Energetics of wind-driven barotropic variability in the...

Energetics of wind-driven barotropic variability in theSouthern Ocean

by Wilbert Weijer12 and Sarah T Gille1

ABSTRACTThis study addresses the energetics of the Southern Ocean in response to high-frequency wind

forcing A constant-density multi-layer model is forced with a band of stochastically varying windstress The focus is on the interplay between the surface layer and the interior circulation

In line with previous examinations it is concluded that the interior ocean is not directly energizedby the wind work but rather through the work done by the pressure field The spatial and temporalcharacteristics of these terms differ substantially Although the wind work may be negative inextensive regions of the World Ocean the pressure work energizes the interior circulation almosteverywhere

For low-frequency variability the total work done by the wind and pressure on the barotropic flowis comparable but discrepancies may arise for high-frequency variability A mechanism is identifiedthrough which kinetic energy can leak from the wind-driven surface layer to the barotropic flow

1 Introduction

The energy cycles of the ocean circulation have become a central topic in present-dayoceanographic research Knowledge of the pathways of energy through the ocean systemdown to the smallest dissipative scales of turbulence is important for understanding theoperation of the global overturning circulation (Munk and Wunsch 1998 Wunsch andFerrari 2004) In this paper we focus on one link in the energy cycle the transmission ofwind energy from the surface layer to the ocean interior

The energetics of the surface layer and the oceanrsquos interior are mostly consideredseparately The source term for the interior circulation is usually expressed as sg middot ug where denotes the wind stress vector and ug the geostrophic velocity at thesurface The simplicity of this expression is misleading in the sense that it does not reflectthe complex interplay between the Ekman flow and geostrophic motions through Ekmanpumping In fact Fofonoff (1981) and Wang and Huang (2004a) pointed out that this windwork is actually invested in the surface flow rather than in the oceanrsquos interior circulationIt drives the Ekman transport up-gradient and hence tends to increase the potential energy

1 Physical Oceanography Research Division Scripps Institution of Oceanography University of CaliforniaLa Jolla California 92093-0230 USA

2 Corresponding author email wweijerucsdedu

Journal of Marine Research 63 1101ndash1125 2005

1101

This reservoir of potential energy can then be tapped to generate motions in the oceanrsquosinterior by means of pressure work (eg Gill et al 1974) Thus although sg provides theactual power it is pressure work that converts this energy to a source that can drive theinterior circulation

This view points at an intimate link between the dynamics of the Ekman layer and thebarotropic interior It raises some questions about the energetics of the deep ocean Inparticular (i) what are the spatial and temporal characteristics of the pressure work incomparison to wind work and what implications does this have for determining where theenergy is dissipated (ii) Could energy leak from the Ekman layer energy source(represented as se middot ue where ue is the Ekman velocity at the surface) to the interiorcirculation

The Southern Ocean receives substantially more wind energy than other basins (Oort etal 1994 Wunsch 1998 Watanabe and Hibiya 2002 Alford 2003 Wang and Huang2004ab) A small continental area and the possibility of circumpolar circulation provideideal conditions for intense air-sea coupling In addition Southern Ocean winds areexceptionally strong and extremely variable Their spectra contain energy over a broadrange of frequencies spanning low-frequency climatic fluctuations (including the AntarcticOscillation (AAO) Thompson and Wallace 2000) storm scales sub-inertial and super-inertial variability Low-frequency variability typically contains much more energy thanhigh-frequency fluctuations so the spectra are essentially red In the frequency rangebetween 10 and 90 cycles per year (cpy) these wind stresses have spectral slopes that areestimated to be between 05 and 14 depending on the wind product (Gille 2005) Between100 and 363 cpy spectral slopes are slightly steeper between 07 and 25 In view of thisuncertainty it is important to know how sensitive the deep ocean is to the high-frequencyspectral content of the wind forcing

Although the results of Wunsch (1998) suggest that the high-frequency part of the windforcing may not contribute substantially to the global energy budget of the interiorcirculation it is generally believed that direct wind forcing is one of the main sources ofeddy-kinetic energy in the ocean (Frankignoul and Muller 1979 Muller and Frankignoul1981) Stammer et al (2001) concluded that this mechanism is especially effective inweakly stratified areas where the contribution of baroclinic instability mechanisms to theeddy field is small Large et al (1991) found that components of the wind (curl) forcingwith frequencies up to 13 cpd (cycles per day) had a noticeable impact on the response of amulti-layer quasi-geostrophic model This time scale is associated with a cut-off frequencyfor the excitation of Rossby waves (Willebrand et al 1980)

In this study we will address the energetics of barotropic variability of the SouthernOcean in response to high-frequency wind forcing We will sketch the pathway of energyfrom the input area to the dissipative regions distinguishing between the roles played bythe Ekman and barotropic circulations As far as we know this may be the first study toaddress the energetics of high-frequency wind-forced variability explicitly discerning theroles of the Ekman layer and the interior circulation

1102 [63 6Journal of Marine Research

In the next section the model specifications will be detailed while Section 3 willintroduce the framework in which the model results will be discussed In Section 4 theenergetics of a standard run will be explored in detail while Section 5 discusses thesensitivity of the response to the stochastic characteristics of the forcing A discussion(Section 6) and summary and conclusion section (7) will conclude this paper

2 The model

Our aim is to increase our qualitative understanding of energy redistributions in theSouthern Ocean Therefore we have chosen a simplified model set-up that allows for aclear analysis of the relevant processes The critical simplifying assumption here is thechoice of wind stress forcing Ignoring the complex wavenumber-frequency spectrum oftrue Southern Ocean winds we force the model using a stochastic zonal wind stress with afixed large-scale spatial pattern The amplitude is determined by a high-frequencystochastic time series with red spectrum We will focus on high-frequency variability(frequencies larger than roughly 01 cpd) This choice allows us to consider the SouthernOcean as a constant-density linear system with no background circulation The use of asimple model is justified for this paper because the objective is to elucidate the interplaybetween the wind-driven layer and the oceanrsquos barotropic interior

For a detailed description of the model set-up we refer to Weijer and Gille (2005) Themodel is a constant-density version of the fully 3-D MITgcm (Marshall et al 1997ab)This choice allows us to make a clear distinction between Ekman transports and interiorbarotropic flows and is motivated by our desire to study their respective roles in theenergetics of the ocean response In shallow-water models (eg Hirose et al 2001) theEkman drift and the interior circulation are merged into one single layer blurring anydistinction between the dynamics of interior and surface layer flows This basically spreadsthe Ekman layer over the entire depth of the ocean allowing it to interact with bottomtopography and to affect velocity-dependent frictional closures such as bottom drag Theuse of a free-surface primitive-equations model also has advantages compared withquasi-geostrophic models (eg McWilliams et al 1978) which only consider balancedmotions and do not account for the Ekman circulation and its energetics

