping Adiabatic of the wind-driven signal, although we ®nd that this is 1 1021.5 not the case here. 2 1023.5 3 1024.5 Of course, the response at the sea surface is not the 4 1026.0 only feature of the model ocean that changes when dia- 5 1026.75 batic processes are included. There are profound dif- 6 1027.25 ferences in the density structure (Fig. 2). In run 1, the 7 1027.25 density does not show a signi®cant seasonal cycle. In Diabatic addition, there are regions where the isopycnals are 1 (mixed layer) 1022.8 packed together (see for instance Fig. 2c at 140ЊE). Wa- 2 (buffer layer) 1022.9 ter can be moved to different parts of the basin within 3 1023.5 4 1024.5 the same layer but cannot cross layer interfaces. Con- 5 1025 vergences and divergences control the density structure 6 1025.5 and create regions with large vertical density gradients. 7 1026 In run 2, however, the density structure must be com- 8 1026.5 9 1027.5 patible with the surface buoyancy forcing. Density sec- 10 1028 tions from the Levitus et al. climatology show layer thicknesses that are more uniform with longitude, with isopycnals depressed in the west, showing reasonable and annual mean . For run 3 and run 4, the agreement with run 2. The mixed layer tends to be too models were run for several years from the spunup ver- deep in the model, most likely owing to incompatible sion of run 2. This ensures that the mean density struc- heat ¯uxes and a lack of density resolution in the ther- ture is similar to that in run 2. In general, the model mocline, especially at low latitudes and the Kraus±Turn- runs do not reproduce the smallest zonal scales seen in er mixed layer model formulation (Ladd and Thompson the observations (Chelton and Schlax 1996; Stammer 2001). 1997; Stammer et al. 1996). This occurs for two reasons: Run 2 results are very robust to change of interior one is that the model is low resolution and the other is mixing values. When the interior diapycnal mixing is that there may not be suf®cient energy at small scales 10 times larger, and 10 times smaller, the SSH is very in the wind ®elds. similar. It appears that it is the mixed layer evolution We ®rst examine the Rossby waves in runs 1 and 2. that plays a large role in how the Rossby waves prop- By Rossby waves, we mean the ®rst baroclinic mode agate and evolve in the model rather than interior dia- Rossby waves, identi®able by their westward phase pycnal mixing. Two additional runs were done to show propagation and distinct phase speed. A comparison that the relaxation part of the buoyancy ¯ux as well as shows that Rossby waves propagate more freely in run thickness diffusion have very little impact on the evo- 1 at all latitudes and the response is larger (Fig. 1). For lution of the SSH. both model runs, the rms SSH is lower than in the ob- In run 3 with heating only, we ®nd that the rms SSH servations at low latitudes and comparable to the ob- is relatively small, ensuring that the differences that we servations at midlatitudes. VKT showed that poleward see between runs 1 and 2 are not simply because of a of 25ЊN, the barotropic topographic Sverdrup balance seasonal steric response (Fig. 3). There is however a is important and this part of the response appears to be striking difference between run 2 and run 4 from 18ЊN well simulated in the model. In run 2 Rossby wave to 38ЊN where the run 4 response is almost twice as generation and propagation is inhibited relative to that large as the run 2 response. Elsewhere, the two simu- in run 1 at midlatitudes. In fact, the rms model response lations are very similar, and both give smaller responses in run 1 is as large or larger than in the observations at when compared to run 1. midlatitudes, despite the lack of a seasonal steric re- sponse. This result echoes that found by Qiu et al. (1997) 4. Modal decomposition and the forcing of Rossby in a one-dimensional wave equation where mixing was waves by diapycnal processes needed to better reproduce the observations. The rms SSH is about a factor of 2 smaller than the So the question becomes, what is it about midlatitudes observations in run 2 when the comparison is made after when the seasonal cycle in heating and cooling is in- applying a correction for non-Boussinesq effects sug- cluded that causes the changes in the oceanic response? gested by Greatbatch (1994) (see also Stammer 1997), We know from previous work that north of about 40ЊN, proportional to the area-averaged heating. In the higher- the barotropic and steric response dominate on seasonal resolution simulations of Stammer et al. (1996), the rms time scales (VKT) and neither of these responses should response is also a factor of 2 too small when compared be in¯uenced by diapycnal pumping that results from with the observations. This result suggests that higher the entrainment/detrainment cycle of the mixed layer. 3660 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 N: (a)±(d) adiabatic simulation (run 1) (contour interval 10 cm), and Њ , and 40 Њ ,30 Њ ,20 Њ (e)±(h) diabatic simulation (run 2) (contour interval 5 cm). . 1. Time±longitude plots model of SSH for years 1993±96 at 10 IG F DECEMBER 2002 NOTES AND CORRESPONDENCE 3661 ) Continued .1.( IG F 3662 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

