notes and correspondence buoyancy and mixed layer …

DECEMBER 2002 3657N O T E S A N D C O R R E S P O N D E N C E

q 2002 American Meteorological Society

NOTES AND CORRESPONDENCE

Buoyancy and Mixed Layer Effects on the Sea Surface Height Response in anIsopycnal Model of the North Pacific

LUANNE THOMPSON

School of Oceanography, 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 Pacific is used to demonstrate that the seasonal cycle of heating and coolingand 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 flux to examine the interactionof nearly annual wind-driven Rossby waves with the seasonal mixed layer cycle. Comparison among four modelruns, one adiabatic (without diapycnal mixing or explicit mixed layer dynamics), one diabatic (with diapycnalmixing and explicit mixed layer dynamics), one with the seasonal cycle of heating only, and one with onlyvariable winds suggests that mixed layer entrainment changes the structure of the response substantially, par-ticularly at midlatitudes. Specifically, the mixed layer seasonal cycle works against Ekman pumping in theforcing of first-mode Rossby waves between 178 and 288N. South of there the mixed layer seasonal cycle haslittle influence on the Rossby waves, while in the north, seasonal Rossby waves do not propagate. To examinethe first baroclinic mode response in detail, a modal decomposition of the numerical model output is done. Inaddition, a comparison of the forcing by diapycnal pumping and Ekman pumping is done by a projection ofEkman pumping and diapycnal velocities on to the quasigeostrophic potential vorticity equation for each verticalmode. The first 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 confirmed by doing a comparison of the modal decomposition in the four runs described above, and bycalculation of the first baroclinic mode Rossby wave response using the one-dimensional Rossby wave equation.

1. Introduction

The TOPEX/Poseidon altimeter observations of SSH(sea surface height) show a variety of responses to at-mospheric forcing and give information about how theocean adjusts to this forcing. The SSH observations canbe explained primarily by the adjustment of the oceanto seasonal variations of wind and heat-flux forcing

Corresponding author address: Dr. LuAnne Thompson, School ofOceanography, University of Washington, Box 355351, Seattle, WA98195-5351.E-mail: [email protected]

(Chelton and Schlax 1996; Vivier et al. 1999, VKThereafter; Stammer 1997; Stammer et al. 1996, Plate 3).Studies in the North Pacific show that the steric responseto heating is large at high latitudes (VKT; Stammer1997). A quasi-steady topographic Sverdrup balance isalso found over the North Pacific to latitudes as low as258N. Equatorward of that, the flat-bottomed Sverdrupbalance is important (VKT). Forced first-mode baro-clinic Rossby waves are seen to propagate south of about408N.

A feature of the observed Rossby waves not explainedby the simple models is apparent damping of Rossbywaves as they propagate from east to west across the

3658 VOLUME 32J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

basin. Qiu et al. (1997) show that Laplacian or bihar-monic mixing should be added to the simple long-wavemodel for better agreement with SSH observations. Thisform of damping represents lateral mixing by eddies,and is scale selective. Another possible form of dampingis Newtonian cooling proportional to SSH in a reducedgravity model. This formulation could represent simplethermal damping of the SSH, assuming SSH is propor-tional to SST. Newtonian damping is also interpretableas potential vorticity mixing, indicative of vertical mix-ing processes in the ocean. This suggests that diabaticeffects may play a role in the dynamics of the responseof the ocean to variability in wind forcing.

In reduced gravity models of the ocean’s response towind forcing (Qiu et al. 1997), many physical processesare ignored; these include the coupling of the Rossbywaves to bottom topography (Killworth and Blundell1999), coupling among different vertical modes, andvertical mixing processes. More sophisticated numericalmodels include these effects but are difficult to interpret.Here we focus on the influence of diapycnal processes,primarily the seasonal cycle of entrainment and detrain-ment of the mixed layer, on the large-scale response ofthe ocean to both wind and heat-flux forcing by doinga series of model runs with different forcing configu-rations. The possibility of forcing of Rossby waves bydiapycnal processes has been explored previously in thecontext of decadal variability (Liu 1999). With simpli-fied geometry and forcing, Liu finds that Ekman pump-ing forces the first baroclinic mode, while heating andcooling at the surface primarily forces the second bar-oclinic mode. Here we quantify these results in a modelthat has realistic forcing to determine if vigorous mixingassociated with the seasonal evolution of the mixed lay-er influences the first baroclinic mode evolution and thusthe SSH in the ocean.