The wind forcing is applied as a body force in the upper layer of the model Only a zonalcomponent is used x( y tn) 0(tn)T( y) here 0 is a basic wind stress amplitude(taken to be 01 Nm2) (tn) a stochastic time series with a red spectrum and T( y) thespatial profile T is chosen to be zonally constant but it has a Gaussian profile in themeridional direction with a maximum at 53S and a 10deg half-width This profile ischosen to reflect the zonally-averaged mean zonal wind stress from climatologies (egGille 2005)

A stochastic time series (tn) is used as described in Weijer (2005) It is constructed as asummation of N frequency components equally distributed between fmin and fmax Thephases i of the frequency components contain a random variable which allows the timeseries to continue infinitely without repetition The amplitudes i of the components are

2005] 1103Weijer amp Gille Wind-driven barotropic variability

inversely proportional to the corresponding frequency through which is the slope of thewind stress power spectrum One of the advantages of this method is that it allows forrestarting model runs without rendering the stochastic time series discontinuous Only thephase of each frequency component needs to be known to continue the time series withoutdistorting the spectral characteristics of particularly the low frequencies

In this study N 100 frequency components are taken The frequencies range fromfmin 014 cpd to fmax 36 cpd corresponding to periods between 7 days and40 minutes (the minimum period resolved by the 20 minute time step) We present theresults of 4 integrations characterized by different values of The runs will be identifiedas R where is 05 10 15 and 20 (Table 1) These values reflect the range of spectralslopes from different wind products (Gille 2005) For all the experiments the amplitude ofthe lowest frequency component 1 is kept fixed at 01 to account for the fact that thewind products agree reasonably well for low-frequency variability Taken with theimposed value of this determines the amplitudes of the other frequency componentsThis approach implies that the standard deviation 0 of the series depends on (see Table1) the steepest spectrum with 20 has the smallest variance and contains the leastoverall energy (Fig 1)

From observations in the Drake Passage (eg Peterson 1988) it is clear that thevariability of the Southern Ocean is highly barotropic for sub-seasonal time scalesKoblinsky (1990) argues that the specific distribution of contours of fH in the SouthernOcean makes it more susceptible to barotropic variability than other ocean basins Inaddition the internal Rossby radius of deformation is small in this area (about 20 kmChelton et al 1998) A simple estimate based on Veronisrsquo (1956) partition equation (hisEq 26) and Chelton et alrsquos (1998) estimate of the Rossby radius suggests that equiparti-tion of energy between the baroclinic and internal modes takes place for length scales ofthe order of 100ndash200 km For larger scales of forcing most of the energy is absorbed bythe barotropic mode The large spatial scales and short time scales of the forcing used inthis study allow us to expect a predominantly barotropic response Note however thatsmall-scale flows induced by bathymetry may violate the Rossby radius argument and inreality may lead to some baroclinic response But in view of the high frequenciesconsidered here we expect the model response to be qualitatively robust

The small spatial scales and fast temporal scales also allow us to neglect the nonlinearmetric and advective terms and to ignore any background circulation The importance of

Table 1 Experiment identifiers and spectral slopes of the four wind stress time series used andtheir standard deviation

ID

R05 05 0248R10 10 0119R15 15 0085R20 20 0073


advective terms with respect to the Coriolis term is measured by the Rossby number U0fL The spatial scales of the disturbances under consideration are of the order of theexternal Rossby radius (L (2 106 m)) so a strong jet with U0 05 m s1 (Nowlinand Klinck 1986) results in (103) On time scales of roughly 5 days the advectiveterms are an order of magnitude smaller than the tendency term

The model domain comprises the entire Southern Ocean from 80S to the equator Aspherical grid is used with 180 80 grid points in the zonal and meridional directionsyielding a horizontal resolution of 2deg 1deg Bathymetry is based on the ETOPO-2 data setwhich is interpolated onto the model grid and smoothed with a Laplace filter Themaximum depth is set at 4000 m The 15 grid points in the vertical are unequallydistributed with vertical grid spacing ranging from 62 m for the uppermost layer to almost600 m for the deepest layer The depth of the uppermost model layer is an order ofmagnitude larger than a characteristic e-folding depth of classical Ekman theory so thecomplexities of the Ekman spiral are not nearly resolved Instead the surface layer in thismodel effectively acts as a slab-model often used to study wind-driven inertial motions(Pollard and Millard 1970 DrsquoAsaro 1985 Alford 2003)

In Weijer and Gille (2005) the free-surface formulation of Dukowicz and Smith (1994)was used in fully implicit mode This approach is known to damp surface gravity waves tosome extent In fact it provided enough damping for the system to run with stress-free

Figure 1 Spectra of the four different wind stress time series that are used Black line denotesstandard forcing time series with spectral slope 15 Gray lines are for values of 05 10 and20 Error bar at the low-frequency end of the curves indicates the 95 confidence interval


boundaries so that only quadratic bottom friction was applied In this study we areconcerned with the energetics of the flow and a semi-implicit free-surface formulation isused This approach conserves potential energy exactly in our constant-density modelNo-slip lateral boundaries are used to provide the system with a sink of energy

Although complex algorithms have been developed to parameterize the conversion ofbarotropic motions to internal waves (eg Jayne and St Laurent 2001) we will adopt theconventional quadratic law to represent bottom drag and we use a Laplacian eddyviscosity formulation for internal friction Coefficients of horizontal and vertical eddyviscosity are Ah 10 105 m2 s1 and Av 10 103 m2 s1 respectively The dragcoefficient for the quadratic friction law is taken to be Cd 00014 The models areintegrated for 455 days with a time step of 20 minutes

3 Basic energetics and sensitivity to

In this section we explore the sensitivity of the wind-driven energetics to the spectralslope of the forcing First the time-averaged balances of potential and kinetic energy willbe derived accounting for the circulation in the surface layer and the barotropic interiorseparately

We start off with the barotropic linear Navier-Stokes equations in terms of velocity u (u v) and sea surface elevation They represent the leading-order dynamics of thenumerical model

ut

f k u g 13z

0Hm u (1)

t middot Hu (2)

As in the numerical model the wind forcing is applied as a body forcing in the surfacelayer of depth Hm the vertical structure function 13( z) is 1 in the surface layer and zerobelow H is the total water depth k is the unit vector in the vertical and denotes afriction operator which will be left unspecified here

