energetics of the global ocean: the role of … · energetics of the global ocean: the role of...

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109

Chapter 4

ENERGETICS OF THE GLOBAL OCEAN:THE ROLE OF MESOSCALE EDDIES

HIDENORI AIKIApplication Laboratory,

Japan Agency for Marine–Earth Science and Technology, Yokohama, Japan

XIAOMING ZHAICentre for Ocean and Atmospheric Sciences,

School of Environmental Sciences, University of East Anglia, Norwich, UK

RICHARD J. GREATBATCH

GEOMAR Helmholtz-Zentrum fur Ozeanforschung, Kiel, Germany

This article reviews the energy cycle of the global ocean circulation, focusing on the role of baroclinicmesoscale eddies. Two of the important effects of mesoscale eddies are: (i) the flattening of theslope of large-scale isopycnal surfaces by the eddy-induced overturning circulation, the basis for theGent–McWilliams parametrization; and (ii) the vertical redistribution of the momentum of basicgeostrophic currents by the eddy-induced form stress (the residual effect of pressure perturbations),the basis for the Greatbatch–Lamb parametrization. While only point (i) can be explained usingthe classical Lorenz energy diagram, both (i) and (ii) can be explained using the modified energydiagram of Bleck as in the following energy cycle. Wind forcing provides an input to the mean KE,which is then transferred to the available potential energy (APE) of the large-scale field by thewind-induced Ekman flow. Subsequently, the APE is extracted by the eddy-induced overturningcirculation to feed the mean KE, indicating the enhancement of the vertical shear of the basiccurrent. Meanwhile, the vertical shear of the basic current is relaxed by the eddy-induced form stress,taking the mean KE to endow the eddy field with an energy cascade. The above energy cycle is usefulfor understanding the dynamics of the Antarctic Circumpolar Current. On the other hand, whilethe source of the eddy field energy has become clearer, identifying the sink and flux of the eddy fieldenergy in both physical and spectral space remains major challenges of present-day oceanography.A recent study using a combination of models, satellite altimetry, and climatological hydrographicdata shows that the western boundary acts as a “graveyard” for the westward-propagating eddies.

1. Introduction

The mechanical energy input to the ocean byatmospheric winds is a major energy sourcefor driving the large-scale ocean circulationand maintaining the abyssal stratification. Thewind power input to the general circulation

in the global ocean is estimated to be O(1)TW (1TW =1012 watts, Wunsch, 1998; Wunschand Ferrari, 2004; Hughes and Wilson, 2008,Scott and Xu, 2009), with a significant frac-tion being supplied by the synoptic winds (Zhaiet al. 2012). It is now clear that regions wherepower is actually injected into the geostrophic

Indo-Pacific Climate Variability and Predictabilityedited by Swadhin Kumar Behera and Toshio YamagataCopyright c© 2016 by World Scientific Publishing Co.


110 H. Aiki, X. Zhai and R. J. Greatbatch

circulation can differ significantly from thoseplaces where power is generated, owing to thelateral energy transfer in the Ekman layer(Kuhlbrodt et al. 2007; Roquet et al. 2011).The energy flux into the geostrophic interiorcan be viewed as work done by the Ekman flowagainst the horizontal pressure gradient, therebyhelping to maintain, by means of Ekman pump-ing/suction, the observed thermocline struc-ture and sea surface height anomaly associatedwith the gyre circulations. The available poten-tial energy (APE) built up by the large-scaleEkman pumping/suction of the main thermo-cline is subsequently released by the generationof mesoscale eddies through baroclinic instabil-ity of the mean flow (Charney 1947; Eady 1949,Gill et al. 1974).

The ocean general circulation models(OGCMs) used for climate projections do not,in general, resolve mesoscale eddies explicitly,requiring that their effect be parametrized.An important impact of mesoscale eddies isan eddy-induced transport (analogous to theStokes drift associated with surface gravitywaves). There are two alternative approaches toparametrizing this transport. The first approachfollows Gent and McWilliams (1990; hereafterGM90) and Gent et al. (1995), and is referredto as the tracer approach. In this approach,the momentum equation carried by a modelsolves for the development of the Eulerianmean velocity, with an eddy-induced advectionterm being added to the tracer equation of themodel. This eddy-induced, additional velocityis parametrized in such a way as to releasethe APE stored in the large-scale density field,mimicking the principal aspect of baroclinicinstability. A basis for the tracer approach isthe classical energy diagram of Lorenz (1955)

that is derived from equations averaged inz coordinates.

The second approach follows Greatbatch andLamb (1990; hereafter GL90) and Greatbatch(1998), and is referred to as the momentumapproach. In this approach, the momentumequation carried by a model solves for the devel-opment of the total transport velocity (i.e. thesum of the Eulerian mean velocity and theeddy-induced additional velocity), with an eddy-induced vertical stress term being added to themomentum equation of the model (with no addi-tional vertical diffusion applied for tracers).1

This eddy-induced additional vertical stress isparametrized in such a way as to verticallyredistribute the geostrophic momentum of thelarge-scale current, mimicking another aspect ofbaroclinic instability. A basis for the momentumapproach is the energy diagram of Bleck (1985)that is derived from equations averaged in den-sity coordinates, and is fundamentally differ-ent from the classical energy diagram of Lorenz(1955).

The additional vertical stress term in theGL90 parametrization represents the layerthickness form stress (the residual effect of pres-sure perturbations) that originates from thethickness-weighted-mean (TWM) momentumequations in density coordinates (Andrews 1983;de Szoeke and Bennett 1993). The layer thick-ness form stress is closely related to theoriesfor the dynamics of the Antarctic CircumpolarCurrent (ACC) that apply in the latitude bandof the Drake Passage, where there are no con-tinental boundaries available to establish thetraditional Sverdrup balance (cf. Nowlin andKlinck 1986; Gnanadesikan and Hallberg 2000;Hughes and De Cuevas 2001). It is generallythought that the layer thickness form stress

1The advection term of the tracer equations of the model is written in terms of the total transport velocity (i.e.the sum of the Eulerian mean velocity and the eddy-induced additional velocity). It should be noted that the totaltransport velocity is available as a prognostic quantity in the model adopting the momentum approach. See Eqs. (4)and (8b).


Energetics of the Global Ocean: The Role of Mesoscale Eddies 111

is responsible for transferring the wind-inducedmomentum from the upper layers of the oceanto the bottom layers so that in the zonalmean the wind stress at the surface is bal-anced by the topographic form stress associ-ated with the pressure difference across ridges(cf. Johnson and Bryden 1989; Rintoul et al.2001; Olbers and Visbeck 2005). Using theGL90 parametrization, the layer thickness formstress at the middepths of the Southern Oceanmay be scaled (see the end of Appendix A)as (0.4 m2/s)(1000 kg/ m3)(1.0 m/s)/(2000 m) =0.2 N/ m2, which is of the same order of magni-tude as the zonally averaged wind stress overthe Southern Ocean. Locally, however, the layerthickness form stress can be much larger, asfound in a diagnosis of the output of an eddyingOGCM simulation by Aiki and Richards (2008).These authors found that the layer thicknessform stress in the region of the Drake Passagewas as large as 4.0 N/ m2. One way to examinethe above-mentioned theory for the dynamics ofthe ACC, as well as the machinery of the GL90parametrization, is to use the energy diagramof Bleck (1985) which contains an energy con-version path associated with the layer thicknessform stress. This energy conversion path is notexplicitly contained in the classical energy dia-gram of Lorenz (1955), and is not suitable forexamining the role of the layer thickness formstress in the ACC.

Since the 2000s, the output of a series ofeddy-resolving simulations for the global oceancirculation has become available to the com-munity. However, identifying the life cycle ofmesoscale eddies remains one of the major chal-lenges of present-day oceanography. While thesource of the eddy field energy has becomeclearer, the sink and flux of the eddy field energyare yet to be characterized in both physicaland spectral space (e.g. Vallis and Hua 1987;Klein et al. 2008; Qiu et al. 2008). The eddyenergy flux in physical space is closely relatedto the westward motion of planetary eddies (e.g.Yamagata 1982; Williams and Yamagata 1984;

Cushman-Roisin et al. 1990; Eden et al. 2007),with the consequence that the western bound-ary is a major sink for eddy energy (Zhai et al.2010), suggesting that this region is also oneof enhanced diabatic mixing in the ocean (e.g.Tandon and Garrett 1996; Eden and Greatbatch2008; Walter et al. 2005; Stoeber et al. 2009).