FIG. 2. Layer interfaces at 22ЊN for winter and summer: (a) and (b) run 2, (c) and (d) run 1, and (e) and (f) Levitus et al. (1994) observations. Mixed layer base is shown in (a) and (b) and (e) and (f) as a dark line. DECEMBER 2002 NOTES AND CORRESPONDENCE 3663

for the bottom layer, where m is the number of layers. In this framework, the streamfunction is related to in- terface displacements by

nϪ1 f ␾ ϭ ghˆ ϩ fhˆ . (4) n 0 ͸ nn iϭ1

Here Hn is the mean thickness of layer n, ␾n is the streamfunction deviation from the mean streamfunction

in layer n, and gn is the reduced gravity between layer n and layer n Ϫ 1 taken from the mean density pro®le.

Also, hˆ n is the interface displacement of the bottom of layer n from its mean position. In matrix notation, (1)±(3) can be written 1 L⌽ ϭϪ ⌽. (5) FIG. 3. Rms SSH anomaly (cm) zonally averaged. Solid line is run c2 1, dot±dashed is run 2, dashed line is run 3, and small dotted line is run 4. Here L is a matrix containing the operators in the ®rst two terms of (1) and (2) and the ®rst term in (3) and ⌽ is a vector containing the vertical structure of the However, the baroclinic modes could be in¯uenced by streamfunction for a particular mode. Equation (5) is diapycnal pumping (Liu 1999). To answer the above solved as an eigenvalue problem for cϪ2. The eigen- question, a modal decomposition is done to determine vectors are sorted by their phase speeds, with the most the relative contributions to the SSH from each vertical rapid phase speed being the barotropic mode. In run 2, mode and then to examine the response of the ®rst bar- for the ®rst two baroclinic modes the structure is smooth oclinic mode under different forcing conditions. This and shows no obvious discontinuities that would suggest allows the separation of the seasonal heating and bar- lack of vertical resolution (Fig. 4), while the third bar- otropic response from the Rossby waves. oclinic mode starts to show the ®nite number of layers To do the modal decomposition, the annual mean in the calculation. Note that the density resolution is strati®cation (layer thicknesses) and densities are used concentrated in the upper 500 m. The phase speed of at each point. Care must be taken in constructing the the ®rst baroclinic mode gravity wave shows reasonable annual mean strati®cation when layers vanish. Using correspondence with other calculations (cf. Killworth et the model results, a mean distribution of layer thick- al. 1997), although it is slightly larger than their estimate nesses is constructed such that each layer is of ®nite based on the Levitus et al. data. thickness, removing layers when they are less than 10 Since the higher vertical modes are not well repre- m from the mean ®eld by combining layers. This results sented by the limited vertical resolution in the model, in some regions of the model ocean where fewer than we will concentrate our discussion on the response in 10 layers are used for the modal calculations. Interface the barotropic and the ®rst two baroclinic modes. We deviations are constructed to be consistent with this al- have ignored the impact of vertical shear on the modal gorithm. We do the model decomposition locally, as- structure and phase speeds. According to the calculation suming that the motions are linear, and thus make the of Killworth et al. (1997), the modal structure for the WKB approximation. A dynamically consistent way to ®rst baroclinic mode is relatively unin¯uenced by the decompose the response in the model is to use the ver- zonal vertical shear, although the phase speed can differ tical structure equation [see section 2 of Killworth et from the traditional phase speed by as much as factor al. (1997) for the continuously strati®ed version of this]. of 2. For the analysis done here, we do not use the phase In the layered framework, the vertical structure equation speed directly, so our results should be relatively robust and the boundary conditions reduce to to the exclusion of the mean ¯ow ®eld, particularly for ␾ Ϫ ␾␾ 1 the ®rst baroclinic mode. In addition, with the inclusion 12ϩϩ 1 ␾ ϭ 0 (1) of the mean ¯ow, the modes are no longer complete or 2 1 Hg11 Hg 1 c orthogonal. for the top layer, First, we decompose the SSH response using the ver- tical structure of the streamfunction anomalies to de- (␾ Ϫ ␾ )(␾ Ϫ ␾ )1 nnϩ1 ϩϩnnϪ1 ␾ ϭ 0 (2) termine how each mode contributes to the SSH varia- 2 n Hgnn Hg nnϪ1 c tions. Since the vertical modes are orthogonal and com- plete by construction, we can write the streamfunction for the nth interior layer, and in any layer and at any point as a sum of contributions (␾ Ϫ ␾ )1 from each vertical mode: mmϪ1 ϩ ␾ ϭ 0 (3) 2 m HgmmϪ1 c ⌿ ϭ Ra, (6) 3664 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

