energetics of the global ocean: the role of layer ... · understanding the role of mesoscale eddies...

Energetics of the Global Ocean: The Role of Layer-Thickness Form Drag

HIDENORI AIKI

International Pacific Research Center, University of Hawaii at Manoa, Honolulu, Hawaii, and Frontier Research Center for GlobalChange, Japan Agency for Marine–Earth Science and Technology, Yokohama, Japan

KELVIN J. RICHARDS

International Pacific Research Center, University of Hawaii at Manoa, Honolulu, Hawaii

(Manuscript received 4 April 2007, in final form 9 January 2008)

ABSTRACT

Understanding the role of mesoscale eddies in the global ocean is fundamental to gaining insight into thefactors that control the strength of the circulation. This paper presents results of an analysis of a high-resolution numerical simulation. In particular, the authors perform an analysis of energetics in densityspace. Such an approach clearly demonstrates the role of layer-thickness form drag (residual effects ofhydrostatic pressure perturbations), which is hidden in the classical analysis of the energetics of flows. Forthe first time in oceanic studies, the global distribution of layer-thickness form drag is determined. Thisstudy provides direct evidence to verify some basic characteristics of layer-thickness form drag that haveoften been assumed or speculated about in previous theoretical studies. The results justify most of theprevious assumptions and speculations, including those associated with (i) the presence of an oceanic energycycle explaining the relationship between layer-thickness form drag and wind forcing, (ii) the manner inwhich layer-thickness form drag removes the energy of vertically sheared geostrophic currents, and (iii) thereason why the work of layer-thickness form drag nearly balances the work of eddy-induced overturningcirculation in each vertical column. However, the result of the analysis disagrees with speculation inprevious studies that the layer-thickness form drag in the Antarctic Circumpolar Current is the agent thattransfers the wind-induced momentum near the sea surface downward to the bottom layers. The authorspresent a new interpretation: the layer-thickness form drag reduces (and thereby cancels) the vertical shearresulting from the eddy-induced overturning circulation (rather than the vertical shear resulting from thesurface wind stress). This interpretation is consistent with the results of the energy analysis conducted in thisstudy.

1. Introduction

The present study extends the theory of Bleck (1985,hereafter B85), Iwasaki (2001, hereafter Iw01), andAiki and Yamagata (2006, hereafter AY06) to theanalyses of currents and eddies simulated by a high-resolution ocean general circulation model (OGCM).This is the first attempt in oceanic studies to determinethe global distribution of energy conversion done byresidual effects of hydrostatic pressure perturbations(called the layer-thickness form drag, as detailed in sec-tion 2b). In the present study this drag is associated with

the adiabatic effects of transient eddies. Although thelayer-thickness form drag has often been assumed tovertically redistribute geostrophic momentum, it andthe associated energetics have not been investigated onthe basis of either the output from high-resolution nu-merical simulations or observations of the ocean. Theeffects of the Reynolds stress, diabatic processes, andthe form drag caused by the bottom topography werenot investigated in the present study. The context of thepresent study is also relevant to atmospheric dynamicsin that the sum of the layer-thickness form drag and theReynolds stress divergence corresponds to the diver-gence of a pressure-based Eliassen–Palm flux (An-drews 1983, hereafter A83; Lee and Leach 1996, here-after LL96; Iw01; Tanaka et al. 2004). As an atmo-spheric and zonal-mean counterpart of the presentstudy, Uno and Iwasaki (2006) have recently analyzed

Corresponding author address: Hidenori Aiki, IPRC/SOEST,University of Hawaii, 1680 East West Road, POST Bldg., 4thFloor, Honolulu, HI 96822.E-mail: [email protected]

VOLUME 38 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y SEPTEMBER 2008

DOI: 10.1175/2008JPO3820.1

© 2008 American Meteorological Society 1845

JPO3820

an output from an atmospheric general circulationmodel (AGCM) and shown a cascade-type energy con-version.