This article is organized as follows. In Sec. 2we explain the energy cycle for large-scale flowsin the global ocean, using the energy diagram ofBleck (1985), which includes the role of layerthickness form stress. In Sec. 3 we explain(the state-of-the-understanding regarding) thesources, sinks, and fluxes of the mesoscale eddyenergy, using the energy diagram of Lorenz(1955). Section 4 provides a summary. The alter-native use of the two energy diagrams in Secs. 2and 3 is due to a tradeoff between the physi-cal convenience (to distinguish the adiabatic anddiabatic processes) of the TWM framework indensity coordinates and the mathematical con-venience of the Eulerian mean framework inz coordinates.

2. Maintenance of the MeanField Energy

Despite several theoretical studies in variousresearch areas of atmosphere and ocean dynam-ics (cf. Rhines and Young 1982; Andrews 1983;Johnson and Bryden 1989), the vertical redis-tribution of momentum by layer thickness formstress and associated energy conversions havebeen little investigated based on output fromhigh-resolution OGCMs. This is more or lessa result of the four-box energy diagram ofLorenz (1955) being exclusively used in previousstudies. For example, the pioneering paper byHolland and Lin (1975) included a diagnosis ofthe energetics of eddies and wind-driven circula-tion as simulated by a two-layer model. Hollandand Lin (1975) used an energy diagram whichis similar to that of Lorenz (1955) and hencethere is no term representing the layer thickness



form stress. As will be shown in this section, it ispossible to revise the definition of the mean andeddy kinetic energies (KEs) in Holland and Lin(1975), with the consequence that the revisedenergy diagram involves an energy conversionterm representing the role of the layer thicknessform stress.

2.1. Revising the Energy Diagram

of Holland and Lin (1975)

We consider a standard two-layer model consist-ing of incompressible water of uniform density ineach layer. Let hi be the thickness of each layer

(i = 1 and 2 represent the upper and lower layersrespectively), and Vi = (ui, vi) be the horizon-tal velocity vector in each layer. Table 1 presentsa list of the symbols used in the text. The seasurface is assumed to be rigid and the bottomdepth Hb(x, y) > 0 may vary in the horizontalspace, and hence h1 + h2 = Hb (Fig. 1). APEand KE in each layer are given by

P =ρ0

2g∗(H1 − h1)2, (1a)

Ki =ρ0

2hi|Vi|2, (1b)

where ρ0 is the reference density of sea water,g∗ = g(ρ2 − ρ1)/ρ0 is the reduced gravity

Table 1. List of symbols, where A is an arbitrary quantity.

hi Thickness of ith layer (>0)

Ai Unweighted time mean in ith layerbAi ≡ hiAi/hi Thickness-weighted time mean in ith layer

A′′′i ≡ Ai − Ai Deviation from the unweighted mean, compared in ith layer (A′′′

i = 0)

A′′i ≡ Ai − bAi Deviation from the thickness-weighted mean, compared in ith layer (hiA′′

i = 0)

Aiτ = ∂Ai/∂τ Time gradient in ith layer

∇Ai Lateral gradient in ith layer

zρ = ∂z/∂ρ Thickness in density coordinates (<0)

A Unweighted time mean in density coordinatesbA ≡ zρA/zρ Thickness-weighted time mean in density coordinates

A′′′ ≡ A − A Deviation from the unweighted mean, compared at fixed ρ (A′′′ = 0)

A′′ ≡ A − bA Deviation from the thickness-weighted mean, compared at fixed ρ (zρA′′ = 0)

Aτ = ∂A/∂τ Time gradient in density coordinates [∂τ = ∂t + (zτ )∂z ]

∇A Lateral gradient in density coordinates [∇ = ∇2 + (∇z)∂z ]

Az

Eulerian time mean in z coordinates

A′ ≡ A − Az

Deviation from the Eulerian mean, compared at fixed z (A′z = 0)

At = ∂A/∂t Time gradient in z coordinates

∇2A Horizontal gradient in z coordinates

∇3A = (∇2A, ∂A/∂z) Three-dimensional gradient in z coordinates

V Horizontal component of velocity

w Vertical component of velocity

U = (V, w) Three-dimensional velocity

( bV, bw) Thickness-weighted-mean velocity

( bV, zτ + bV · ∇z) Total transport velocity

(VB , wB) Bolus velocity ≡ ( bV − V, zτ + bV · ∇z − w)

(Vqs, wqs) Quasi-Stokes velocity ≡ ( bV − Vz, zτ + bV · ∇z − wz)

ρgb Global background density (∇2ρgb = 0)

p ≡ Rz g(ρ − ρgb) dz Hydrostatic pressure

φ ≡ Rz

gρ dz + gρz Montgomery potential (∇φ = ∇ Rz

gρ dz + gρ∇z = ∇2R

zgρ dz = ∇2p)



wind-inducedEkman velocity

thermocline

continental-dragEkman velocity

mixedlayer

mixedlayer

upper layer

lower layer

eastward wind westward wind eastward wind

eddy-inducedvelocity

z

y

z=0

z=-h

z=-H

z=-H

b

1

1

h1

h2

Fig. 1. Schematic of the meridional section of the global ocean.

associated with the difference of density ρi inthe upper and lower layers, and H1 is the refer-ence thickness of the upper layer. As in Hollandand Lin (1975), we use a low-pass time filter todecompose the layer thickness into the mean andperturbation components, resulting in the meanand eddy APEs being defined as

hi ≡ hi + h′′′i , (2a)

P =ρ0

2g∗(H1 − h1)2︸︷︷︸

Pmean

+ρ0

2g∗h′′′

12

︸︷︷︸P eddy

, (2b)

(for i = 1, 2), where the overbar and the tripleprime symbols denote the time filter and theassociated deviation in each layer.2 On theother hand, KE in each layer is a third-momentquantity, as shown in Eq. (1b). Central to ourargument is the question of how to separateKE into that associated with mean and eddy

components. Holland and Lin [1975; see theirEq. (14)] defined the mean KE as (ρ0/2)hi|Vi|2and the eddy KE as (ρ0/2)(hi|Vi|2 − hi|Vi|2),the latter of which is not a positive-definiteexpression. A clearer definition of the mean andeddy KEs is given by Bleck (1985) as follows:

Vi ≡ Vi + V′′i , (3a)

Ki =ρ0

2hi|Vi|2︸︷︷︸Kmean

i

+ρ0

2hi|V′′

i |2︸︷︷︸Keddy

i

, (3b)

(for i = 1, 2), where A ≡ hA/h is the TWMand A′′ ≡ A − A is the associated deviation ofan arbitrary quantity A. It should be noted thathA′′ = 0. Also, note that each of the mean andeddy KEs in (3b) is clearly a positive-definitequantity. The TWM velocity Vi ≡ hiVi/hi

is written as the sum of the unweighted meanvelocity Vi and the so-called bolus velocity

2The mathematical symbols of the present study are also applicable to equations in density coordinates for a continu-ously stratified fluid (Table 1). It should be noted that the deviation from the unweighted mean in density coordinates(A′′′ ≡ A − A compared at constant density) is slightly different from the deviation from the Eulerian mean inz coordinates (A′ ≡ A− A

zcompared at constant depth, to be used in the next section) for an arbitrary quantity A.



VB ≡ h′′′i V′′′

i /hi:

Vi = Vi + VBi . (4)

Usually, the unweighted mean velocity repre-sents the large-scale geostrophic circulation aswell as the Ekman flow at the top and bot-tom of the ocean, with the bolus velocity repre-senting the eddy-induced circulation. Equation(3b) states that the mean KE, as defined by(ρ0/2)hi|Vi|2, includes the effect of the bolusvelocity. Because of this modified definition forthe mean and eddy KEs, the energy diagramshown below is different from the Lorenz (1955)energy diagram.

2.2. Exact Energy Equations for a

Two-Layer Ocean Model

Equations for the thickness and the horizontalvelocity vector in each layer are written by

∂τhi + ∇ · (hiVi) = 0, (5a)

ρ0(∂τ + Vi · ∇)Vi + ρ0fz× Vi = −∇φi, (5b)

φ2 = φ1 + ρ0g∗(H1 − h1), (5c)

where ∂τ and ∇ are the temporal and lateralderivative operators in each layer (i.e. operatorsin density coordinates), f is the Coriolis param-eter, and φi is the anomaly of the Montgomerypotential (MP) in each layer.3 Using Eqs. (5a)–(5c), one can derive equations for the APE, KE,and the MP flux divergence in each layer;

∂τP = −ρ0g∗(H1 − h1)∂τh1

= φ1∂τh1 + φ2∂τh2, (6a)

∂τKi + ∇ · (ViKi) = −hiVi · ∇φi, (6b)

∇ · (φihiVi) = −φi∂τhi + hiVi · ∇φi, (6c)

where Eq. (5c) and ∂τh1 = −∂τh2 (the rigid lidapproximation) have been used to derive (6a).All terms on the right hand side cancel out onceEqs. (6a)–(6c) are summed to yield a conserva-tion equation for P + K1 + K2.