The directly wind-forced motion denoted by us is assumed homogeneous in the surfacelayer and hence represents an average of the Ekman drift rather than the Ekman velocity atthe surface of the ocean Underneath this surface layer the flow is to a large extentdepth-independent and pressure-driven for our choice of frictional parameters horizontalviscosity is the dominant frictional process so deviations in the bottom boundary layer aresmall This allows us to express the velocity field in terms of us and a homogeneous interiorcontribution ub as u( x y z) ub( x y) 13( z)us( x y) The (linear) momentumequations Eq (1) can now be decomposed to describe the wind- and pressure-driven flowcomponents

us

t f k us

0Hm us (3)


ub

t f k ub g ub (4)

where it is understood that Eq (3) is only defined for the surface layerVertically-integrated kinetic energy defined as E 1

20 H

0 u udz can bedecomposed accordingly into a barotropic part a surface layer component and a mixedcontribution

Es 1

20Hmus middot us (5a)

Eb 1

20Hub middot ub (5b)

Ebs 0Hmub middot us (5c)

Their time-rate of change can be derived by multiplying Eq (3) by us Eq (4) by ub andtaking the sum of ub times Eq (3) and us times Eq (4)

Es

t ss ds (6a)

Eb

t pb db (6b)

Ebs

t ps sb dbs (6c)

where

ps 0gHmus middot (7a)

pb 0gHub middot (7b)

ss us middot (7c)

sb ub middot (7d)

ds 0Hmus middot us (7e)

db 0Hub middot ub (7f)

dbs 0Hmub middot us us middot ub (7g)

In our stochastically forced model we are interested in the time-averaged energetics whichcan be written as a double time-integral of the source and sink terms

Es Ss Ds (8a)


Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)

where (1) 0 dt being the period of integration and Pb 0

t pbdt etc Inthe rest of this manuscript the brackets will be dropped for convenience but time-averagedvalues are still implied unless otherwise noted The terms p s and d denote the rate of workby pressure forces wind stress and friction respectively Their capital letter counterpartsdenote the average of the total energy exchanged by the corresponding terms since the startof the integration

Potential energy is defined as U 120g(2 H2) and its time-derivative is given by

U

t mb ms (9)

where

mb 0g middot Hub (10a)

ms 0g middot Hmus (10b)

As before we will consider time-averaged quantities

U Mb Ms (11)

Again the capitals denote the average of the total energy exchanged by the correspondingterm and the brackets will be dropped in the following

Comparing the pressure work terms p and the divergence terms m we see that they areboth components of the divergence of the energy fluxes gb 0gHub and gs 0gHmus In fact the spatial integrals of gb and gs are zero due to the no-normalflow condition on the boundaries This means that the spatial integrals of mb and pb and ofms and ps add up to zero Note that

p middot g m (12)

Figure 2a shows the input of kinetic energy by the wind stress as a function of the spectralpower of the forcing The energy input in the Ekman layer (Ss) is strongly affected by thespectral characteristics of the forcing A factor of 40 difference is achieved between R05

and R20 In addition there is a strong factor 20 change in kinetic energy of the Ekmandrift itself (Es in Fig 2b)

This contrasts with the wind work on the interior flow (Sb) and the kinetic energy of theinterior flow (Eb) Sb is basically insensitive to the value of but Eb decreases by a factorof 24 as increases One may then wonder how the interior flow is energized In the nextsections we will look at the balances and redistributions of energy first for the standard runR15 and then for cases with other values of


4 Energetics of R15

a Time-averaged distributions of energy

Figure 3 shows the time-averaged fields of kinetic energy and its components Es Eband Ebs The distribution of kinetic energy is concentrated in the forcing region Thesurface flow contribution Es is distributed homogeneously in a broad band which isobstructed only by the tip of South America The kinetic energy of the abyssal flow onthe other hand is much more patchy The highest values of Eb are found where the flowhas to squeeze through gaps in between bathymetric features In contrast to thequadratic quantities Es and Eb the mixed-mode kinetic energy Ebs has no definite signYet Fig 3d shows that it is negative almost anywhere According to its definition Eq(5c) this reflects the fact that the interior and surface circulations consistently flow inthe opposite direction with the barotropic flow compensating for the Ekman transportEbs shows up as a broad and weak band in the forcing area and has a few isolatedmaxima above the Kerguelen Plateau the Macquarie Ridge and around the tip ofSouth America

Figure 2 (a) Kinetic energy input and (b) energy content as a function of spectral slope of theforcing time series Source terms are plotted for the Ekman (black line) and barotropic (gray)modes and are surface-integrated They represent the terms as they appear in the balance oftime-averaged kinetic energy and hence are a double integral over time (Eq (8)) Energy contentsrepresent time-averaged kinetic energy of the surface layer (Es black) barotropic (Eb dark gray)and mixed (Ebs light gray) modes as well as potential energy (U black dashed) Note that Ebs

is negative


b Energy balances

Figure 4a shows the zonally-averaged terms in the (time-averaged) kinetic energybalance Eqs (8) These terms add up to the zonally-averaged kinetic energy which wouldbe indistinguishable from zero in this plot The dominant terms reflect the energetics of theEkman balance in which the kinetic energy input by the wind stress (Ss) is locally

Figure 3 Time-averaged fields of (a) kinetic energy E and its constituents (b) Es (c) Eb and (d)10Ebs Black dashed line denotes axis of maximum wind stress forcing thin black lines are 5001500 2500 and 3500 m isobaths


dissipated in the mixed layer by turbulent processes (Ds) the remaining terms are basicallyof the same order of magnitude and will be discussed in the following paragraphs

i Wind stress and pressure work done on the interior flow As observed by Fofonoff(1981) and Wang and Huang (2004a) the energy input by the wind stress into geostrophic

a)

b)

Figure 4 (a) Zonally averaged source and sink terms of the time-averaged kinetic energy Shown arethe contributions by the wind stress (S dash-dotted lines) viscous dissipation (D dashed) andpressure work (P solid) to the surface (black) mixed (blue) and barotropic mode (red) kineticenergies as in Eqs (8) Vertical dotted line denotes the latitude of maximum wind stress at 53S(b) As Figure 4 but now for the differences R10 R20 Only the terms contributing to Ebs andEb are plotted


flow is mostly used to push the Ekman transport up-pressure This can be seen by rewritingthe pressure work term ps as

ps 0gHmvsy 0gx

0 ffug

g ug (13)

where ug denotes the zonal geostrophically balanced surface velocity Indeed Figure 4ashows that the wind work Sb and pressure work Ps nearly cancel The difference isaccounted for by a friction term Dbs which will later be shown to play an important rolefor smaller values of