The outline of the paper is as follows. The numericalmodel used in this study is described in section 2. Fourmodel runs are discussed in section 3. The dynamics atwork are explored in the context of a quasigeostrophicmodal decomposition of the oceanic response in section4. Consequences of the results for the interpretation ofthe response of the ocean to seasonal forcing are dis-cussed in section 5.

2. The isopycnal model formulation

The model used for this study is the Hallberg Iso-pycnal Model (Hallberg 2000; Hallberg and Rhines1996; Ladd and Thompson 2001), an isopycnal modelsimilar to MICOM [New et al. (1995) and referencestherein]. The model includes diapycnal mixing and anembedded mixed layer. The mixed layer has Kraus–Turner physics with additional shear-dependent mixingbelow the mixed layer, and the entrainment algorithmused in MICOM. However, detrainment occurs througha buffer layer, an additional layer with variable density.The detrainment method is described in detail in the

appendix and has been used to study the seasonal cycleand water mass formation (Ladd and Thompson 2001).

To maintain the model’s free surface while increasingthe time step, the value of gravity at the free surface is10 times smaller than the actual value. The SSH re-sponse is 10 times larger than what it would be in theocean so that in all plots of SSH from the model, wedivide the magnitudes by 10. High-frequency barotropicmotions will not be well represented in the model, owingto both the artificial value of g as well as the closeddomain. An experiment with the true value of g wasdone and compared against the control run describedbelow for year 1992. The response is quantitatively sim-ilar, with rms SSH locally no more than 10% larger.The low-frequency (annual period) motions of interestare very well represented by the approximation usedhere. In particular, quasi-steady barotropic topographicSverdrup response on the annual period is well repro-duced in the model, as demonstrated by VKT.

The model domain is the North Pacific between 138Sand 608N. Resolution is 28 in each direction. The hor-izontal mixing is biharmonic (with the coefficient set at2 3 1015 m4 s21), which favors mixing at smaller lateralscales. Thus, the solution is less viscously controlledthan one with the numerically required value of Lapla-cian mixing. The diapycnal mixing coefficient is 1 31025 m2 s21 unless otherwise noted. There is also hor-izontal thickness diffusion of 20 m2 s21, required fornumerical stability. Bathymetry is taken from ETOPO60. The model is forced by National Centers for En-vironmental Prediction (NCEP) daily wind fields, usingperpetual 1992 winds to spin up the model for 20 yearsfrom the September initial conditions based on Levituset al. (1994) and Levitus and Boyer (1994). Year 1992is used for sensitivity studies. The buoyancy forcing isa combination of da Silva et al. (1994) heat fluxes anda relaxation to Levitus et al. sea surface buoyancy. Thetimescale of the relaxation is 100 days for a mixed layer100 m thick (Rothstein et al. 1998). The coefficient ofthermal expansion varies with location and season sothat the equation of state is nonlinear in temperature.Two different density layer configurations are used (Ta-ble 1). In the first case, an adiabatic simulation is donewith 7 isopycnal layers. In the second, 8 isopycnal layersare used along with a buffer layer and a mixed layer,giving a total of 10 layers.

3. The role of diapycnal and mixed layer processes

To elucidate the role of diapycnal processes in theresponse of the ocean to seasonal forcing, we presentthe results from four model runs, run 1: wind forcingbut no diapycnal processes either in the mixed layer orthe interior and no heat flux forcing; run 2: heat flux,wind forcing, and interior diapycnal mixing; run 3: windforcing but heat flux is given by relaxation to the an-nually averaged SST and wind mixing is turned off inthe mixed layer; and run 4: seasonal heat flux forcing


TABLE 1. Simulations.

Layer Density

Adiabatic

1234567

1021.51023.51024.51026.01026.751027.251027.25

Diabatic1 (mixed layer)2 (buffer layer)3456789

10

1022.81022.91023.51024.510251025.510261026.51027.51028

and annual mean wind stress. For run 3 and run 4, themodels were run for several years from the spunup ver-sion of run 2. This ensures that the mean density struc-ture is similar to that in run 2. In general, the modelruns do not reproduce the smallest zonal scales seen inthe observations (Chelton and Schlax 1996; Stammer1997; Stammer et al. 1996). This occurs for two reasons:one is that the model is low resolution and the other isthat there may not be sufficient energy at small scalesin the wind fields.