We now consider equations for the generalcirculation. Using (5a), we rewrite the momen-tum equation (5b) to a flux divergence form:

ρ0[∂τ (hiVi) + ∇ · (hiViVi) + fz × hiVi]

= −hi∇φi. (7)

Application of a low-pass temporal filter toEqs. (5a) and (7) yields

∂τ hi + ∇ · (hiVi) = 0, (8a)

ρ0[∂τ (hiVi) + ∇ · (hiViVi) + fz × hiVi]

= −hi∇φi − h′′′i ∇φ′′′

i − ρ0∇ · (hiV′′i V

′′i ),(8b)

φ2 = φ1 + ρ0g∗(H1 − h1),

φ′′′2 = φ′′′

1 − ρ0g∗h′′′

1 , (8c)

where the Reynolds stress term −ρ0∇·(hV′′i V

′′i )

is associated with only the lateral redistributionof momentum. The form stress term −h′′′

i ∇φ′′′i ,

on the other hand, can be rewritten as

−h′′′1 ∇φ′′′

1 = −(1/2)h′′′1 ∇(φ′′′

1 + φ′′′2 )

− (1/2)∇P eddy, (9a)

−h′′′2 ∇φ′′′

2 = +(1/2)h′′′1 ∇(φ′′′

1 + φ′′′2 )

− (1/2)∇P eddy, (9b)

where Eq. (8c) and h′′′1 + h′′′

2 = 0 have beenused to derive Eq. (9b), and P eddy is given byEq. (2b). The first terms of Eqs. (9a) and (9b)

3Let ps be the sea surface pressure and z (< 0) the water depth. At all depths in the upper layer (i.e. −h1 < z < 0),the MP is written by [ps − ρ1gz] + ρ1gz = ps. At all depths in the lower layer (i.e. −Hb < z < −h1), the MP iswritten by [ps + ρ1gh1 + ρ2g(−h1 − z)] + ρ2gz = ps − (ρ2 − ρ1)gh1 = ps − ρ0g∗h1. The difference of the MP in theupper and lower layers is −ρ0g∗h1, and hence its anomaly is written by ρ0g∗(H1 − h1) = φ2 − φ1, which is Eq. (5c).



represent the transfer of momentum betweenthe upper and lower layers. GL90 suggested toparametrize these terms as a vertical viscosityterm, noting that mesoscale eddies relax the ver-tical shear of the basic current in thermal windbalance. The first terms of Eqs. (9a) and (9b) arealso closely related to the pressure-based expres-sion of the vertical component of the Eliassen–Palm flux (Andrews 1983; Lee and Leach 1996;Greatbatch 1998; Aiki et al. 2015). The second

terms on the right hand side of Eqs. (9a) and(9b) may be regarded as the lateral gradientof an eddy-induced pressure. Aiki and Richards(2008) have confirmed that the net effect of thelateral gradient term is one order in terms ofmagnitude smaller than the vertical redistribu-tion term.

Using Eqs. (8a)–(8c), one can derive equa-tions for the mean APE, the mean KE, and themean MP flux divergence in each layer:

∂τPmean = −ρ0g∗(H1 − h1)∂τh1 = φ1∂τh1 + φ2∂τh2, (10a)

∂τKmeani + ∇ · (ViK

meani ) = −hi(Vi + VB

i︸︷︷︸bVi

) · ∇φi − Vi · h′′′i ∇φ′′′

i︸︷︷︸#1

− ρ0Vi · [∇ · (hiV′′i V

′′i )]︸︷︷︸

#2

, (10b)

∇ · (φihiVi) = −φi∂τhi + hi (Vi + VBi )︸︷︷︸

bVi

·∇φi, (10c)

where Eq. (8c) and ∂τh1 = −∂τh2 have beenused to derive Eq. (10a). The last two terms ofEq. (10b) represent the roles of the form stressand the Reynolds stress, and account for interac-tion with the eddy field energy. The other termson the right hand side cancel out once the sum ofEqs. (10a)–(10c) is taken to yield an equation forthe mean field energy (Pmean+Kmean

1 +Kmean2 ).

It should also be noted that the first term onthe right hand side of Eq. (10b), as well as

the last term of Eq. (10c), have been writtenusing Eq. (4) as −hiVi · ∇φi = −hi(Vi +VB

i ) · ∇φi, which will prove useful later in thearticle.

Finally, we consider equations for mesoscaleeddies. Subtraction of Eqs. (10a)–(10c) from thelow-pass-filtered version of Eqs. (6a)–(6c) yieldsequations for the eddy APE, the eddy KE, andthe MP flux divergence in each layer:

∂τP eddy = ρ0g∗h′′′

1 ∂τh′′′1 = φ′′′

1 ∂τh′′′1 + φ′′′

2 ∂τh′′′2 , (11a)

∂τKeddyi + ∇ · (ViK

eddyi + V′′

i Ki) = −hiV′′i · ∇φ′′

i + ρ0Vi · [∇ · (hiV′′i V

′′i )]︸︷︷︸

#2

, (11b)

∇ · (φ′′′i h′′′

i Vi + φ′′i hiV′′

i ) = −φ′′′i ∂τh′′′

i + Vi · h′′′i ∇φ′′′

i︸︷︷︸#1

+ hiV′′i · ∇φ′′

i . (11c)

An energy diagram for Eqs. (10a)–(11c) is illus-trated in Fig. 2a. In order to clarify the struc-ture of the energy cycle, equations for the energyinteraction, (10c) and (11c), are here written

separately from the budget of the APE andKE (in contrast to traditional oceanic studies,where equations for the pressure flux divergence



(a)

(b) Pmean

Kmean

Peddy

Keddy

g(ρz − ρgb)wz

−Vz · ∇2p

z −V′ · ∇2p′z

gρ′w′z

−ρ0Vz · [∇3 · (U′V′z)]

Ψqs · g∇2ρz − wz

zPeddy

Pmean

Kmean

P eddy

Keddy

φ∂τh

−hV · ∇φ −hVB · ∇φ −hV′′ · ∇φ′′

φ′′′∂τh′′′

−V · h′′′∇φ′′′

−ρ0V · [∇ · (hV′′V′′)]

Fig. 2. A four-box energy diagram based on energy equations averaged in (a) density coordinates as shown byEqs. (10a)–(11c) and (b) z coordinates as shown by Eqs. (17a)–(19c). (a) is appropriate to the approach taken byGreatbatch and Lamb (1990), and (b) to the approach taken by Gent and McWilliams (1990). The cyan arrowsrepresent the energy conversion route associated with the wind-induced Ekman flow. The green arrows represent theenergy conversion route associated with the continental drag Ekman flow. The red arrows represent the main energyconversion route associated with baroclinic mesoscale eddies. The yellow arrows represent the energy conversion routeassociated with the Reynolds stress. Note that ∇φ = ∇2p (the lateral gradient of the MP in density coordinates isidentical to the horizontal gradient of hydrostatic pressure in z coordinates).

have been merged into equations for the KE).In the eddy field of Fig. 2a, the quantity −V ·h′′′∇φ′′′ is independently connected to boththe eddy APE and the eddy KE; this is dueto Eq. (11c). In particular, the direct connec-tion between the eddy APE and the mean KEinvolves both the density surface perturbationand the layer thickness form stress. The situa-tion in the eddy field is consistent with the resultof Bleck (1985) and Iwasaki (2001) derivedfrom the mass-weighted-mean equations for acontinuously stratified non-Boussinesq fluid [see

Chassignet and Boudra (1988) and Røed (1999)for oceanic applications]. In addition to the formstress term, the Reynolds stress term connectsthe mean and eddy KEs, which is as in theLorenz (1955) energy diagram and is relevant tothe role of relative vorticity in barotropic insta-bility, baroclinic instability, and geostrophicturbulence (cf. Thompson 2010; Berloff andKamenkovich 2013; Chapman et al. 2015).