FIG. 4. Nondimensional vertical structure for the ®rst four vertical modes from run 2 in (a)±(d) streamfunction and (e)±(h) interface displacement at 22ЊN, 180Њ. where R is the matrix whose columns are made up of VKT while the second baroclinic mode is larger in the the eigenvectors ⌽ that represent the vertical structure Kuroshio Extension and subpolar gyre (not shown). of each mode and a is a vector that represents the time- The modal decomposition for runs 2, 3, and 4 shows dependent magnitude of each mode. The vector ⌿ con- the destructive interference of the Ekman pumping and tains the model streamfunction as a function of time. diabatic pumping response (Fig. 6). The ®rst baroclinic At each time step a is found from (6) and the model mode response in run 4 is larger than the response in streamfunction deviations using run 2 or in run 3. This suggests that the wind-driven response and the diabatic response destructively inter- a ϭ RϪ1⌿. (7) fere to lower the total (run 2) response. This interference To determine the contribution to the sea surface height is largest between 20Њ and 35ЊN. In contrast, for the for each mode, we ®nd how much each vertical mode second baroclinic mode there is constructive interfer- contributes to the stream function in layer 1. There the ence poleward of 25ЊN. Note that the second baroclinic streamfunction is related to the SSH (␩)by mode response is larger than the ®rst baroclinic mode response under thermodynamic forcing alone (run 3) gg ␩ ϭ ␺ ϭ Ra. (8) since its vertical structure projects better onto the ther- 11͸ jj ff00jϭ1,m modynamic forcing. To study how each mode contributes to the SSH, we To determine how the diapycnal processes in¯uence each vertical mode in the model, a dynamical interpre- examine R1j aj for j ϭ 1, 3 (the barotropic and ®rst two baroclinic modes). Each of the vertical modes contrib- tation must be made. To do this, we use quasigeostrophic utes different spatial and temporal structure to the SSH theory. We have assumed that the motion is linear and response. The barotropic mode is quite noisy, but does has large horizontal length scales so that the relative vor- show structure consistent with a barotropic Sverdrup- ticity can be ignored. Mixing is written as a diapycnal like response as a zonally uniform and nonpropagating ¯ux at the top and bottom of each layer. The continuity signal (Fig. 5a). The ®rst baroclinic mode shows prop- equation for each layer can then be written as agation and is the mode that we identify with Rossby ץ .waves that appear so strongly in the observations (Fig (u h ) ϭ w Ϫ w , (9)´ ١ h ϩ t nnnnϪ1 nץ -5b). The second baroclinic modes have slower propa gation and a weaker response (Fig. 5c). The contribution to the SSH from the ®rst baroclinic mode shows western where hn is the thickness of layer n and wn is the diabatic intensi®cation and has similar structure to that found by velocity or ¯ux across the interface at the bottom of DECEMBER 2002 NOTES AND CORRESPONDENCE 3665

FIG. 6. Zonally averaged rms SSH anomaly (cm) from each vertical N to SSH anomaly for year 1992 from run 2 for each of the vertical modes: (a) barotropic mode (contour interval 1.25 cm),

Њ mode from run 2 (solid), run 3 (dashed), and run 4 (dotted): (a) ®rst baroclinic mode and (b) second baroclinic mode.

layer n. The linear layer quasigeostrophic potential vor- ticity (PV) equations are (b) ®rst baroclinic mode (contour interval 1.25 cm), and (c) second baroclinic mode (contour interval 0.375 cm).