We first examine the Rossby waves in runs 1 and 2.By Rossby waves, we mean the first baroclinic modeRossby waves, identifiable by their westward phasepropagation and distinct phase speed. A comparisonshows that Rossby waves propagate more freely in run1 at all latitudes and the response is larger (Fig. 1). Forboth model runs, the rms SSH is lower than in the ob-servations at low latitudes and comparable to the ob-servations at midlatitudes. VKT showed that polewardof 258N, the barotropic topographic Sverdrup balanceis important and this part of the response appears to bewell simulated in the model. In run 2 Rossby wavegeneration and propagation is inhibited relative to thatin run 1 at midlatitudes. In fact, the rms model responsein run 1 is as large or larger than in the observations atmidlatitudes, despite the lack of a seasonal steric re-sponse. This result echoes that found by Qiu et al. (1997)in a one-dimensional wave equation where mixing wasneeded to better reproduce the observations.

The rms SSH is about a factor of 2 smaller than theobservations in run 2 when the comparison is made afterapplying a correction for non-Boussinesq effects sug-gested by Greatbatch (1994) (see also Stammer 1997),proportional to the area-averaged heating. In the higher-resolution simulations of Stammer et al. (1996), the rmsresponse is also a factor of 2 too small when comparedwith the observations. This result suggests that higher

horizontal or vertical resolution is needed or the windsthat forced the model had too low amplitude. In addition,the relaxation part of the heat flux could cause dampingof the wind-driven signal, although we find that this isnot the case here.

Of course, the response at the sea surface is not theonly feature of the model ocean that changes when dia-batic processes are included. There are profound dif-ferences in the density structure (Fig. 2). In run 1, thedensity does not show a significant seasonal cycle. Inaddition, there are regions where the isopycnals arepacked together (see for instance Fig. 2c at 1408E). Wa-ter can be moved to different parts of the basin withinthe same layer but cannot cross layer interfaces. Con-vergences and divergences control the density structureand create regions with large vertical density gradients.In run 2, however, the density structure must be com-patible with the surface buoyancy forcing. Density sec-tions from the Levitus et al. climatology show layerthicknesses that are more uniform with longitude, withisopycnals depressed in the west, showing reasonableagreement with run 2. The mixed layer tends to be toodeep in the model, most likely owing to incompatibleheat fluxes and a lack of density resolution in the ther-mocline, especially at low latitudes and the Kraus–Turn-er mixed layer model formulation (Ladd and Thompson2001).

Run 2 results are very robust to change of interiormixing values. When the interior diapycnal mixing is10 times larger, and 10 times smaller, the SSH is verysimilar. It appears that it is the mixed layer evolutionthat plays a large role in how the Rossby waves prop-agate and evolve in the model rather than interior dia-pycnal mixing. Two additional runs were done to showthat the relaxation part of the buoyancy flux as well asthickness diffusion have very little impact on the evo-lution of the SSH.

In run 3 with heating only, we find that the rms SSHis relatively small, ensuring that the differences that wesee between runs 1 and 2 are not simply because of aseasonal steric response (Fig. 3). There is however astriking difference between run 2 and run 4 from 188Nto 388N where the run 4 response is almost twice aslarge as the run 2 response. Elsewhere, the two simu-lations are very similar, and both give smaller responseswhen compared to run 1.

4. Modal decomposition and the forcing of Rossbywaves by diapycnal processes

So the question becomes, what is it about midlatitudeswhen the seasonal cycle in heating and cooling is in-cluded that causes the changes in the oceanic response?We know from previous work that north of about 408N,the barotropic and steric response dominate on seasonaltime scales (VKT) and neither of these responses shouldbe influenced by diapycnal pumping that results fromthe entrainment/detrainment cycle of the mixed layer.


FIG

.1.

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plot

sm

odel

ofS

SH

for

year

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93–9

6at

108,

208,

308,

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408N

:(a

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urin

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FIG

.1.

(Con

tinu

ed)


FIG. 2. Layer interfaces at 228N 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.


FIG. 3. Rms SSH anomaly (cm) zonally averaged. Solid line is run1, dot–dashed is run 2, dashed line is run 3, and small dotted line isrun 4.

However, the baroclinic modes could be influenced bydiapycnal pumping (Liu 1999). To answer the abovequestion, a modal decomposition is done to determinethe relative contributions to the SSH from each verticalmode and then to examine the response of the first bar-oclinic mode under different forcing conditions. Thisallows the separation of the seasonal heating and bar-otropic response from the Rossby waves.