Despite several theoretical studies in various re-search areas of atmosphere and ocean dynamics (cf.Rhines 1979; A83; Johnson and Bryden 1989; Cush-man-Roisin et al. 1990; Greatbatch 1998, hereafterG98) the vertical mixing of momentum by layer-thickness form drag and associated energy conversionshave been little investigated (or confirmed) on the basisof output from high-resolution OGCMs. This is moreor less a result of the classical energy diagram of Lorenz(1955, hereafter L55) being exclusively used in previousstudies. The role of layer-thickness form drag cannot beexplained by the energy diagrams of L55 and Plumb(1983); rather, it requires an adiabatic mean four-boxenergy diagram that was derived by B85, Iw01, andAY06. This special feature of the adiabatic mean en-ergy diagram is attributed to its use of modified defini-tions of the mean and eddy kinetic energies (B85;Iw01); these are given by a thickness-weighted meanformulation of an inviscid stratified fluid in density (orisentropic) coordinates. Although it was a problem inthe formulation of B85, the boundary condition of theadiabatic mean energy equations is robust and straight-forward even in the presence of density surfaces inter-secting (i.e., outcropping at) the top and bottom bound-aries of the ocean, as recently shown by AY06 usingonly no-normal-flow boundary conditions of the totaltransport velocity (defined in section 2c) and the rawvelocity.

AY06 stated that understanding the adiabatic meanenergy diagram is fundamental to introducing a param-eterization of layer-thickness form drag in OGCMs(McWilliams and Gent 1994; Krupitsky and Cane 1997;G98; Greatbatch and McDougall 2003; Ferreira andMarshall 2006). The energetics of layer-thickness formdrag are important in a number of aspects of oceandynamics, such as (i) the dynamics of the AntarcticCircumpolar Current (e.g., Johnson and Bryden 1989),(ii) the ventilated thermocline theory (e.g., Luyten et al.1983), (iii) mesoscale eddy parameterization in coarse-resolution OGCMs (e.g., G98), and (iv) the dynamics ofbaroclinically coupled coherent eddies (e.g., Cushman-Roisin et al. 1990; Aiki and Yamagata 2000, 2004). Thepresent study addresses the issue of the Antarctic Cir-cumpolar Current, and the results of our energy analy-sis resolve a confusion in previous studies, as explainedbelow.

The layer-thickness form drag has received consider-able attention in previous investigations of the Antarc-tic Circumpolar Current regarding the vertical transferof momentum and its relationship with the wind stress

applied at the sea surface. The status of modern ocean-ography is well represented by the debate between Ol-bers (1998) and Warren et al. (1996, 1998), as explainedbelow. There has been a series of attempts to prove thatthe layer-thickness form drag is an agent for transfer-ring the wind-induced momentum near the sea surfaceto the bottom layers (McWilliams et al. 1978; Johnsonand Bryden 1989; Olbers 1998; Olbers and Ivchenko2001). This requires the vertical flux (the form stress) ofmomentum associated with the layer-thickness formdrag to connect the bottom of the Ekman layer (or thesurface mixed layer) and the top of the deep layer in-teracting with the bottom topography, with a constant(divergence-free) vertical profile of the momentum flux(the form stress) between the two depths (see Fig. 2 ofOlbers and Visbeck 2005). On the other hand, Warrenet al. (1996, 1998) have pointed out that this view of thevertical momentum transfer may be incorrect because itappears to come from intuitions based on nonrotationalfluid dynamics. Because the ocean is a rotational strati-fied fluid, the wind stress applied at the sea surfaceshould induce Ekman transports (flowing perpendicu-lar to the direction of the wind stress) such that themomentum equation is balanced in the thin Ekmanlayer at the sea surface, resulting in no wind-inducedmomentum being transferred downward.

Moreover, the layer-thickness form drag in some pre-vious studies (Ivchenko et al. 1996; Stevens andIvchenko 1997; Olbers and Ivchenko 2001; Marshalland Radko 2003) is misleading in that (i) this form dragis disguised as the Coriolis force associated with aneddy-induced velocity and (ii) the pressure field is notanalyzed in these studies, which is a custom originatingin the transformed Eulerian mean (TEM) theory andassociated formulation of a velocity-based Eliassen–Palm flux (Andrews and McIntyre 1976). In contrast tothe TEM theory, the layer-thickness form drag used inthe present study is based on the pressure field and thusis a formal approach (as in Killworth and Nanneh 1994;LL96; Greatbatch and McDougall 2003; AY06).