␺ 1fץ␺21Ϫ ␺␺ 1 ץ Ϫϩ␤ ϭ (wEkϪ w 1) (10) xH 1ץtHg΂΃11 Hg 1ץ for layer 1,

␺ץ ␺ Ϫ ␺␺Ϫ ␺ ץ nϩ1 nnϩϩϪ1 nn␤ xץ tHg΂΃nn Hg nnϪ1ץ

. 5. Time±longitude plots for contributions at 22 f IG ϭ (w Ϫ w ) (11) F nϪ1 n Hn for interior layers, and 3666 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

␺ fץ␺mϪ1 Ϫ ␺mm ץ ϩ ␤ ϭ wmϪ1 (12) xHmץ tHg΂΃mmϪ1ץ for the bottom layer. In setting up (10)±(12), we have assumed that the Ekman layer is much thinner than the mixed layer so that the mean mixed layer can be treated as layer one in the above equations. This means that the Ekman pumping acts on the top of the ®rst layer. This is not strictly accurate since in the model the wind acts as a body force in the mixed layer; thus the mixed layer and Ekman layer are the same depth. Treating the mixed layer with separate dynamics from the interior layers would introduce additional complexity and is not war- ranted here. Note that Ekman pumping in the layered model is not a diabatic vertical velocity in that it does not cause ¯uid to pass between layers; thus it does not appear in the diapycnal velocity diagnosed from the model. It does, however, directly modify PV in the top layer. In contrast, the diapycnal velocity causes mass redistribution between the layers, and affects the PV in each layer. In this case, most of the diapycnal mixing comes about from entrainment and detrainment at the base of the mixed layer. The interior diapycnal velocities that are diagnosed from the model are assigned to the appropriate interface when layers vanish using the same algorithm as described above for the construction of the interface deviations. We also continue to ignore vertical shear in the background ¯ow ®eld. The PV equations for each layer (10)±(12) can be written ⌿ץ ⌿ץ L ϩ ␤ ϭ F, (13) xץ tץ where ⌿ is a vector representing the streamfunction in each layer and F is the rhs of (10)±(12). To transform (13) to PV equations for each vertical mode, we ®rst note that we can write (5) as FIG. 7. Rms contribution to the forcing of the baroclinic modes by LR ϭϪRC, (14) Ekman pumping (solid) and diabatic pumping (dashed) in run 2: (a) ®rst baroclinic mode and (b) second baroclinic mode. where C is a matrix containing the eigenvalues (c Ϫ2 ) of (5) as elements of the diagonal. Then we can write (14) as and forcing of the potential vorticity for each vertical HLR ϭϪHRC, (14) mode, in terms of the vertical divergences (F) where now the jth element of RTH⌿ is the potential vorticity where H is a matrix whose diagonal elements are the of the jth vertical mode. We have assumed that R and thickness of each layer. If we then take the transpose H are independent of time and that they vary slowly in of each side of (14), we ®nd that the zonal direction. RHLTTϭϪCR H (15) The rms contribution of the Ekman pumping and dia- pycnal velocities to the PV forcing of each vertical mode since HL is symmetric (this is another way of saying as calculated from the rhs of (16) can then be compared that the operator is self-adjoint). Thus, we can transform (Fig. 7). For the ®rst baroclinic mode, the Ekman pump- T (13) by multiplying by R H and using (15) to get ing dominates everywhere, while diapycnal pumping ⌿ plays an important role between 30Њ and 40ЊN near theץ ⌿ץ ϪCRT H ϩ ␤RHT ϭ RHFT . (16) Kuroshio Extension (Fig. 7a). The diabatic pumping is xץ tץ large just in the region where there is the largest dif- We have created an equation that de®nes the propagation ference between the wind only run and the control run DECEMBER 2002 NOTES AND CORRESPONDENCE 3667