To do the modal decomposition, the annual meanstratification (layer thicknesses) and densities are usedat each point. Care must be taken in constructing theannual mean stratification when layers vanish. Usingthe model results, a mean distribution of layer thick-nesses is constructed such that each layer is of finitethickness, removing layers when they are less than 10m from the mean field by combining layers. This resultsin some regions of the model ocean where fewer than10 layers are used for the modal calculations. Interfacedeviations are constructed to be consistent with this al-gorithm. We do the model decomposition locally, as-suming that the motions are linear, and thus make theWKB approximation. A dynamically consistent way todecompose the response in the model is to use the ver-tical structure equation [see section 2 of Killworth etal. (1997) for the continuously stratified version of this].In the layered framework, the vertical structure equationand the boundary conditions reduce to

f 2 f f 11 2 11 1 f 5 0 (1)12H g H g c1 1 1

for the top layer,

(f 2 f ) (f 2 f ) 1n n11 n n211 1 f 5 0 (2)n2H g H g cn n n n21

for the nth interior layer, and

(f 2 f ) 1m m21 1 f 5 0 (3)m2H g cm m21

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

n21

f f 5 gh 1 f h . (4)On 0 n ni51

Here Hn is the mean thickness of layer n, fn is thestreamfunction deviation from the mean streamfunctionin layer n, and gn is the reduced gravity between layern and layer n 2 1 taken from the mean density profile.Also, hn is the interface displacement of the bottom oflayer n from its mean position.

In matrix notation, (1)–(3) can be written

1LF 5 2 F. (5)

2c

Here L is a matrix containing the operators in the firsttwo terms of (1) and (2) and the first term in (3) andF is a vector containing the vertical structure of thestreamfunction for a particular mode. Equation (5) issolved as an eigenvalue problem for c22. The eigen-vectors are sorted by their phase speeds, with the mostrapid phase speed being the barotropic mode. In run 2,for the first two baroclinic modes the structure is smoothand shows no obvious discontinuities that would suggestlack of vertical resolution (Fig. 4), while the third bar-oclinic mode starts to show the finite number of layersin the calculation. Note that the density resolution isconcentrated in the upper 500 m. The phase speed ofthe first baroclinic mode gravity wave shows reasonablecorrespondence with other calculations (cf. Killworth etal. 1997), although it is slightly larger than their estimatebased on the Levitus et al. data.

Since the higher vertical modes are not well repre-sented by the limited vertical resolution in the model,we will concentrate our discussion on the response inthe barotropic and the first two baroclinic modes. Wehave ignored the impact of vertical shear on the modalstructure and phase speeds. According to the calculationof Killworth et al. (1997), the modal structure for thefirst baroclinic mode is relatively uninfluenced by thezonal vertical shear, although the phase speed can differfrom the traditional phase speed by as much as factorof 2. For the analysis done here, we do not use the phasespeed directly, so our results should be relatively robustto the exclusion of the mean flow field, particularly forthe first baroclinic mode. In addition, with the inclusionof the mean flow, the modes are no longer complete ororthogonal.

First, we decompose the SSH response using the ver-tical structure of the streamfunction anomalies to de-termine how each mode contributes to the SSH varia-tions. Since the vertical modes are orthogonal and com-plete by construction, we can write the streamfunctionin any layer and at any point as a sum of contributionsfrom each vertical mode:

C 5 Ra, (6)


FIG. 4. Nondimensional vertical structure for the first four vertical modes from run 2 in (a)–(d)streamfunction and (e)–(h) interface displacement at 228N, 1808.

where R is the matrix whose columns are made up ofthe eigenvectors F that represent the vertical structureof each mode and a is a vector that represents the time-dependent magnitude of each mode. The vector C con-tains the model streamfunction as a function of time.At each time step a is found from (6) and the modelstreamfunction deviations using

21a 5 R C. (7)

To determine the contribution to the sea surface heightfor each mode, we find how much each vertical modecontributes to the stream function in layer 1. There thestreamfunction is related to the SSH (h) by

g gh 5 c 5 R a . (8)O1 1 j jf f j51,m0 0

To study how each mode contributes to the SSH, weexamine R1j aj for j 5 1, 3 (the barotropic and first twobaroclinic modes). Each of the vertical modes contrib-utes different spatial and temporal structure to the SSHresponse. The barotropic mode is quite noisy, but doesshow structure consistent with a barotropic Sverdrup-like response as a zonally uniform and nonpropagatingsignal (Fig. 5a). The first baroclinic mode shows prop-agation and is the mode that we identify with Rossbywaves that appear so strongly in the observations (Fig.5b). The second baroclinic modes have slower propa-gation and a weaker response (Fig. 5c). The contributionto the SSH from the first baroclinic mode shows westernintensification and has similar structure to that found by

VKT while the second baroclinic mode is larger in theKuroshio Extension and subpolar gyre (not shown).