To summarize, although it has been lit-tle mentioned in previous studies, the simplestframework for understanding the role of layer



thickness form stress and associated energy con-versions is a classical two-layer model with amodified definition of the mean and eddy KEs.Extension of the above formulation to a con-tinuously stratified fluid with arbitrary bottomtopography is straightforward as done by Aikiand Yamagata (2006) and Aiki and Richards(2008). These authors have further shown thatthe energy equations are robust in the presenceof the ocean boundaries; in particular, the struc-ture of the energy diagram is unchanged evenin the presence of (i) finite-amplitude perturba-tions associated with mesoscale eddies, as well as(ii) density surface outcropping (i.e. intersect-ing) at the top and bottom boundaries of theocean.

2.3. A Model Diagnosis in Density

Coordinates

In order to illustrate the energy cycle in the newenergy diagram, we use a set of three-day snap-shots from a high-resolution (0.1◦ × 0.1◦) near-global hindcast simulation. This simulation iscalled the OFES (OGCM For the Earth Sim-ulator) hindcast run, and was forced by dailymean wind stress from the QuikSCAT data anddaily-mean heat and fresh water fluxes fromthe NCEP/NCAR reanalysis (Masumoto 2010).A set of three-day snapshots archived through-out the years 2004–2007 of the OFES hind-cast run was used for the diagnosis (a total of483 three-dimensional snapshots of the globalocean). These snapshots were originally in z

coordinates. The model has 54 depth levels,with the discretization varying from 5 m at thesurface to 330 m at the maximum depth of6065m. Each snapshot was first mapped ontodensity coordinates in each vertical column. Asin Aiki and Richards (2008), we use 80 den-sity layers defined by the potential density refer-enced to sea surface pressure. This density waschosen in an attempt to overview the globalocean with most of the KE distributed abovethe main thermocline. In the present diagnosis,

the low-pass temporal filter (overbar) was setto a four-year mean for 2004–2007. We havedetermined the global distribution of energyconversion terms concerning the budget of themean KE in Eq. (10b) and Fig. 2a. These are−hV · ∇φ, −hVB · ∇φ, −V · h′′′∇φ′′′, and−ρ0V · [∇ · (hV′′V′′)], for each of which thedepth integral in each vertical column is plottedin Fig. 3 with negative (positive) values indicat-ing a decrease (increase) in the mean KE.

Whilst significant positive and negative signsare evident in the work of the isopycnal meanvelocity, −hV · ∇φ (Fig. 3a), the working ratein the model, integrated globally, is −0.77 TW(negative), which indicates conversion of themean KE to the mean APE. This quantitymainly reflects the wind-induced Ekman flowsteepening the slope of isopycnal surfaces nearthe sea surface. Strictly speaking, the work ofthe isopycnal mean velocity includes also theeffects of continental-drag Ekman flow (associ-ated with the sum of the horizontal friction termand the bottom friction term in the model),which is represented by the positive signals nearthe western boundaries of the Pacific, Atlantic,and Indian Oceans, in particular the GreenlandSea, the upstream of the Kuroshio and the GulfStream, and the Agulhas Current. These posi-tive signals indicate conversion of the mean APEto the mean KE, which is then dissipated bythe boundary friction at the continental slopeof the ocean. (See Aiki et al. 2011b for theglobal distribution of the energy dissipation bythe boundary friction in the 0.1◦ × 0.1◦ OFESsimulation; the working rate by the boundaryfriction integrates to −0.32 TW in the mod-elled global ocean, not shown. See also Trossmanet al. 2013.)

The work done by the bolus velocity,−hVB · ∇φ, is shown in Fig. 3b. This quan-tity is clearly positive over each region of higheddy activity in the global ocean: the mean APEis extracted to provide an input to the meanKE when the eddy-induced overturning circu-lation relaxes the slope of isopycnal surfaces.



Fig. 3. The depth integral of the energy conversion

rates (W/ m2) associated with (a) the unweighted meanvelocity −h V · ∇φ representing the effect of the Ekmanflow, (b) the bolus velocity −hVB · ∇φ representing theeffect of the eddy-induced overturning circulation, (c)the layer thickness form stress − bV·h′′′∇φ′′′ representingthe effect of the vertical redistribution of momentum byeddies, and (d) the Reynolds stress −ρ0

bV · [∇·(hV′′V′′)]representing the effect of the lateral redistribution ofmomentum by eddies. The sign is referenced to the bud-get of the mean kinetic energy, positive indicating anincrease in the mean kinetic energy. The estimate isbased on the three-day snapshots of the OFES hindcastsimulation with application of a four-year mean in den-sity coordinates.

The working rate in the modeled global oceanintegrates to +0.87 TW. The fact that the globaldistribution is mostly positive indicates that thisquantity provides a suitable basis for mesoscaleeddy parametrization in climate ocean models,and is consistent with GM90.

The work done by the layer thickness formstress, −V · h′′′∇φ′′′, is shown in Fig. 3c, wherenegative values indicate an energy cascade tothe eddy field associated with the relaxationof the vertical shear of the geostrophic cur-rents. The horizontal distribution of the workdone by the layer thickness form stress is partlyanticorrelated with the work done by the bolusvelocity in Fig. 3b. Nevertheless, there are alsopositive signals, for example in the KuroshioExtension region. The working rate in the mod-eled global ocean integrates to −0.48 TW. Notethat this is the combined effect of the first andsecond terms on the right hand side of Eqs. (9a)and (9b). If we had plotted the work done byonly the first (the vertical redistribution) term ofEqs. (9a) and (9b) which is associated with theGL90 parametrization, the sign is clearly neg-ative in each region of the global ocean, as isconfirmed by Aiki and Richards (2008).

The work done by the Reynolds stress,−ρ0V · [∇ · (hV′′V′′)], is shown in Fig. 3d.The working rate in the model-global oceanintegrates to −0.15 TW, suggesting an energycascade to the eddy field associated with therelaxation of the lateral shear of the geostrophiccurrents. This might indicate the effect ofbarotropic instability, for example at the west-ern boundary of the Indian Ocean, and in theKuroshio and the Gulf Stream regions. However,in the Gulf Stream extension region, the signalis clearly positive, indicating acceleration of themean current by the eddies (Held and Andrews1983; Greatbatch et al. 2010; Waterman andJayne 2011), in which the meridional radiationof Rossby waves could play a role (Thompson1971).

To summarize, the equilibrium of the meanKE in the wind-driven ocean circulation is



maintained by the following energy cycle. Windforcing provides an input to the mean KE,which is then transferred to the mean APEby the wind-induced Ekman flow (the role ofthe isopycnal mean velocity V). Nevertheless,the net mean APE does not change, becausethe eddy-induced overturning circulation (therole of the bolus velocity VB) extracts some ofthe mean PE and feeds the mean KE, which issubsequently drained by the work of the layerthickness form stress term −h′′′∇φ′′′, endow-ing the eddy field with an energy cascade. Theresult of the above diagnosis is complementaryto Aiki and Richards (2008), in that (i) wealso identify the work of the Reynolds stressterm, (ii) our statistics are based on a four-yearmean of three-day snapshots instead of monthlymeans of three-day snapshots, and (iii) we usethe output of a hindcast simulation forced bydaily-mean atmospheric forcing instead of a cli-matological simulation forced by monthly-meanatmospheric forcing.

3. Maintenance of the EddyField Energy

Identifying the life cycle of mesoscale eddiesis one of the major challenges of present-dayoceanography. Theories for mesoscale eddiesdevelop through diagnoses of field observations,satellite data, and numerical simulations. Apractical approach is to work in z coordinates(where z is the geopotential height). This is atradeoff between the mathematical convenienceof the Eulerian mean framework in z coordinatesand the physical convenience (to distinguish theadiabatic and diabatic processes) of the TWMframework in density coordinates. Also, manyof the widely used numerical models are for-mulated in z coordinates, an example being theOFES model discussed in the previous section.