If we neglect this frictional coupling between the surface circulation and the interior it isthe pressure work Pb that actually drives the interior circulation rather than the wind workSb This allows the system to release the energy at places where it may be more efficientlydissipated Figure 5 shows that Sb has a mainly zonal structure with highest values in theregion of maximum wind stress forcing In addition there are large areas (mainly in themid-latitude basins) where wind stress extracts energy from the flow The term Pbhowever is much more patchy and more consistently positive throughout the entiredomain it shows strong correlations with topographic features and naturally through Eq(8b) with the dissipation term Db In the areas where Pb is negative viscous frictionlocally energizes the flow (Db 0) as it extracts momentum from its surroundings

There is also a spectral difference between the work terms Sb and Pb Figure 6 showsthat the spectrum of Eb (no time-averaging implied) contains substantial amounts of

Figure 5 (a) Source term Sb and (b) pressure work term Pb Black lines are the zero-contours


inertial energy The spectrum of Sb (no time-averaging) on the other hand only showsminor elevations in the inertial band the variability in energy input it reflects is obviouslynot enough to explain the enhanced levels of global kinetic energy In contrast the spectralcontent of Pb (no time-averaging) is basically identical to that of Eb

Sb may be an actual source of kinetic energy for the ocean circulation but it cannot beconsidered as the primary source of energy for the interior flow additional conversionsinvolving the Ekman layer take place to transfer energy to the interior flow The actualsource of energy is the work done by pressure and we have shown that both the spatial andthe temporal characteristics of Pb and Sb differ How this energy transfer takes place willbe studied in the next paragraph

ii Energetic coupling of surface and interior flows The following thought experimentmay serve to illustrate the role of the Ekman circulation in the energetics of thewind-driven motions Consider a situation in which the sea surface can be deformed bydivergences in the Ekman transports (Vs) only hence allowing for large-amplitudedeviations (s) that are not affected by compensating motions in the oceanrsquos interior

Figure 6 Spectra of the time series of globally-integrated kinetic energy of the barotropic flow Eband of the source terms Sb and Pb (no time-averaging is implied here) Spectra are an average of 8subsamples of the time series light-gray bars denote 95 confidence level Time series aredetrended and prewhitened The spectrum of Eb is offset by a factor of 400 for clarity


Given our idealized wind stress profile T this would lead to an energy flux gss 0gsVs

[(Tf )2]y (Eq A3 see the Appendix for details) This flux transports energy awayfrom the axis of maximum wind stress (Fig 10) to the latitude bands around 60S and 46Swhere Ekman pumping is strongest Here it is stored as potential energy via a contributionto Ms Mss 0gsVsy [(Tf )y]2 (Eq A5) The energy is extracted from the Ekmanflow which has to do work against a gradient in sea-level s Pss 0gVssy Tf(Tf )yy (Eq A6) At the central axis of the wind stress profile the strongest Ekmantransport flows against the largest gradient in s The dotted lines in Figure 7 show theseterms attenuated by a factor of 10 Clearly the work and divergence terms of the Ekmancomponent (Ps and Ms) reflect in shape and sign the terms Pss and Mss

The interior motion largely compensates for the sea-level deformations induced by theEkman transport and accounts for a sea-level deviation that we could indicate by b (sothat s b) This compensation reduces the work that has to be done by the Ekmantransport (via a term that we could denote by Psb 0gVsby) and the amount ofpotential energy that can be stored (by an amount equal to Msb 0gbVsy) Moreimportantly the pressure field can tap the stored potential energy (Mb) and convert it tokinetic energy of the interior flow (Pb) Only a small residual of potential energy Mremains unused (Fig 7b) since the ocean only allows for small deviations of the seasurface This means that almost the entire supply of potential energy Ms is immediatelyused by the interior flow

Not all of the energy available to the interior flow returns to the central axis of forcing tocompensate for Ps Hence net energy is extracted from the total flow in the central band ofwind stress forcing (Fig 7a negative P between 60S and 46S) while energy is re-investedoutside this band Figure 7c shows the zonal integral of these meridional fluxes G

5 Energetics sensitivity to spectral slope

In the previous section we described the balances and redistributions of energy for thestandard run Here we will consider the differences between the runs with different spectralslopes To find out how an increase in interior kinetic energy Eb can be accounted forwhen the wind work Sb does not change we will compare the energetics of the runs R10

and R20 to see how the flows of energy change when more energy is available at highfrequencies

Figure 4b shows how the terms pertaining to the balances of Ebs and Eb change when is decreased from 20 to 10 This change flattens the spectrum and increases the energy athigh frequencies It confirms that the wind work on the barotropic mode (Sb) hardlychanges Nevertheless the pressure work (Pb) is considerably enhanced as is theassociated dissipation Db This is consistent with the increase in kinetic energy of theinterior flow Eb (Fig 2)

Where does this extra energy come from For the standard run this energy Pb is largelybalanced by the work done against the pressure field by the surface flow Ps for which inturn energy is supplied by the wind stress through Sb Figure 4b shows the corresponding


increase in Ps but it is obviously not balanced by an increase in Sb Instead it is drawnfrom the wind work on the surface layer Ss The term that is responsible is the viscouscoupling between the Ekman and barotropic modes Dbs (Fig 8)

The mixed mode kinetic energy Ebs is a correction term when us and ub are in the samedirection the sum of the barotropic and kinetic energies Eb and Es underestimates the totalkinetic energy and the correction term Ebs is positive In our model Ebs is largely negative(Fig 3d) suggesting that us and ub are in opposite direction in this case they partly cancel

Figure 7 (a) Zonally-averaged pressure work P and (b) divergence contribution to the potentialenergy equation M and (c) zonally integrated meridional flux of energy G Contributions ofsurface layer (dashed) and barotropic interior (dash-dotted) add up to total (solid) Dotted linesdenote the Pss Mss and Gss terms (attenuated by a factor of 10) which would reflect Ps Ms andGs if divergences of the Ekman transport were the only factors able to deform the surface See textfor explanation


and Eb Es overestimates the kinetic energy Indeed coherence analysis shows that thewind- and pressure-driven transport components are consistently 180deg out of phasethroughout the frequency spectrum

Just as the dissipation terms Db and Ds tend to reduce the kinetic energies Eb and EsDbs tends to reduce the magnitude of Ebs so they are of opposite sign in Eq (8c) Thismeans that it cannot be interpreted as a transfer of energy from the Ekman mode to thebarotropic mode Dbs just dissipates kinetic energy of the Ekman mode due to friction withthe barotropic mode and vice versa (see Eq (7g))