The modal decomposition for runs 2, 3, and 4 showsthe destructive interference of the Ekman pumping anddiabatic pumping response (Fig. 6). The first baroclinicmode response in run 4 is larger than the response inrun 2 or in run 3. This suggests that the wind-drivenresponse and the diabatic response destructively inter-fere to lower the total (run 2) response. This interferenceis largest between 208 and 358N. In contrast, for thesecond baroclinic mode there is constructive interfer-ence poleward of 258N. Note that the second baroclinicmode response is larger than the first baroclinic moderesponse under thermodynamic forcing alone (run 3)since its vertical structure projects better onto the ther-modynamic forcing.

To determine how the diapycnal processes influenceeach vertical mode in the model, a dynamical interpre-tation must be made. To do this, we use quasigeostrophictheory. We have assumed that the motion is linear andhas large horizontal length scales so that the relative vor-ticity can be ignored. Mixing is written as a diapycnalflux at the top and bottom of each layer. The continuityequation for each layer can then be written as

]h 1 = · (u h ) 5 w 2 w , (9)n n n n21 n]t

where hn is the thickness of layer n and wn is the diabaticvelocity or flux across the interface at the bottom of


FIG. 6. Zonally averaged rms SSH anomaly (cm) from each verticalmode from run 2 (solid), run 3 (dashed), and run 4 (dotted): (a) firstbaroclinic mode and (b) second baroclinic mode.

FIG

.5.

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rco

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ons

at22

8Nto

SS

Han

omal

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rye

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92fr

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375

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layer n. The linear layer quasigeostrophic potential vor-ticity (PV) equations are

] c 2 c c ]c f2 1 1 12 1 b 5 (w 2 w ) (10)Ek 11 2]t H g H g ]x H1 1 1 1

for layer 1,

] c 2 c c 2 c ]cn11 n n21 n n1 1 b1 2]t H g H g ]xn n n n21

f5 (w 2 w ) (11)n21 nHn

for interior layers, and


FIG. 7. Rms contribution to the forcing of the baroclinic modes byEkman pumping (solid) and diabatic pumping (dashed) in run 2: (a)first baroclinic mode and (b) second baroclinic mode.

] c 2 c ]c fm21 m m1 b 5 w (12)m211 2]t H g ]x Hm m21 m

for the bottom layer.In setting up (10)–(12), we have assumed that the

Ekman layer is much thinner than the mixed layer sothat the mean mixed layer can be treated as layer onein the above equations. This means that the Ekmanpumping acts on the top of the first layer. This is notstrictly accurate since in the model the wind acts as abody force in the mixed layer; thus the mixed layer andEkman layer are the same depth. Treating the mixedlayer with separate dynamics from the interior layerswould introduce additional complexity and is not war-ranted here. Note that Ekman pumping in the layeredmodel is not a diabatic vertical velocity in that it doesnot cause fluid to pass between layers; thus it does notappear in the diapycnal velocity diagnosed from themodel. It does, however, directly modify PV in the toplayer. In contrast, the diapycnal velocity causes massredistribution between the layers, and affects the PV ineach layer. In this case, most of the diapycnal mixingcomes about from entrainment and detrainment at thebase of the mixed layer. The interior diapycnal velocitiesthat are diagnosed from the model are assigned to theappropriate interface when layers vanish using the samealgorithm as described above for the construction of theinterface deviations. We also continue to ignore verticalshear in the background flow field.

The PV equations for each layer (10)–(12) can bewritten

]C ]CL 1 b 5 F, (13)

]t ]x

where C is a vector representing the streamfunction ineach layer and F is the rhs of (10)–(12).