FIG. 8. Ratio of transfer coef®cient. Solid line is ratio of transfer FIG. 9. Correlation coef®cients between ®rst baroclinic mode re- coef®cient of the run 4 to run 2 (zonally averaged), and dashed line sponse in the numerical model run 2 and that from the 1D wave is the ratio of rms SSH of run 4 to run 2 (zonally averaged). equation (solid line), and correlation coef®cient between the the dia- batic pumping response and the Ekman pumping response from the 1D wave equation (dashed line). (cf. Fig. 7 to Fig. 3). For the second baroclinic mode, Ekman pumping and diabatic pumping are coincident and of similar magnitude with the maximum in the Ku- model (Fig. 8). Between 17Њ and 28ЊN, the diapycnal roshio Extension (Fig. 7b). One would not expect a one- pumping response is anticorrelated with the Ekman to-one correspondence between the energy in the modes pumping response, resulting in a total wave ®eld that and the forcing patterns. Rossby wave energy locally is smaller than the wave ®eld forced by Ekman pumping depends on the accumulation of forcing as the Rossby alone. wave propagates to the west. There are several caveats that should be made about We can de®ne a transfer coef®cient for Ekman pump- this analysis. In the modal analysis, we have linearized ing for each vertical mode taken from (16) that deter- about the mean strati®cation, which does not take into mines the magnitude of the forcing for mode given by account the order one changes in the density structure 2 f 0␾n(0)cn . A question arises whether the changes in the that occur throughout the year in the upper ocean. The strati®cation in the various experiments that were dis- one-dimensional model does not do very well in pre- cussed in section 3 could explain the changes in the dicting the response of the full model poleward of 45Њ amplitude of the response rather than diapycnal pump- where the annual Rossby wave is most likely evanes- ing. The ratio of the transfer coef®cient for run 2 and cent, and VKT found very little Rossby wave response. run 4 shows that the expected wind-driven response is In this region we expect that the one-dimensional model approximately the same in both runs (Fig. 8). However, should fail. In addition, in this region the mean ¯ow is the ratio of the response is signi®cantly different at mid- strongly eastward, and the wave propagation character- latitudes, con®rming the role of diapycnal pumping in istic would be signi®cantly modi®ed so that the one- the evolution of the ®rst baroclinic mode. dimensional model would not be valid. There is no guarantee that diapycnal forcing will The higher vertical modes are strongly in¯uenced by cause damping (see, e.g., Liu 1999). To ascertain its diapycnal processes in this analysis, which suggests that effect on the ®rst baroclinic mode more directly, we they would be strongly forced by diapycnal processes, solve the one-dimensional wave equation (16) using the much more so than by Ekman pumping. However, the forcing ®elds calculated from its right-hand side and the higher modes are less well represented in the model, phase speed calculated from the modal analysis. The and the mean velocity ®eld is more important in their equation is solved by an upwind differencing scheme evolution than it is for the ®rst baroclinic mode. The and then is compared against the modal calculation. In effects of the mean ¯ow on the propagation character- general, the amplitude found from the one-dimensional istics of the higher modes will be explored in a sub- model is about a factor of 2 smaller than found from sequent paper. the modal analysis. This can be explained by the errors in the wave equation associated with the presence of 5. Conclusions the mean ¯ow related to the factor-of-2 difference in the phase speed from that of a resting ocean at midlat- Model simulations of the SSH response to wind forc- itudes (as show by Killworth et al. 1997). Barring the ing show that diapycnal processes and mixed layer evo- differences in amplitude, south of 30ЊN, the wave equa- lution are important in the dynamics of the SSH re- tion reproduces the modal time/longitude behavior and sponse, particularly at mid- to high latitudes. Typically it is well correlated with the results from the numerical the propagation of the ®rst baroclinic mode Rossby 3668 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 waves has been studied in simple models without the and M. Schlax for mapped altimetric ®elds. We thank inclusion of diapycnal processes and mixed layer phys- Helma Lindow for some early model runs. Funding was ics, and thus the adjustment of the ocean to seasonal provided by NASA Contract 960888 with the Jet Pro- changes in wind stress has been decoupled from the pulsion Laboratory (TOPEX/Poseidon Extended Mis- seasonal cycle of heating and cooling. The simulations sion Science Working Team) and an NSF grant to show that, in particular, the presence of the seasonal Thompson and Kelly. cycle mixed layer entrainment/detrainment can be im- portant in oceanic adjustment at midlatitudes. In an adi- APPENDIX abatic simulation (run 1), Rossby waves forced near the eastern boundary propagate without reduction in am- The Mixed Layer Formulation plitude, and an obvious seasonal cycle is lost. In con- The model uses a bulk Kraus±Turner (1969) type trast, in run 2 Rossby waves are less vigorous. We are mixed layer, much like that of Bleck et al. (1989) except argue here that the mixed layer seasonal cycle, and the that a variable density buffer layer is used to receive associated diapycnal velocities, is important for chang- the ¯uid detrained from the mixed layer. ing the Rossby waves. The mixed layer depth is determined by energy con- A modal decomposition of the model interface dis- siderations. If there is turbulent kinetic energy available placements shows that, as expected, the barotropic mode to drive mixed layer deepening, it does so entraining dominates the SSH signal at high latitudes, with the ®rst successively from the lightest layers that contain any baroclinic mode important at mid to low latitudes and mass. Alternately, if there is net surface loss of buoy- higher vertical modes not contributing as much. The ancy, the mixed layer may entrain due to free convec- model decomposition shows that the ®rst baroclinic tion. Otherwise, the mixed layer detrains to the Monin± mode is suppressed at midlatitudes by diapycnal pump- Obukhov depthÐthe depth at which wind generated tur- ing. In contrast, the second baroclinic mode is enhanced bulent kinetic energy balances the work required to mix by diapycnal pumping. the surface buoyancy forcing throughout the mixed lay- The results from a one-dimensional wave equation er. are consistent with the modal analysis and show that Fluid that is detrained from the mixed layer goes into the diabatic response and Ekman pumping response are a variable density buffer layer. This avoids the necessity opposite between 17Њ and 28ЊN for the ®rst baroclinic of having a ``fossil mixed layer'' that is appended to mode. The resulting signal is weaker than the Ekman the real mixed layer, as in Bleck et al. (1989). The details pumping signal alone. In addition, the one-dimensional of this buffer layer differ from those of Murdegudde et model has good predictive skill south of 30ЊN, while al. (1995), but the layer is conceptually similar. For north of 45ЊN, the skill is very poor indeed, suggesting example, the scheme described here conserves heat that the linearized potential vorticity equation is not while Murdegudde's does not. If there is an isopycnal valid in this region. A more complete analysis would layer that is intermediate in density between the mixed include the effects of the mean ¯ow on the propagation layer and the buffer layer, it is possible to split the buffer pathways of the waves. layer while maintaining a stable density pro®le. Bleck Higher resolution in the model, both vertical and hor- et al. are careful to insure that the fossil mixed layer izontal, would help with a more faithful representation does not alter the surface temperature or salinity ten- of the observations. In particular, the choice of density dency due to vertical processes. But mixed layer thick- resolution in¯uences how well the strati®cation can be ness, temperature, and salinity tendencies due to hori- represented. The vertical structure in these model sim- zontal within the mixed layer are unavoidably ulations was chosen to maximize resolution in the sub- affected by the presence of a fossil mixed layer. The tropics, which leaves the resolution in the Tropics lack- mixed layer/buffer layer approach used here gives a ing near the surface. This leads to mixed layers that are more accurate representation of the effects of horizontal too deep, and vertical mixing that is unrealistically large. advection on the mixed layer properties. Likewise, the lack of horizontal resolution affects the The speci®c calculation of the one-dimensional mixed northwestern Paci®c, where the Kuroshio Extension is layer evolution is as follows. too far north, and where the mixed layers are too deep. This study shows that Rossby waves, which are tra- 1) If there is a net surface loss of buoyancy from the ditionally thought to be purely adiabatic, can be greatly ocean, the mixed layer becomes denser. in¯uenced by diabatic processes. This has been explored 2) The water column convectively adjusts. Any mass for decadal variability (Liu 1999), but it is on the sea- in layers lighter than the mixed layer is entrained sonal cycle that the dynamic impact of diapycnal ve- into the mixed layer. Penetrative seems locity associated with the seasonal cycle of heating and to be fairly unimportant in the ocean (Marshall and cooling is the largest. Schott 1999), so none of the potential energy re- leased through convective adjustment is retained to Acknowledgments. We acknowledge the TOPEX/Po- drive further entrainment. seidon Extended Mission project at JPL and D. Chelton 3) As prescribed by Kraus and Turner (1969) or Bleck DECEMBER 2002 NOTES AND CORRESPONDENCE 3669