The significance of this term may be illustrated by the example shown in Figure 9suppose that the surface layer is forced by a constant westerly wind Assuming forsimplicity that the flow is damped by Rayleigh friction so that (u) ru the slabmodel of the surface layer predicts a steady Ekman flow with components

us us vs r f

0Hmr2 f 2

(14)

Note that the two velocity components are related through us rvsf Suppose furtherthat a barotropic flow balances the Ekman transport exactly

ub ub vb us vs (15)

Here HmH is the fractional depth of the surface layer This flow is associated with azonal tilt in sea-level

Figure 8 (a) Diagram of the time-mean volume-integrated energies and energy flows as in Eqs (8)and their numerical values for (b) R10 and (c) R20


gx r2 f 2

fvb

0Hm (16)

In this context the balances of energy as given by Eq (6) can be expressed as

Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs

t ps sb dbs 2 0 (17c)

where

r2

0Hmr2 f 2

(18)

The total dissipation in the surface layer can be written as ds dbs db where the lastcontribution is the fraction of the dissipation of the barotropic flow that takes place in the

Figure 9 Set-up of example problem as discussed in the text The surface layer of depth Hm isforced by a constant zonal wind stress which forces an Ekman transport us (us vs) in theupper layer This transport is compensated by a barotropic interior flow ub (ub vb) (usvs) where HmH This flow is driven by a tilt in the sea-level The actual surface layervelocity is now us ub (1 )us


surface layer This sum equals (1 )2 which is a fraction (1 ) smaller than thereduction in the energy input by the wind ss sb (1 )

This example illustrates how the presence of a significant barotropic flow component inthe surface layer affects the source and sink terms of the energy balance differently whenthe surface and interior flows are in opposite direction the dissipation is reduced more thanthe work done by wind stress This leaves a residual of energy that can be used by theEkman flow to perform work against the pressure field (and in this case also by thebarotropic flow to perform work against the wind sb 0) In this way extra energy issupplied to the oceanrsquos interior without increasing the wind work performed on thebarotropic flow (which is negative in this example)

In the latitude band of the Drake Passage no net zonal pressure gradient can be sustainedso this example might appear only to be applicable to the area north of the Drake PassageIn the unbounded latitudes one would expect a flow that is locally controlled by frictionand a barotropic circulation that is perpendicular to the surface flow However Weijer andGille (2005) have shown that the flow in the Southern Ocean interacts strongly withbathymetry and that topographic form stress plays an important role in the momentumbalance Hence pressure gradients can be sustained locally between bathymetric featuresexplaining the fact that even in the unbounded region us and ub are close to 180deg ratherthan 90deg out of phase

6 Discussion

The oceanrsquos response to external forcing is constrained by several balances eachgoverning a different aspect of the flow The energy balance imposes fewer constraints onthe ocean circulation than do the balances of vorticity and momentum In other wordsmany different flow configurations can satisfy the energy balance while fewer configura-tions are consistent with the vorticity or momentum balances This is nicely illustrated bysome additional numerical experiments in which the frictional parameters were variedneither vertical friction nor bottom friction contributes significantly to the kinetic energybalance and reducing the coefficient of horizontal friction only slightly alters the terms inthe energy balance The reason is that the topographic form stress is the dominant retardingmechanism in the momentum balance as shown by Weijer and Gille (2005) Butinterestingly this mechanism does not appear explicitly in the balance of kinetic energy(MacCready et al 2001) A change in for instance the horizontal friction parameter Ah

changes the field of kinetic energy to some extent but this seems to be intended solely tomaintain the energy sink that is required by other processes

Whether or not the wind work sb directly forces the interior circulation is still a matter ofsome confusion in present-day literature (eg Scott 1999 Wunsch and Ferrari 2004)Although sb provides the actual power it is pressure work pb that truly delivers this energyat depth (eg Gill et al 1974) This may clear up a few ambiguities The first is theprinciple that purely geostrophic flow is essentially balanced and to leading ordernon-dissipative (Pedlosky 1987) If the subsurface geostrophic flow were to receive


energy through wind work it would also require a way to dissipate which is at odds withthe conservative nature of purely geostrophic flows

Secondly there are extensive regions in the World Ocean where wind work is negative(Wunsch 1998) At first sight this may seem remarkable since the balance of kineticenergy must be closed locally wherever nonlinear advection terms are negligible And inthe ubiquitous presence of friction there would be no source of energy to keep the water inmotion However we have shown that the spatial distribution of pressure work differssubstantially from the wind work The pressure work is mainly positive throughout theentire ocean so that even in the areas where wind work is negative the pressure field canmaintain flow against friction In addition the spectrum of Pb closely matches thespectrum of Eb with enhanced levels of energy in the inertial band The spectrum of Sb onthe other hand is basically straight and unable to explain the spectrum of Eb Thedifference is especially pronounced for very high frequencies in the inertial range

Thirdly the barotropic mode may be able to extract kinetic energy from the Ekmanmode The sensitivity studies show that a flatter forcing spectrum (smaller ) does notchange the wind work term Sb but increases the level of kinetic energy in the interiorsubstantially Coupling between the interior flow and the Ekman transport (Dbs) reducesthe net dissipation in the Ekman layer yielding a surplus of energy that can be used toenhance Ps and thereby Pb The insensitivity of this process with respect to the frictionparameter Ah suggests that it is rather robust and hence may be a consistent feature insystems where the barotropic velocity is a significant fraction of the Ekman flow

The presence of (near-)inertial energy in the interior ocean is usually ascribed to internalwaves propagating downward from the mixed layer or upward from the bottom (egGarrett and Munk 1979 Fu 1981) Hypotheses involving loss of balance and theresulting excitation of inertial waves (the Rossby adjustment problem eg Veronis 1956Gill 1982) are not generally put forward Veronis and Stommel (1956) already concludedthat in their two-layer model baroclinic inertial oscillations were much stronger than theirbarotropic counterparts Although there are some indications that moving storms mayexcite depth-independent inertial motions (Shay et al 1990) it seems that the lengthscales of synoptic variability in the atmosphere facilitate the excitation of baroclinicinertial motions instead (Ginis and Sutyrin 1995) It is therefore likely that the large spatialscale of our forcing and its static character may have facilitated the excitation ofbarotropic inertial motions in this model