Let U ≡ (V, w) and ∇3 ≡ (∇2, ∂z), where∇2 is the horizontal gradient operator in z coor-dinates (Table 1). Let a low-pass time filterin z coordinates be denoted by an overbar

with superscript z, and A′ ≡ A − Az

(com-pared at constant depth) for an arbitrary quan-tity A. It is obvious that ∇3 ·U′ = 0, andU′ ·n = 0, where n is a unit vector nor-mal to the sloping bottom boundary as well asthe rigid sea surface (i.e. as before, we makethe rigid lid approximation). This is convenientfor constructing theories for the life cycle ofmesoscale eddies. In contrast, mapping the devi-ation in density coordinates (A′′′ ≡ A − A,compared at constant density) onto the meandepth of each isopycnal yields ∇3 ·U′′′ �= 0 andU′′′ ·n �= 0. This concern is the reason whyMcDougall and McIntosh (2001) suggested toreplace the set of the isopycnal mean velocityand the bolus velocity (each of which is three-dimensionally divergent and does not satisfya no-normal-flow boundary condition) in Gentet al. (1995) with the set of the Eulerian meanvelocity and the quasi-Stokes velocity, each ofwhich is three-dimensionally nondivergent andsatisfies a no-normal-flow boundary condition(Appendix B). The exact definition of the quasi-Stokes stream function is

Ψqsexact =

∫ z

−Hb

(V − Vz)dz, (12)

where Hb(x, y) > 0 is the bottom depth. It can beeasily shown that Ψqs

exact vanishes at the top andbottom boundaries of the ocean without rely-ing on extra assumptions [see the pile-up rule ofAiki and Yamagata (2006)], which is in contrastto theories relying on diabatic mixing in the sur-face mixed layer (Plumb and Ferrari 2005; Fer-rari et al. 2008). In this section, we briefly explainenergy equations averaged in z coordinates andthen present the state-of-the-art understandingof the life cycle of mesoscale eddies in the classi-cal Lorenz (1955) energy diagram.

3.1. Energy Equations Averaged

in z-coordinates

Equations for the incompressibility, density, andmomentum in z coordinates (neglecting diabatic



and viscous effects for simplicity) read

∇3 ·U = 0, (13a)

(∂t + U · ∇3)ρ = 0, (13b)

ρ0[(∂t + U · ∇3)V + fz × V] = −∇2p, (13c)

0 = −∂zp − g(ρ − ρgb), (13d)

where p is the combined sea surface and hydro-static pressure (∇2p = ∇φ is understood). Thequantity ρgb = ρgb(z) is the global backgrounddensity which varies only in the vertical direc-tion, and might be obtained by sorting all waterparcels in the global ocean. Potential energy(PE) and KE are here defined by

P = g(ρ − ρgb)z, (14a)

K =ρ0

2|V|2. (14b)

UsingEqs. (13a)–(13d), onemayderive equationsfor PE, KE, and the pressure flux divergence:

∂tP + ∇3 · [U(P + φgb

)] = g(ρ − ρgb)w, (15a)

∂tK + ∇3 · (UK) = −V · ∇2p, (15b)

∇3 · (Up) = V · ∇2p − g(ρ − ρgb)w, (15c)

where φgb

= gρgbz +∫

zgρgb dz is the MP asso-

ciated with the background density. Equation(15a) for PE is exact and applicable to finite-amplitude density variations as well as sloping-bottom boundary conditions (Kang and Fringer2010; Aiki et al. 2011a). For each of P and K, thelow-pass-filtered energy canbewritten as the sumof themeanand eddyfield energies (AppendixC),

Pz

= g(ρz − ρgb)z +g

2ρ′2

z

ρzz︸︷︷︸

Pmean

+g

2ρ′2

z

(−ρzz)︸︷︷︸

Peddy

, (16a)

Kz

=ρ0

2|Vz|2︸︷︷︸

Kmean

+ρ0

2|V′|2z

︸︷︷︸Keddy

. (16b)

As shown below, the exact definition of themean and eddy PEs in Eq. (16a) allows us toderive the mean and eddy PE equations (19a)and (17a) without relying on any approxima-tions, which is in contrast to previous studieswhere the mean and eddy PEs are defined by(g/2)(ρz − ρgb)2/(−ρgb

z ) and (g/2)ρ′2z/(−ρgb

z ),respectively (Boning and Budish 1993; Olberset al. 2012). See Appendix D for details. Using(∂t +U

z ·∇3)ρ′ = −U′ ·∇3ρz −∇3 · (U′ρ′)′ and

(∂t +Uz ·∇3)ρz

z = −(Uz

z ·∇3)ρz −∇3 · (U′ρ′z)z,

one may derive equations for the eddy PE, theeddy KE, and the pressure flux divergence in theeddy field to read

∂tPeddy + ∇3 · (Uz

Peddy) = gρ′w′z +(−Ψqs · g∇2ρ

z + wzzP

eddy)

︸︷︷︸#1

, (17a)

∂tKeddy + ∇3 · [Uz

Keddy + U′Kz] = −V′ · ∇2p′

z+ ρ0V

z · [∇3 · (U′V′z)]︸︷︷︸#2

, (17b)

∇3 · (U′p′z) = V′ · ∇2p′

z − gρ′w′z, (17c)

where the expression (17a) omits the triple andhigher product of perturbation quantities forsimplicity, and

Ψqs = −ρ′V′z

ρzz

+12

ρ′2z

ρzz2 V

z

z, (18)

turns out to be identical to the approximatedexpression of Ψqs

exact in McDougall and McIntosh(2001). They transformed Eq. (12) to Eq. (18)using an approximation z′′′ ∼ −ρ′/ρz

z which isapplicable to only depths away from the top



and bottom boundaries of the ocean. This iswhy Eq. (18) does not vanish at the top andbottom boundaries of the ocean, in contrastto Eq. (12).

Subtraction of Eqs. (17a)–(17c) from thelow-pass-filtered version of Eqs. (15a)–(15c)yields equations for the mean PE, the mean KE,and the pressure flux divergence in the meanfield:

∂tPmean + ∇3 · [Uz

(Pmean + φgb

) + U′ρ′zgz] = g(ρz − ρgb)wz − (−Ψqs · g∇2ρ

z + wzzP

eddy)

︸︷︷︸#1

, (19a)

∂tKmean + ∇3 · (Uz

Kmean) = −Vz · ∇2p

z − ρ0Vz · [∇3 · (U′V′z)]︸︷︷︸

#2

, (19b)

∇3 · (Uzpz) = V

z · ∇2pz − g(ρz − ρgb)wz . (19c)

An energy diagram based on Eqs. (17a)–(19c)is illustrated in Fig. 2b, and it is a slightlyimproved version of the classical Lorenz (1955)energy diagram.

When one is considering baroclinic mesoscaleeddies that extract the mean PE (to feed theeddy PE in the Lorenz diagram), the sign ofΨqs · g∇2ρ

z should be negative. Using integra-tion by parts, the depth integral of −Ψqs ·g∇2ρ

z

approximates that of −VB · ∇φ, the latter ofwhich has already been shown in Fig. 3b andis clearly positive in each region of the globalocean. This result of the model diagnosis justi-fies the principle of the GM90 parametrization,which is to make Ψqs · g∇2ρ

z negative in eachvertical column. There are several possibilitiesfor parametrizing Ψqs. The solution of GM90is Ψqs ∝ −∇2ρ

z, which has been explained inSec. 1 (see also Appendix A). The solution ofAiki et al. (2004) is Ψqs

zz ∝ ∇2ρz, which enables

the stream function to be closed at the top andbottom boundaries without resorting to taper-ing. Ferrari et al. (2010) suggested to blendthe above two solutions. It is also of interestto note that Eq. (12) of Eden et al. (2007) isthe quasi-geostrophic equivalent of Eq. (17a).Eden et al. (2007) interpreted their Eq. (12) asa variance equation in which the term labeled#1 in Eq. (17a) corresponds to the produc-tion term and −gρ′w′z to the dissipation term.

Equation (13) in their paper goes on to providea formula for deriving the GM90 diffusivity forparametrizing the quasi-Stokes velocity.

3.2. Eddy Energy Generation and

Vertical Eddy Energy Flux

The APE built up by the large-scale Ekmanpumping/suction of the main thermocline ishypothesized to be subsequently released by thegeneration of eddies through baroclinic insta-bility of the mean flow (e.g. Gill et al. 1974;Wunsch 1998). This hypothesized energy routehas been supported by a series of idealizednumerical model experiments (e.g. rectangu-lar basin, flat bottom, and simplified surfaceforcing) by Radko and Marshall (2003), whofound that the downward energy flux fromthe Ekman layer into the main thermocline islargely balanced by lateral eddy transfer, withsmall-scale mixing making only a minor con-tribution. Recently, Zhai and Marshall (2013)re-examined this problem using a realistic eddy-permitting model of the North Atlantic Oceandriven by climatological monthly-mean forcing.The model used in their study is the MIT Gen-eral Circulation Model (Marshall et al. 1997).The model spans the Atlantic Ocean between



Fig. 4. (a) The wind power input to the North Atlantic Ocean (W m−2). (b), (c), and (d) show the rate of APEreleased by baroclinic instability (−ρ′w′zg; ×10−5 W m−3) at 61 m, 900 m, and 1875 m, respectively. The color scaleis saturated in order to reveal regions of moderate eddy activity. (Adapted from Zhai and Marshall 2013.)