et al. (1989), turbulent kinetic energy (with a spec- The mixed layer and buffer layer obey the same i®ed surface source) is used to drive mixed layer momentum equations as all of the interior isopycnal entrainment if there is a net surface buoyancy loss layers, except for the inclusion of an additional pres- or if the mixed layer is shallower than the Monin± sure gradient term. When the momentum equations, Obukhov depth, h ϭ 2m u3 /B . Here u*isthe uץ MO 0 * 0 ϫ u) ϫ u ١ surface friction velocity, B 0 is the total surface buoy- ϩ ( fkˆ ϩ t Sץ ancy ¯ux into the ocean and m 0 is a nondimensional constant, 0.5. The entrainment is essentially the same p 1 ϩ gz ϩ u ´ u as described in Bleck et al. The mixed layer entrains ϭϪ١ from successively denser layers until the increase in S ΂΃␳ 2 the potential energy due to the entrainment equals p (␳ ϩ viscosity (A5 ١ Ϫ the surface energy available to drive mixing: ␳2 S h (written in standard notation, with the subscript S TKE ϭ⌬tmu*3 Ϫ 0 [(B ϩ |B |) 0004 emphasizing that the gradients are taken along co- Ά ordinate surfaces, and only the horizontal compo- nents of the velocities are used, consistent with the ϩ n(B00Ϫ |B |)] , (A1) hydrostatic approximation), are evaluated in layers · of nonconstant density, the middle term on the right-