Baroclinicity may complicate the simple picture sketched in this paper in several otherways In a stratified ocean the work term ps can additionally store potential energy bydeflecting isopycnals (Gill et al 1974) Energy stored in this form (often expressed interms of Available Potential Energy Huang 1998) can be tapped by baroclinic motionsThis effect is significant when substantial Ekman pumping velocities are achieved ie forsteady or slowly-varying winds with relatively small spatial scales The high frequenciesand large spatial scale of the forcing used in this study hence seem to warrant the barotropicapproach


However a substantial amount of barotropic energy may be lost to the internal modewhen the flow interacts with topography (eg Holland 1975 Ivchenko et al 1997Simmons et al 2004 Arbic et al 2004) Steady and transient flows over topography maygenerate internal waves that provide a drag on the barotropic flow (Bell 1975) There arebasically two regimes the regime of oscillatory flow and that of (almost) steady flow overvarying bottom topography Simmons et al (2004) estimate that in their global 10-layertide model forced by an M2 tidal frequency only 30 of the barotropic tidal energy wasconverted to the internal mode However oscillatory flow can only excite internal modegravity waves for frequencies above the local inertial frequency At 53deg the latitude wherein our model the wind forcing is strongest this period is 15 hr and hence the absence ofbaroclinicity in our model is likely to underestimate the damping in the super-inertialregime The wind stress forcing applied to our model only contains a small fraction ofenergy in the super-inertial band so baroclinic conversion is probably less important thanin tide models (eg Arbic et al 2004)

When the flow U is almost steady but the topography is bumpy the frequency withwhich the flow encounters ridge crests is given by Uk where k1 is a characteristic lengthscale of the topography (Gill 1982) Again only when this frequency exceeds the localinertial frequency is the generation of internal waves possible A characteristic velocityscale of U 001 m s1 and a minimum resolved topographic wavelength 2x 2 105 m yield an upper bound to the encounter frequency of 31 107 s1 Hence itseems unlikely that the inclusion of stratification in our model would have generated asubstantial baroclinic response

Mesoscale eddies play an important role in the energetics of the Southern OceanIvchenko et al (1997) studied the energetics of the Southern Ocean in the high-resolutionmodel FRAM and found that as in our model the dominant sink of eddy kinetic energy(EKE) was lateral friction The main sources of EKE were found to be transfer from eddypotential energy (resulting from baroclinic instability processes) and non-linear transferfrom the mean circulation (barotropic instability) They ascribe the lack of wind-inducededdy-variability to the wind stress forcing of the model which is by climatological(monthly mean) winds The schematic wind stress in our model does not allow us toestimate this term from our results

7 Summary and conclusion

In this paper we addressed the energetics of the Southern Ocean in response tohigh-frequency wind stress variability The main issue at hand was the interaction betweenthe Ekman flow and the oceanrsquos interior (barotropic) circulation The source of energy forthe interior flow is often expressed as sg ug but as pointed out by Fofonoff (1981)and Wang and Huang (2004a) this term can only provide energy in an indirect way whatis usually thought of as the work done on the interior circulation is actually work done onthe Ekman flow

We discussed the main pathways of energy in a simple barotropic model of the Southern


Ocean forced by a band of stochastically varying winds In our basic experiment theenergy input by the wind stress through sb is used to push the Ekman drift against thepressure field ( ps) The energy is subsequently stored in potential form via ms Thisreservoir of potential energy can then be tapped (mb) and used by the pressure field toperform work on the interior flow ( pb) Frictional processes ultimately remove the energyfrom the system

It was found that enhanced levels of energy at high frequencies (014 cpd) in theforcing may affect the energetics of the interior flow considerably Despite the fact that thewind work on the barotropic flow did not change the time-mean kinetic energy of theinterior flow increased by a factor 24 when reducing the spectral slope of the forcing from 20 to 05 The reason for this is the fact that dissipation is less efficient in removingkinetic energy when a substantial barotropic flow counteracts the Ekman transport in thesurface layer The resulting surplus of energy can then be used by the pressure field toenhance the interior flow

The mechanisms observed in this model may not control all aspects of the oceanresponse to wind forcing in a fully stratified ocean However the fact that the barotropicresponse to wind forcing is sensitive to the spectral slope of the wind stress implies thathigh-frequency winds with realistic spectral slopes are needed to force ocean models inorder to have realistic energetics at high-frequencies

Acknowledgments This research was supported by the NASA Ocean Vector Wind Science TeamJPL contract 1222984 The comments of two anonymous reviewers are gratefully acknowledged

APPENDIX

Having divided the flow into surface and barotropic components we can represent thesea level fluctuations in terms of the divergences of the respective flow components Hencewe can introduce

s 0

t

middot Usdt b 0

t

middot Ubdt (A1)

Here Us Hmus and Ub Hub Recall that we apply only a zonally-constant zonal windstress that can be expressed in terms of a spatial pattern T( y) an amplitude 0 and astochastic time series (t) as introduced before Suppose that the (meridional) Ekmantransport is in balance with the (zonal) wind stress so that Vs x0f Then we havefor s

s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)

The meridional component of the energy flux gs then contains the term 0gsVs whichcan be written as


gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)

The time-integral of this transport is positive since

0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)

This flux transports energy away from the latitude of maximum wind stress (Fig 10)The divergence of this flux gives contributions to the divergence term ms and the

pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)

describes the potential energy change due to Ekman pumping It is almost zero at the axisof maximum wind stress (Fig 10) but displays maxima at the latitudes where themeridional gradient of the wind stress is strongest (60S and 46S) Of course divergences of

Figure 10 Meridional profiles of Gss (black line) Pss (dark gray) and Mss (light gray) Profile ofGss is scaled with a factor of 2 106


the Ekman transport will be largely compensated by convergences of the subsurface flowbut they do not completely erase the signature of the mss term

The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)

measures the work done on the Ekman transport due to pressure fluctuations induced bydivergences of this transport As shown by Fig 10 this term is largely negative meaningthat the wind pushes the Ekman transport up-gradient

REFERENCESAlford M H 2003 Improved global maps and 54-year history of wind-work on ocean inertial

motions Geophys Res Lett 30 1424 doi1010292002GL016614Arbic B K S T Garner R W Hallberg and H L Simmons 2004 The accuracy of surface

elevations in forward global barotropic and baroclinic tide models Deep-Sea Res II 513069ndash3101

Bell T H 1975 Lee waves in stratified flows with simple harmonic time dependence J FluidMech 67 705ndash722

Chelton D B R A DeSzoeke M G Schlax K El Naggar and N Siwertz 1998 Geographicalvariability of the first baroclinic Rossby radius of deformation J Phys Oceanogr 28 433ndash460

DrsquoAsaro E A 1985 The energy flux from the wind to near-inertial motions in the surface mixedlayer J Phys Oceanogr 15 1043ndash1059