14◦S and 74◦N, and has a horizontal resolutionof 1/10◦ × 1/10◦. Readers are referred to Zhaiand Marshall (2013) for the model details.

Figure 4a shows the time-mean wind powerinput to the ocean in their model. As in pre-vious studies (e.g. Zhai and Greatbatch 2007;Roquet et al. 2011), the wind stress appearsto spin up the subtropical gyre at its north-ern and southern gyre boundaries, with littleenergy input in the gyre interior. However, recallthat power injected into the Ekman layer is firstredistributed laterally by the Ekman transportbefore being pumped into the geostrophic inte-rior (Roquet et al. 2011). Therefore, the interiorof the subtropical gyre is not a “desert” for wind

power input. As noted by Zhai and Marshall(2013), the hot spot of wind energy input inthe Caribbean Sea appears to be a robust fea-ture. It will be interesting to see what fraction ofthis energy input is dissipated locally, and howmuch is exported out of the Gulf of Mexico. Inte-grating over the North Atlantic Ocean, the totalwind energy input is found to be about 0.14 TW.

Figures 4b–d show the rate at which theAPE in the model is released through baro-clinic instability, i.e. −gρ′w′z [see Eqs. (17a)and (17c)], at different depths. Positive valuesof −gρ′w′z are associated with dense fluid sink-ing and buoyant fluid rising, therefore releasingthe APE stored in the mean stratification. It is



remarkable to see that the values are overwhelm-ingly positive, especially away from the tropicalregion, meaning that the eddies act as a sink forthe APE built up by the large-scale wind actionalmost everywhere in the model. Interestingly,besides the western boundary region where theeddies are known to have a major role to play,the eddies are also found to systematically flat-ten isopycnals in regions such as the eastern andsouthern rims of the subtropical gyre (Fig. 4b).Integrating over the North Atlantic Ocean, therate of APE extracted by baroclinic instabilityis found to be about 0.11 TW, approximately80% of the total wind energy input. The model-ing study by Zhai and Marshall (2013) thus addsfurther support to the hypothesis that the wind

energy input to the gyre circulations is largelybalanced by eddy generation through baroclinicinstability — consistent with our earlier findingwhen discussing Fig. 3 (see also Fig. 4 of Edenet al. 2007).

As noted by Zhai and Marshall (2013), theeddy KE generation, −gρ′w′z, is mainly con-fined to the upper ocean in the subtropicalgyre, but has a much deeper structure in thesubpolar gyre, with its maximum strength atabout 2000m depth on average. This differencein the vertical structure of eddy KE produc-tion has implications for the vertical fluxes ofeddy energy. Figure 5 shows the vertical eddyenergy flux, p′w′z, at different depths and lati-tudes, which represents the redistribution of

Fig. 5. The vertical eddy energy flux (p′w′z ; ×10−2 W m−2) at (a) 61 m, (b) 900 m, (c) 39◦N, and (d) 59◦N.(Adapted from Zhai and Marshall 2013.)



eddy energy in the vertical [see Eq. (17c)]. Largedownward eddy energy fluxes are found nearthe surface in the North Brazil Current andCaribbean Sea, and deeper down in the watercolumn in the western boundary current and itsextension. In contrast, the vertical eddy energyflux is mostly upward in the subpolar region,especially in the Labrador Sea. This differencein the direction of p′w′z is further illustrated inFigs. 5c and 5d, where p′w′z is mainly down-ward under the Gulf Stream at 39◦N, but pre-dominantly upward in the Labrador Basin andIrminger Basin at 59◦N — consistent with thedepths of eddy KE sources at these two lati-tudes. The vertical eddy energy flux has receivedrelatively little attention to date, although theequivalent vertical energy flux associated withinternal waves/tides has been widely diagnosedin both observations and numerical models. Thisissue apparently merits further investigation.

3.3. The Eddy Energy Sink and Its

Large-Scale Effect

In equilibrium, the energy flux into the eddyfield has to be dissipated. However, our under-standing of the fate of ocean eddies remainsrather poor. Potential candidate processes thatmay remove eddy energy from the ocean include:bottom frictional dissipation (e.g. Sen et al.2008; Wright et al. 2012), damping by air–seamomentum and heat fluxes (e.g. Duhaut andStraub 2006; Zhai and Greatbatch 2006, 2007;Greatbatch et al. 2007; Shuchburgh et al. 2011),generation of lee waves over rough topogra-phy (e.g. Marshall and Naveira Garabato 2008;Nikurashin and Ferrari 2011; Scott et al. 2011),the “Neptune effect” along sloping topogra-phy (e.g. Holloway 1992; Dukowicz and Great-batch 1999; Greatbatch and Li 2000; Adcockand Marshall 2000), and interactions with theinternal wave field through loss of balance andRossby wave deformation (e.g. Straub 2003;

Molemaker et al. 2005; Buhler and McIntyre2005). To date, there is still no consensuson which mechanism(s) provides the dominanteddy energy sink.

Ocean eddies are observed to propagate west-ward at long Rossby wave speeds (e.g. Cheltonet al. 2007), but what happens to the eddies whenthey encounter the western boundary has beenunclear. Recently, using a combination of mod-els, satellite altimetry, and climatological hydro-graphic data, Zhai et al. (2010) showed that thewestern boundary acts as a “graveyard” for thewestward-propagating eddies. The model used isa nonlinear reduced-gravity model on a β plane,with a lateral resolution of 3.5 km. Model inte-grations initialized with both a single eddy anda random sea of eddies have been conducted toexamine the eddy energy budget near the westernboundary. These model experiments suggest thefollowing picture: upon encountering the westernboundary, the available potential energy asso-ciated with the eddies is converted into kineticenergy of the reflected short Rossby waves andsmaller eddies, the majority of which is dissi-pated near the western boundary. In the reduced-gravity model, there is only one baroclinic modeand the only energy sink for eddies is lateralviscous dissipation, whereas in the ocean muchof the eddy energy is likely scattered into high-wave-number vertical modes that rapidly dissi-pate (e.g. Dewar and Hogg 2010).

Using satellite altimetry and climatologi-cal hydrographic data, Zhai et al. (2010) com-puted the divergence of the depth-integratedlinear eddy energy fluxes4 associated with thefirst baroclinic mode, which is presumably bal-anced by eddy energy sources and sinks. Read-ers are referred to Zhai et al. (2010) for adetailed method description and error analy-sis (see Kunze et al. 2002 for an application ofa similar method to diagnosing internal waveenergy dissipation). Figure 6 shows the sourcesand sinks of eddy energy in the first baroclinic

4This comes from vertically integrating the term on the left hand side of Eq. (17c).



Fig. 6. Sources (red) and sinks (blue) of eddy energy in the first baroclinic mode. Regions shallower than 300 m deepare shaded in light gray and excluded in the calculation. The color scale is saturated to reveal regions of relativelymoderate eddy energy sources and sinks. (Adapted from Zhai et al. 2010.)

mode. Note that regions shallower than 300mdeep are shaded in light gray and excluded inthe calculation. The most striking features arethe ubiquitous eddy energy source (in red) inthe interior and the eddy energy sink (in blue)near the western boundary in each ocean basin.Therefore, in agreement with the model results,the satellite altimetry data also point to thewestern boundaries as an important region ofenergy loss for ocean eddies. Furthermore, thetotal eddy energy sink near the western bound-aries poleward of 10◦ of latitude is estimatedto be approximately 0.1–0.3TW, representing asignificant fraction of wind power input to theocean general circulation. Wright et al. (2012)have recently estimated the bottom dissipationof eddy energy at the Atlantic zonal boundariesusing ocean current meter data and found thatdissipation at the western boundary is signifi-cantly larger than that at the eastern bound-ary, adding support to the idea that the westernboundary acts as a “graveyard” for ocean eddyenergy.

Some of the dissipating processes thatremove eddy energy may lead to enhanceddiapycnal mixing in the western boundaryregions (i.e. the dissipated eddy energy is

converted back to the mean PE), which canhave important implications for the large-scaleocean circulation and climate. Saenko et al.(2012) recently conducted a suite of sensitiv-ity experiments with an ocean general circula-tion model assuming that the eddy energy thatconverges at the western boundary is scatteredinto high-wave-number vertical modes, resultingin locally enhanced diapycnal mixing (Dewarand Hogg 2010). When diapycnal mixing intheir model is maintained only by the tidalenergy dissipation (tidal experiment; Fig. 7a),the abyssal overturning circulation and strati-fication are too weak. When diapycnal mixingin their model is maintained by both the tidaland eddy energy dissipation (tidal+eddy exper-iment; Fig. 7c), the abyssal overturning circu-lation and stratification become stronger andcloser to the observations. Furthermore, mixingassociated with the eddy energy dissipation isfound to be able to drive a relatively strong over-turning in the abyss (∼5 Sv) on its own (eddyexperiment; Fig. 7b). These sensitivity exper-iments highlight the importance of includingmixing associated with the eddy energy dissi-pation especially near the western boundary infuture ocean climate models.