To transform (13) to PV equations for each verticalmode, we first note that we can write (5) as

LR 5 2RC, (14)

where C is a matrix containing the eigenvalues (c 22 )of (5) as elements of the diagonal. Then we can write(14) as

HLR 5 2HRC, (14)

where H is a matrix whose diagonal elements are thethickness of each layer. If we then take the transposeof each side of (14), we find that

T TR HL 5 2CR H (15)

since HL is symmetric (this is another way of sayingthat the operator is self-adjoint). Thus, we can transform(13) by multiplying by RTH and using (15) to get

]C ]CT T T2CR H 1 bR H 5 R HF. (16)

]t ]x

We have created an equation that defines the propagation

and forcing of the potential vorticity for each verticalmode, in terms of the vertical divergences (F) wherenow the jth element of RTHC is the potential vorticityof the jth vertical mode. We have assumed that R andH are independent of time and that they vary slowly inthe zonal direction.

The rms contribution of the Ekman pumping and dia-pycnal velocities to the PV forcing of each vertical modeas calculated from the rhs of (16) can then be compared(Fig. 7). For the first baroclinic mode, the Ekman pump-ing dominates everywhere, while diapycnal pumpingplays an important role between 308 and 408N near theKuroshio Extension (Fig. 7a). The diabatic pumping islarge just in the region where there is the largest dif-ference between the wind only run and the control run


FIG. 8. Ratio of transfer coefficient. Solid line is ratio of transfercoefficient of the run 4 to run 2 (zonally averaged), and dashed lineis the ratio of rms SSH of run 4 to run 2 (zonally averaged).

FIG. 9. Correlation coefficients between first baroclinic mode re-sponse in the numerical model run 2 and that from the 1D waveequation (solid line), and correlation coefficient between the the dia-batic pumping response and the Ekman pumping response from the1D wave equation (dashed line).

(cf. Fig. 7 to Fig. 3). For the second baroclinic mode,Ekman pumping and diabatic pumping are coincidentand of similar magnitude with the maximum in the Ku-roshio Extension (Fig. 7b). One would not expect a one-to-one correspondence between the energy in the modesand the forcing patterns. Rossby wave energy locallydepends on the accumulation of forcing as the Rossbywave propagates to the west.

We can define a transfer coefficient for Ekman pump-ing for each vertical mode taken from (16) that deter-mines the magnitude of the forcing for mode given byf 0fn(0) . A question arises whether the changes in the2cn

stratification in the various experiments that were dis-cussed in section 3 could explain the changes in theamplitude of the response rather than diapycnal pump-ing. The ratio of the transfer coefficient for run 2 andrun 4 shows that the expected wind-driven response isapproximately the same in both runs (Fig. 8). However,the ratio of the response is significantly different at mid-latitudes, confirming the role of diapycnal pumping inthe evolution of the first baroclinic mode.

There is no guarantee that diapycnal forcing willcause damping (see, e.g., Liu 1999). To ascertain itseffect on the first baroclinic mode more directly, wesolve the one-dimensional wave equation (16) using theforcing fields calculated from its right-hand side and thephase speed calculated from the modal analysis. Theequation is solved by an upwind differencing schemeand then is compared against the modal calculation. Ingeneral, the amplitude found from the one-dimensionalmodel is about a factor of 2 smaller than found fromthe modal analysis. This can be explained by the errorsin the wave equation associated with the presence ofthe mean flow related to the factor-of-2 difference inthe phase speed from that of a resting ocean at midlat-itudes (as show by Killworth et al. 1997). Barring thedifferences in amplitude, south of 308N, the wave equa-tion reproduces the modal time/longitude behavior andit is well correlated with the results from the numerical

model (Fig. 8). Between 178 and 288N, the diapycnalpumping response is anticorrelated with the Ekmanpumping response, resulting in a total wave field thatis smaller than the wave field forced by Ekman pumpingalone.

There are several caveats that should be made aboutthis analysis. In the modal analysis, we have linearizedabout the mean stratification, which does not take intoaccount the order one changes in the density structurethat occur throughout the year in the upper ocean. Theone-dimensional model does not do very well in pre-dicting the response of the full model poleward of 458where the annual Rossby wave is most likely evanes-cent, and VKT found very little Rossby wave response.In this region we expect that the one-dimensional modelshould fail. In addition, in this region the mean flow isstrongly eastward, and the wave propagation character-istic would be significantly modified so that the one-dimensional model would not be valid.

The higher vertical modes are strongly influenced bydiapycnal processes in this analysis, which suggests thatthey would be strongly forced by diapycnal processes,much more so than by Ekman pumping. However, thehigher modes are less well represented in the model,and the mean velocity field is more important in theirevolution than it is for the first baroclinic mode. Theeffects of the mean flow on the propagation character-istics of the higher modes will be explored in a sub-sequent paper.