where ⌬t is the time increment, h0 is the mixed layer hand side does not vanish. thickness at the start of the entrainment, and n is The average through the layer of this term is another nondimensional constant, here 0.03. It is easy to show that the potential energy increase due to an pgh (␳, (A6 ١ ␳ ഠ Ϫ ␩ ϩ ١ Ϫ increment ⌬h of entrainment by the mixed layer is 2 SS ␳␳0 ΂΃2 1 ⌬PE ϭ h ⌬h(b Ϫ b ), (A2) where ␩ is the height of the interface above the layer 2 0MLEnt and h is the thickness of the layer. The natural ex- pression of this term on a C grid is not obvious, but where bML is the mixed layer buoyancy and bEnt is the buoyancy of the entrained ¯uid. Entrainment a ®nite-element interpretation suggests the discreti- continues until ⌬PE ϭ TKE. The surface TKE avail- zation able to drive mixing does not decay with depth in ␳ץgh the current model. Ϫ ␩ ϩ xΗSץ If the mixed layer is deeper than the Monin±Obu- ␳ 0 ΂΃2 (4 khov depth, the mixed layer detrains into the buffer ghhϩϪϩ ␩ ϩϪh ϩ ␩ Ϫϩh ␳ ϩϪ ␳ Ϫ layer until the mixed layer thickness agrees with the Ϫ , (A7) ഠ ϩϪ Monin±Obukhov depth and the mixed layer buoy- ␳ 0 h ϩ h ⌬x ancy increases due to any net positive buoyancy forc- where the superscripts ϩ and Ϫ indicate the points ing. to the east and west of the velocity point where this 5) If there is an isopycnal layer that is intermediate in term is being evaluated. This discretization is used density between the buffer layer and the mixed layer in the mixed layer calculations. (and there is any ¯uid in the buffer layer), the buffer layer is detrained. It is partitioned into a portion that matches the density of the intermediate density layer REFERENCES and a portion that matches the density of the next denser layer. If layer k is intermediate in density Bleck, R., H. P. Hansen, D. Hu, and E. B. Kraus, 1989: Mixed layer± between the buffer layer and the mixed layer, the interaction in a three-dimensional isopycnic coor- amount of ¯uid that goes into layers k and k ϩ 1 is, dinate model. J. Phys. Oceanogr., 19, 1417±1439. Chelton, D. B., and M. G. Schlax, 1996: Global observations of respectively, oceanic Rossby waves. Science, 272, 234±238. ␳ Ϫ ␳ da Silva, A. M., C. C. Young, and S. Levitus, 1994: Algorithms and Bk Procedures. Vol. 1, Atlas of Surface Marine Data 1994, NOAA ␻kϩ1 ϭ hB and ␳kϩ1 Ϫ ␳k Atlas NESDIS 6, 51 pp. Greatbatch, R. J., 1994: A note on the representation of steric sea ␳kϩ1 Ϫ ␳B level in models that conserve volume rather than mass. J. Geo- ␻kBϭ h . (A3) phys. Res., 99, 12 767±12 771. ␳kϩ1 Ϫ ␳k Hallberg, R., 2000: Time integration of diapycnal diffusion and Rich- Whenever ¯uid moves across interfaces between lay- ardson number dependent mixing in isopycnal coordinate ocean models. Mon. Wea. Rev., 128, 1402±1419. ers (except during the partitioning of the buffer ÐÐ, and P. Rhines, 1996: Buoyancy-driven circulation in an ocean layer), it does so with the velocity and tracer con- basin with isopycnals intersecting the sloping boundary. J. Phys. centrations of the layer from where it came. Oceanogr., 26, 913±940. 3670 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32

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