Fig. 7. Mean overturning circulation (Sv) below about1 km depth in (a) tidal, (b) eddy, and (c) tidal+eddyexperiments. Negative values correspond to counter-clockwise circulation. Diapycnal mixing is maintainedby dissipation of tidal energy, eddy energy and a com-bination of tidal and eddy energy in the tidal, eddy,and tidal+eddy experiments, respectively. (From Saenkoet al. 2012.)

4. Summary

We have reviewed the state-of-the-understanding of the energy cycle of the globalocean circulation, focusing on the role of baro-clinic mesoscale eddies. The presence of twoindependent but equivalent prescriptions for

the energy cycle — one based on the energydiagram of Lorenz (1955) and the other basedon the energy diagram of Bleck (1985) — pro-vides complementary ways of understanding therole of mesoscale eddies in ocean circulation the-ories, such as parametrization used in climateOGCMs, the dynamics of the Antarctic Cir-cumpolar Current (ACC), and the life cycle ofmesoscale eddies. Two of the important effectsof mesoscale eddies are: (i) the flattening of theslope of large-scale isopycnal surfaces by theeddy-induced overturning circulation, the basisfor the GM90 parametrization; and (ii) the ver-tical redistribution of the momentum of basicgeostrophic currents by the layer thickness formstress (the residual effect of pressure perturba-tions), the basis for the GL90 parametrization.While only point (i) can be explained using theclassical Lorenz (1955) energy diagram, bothpoint (i) and point (ii) can be explained usingthe energy diagram of Bleck (1985), which isbased on the modified definitions of the meanand eddy KEs in density coordinates or layermodels.

With the role of the layer thickness formstress in the ACC in mind, the energy cycle ofthe wind-driven circulation in the global oceanis explained as follows. Wind forcing providesan input to the mean KE, which is then trans-ferred to the available potential energy (APE) ofthe large-scale field by the wind-induced Ekmanflow. In some regions the APE is extracted bythe eddy-induced overturning circulation to feedthe mean KE, as expressed by −hVB · ∇φ inFig. 2a. This route is found to be in the meanfield, and it provides an input to the mean KE(i.e. acceleration of the mean current), in con-trast to the corresponding conversion term inthe classical Lorenz energy diagram, as repre-sented by Ψqs ·g∇2ρ

z in Fig. 2b. It is also notedin Fig. 2a that the mean KE (ρ0/2)h|V|2 willleak to the eddy field by −V · h′′′∇φ′′′, a resultof the vertical redistribution of momentum (i.e.deceleration of the mean current) by the layerthickness form stress.



While the source of the eddy field energyhas become clearer, identifying the life cycle ofmesoscale eddies remains one of the major chal-lenges of present-day oceanography. The sinkand flux of the eddy field energy are yet tobe characterized in both physical and spectralspace. The eddy energy flux in physical space isclosely related to the westward motion of plane-tary eddies, with the consequence that the west-ern boundary is a major sink for eddy energy.Potential candidate processes that may removeeddy energy from the ocean include: bottom fric-tional dissipation, damping by air–sea momen-tum and heat fluxes, and interactions with theinternal gravity wave field through loss of bal-ance and Rossby wave deformation.

Acknowledgments

H. A. would like to express the deepest grat-itude to his supervisor, Emeritus ProfessorToshio Yamagata, for introducing the wonderof mesoscale eddies and the valuable guidanceat the University of Tokyo and JAMSTEC. Healso thanks Prof. Masaki Ishiwatari for helpfulcomments on Appendix D. R. J. G. is grate-ful to Prof. Yamagata for his friendship of over30 years and to GEOMAR for continuing sup-port. X. Z. would like to thank David Marshallfor the many helpful discussions.

Appendix A. Tracer and MomentumApproaches for the Mesoscale EddyParametrization

Most OGCMs currently maintained at theworld’s climate centers adopt the tracer

approach. Indeed, the inclusion of the eddy-induced advection of tracers represented a majoradvance in ocean modeling, allowing coarse-resolution OGCMs used in climate studies tomaintain a sharper and more realistic ther-mocline and removing spurious overturningcells in the Southern Ocean (Danabasoglu andMcWilliams 1994; Griffies et al. 2000; Gent2011). The GM90 parametrization presents aneddy-induced velocity based on the principlethat mesoscale eddies flatten the large-scaleslope of isopycnals, which may be interpreted asa lateral diffusion of the depth of each isopyc-nal surface (Fig. 1). Letting κ denote this dif-fusion coefficient, the eddy-induced overturn-ing stream function associated with the GM90parametrization can be written as: −κ timesthe slope of the isopycnals [cf. Eq. (18) ofGent et al. (1995)].

The momentum approach is much less widelyused in the community, despite being advo-cated by a number of authors (Greatbatch 1998;Wardle and Marshall 2000; Zhao and Vallis2008). An example applied to a global oceanmodel is the study by Ferreira and Marshall(2006). Let ν denote the coefficient of theGL90 viscosity; then the vertical flux of momen-tum associated with the layer thickness formstress is parametrized by: ρ0ν times the verti-cal shear of the total transport velocity. Thetwo approaches, GM90 and GL90, are equivalentwhen the main component of the total trans-port velocity is the geostrophic Eulerian meanvelocity5 and the coefficients of the GM90 dif-fusivity and the GL90 viscosity are related asκ/N2 = ν/f2 where N =

√(g/ρ0)(−∂zρ

z) isthe buoyancy frequency. Using the former con-dition, GL90 and McWilliams and Gent (1994)

5The total transport velocity may consist of three kinds of velocity: (i) the geostrophic component of the Eulerianmean velocity, (ii) the Ekman component of the Eulerian mean velocity, and (iii) the eddy-induced additional trans-port velocity (the quasi-Stokes velocity). The Ekman velocity is almost absent in the midlatitude ocean interior, i.e.outside the equatorial region and the surface and bottom boundary layers, and the eddy-induced velocity is usuallysmall compared to the geostrophic velocity of the large-scale flow (Aiki and Yamagata 2006).



have shown that

∂

∂z

(ν

∂V∂z

� ∂

∂z

(ν

∂Vz

∂z

= −z× ∂

∂z

ν

f

g∇2ρz

ρ0

)

= fz× ∂

∂z

νN2

f2

∇2ρz

∂zρz

)

= fz× ∂

∂z

[−κ

∇2ρz

−∂zρz

)], (20)

where the last line may be interpreted as theCoriolis force associated with the eddy-inducedvelocity as given by the GM90 parametriza-tion. A typical value for the GM90 diffusiv-ity is κ = 1000 m2/s (i.e. operating laterallyto flatten the slope of isopycnals), and a typi-cal value for N2/f2 at midlatitudes is 400, forwhich the corresponding GL90 viscosity is ν =0.4 m2/s — several orders of magnitude greaterthan the coefficient of vertical eddy viscosityassociated with three-dimensional turbulence.Then, in order to scale the eddy-induced verticalstress in the ACC (Sec. 1), we have assumed thevertical shear of the geostrophic Eulerian meanvelocity to be (1.0 m/s)/(2000 m).