5. Conclusions

Model simulations of the SSH response to wind forc-ing show that diapycnal processes and mixed layer evo-lution are important in the dynamics of the SSH re-sponse, particularly at mid- to high latitudes. Typicallythe propagation of the first baroclinic mode Rossby


waves has been studied in simple models without theinclusion of diapycnal processes and mixed layer phys-ics, and thus the adjustment of the ocean to seasonalchanges in wind stress has been decoupled from theseasonal cycle of heating and cooling. The simulationsshow that, in particular, the presence of the seasonalcycle mixed layer entrainment/detrainment can be im-portant in oceanic adjustment at midlatitudes. In an adi-abatic simulation (run 1), Rossby waves forced near theeastern boundary propagate without reduction in am-plitude, and an obvious seasonal cycle is lost. In con-trast, in run 2 Rossby waves are less vigorous. We areargue here that the mixed layer seasonal cycle, and theassociated diapycnal velocities, is important for chang-ing the Rossby waves.

A modal decomposition of the model interface dis-placements shows that, as expected, the barotropic modedominates the SSH signal at high latitudes, with the firstbaroclinic mode important at mid to low latitudes andhigher vertical modes not contributing as much. Themodel decomposition shows that the first baroclinicmode is suppressed at midlatitudes by diapycnal pump-ing. In contrast, the second baroclinic mode is enhancedby diapycnal pumping.

The results from a one-dimensional wave equationare consistent with the modal analysis and show thatthe diabatic response and Ekman pumping response areopposite between 178 and 288N for the first baroclinicmode. The resulting signal is weaker than the Ekmanpumping signal alone. In addition, the one-dimensionalmodel has good predictive skill south of 308N, whilenorth of 458N, the skill is very poor indeed, suggestingthat the linearized potential vorticity equation is notvalid in this region. A more complete analysis wouldinclude the effects of the mean flow on the propagationpathways of the waves.

Higher resolution in the model, both vertical and hor-izontal, would help with a more faithful representationof the observations. In particular, the choice of densityresolution influences how well the stratification can berepresented. The vertical structure in these model sim-ulations was chosen to maximize resolution in the sub-tropics, which leaves the resolution in the Tropics lack-ing near the surface. This leads to mixed layers that aretoo deep, and vertical mixing that is unrealistically large.Likewise, the lack of horizontal resolution affects thenorthwestern Pacific, where the Kuroshio Extension istoo far north, and where the mixed layers are too deep.

This study shows that Rossby waves, which are tra-ditionally thought to be purely adiabatic, can be greatlyinfluenced by diabatic processes. This has been exploredfor decadal variability (Liu 1999), but it is on the sea-sonal cycle that the dynamic impact of diapycnal ve-locity associated with the seasonal cycle of heating andcooling is the largest.

Acknowledgments. We acknowledge the TOPEX/Po-seidon Extended Mission project at JPL and D. Chelton

and M. Schlax for mapped altimetric fields. We thankHelma Lindow for some early model runs. Funding wasprovided by NASA Contract 960888 with the Jet Pro-pulsion Laboratory (TOPEX/Poseidon Extended Mis-sion Science Working Team) and an NSF grant toThompson and Kelly.

APPENDIX

The Mixed Layer Formulation

The model uses a bulk Kraus–Turner (1969) typemixed layer, much like that of Bleck et al. (1989) exceptthat a variable density buffer layer is used to receivethe fluid detrained from the mixed layer.

The mixed layer depth is determined by energy con-siderations. If there is turbulent kinetic energy availableto drive mixed layer deepening, it does so entrainingsuccessively from the lightest layers that contain anymass. Alternately, if there is net surface loss of buoy-ancy, the mixed layer may entrain due to free convec-tion. Otherwise, the mixed layer detrains to the Monin–Obukhov depth—the depth at which wind generated tur-bulent kinetic energy balances the work required to mixthe surface buoyancy forcing throughout the mixed lay-er.

Fluid that is detrained from the mixed layer goes intoa variable density buffer layer. This avoids the necessityof having a ‘‘fossil mixed layer’’ that is appended tothe real mixed layer, as in Bleck et al. (1989). The detailsof this buffer layer differ from those of Murdegudde etal. (1995), but the layer is conceptually similar. Forexample, the scheme described here conserves heatwhile Murdegudde’s does not. If there is an isopycnallayer that is intermediate in density between the mixedlayer and the buffer layer, it is possible to split the bufferlayer while maintaining a stable density profile. Blecket al. are careful to insure that the fossil mixed layerdoes not alter the surface temperature or salinity ten-dency due to vertical processes. But mixed layer thick-ness, temperature, and salinity tendencies due to hori-zontal advection within the mixed layer are unavoidablyaffected by the presence of a fossil mixed layer. Themixed layer/buffer layer approach used here gives amore accurate representation of the effects of horizontaladvection on the mixed layer properties.