Dukowicz J K and R D Smith 1994 Implicit free-surface method for the Bryan-Cox-Semtnerocean model J Geophys Res 99 7991ndash8014

Fofonoff N P 1981 The Gulf Stream in Evolution of Physical Oceanography Scientific Surveys inHonor of Henry Stommel B A Warren and C Wunsch eds MIT Press Cambridge MA112ndash139

Frankignoul C and P Muller 1979 Quasi-geostrophic response of an infinite -plane ocean tostochastic forcing by the atmosphere J Phys Oceanogr 9 104ndash127

Fu L-L 1981 Observations and models of inertial waves in the deep ocean Rev Geophys SpacePhys 19 141ndash170

Garrett C and W Munk 1979 Internal waves in the ocean Ann Rev Fluid Mech 11 339ndash369Gill A E 1982 Atmosphere-Ocean Dynamics Academic Press NY 662 ppGill A E J S A Green and A J Simmons 1974 Energy partition in the large-scale ocean

circulation and the production of mid-ocean eddies Deep-Sea Res 21 499ndash528Gille S T 2005 Statistical characterization of zonal and meridional ocean wind stress J Atmos

Ocean Tech 22 1353ndash1372Ginis I and G Sutyrin 1995 Hurricane-generated depth-averaged currents and sea surface

elevation J Phys Oceanogr 25 1218ndash1242Hirose N I Fukumori and V Zlotnicki 2001 Modeling the high-frequency barotropic response of

the ocean to atmospheric disturbances Sensitivity to forcing topography and friction J GeophysRes 106 30987ndash30995

Holland W R 1975 Energetics of baroclinic oceans in Numerical Models of the Ocean Circulation5 National Academy of Sciences Washington DC 168ndash177

Huang R X 1998 Mixing and available potential energy in a Boussinesq ocean J Phys Oceanogr28 669ndash678


Ivchenko V O A M Treguier and S E Best 1997 A kinetic energy budget and internalinstabilities in the Fine Resolution Antarctic Model J Phys Oceanogr 27 5ndash22

Jayne S R and L C St Laurent 2001 Parameterizing tidal dissipation over rough topographyGeophys Res Lett 28 811ndash814

Koblinsky C J 1990 The global distribution of fH and the barotropic response of the ocean JGeophys Res 95 3213ndash3218

Large W G W R Holland and J C Evans 1991 Quasi-geostrophic ocean response to real windforcing the effects of temporal smoothing J Phys Oceanogr 21 998ndash1017

MacCready P G Pawlak K A Edwards and R McCabe 2001 Form drag on ocean flows in NearBoundary Processes and Their Parameterization Proc lsquoAha Hulikorsquoa Hawaiian Winter Work-shop University of Hawaii at Monoa Honolulu HI 119ndash130

Marshall J A Adcroft C Hill L Perelman and C Heisey 1997a A finite-volume incompressibleNavier-Stokes model for studies of the ocean on parallel computers J Geophys Res 1025753ndash5766

Marshall J C Hill L Perelman and A Adcroft 1997b Hydrostatic quasi-hydrostatic andnon-hydrostatic ocean modelling J Geophys Res 102 5733ndash5752

McWilliams J C W R Holland and J H S Chow 1978 A description of numerical AntarcticCircumpolar Currents Dyn Atmos Oceans 2 213ndash291

Muller P and C Frankignoul 1981 Direct atmospheric forcing of geostrophic eddies J PhysOceanogr 11 287ndash308

Munk W and C Wunsch 1998 Abyssal recipes II energetics of tidal and wind mixing Deep-SeaRes I 45 1977ndash2010

Nowlin W D and J M Klinck 1986 The physics of the Antarctic Circumpolar Current RevGeophys 24 469ndash491

Oort A H L A Anderson and J P Peixoto 1994 Estimates of the energy cycle of the oceans JGeophys Res 99 7665ndash7688

Pedlosky J 1987 Geophysical Fluid Dynamics Springer-Verlag NY 710 ppPeterson R G 1988 On the transport of the Antarctic Circumpolar Current through Drake Passage

and its relation to wind J Geophys Res 93 13993ndash14004Pollard R T and R C Millard 1970 Comparison between observed and simulated wind-generated

inertial oscillations Deep-Sea Res 17 813ndash821Scott R B 1999 Mechanical energy flux to the surface geostrophic flow using TOPEXPoseidon

data Phys Chem Earth (A) 24 399ndash402Shay L K S W Chang and R L Elsberry 1990 Free surface effects on the near-inertial ocean

current response to a hurricane J Phys Oceanogr 20 1405ndash1424Simmons H L R W Hallberg and B K Arbic 2004 Internal wave generation in a global

baroclinic tide model Deep-Sea Res II 51 3043ndash3068Stammer D C Boning and C Dieterich 2001 The role of variable wind forcing in generating eddy

energy in the North Atlantic Prog Oceanogr 48 289ndash311Thompson D W J and J M Wallace 2000 Annular modes in the extratropical circulation Part I

Month-to-month variability J Clim 13 1000ndash1016Veronis G 1956 Partition of energy between geostrophic and non-geostrophic ocean motions

Deep-Sea Res 3 157ndash177Veronis G and H Stommel 1956 The action of variable wind stresses on a stratified ocean J Mar

Res 15 43ndash75Wang W and R X Huang 2004a Wind energy input to the Ekman layer J Phys Oceanogr 34

1267ndash12752004b Wind energy input to the surface waves J Phys Oceanogr 34 1276ndash1280


Watanabe M and T Hibiya 2002 Global estimates of the wind-induced energy flux to inertialmotions in the surface mixed layer Geophys Res Lett 29 doi1010292001GL014422

Weijer W 2005 High-frequency wind forcing of a channel model of the ACC Normal modeexcitation Ocean Model 9 31ndash50

Weijer W and S T Gille 2005 Adjustment of the Southern Ocean to wind forcing on synoptic timescales J Phys Oceanogr 35 2076ndash2089

Willebrand J S G H Philander and R C Pacanowski 1980 The oceanic response to large-scaleatmospheric disturbances J Phys Oceanogr 10 411ndash429

Wunsch C 1998 The work done by the wind on the oceanic general circulation J Phys Oceanogr28 2332ndash2340

Wunsch C and R Ferrari 2004 Vertical mixing energy and the general circulation of the oceansAnn Rev Fluid Mech 36 281ndash314

Received 2 March 2005 revised 19 July 2005


This reservoir of potential energy can then be tapped to generate motions in the oceanrsquosinterior by means of pressure work (eg Gill et al 1974) Thus although sg provides theactual power it is pressure work that converts this energy to a source that can drive theinterior circulation