Appendix B. The Quasi-StokesVelocity and Gent andMcWilliams (1990)

In density coordinates for a continuously strati-fied fluid, Eq. (8a) may be rewritten as

zρτ + ∇ · (zρV) = 0, (21a)

where z is the height of an isopycnal surfaceand zρ = ∂z/∂ρ is the thickness between adja-cent isopycnal surfaces. The subscript τ and thesymbol ∇ are the temporal and lateral gradientoperators in density coordinates, respectively,and A is the unweighted isopycnal mean and

A ≡ zρA/zρ is the TWM for an arbitrary quan-tity A (Table 1). Mapping the above equationto z coordinates referencing the low-pass filteredheight of each isopycnal surface yields

∇2 · V + ∂z(zτ + V · ∇z) = 0, (21b)

where ∇2 in the horizontal gradient operator inz coordinates (Table 1), and ∇ = ∇2 + (∇z)∂z

has been used (de Szoeke and Bennett 1993).6

The three-dimensional velocity (V, zτ + V · ∇z)has been referred to as either the total trans-port velocity (Aiki and Yamagata 2006; Aiki andRichards 2008) or the tracer transport velocity(Greatbatch and McDougall 2003), in order todistinguish it from the TWM velocity (V, w).With Eq. (21b) in mind, the bolus velocityis defined as the difference between the totaltransport velocity and the unweighted isopycnalmean velocity to read

VB ≡ V − V, (22a)

wB ≡ zτ + V · ∇z − w, (22b)

which leads to ∇2 · VB + ∂zwB = −(∇2 · V +

∂zw) �= 0. Again with Eq. (21b) in mind, thequasi-Stokes velocity is defined as the differencebetween the total transport velocity and theEulerian mean velocity to read

Vqs ≡ V − Vz, (23a)

wqs ≡ zτ + V · ∇z − wz, (23b)

which leads to ∇2 ·Vqs + ∂zwqs = −(∇2 ·Vz

+∂zw

z) = 0. Thus, one can introduce an overturn-ing vector stream function for the quasi-Stokesvelocity, as in (12), to read Vqs = ∂zΨ

qsexact and

wqs = −∇2 · Ψqsexact (McDougall and McIntosh

2001). Substitution of these into (23b) yields

zτ + Vz · ∇z − wz = −∇ · Ψqs

exact, (24)

where the advection term on the left-hand sideis based on the Eulerian mean velocity and

6Jacobson and Aiki (2006) presented an alternative proof for Eq. (21b) using a one-dimensional analog of the hybridLagrangian–Eulerian coordinates of Andrews and McIntyre (1978).



the right-hand side has been written using thelateral-divergence operator in density coordi-nates. Substitution of the GM90 stream func-tion (see Appendix A) into the last term ofEq. (24) yields ∇·(κ∇z), which is why the GM90parametrization is formally referred to as “thelateral diffusion of the depth of each isopycnalsurface” (Gent et al. 1995; Gent 2011) ratherthan the so-called thickness diffusion.

Appendix C. Interpreting the MeanPotential Energy of the PresentStudy

Using a Taylor expansion in the vertical direc-tion, we express the Eulerian mean of anarbitrary quantity A (averaged in z coordi-nates) in terms of quantities averaged in densitycoordinates:

Az

= A + (−z′′′)(Az) + (−z′′′)2(Azz)/2 + · · · .

= A − z′′′(Aρ/zρ) + z′′′2[(Aρ/zρ)ρ/zρ]/2 + · · · .

� A − z′′′[(Aρ + A′′′ρ )/zρ](1 − z′′′ρ /zρ) + (z′′′2/2)(Aρ/zρ)ρ/zρ + · · · .

= A − z′′′A′′′ρ /zρ + [(z′′′2/2)ρAρ/zρ + (z′′′2/2)(Aρ/zρ)ρ]/zρ + · · · .

= A − z′′′A′′′ρ /zρ + [(z′′′2/2)Aρ/zρ]ρ/zρ + · · · .

= A − z′′′ρ A′′′/zρ − z′′′A′′′ρ /zρ + [(z′′′2/2)Aρ/zρ]ρ/zρ + · · · .

= A + [−z′′′A′′′ + (z′′′2/2)Aρ/zρ]ρ/zρ + · · · .

= A + (∂/∂z)[−z′′′A′′′ + (z′′′2/2)Aρ/zρ] + · · · , (25)

where z = z + z′′′ is the height of an isopy-cnal surface and zρ = ∂z/∂ρ is the thicknessbetween adjacent isopycnal surfaces. The tripleand higher product of perturbation quantitieshave been omitted in Eq. (25). The depth inte-gral of Eq. (25) yields

∫ 0

−HbA

zdz =

∫ 0

−HbAdz,

which is consistent with the pile-up rule (Aikiand Yamagata 2006). In other words, the depthintegral of the second term on the last line ofEq. (25) vanishes with z′′′ = 0 at z = 0 and−Hb, no matter how the bottom is sloped. Wesubstitute A = gρz into Eq. (25):

gρzz = gρzz

= gzρρz/zρ + g[−z′′′(ρz)′′′ + (z′′′2/2)(ρz)ρ/zρ]z + · · · .

= gρzρz/zρ + g[−ρz′′′2 + (z′′′2/2)(ρz)ρ/zρ]z + · · · .

= gρ(z + z′′′ρ z′′′/zρ) + g[−ρz′′′2 + (z′′′2/2)(z/zρ + ρ)]z + · · · .

= gρz + gρ(z′′′2/2)ρ/zρ + g[−ρ(z′′′2/2) + (z′′′2/2)z/zρ]ρ/zρ + · · · .

= gρz − g(z′′′2/2)/zρ + [g(z′′′2/2)z/zρ]z + · · · .

= gρz − (gρz/2)z′′′2 + [(gzρz/2)z′′′2]z + · · · , (26)



(a) raw density (b) Eulerian mean density (c) mean-height density

t

z

tt

zz

Fig. 8. Views of (a) the raw density ρ(z, t), showing thevertical fluctuation of a density surface in a two-densityfluid; (b) the Eulerian mean density ρz(z, t), which isgiven by the fixed-height temporal average; and (c) themean-height density ρ(z, t) = ρ(z, t), which is a z coor-dinate expression of the adiabatically low-pass-filteredlayer interface, given by the average of the interfaceheight between the two density layers. A darker shadeindicates higher density. (Adapted from Aiki and Yama-gata 2006.)

where the third line has been derived usingρ = ρ+ρ′′ = ρ and the last line has been derivedusing ρz = 1/zρ. The quantity ρ is referred to asthe mean-height density in Aiki and Yamagata(2006), as it is given by the distribution of thetime-mean height z of each isopycnal surface, asshown in Fig. 8. Usually the mean-height den-sity ρ represents a sharper stratification than theEulerian mean density ρz.

Using Eq. (26), we interpret the mean PE inEq. (16a) as

Pmean = g(ρz − ρgb)z + (gρzz/2)(ρ′/ρz

z)2z

� g(ρz − ρgb)z + (gρz/2)z′′′2

= g(ρ − ρgb)z + [(gzρz/2)z′′′2]z ,

(27)

where the first term on the last line is basedon the mean-height density ρ. The second termon the last line of Eq. (27) vanishes when thedepth integral is taken in each vertical column,

no matter how the bottom is sloped. To sum-marize, gρz is at the heart of the mean PE inEq. (16a). While the total (mean plus eddy) PEis associated with the Eulerian mean density ρz,the eddy PE is associated with the density dif-ference (ρz − ρ).

Appendix D. Approximate Equationsfor the Available Potential Energyin Previous Studies

An Eulerian approximation for the availablepotential energy (APE) has been widely usedin previous studies,

g

2(ρ − ρgb)2

(−∂zρgb)

, (28)

where the global background density ρgb =ρgb(z) varies only in the vertical direction.Application of an Eulerian time mean to (28)yields

g

2(ρ − ρgb)2

z

(−∂zρgb)

=g

2(ρz − ρgb)2

(−∂zρgb)︸︷︷︸

Amean

+g

2ρ′2

z

(−∂zρgb)︸︷︷︸

Aeddy

,

(29)

where the symbols Amean and Aeddy representtraditional definitions for the mean APE andthe eddy APE, respectively (Boning and Budish1993). However, it is rather difficult to deriveexact prognostic equations for the mean APEand the eddy APE in Eq. (29). A compromisein the previous studies (Olbers et al. 2012; vonStorch et al. 2012) is to approximate Eq. (13b)as (∂t + V · ∇2)ρ + w∂zρ

gb = 0, and then deriveprognostic equations for the mean APE and theeddy APE to read

(∂t + Vz · ∇2)Amean + g

V′ · ∇2[ρ′(ρz − ρgb)]z

(−∂zρgb)

= −ρ′V′z

∂zρgb

· g∇2ρz + gwz(ρz − ρgb), (30a)

(∂t + Vz · ∇2)Aeddy + g

ρ′V′ · ∇2ρ′z

(−∂zρgb)

=ρ′V′z

∂zρgb

· g∇2ρz + gw′ρ′

z, (30b)



which contain three concerns. First, the advec-tion term of Eqs. (30a) and (30b) is based ononly the horizontal component of velocity, whichis not suitable for calculating the volume budgetin a domain with a sloping bottom. Second thebaroclinic conversion term [i.e. the first term onthe right hand side of each of (30a) and (30b)]depends on the global background density ρgb.Third, the definition of the eddy APE in Eq. (29)depends on the global background density ρgb.The above three concerns have been resolved inthe PE equations (17a) and (19a) of the presentstudy.

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