The specific calculation of the one-dimensional mixedlayer evolution is as follows.

1) If there is a net surface loss of buoyancy from theocean, the mixed layer becomes denser.

2) The water column convectively adjusts. Any massin layers lighter than the mixed layer is entrainedinto the mixed layer. Penetrative convection seemsto be fairly unimportant in the ocean (Marshall andSchott 1999), so none of the potential energy re-leased through convective adjustment is retained todrive further entrainment.

3) As prescribed by Kraus and Turner (1969) or Bleck


et al. (1989), turbulent kinetic energy (with a spec-ified surface source) is used to drive mixed layerentrainment if there is a net surface buoyancy lossor if the mixed layer is shallower than the Monin–Obukhov depth, hMO 5 2m0 /B0. Here u* is the3u*surface friction velocity, B0 is the total surface buoy-ancy flux into the ocean and m0 is a nondimensionalconstant, 0.5. The entrainment is essentially the sameas described in Bleck et al. The mixed layer entrainsfrom successively denser layers until the increase inthe potential energy due to the entrainment equalsthe surface energy available to drive mixing:

h03TKE 5 Dt m u* 2 [(B 1 |B |)0 0 05 4

1 n(B 2 |B |)] , (A1)0 0 6where Dt is the time increment, h0 is the mixed layerthickness at the start of the entrainment, and n isanother nondimensional constant, here 0.03. It is easyto show that the potential energy increase due to anincrement Dh of entrainment by the mixed layer is

1DPE 5 h Dh(b 2 b ), (A2)0 ML Ent2

where bML is the mixed layer buoyancy and bEnt isthe buoyancy of the entrained fluid. Entrainmentcontinues until DPE 5 TKE. The surface TKE avail-able to drive mixing does not decay with depth inthe current model.

4) If the mixed layer is deeper than the Monin–Obu-khov depth, the mixed layer detrains into the bufferlayer until the mixed layer thickness agrees with theMonin–Obukhov depth and the mixed layer buoy-ancy increases due to any net positive buoyancy forc-ing.

5) If there is an isopycnal layer that is intermediate indensity between the buffer layer and the mixed layer(and there is any fluid in the buffer layer), the bufferlayer is detrained. It is partitioned into a portion thatmatches the density of the intermediate density layerand a portion that matches the density of the nextdenser layer. If layer k is intermediate in densitybetween the buffer layer and the mixed layer, theamount of fluid that goes into layers k and k 1 1 is,respectively,

r 2 rB kv 5 h andk11 B r 2 rk11 k

r 2 rk11 Bv 5 h . (A3)k B r 2 rk11 k

Whenever fluid moves across interfaces between lay-ers (except during the partitioning of the bufferlayer), it does so with the velocity and tracer con-centrations of the layer from where it came.

The mixed layer and buffer layer obey the samemomentum equations as all of the interior isopycnallayers, except for the inclusion of an additional pres-sure gradient term. When the momentum equations,

]u1 ( f k 1 = 3 u) 3 uS]t

p 15 2= 1 gz 1 u · uS1 2r 2

p2 = r 1 viscosity (A5)S2r

(written in standard notation, with the subscript Semphasizing that the gradients are taken along co-ordinate surfaces, and only the horizontal compo-nents of the velocities are used, consistent with thehydrostatic approximation), are evaluated in layersof nonconstant density, the middle term on the right-hand side does not vanish.The average through the layer of this term is

p g h2 = r ø 2 h 1 = r, (A6)S S2 1 2r r 20

where h is the height of the interface above the layerand h is the thickness of the layer. The natural ex-pression of this term on a C grid is not obvious, buta finite-element interpretation suggests the discreti-zation

g h ]r2 h 11 2 )r 2 ]x0 S

1 2 1 2 2 1 1 2g h h 1 h h 1 h h r 2 rø 2 , (A7)

1 2r h 1 h Dx0

where the superscripts 1 and 2 indicate the pointsto the east and west of the velocity point where thisterm is being evaluated. This discretization is usedin the mixed layer calculations.

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notes and correspondence buoyancy and mixed layer …

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