This view points at an intimate link between the dynamics of the Ekman layer and thebarotropic interior It raises some questions about the energetics of the deep ocean Inparticular (i) what are the spatial and temporal characteristics of the pressure work incomparison to wind work and what implications does this have for determining where theenergy is dissipated (ii) Could energy leak from the Ekman layer energy source(represented as se middot ue where ue is the Ekman velocity at the surface) to the interiorcirculation

The Southern Ocean receives substantially more wind energy than other basins (Oort etal 1994 Wunsch 1998 Watanabe and Hibiya 2002 Alford 2003 Wang and Huang2004ab) A small continental area and the possibility of circumpolar circulation provideideal conditions for intense air-sea coupling In addition Southern Ocean winds areexceptionally strong and extremely variable Their spectra contain energy over a broadrange of frequencies spanning low-frequency climatic fluctuations (including the AntarcticOscillation (AAO) Thompson and Wallace 2000) storm scales sub-inertial and super-inertial variability Low-frequency variability typically contains much more energy thanhigh-frequency fluctuations so the spectra are essentially red In the frequency rangebetween 10 and 90 cycles per year (cpy) these wind stresses have spectral slopes that areestimated to be between 05 and 14 depending on the wind product (Gille 2005) Between100 and 363 cpy spectral slopes are slightly steeper between 07 and 25 In view of thisuncertainty it is important to know how sensitive the deep ocean is to the high-frequencyspectral content of the wind forcing

Although the results of Wunsch (1998) suggest that the high-frequency part of the windforcing may not contribute substantially to the global energy budget of the interiorcirculation it is generally believed that direct wind forcing is one of the main sources ofeddy-kinetic energy in the ocean (Frankignoul and Muller 1979 Muller and Frankignoul1981) Stammer et al (2001) concluded that this mechanism is especially effective inweakly stratified areas where the contribution of baroclinic instability mechanisms to theeddy field is small Large et al (1991) found that components of the wind (curl) forcingwith frequencies up to 13 cpd (cycles per day) had a noticeable impact on the response of amulti-layer quasi-geostrophic model This time scale is associated with a cut-off frequencyfor the excitation of Rossby waves (Willebrand et al 1980)

In this study we will address the energetics of barotropic variability of the SouthernOcean in response to high-frequency wind forcing We will sketch the pathway of energyfrom the input area to the dissipative regions distinguishing between the roles played bythe Ekman and barotropic circulations As far as we know this may be the first study toaddress the energetics of high-frequency wind-forced variability explicitly discerning theroles of the Ekman layer and the interior circulation



2 The model











ID

R05 05 0248R10 10 0119R15 15 0085R20 20 0073












ut

f k u g 13z

0Hm u (1)

t middot Hu (2)



us

t f k us

0Hm us (3)


ub

t f k ub g ub (4)


20 H


Es 1


Eb 1




Es

t ss ds (6a)

Eb

t pb db (6b)

Ebs

t ps sb dbs (6c)

where



ss us middot (7c)

sb ub middot (7d)





Es Ss Ds (8a)


Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)




U

t mb ms (9)

where




U Mb Ms (11)



p middot g m (12)





4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)





















































2 The model











ID

R05 05 0248R10 10 0119R15 15 0085R20 20 0073












ut

f k u g 13z

0Hm u (1)

t middot Hu (2)



us

t f k us

0Hm us (3)


ub

t f k ub g ub (4)


20 H


Es 1


Eb 1




Es

t ss ds (6a)

Eb

t pb db (6b)

Ebs

t ps sb dbs (6c)

where



ss us middot (7c)

sb ub middot (7d)





Es Ss Ds (8a)


Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)




U

t mb ms (9)

where




U Mb Ms (11)



p middot g m (12)





4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)

























































ID

R05 05 0248R10 10 0119R15 15 0085R20 20 0073












ut

f k u g 13z

0Hm u (1)

t middot Hu (2)



us

t f k us

0Hm us (3)


ub

t f k ub g ub (4)


20 H


Es 1


Eb 1




Es

t ss ds (6a)

Eb

t pb db (6b)

Ebs

t ps sb dbs (6c)

where



ss us middot (7c)

sb ub middot (7d)





Es Ss Ds (8a)


Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)




U

t mb ms (9)

where




U Mb Ms (11)



p middot g m (12)





4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)






























































ut

f k u g 13z

0Hm u (1)

t middot Hu (2)



us

t f k us

0Hm us (3)


ub

t f k ub g ub (4)


20 H


Es 1


Eb 1




Es

t ss ds (6a)

Eb

t pb db (6b)

Ebs

t ps sb dbs (6c)

where



ss us middot (7c)

sb ub middot (7d)





Es Ss Ds (8a)


Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)




U

t mb ms (9)

where




U Mb Ms (11)



p middot g m (12)





4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































ub

t f k ub g ub (4)


20 H


Es 1


Eb 1




Es

t ss ds (6a)

Eb

t pb db (6b)

Ebs

t ps sb dbs (6c)

where



ss us middot (7c)

sb ub middot (7d)





Es Ss Ds (8a)


Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)




U

t mb ms (9)

where




U Mb Ms (11)



p middot g m (12)





4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































Eb Pb Db (8b)

Ebs Ps Sb Dbs (8c)




U

t mb ms (9)

where




U Mb Ms (11)



p middot g m (12)





4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































4 Energetics of R15




is negative


b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































b Energy balances






a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)






















































a)

b)




ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)





















































ps 0gHmvsy 0gx

0 ffug

g ug (13)




























us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)










































































us us vs r f

0Hmr2 f 2

(14)


ub ub vb us vs (15)




gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































gx r2 f 2

fvb

0Hm (16)


Es

t ss ds 0 (17a)

Eb

t pb db 0 (17b)

Ebs


where

r2

0Hmr2 f 2

(18)







6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)























































6 Discussion






















APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)





































































APPENDIX


s 0

t

middot Usdt b 0

t

middot Ubdt (A1)


s 0

t

Vsydt 1

0

0

t x

f y

dt 0

0T

fy

0

t

tdt (A2)



gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































gss 0gsVs g0

2

20T

f2

y

0

t

dt (A3)


0

t dtdt 1

2 0

t d

dt dt 2dt 1

2 dt 2

(A4)


pressure work ps

mss 0gsVsy g0

2

0T

fy

2

0

t

dt (A5)





The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)





















































The term

pss 0gVssy g0

2

0

T

f T

fyy

0

t

dt (A6)




















































Energetics of wind-driven barotropic variability in the...

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