The main result of the present study is given in sec-tion 3, where we analyze the output from a high-resolution (0.1° � 0.1°) global OGCM based on a der-ivation of energy equations in density coordinates (seeappendix A or AY06). The purpose of the analysis is toverify some basic characteristics of layer-thickness formdrag that were often assumed or speculated about inprevious studies, as explained below. Section 3a com-pares energy conversions caused by the layer-thicknessform drag, the eddy-induced overturning circulation,the wind-induced Ekman transport, and the wind forc-ing. This is intended to verify the presence of an energycycle as suggested by AY06 for the wind-driven ocean

1846 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 38

circulation, whose dependence on the time scale of alow-pass filter is investigated in section 3b. Then weinvestigate some characteristics of layer-thickness formdrag associated with this energy cycle. To investigatehow the layer-thickness form drag presents a sink of themean kinetic energy, section 3c examines which of thevertical and horizontal redistributions of momentum bylayer-thickness form drag is predominant in terms ofenergy conversions in the global ocean. The result ofthis analysis confirms the most fundamental assumptionin previous studies, namely that the layer-thicknessform drag redistributes momentum mainly in the ver-tical direction. To investigate why the work done bylayer-thickness form drag nearly balances the workdone by the eddy-induced overturning circulation ineach vertical column, section 3d examines the presenceof the geostrophic balance of the layer-thickness formdrag in the global ocean. The result of this analysisjustifies the frequent use of the assumption of the eddygeostrophic balance in previous studies (such as the useof the disguised form drag in the TEM theory). Theadiabatic aspects of oceanic dynamics confirmed aboveare summarized in section 4 using a set of planetarygeostrophic equations, including the energetics oflayer-thickness form drag. This set of equations issimple but includes eddy effects, and it has a lot ofpotential to advance the understanding of the role oflayer-thickness form drag in various research areas ofocean dynamics.

The primitive and energy equations in density coor-dinates used in sections 3 and 4 are systematically de-scribed in section 2, which serves to clarify the defini-tions of terms and symbols used in the present study.The paper concludes with a summary in section 5.

2. Primitive equations

This section summarizes the momentum, density, andenergy equations for a continuously stratified fluid indensity coordinates with inviscid, incompressible, hy-drostatic, and Boussinesq approximations. Diabaticmixing, thermobaricity, friction, and boundary forcingare excluded from the present formulation for simplic-ity.

The explicit use of density coordinates in the presentstudy is intended to demonstrate how density surfacesare handled near the top and bottom of the ocean.Previous formulations of density coordinates (B85;Killworth and Nanneh 1994) assume that unused den-sity surfaces are condensed at the top and bottom of theocean with infinitesimal thickness (i.e., the buoyancyfrequency is infinite there). This problem is fixed in thepresent study by adopting a formulation similar to that

in A83 to achieve realistic (i.e., nonhomogeneous) dis-tributions of density and other tracers at the top andbottom boundaries (section 2a). Elaborating the struc-ture of density coordinates near the top and bottomboundaries of the ocean is fundamental also to clarify-ing the definition of the isopycnal (unweighted) meanin the presence of density surfaces that intersect (out-crop at) the top and bottom boundaries of the ocean(section 2b).

The other aspects of the derivation of the adiabaticmean equations are consistent with those in AY06, in-cluding the no-normal-flow boundary condition of thetotal transport velocity explained by using an integralidentity (section 2c) and the use of energy equationsbased on modified definitions of the mean and eddykinetic energies. Comparisons with the formulationsused in previous studies are given in section 2d. Char-acteristics of the energy equations and the adiabaticmean four-box energy diagram are briefly explained insection 2e.

a. Density coordinates

Let (u, �, w) be the three-dimensional velocity inCartesian coordinates (often called z coordinates inoceanic studies) spanned by a set of independent vari-ables (xc, yc, zc, and tc), with the horizontal axes (xc, yc)extended to those in spherical coordinates in section 3.The equations for potential density �, incompressibility,and the horizontal velocity V � (u, �) are

�tc � V � �c� � w�zc � 0, �1�

�c � V � wzc � 0, and �2�

�0�Vtc � V � �cV � wVzc � f � V� � �cp, �3�

where c � (�/�xc, �/�yc) is the horizontal gradient op-erator, �0 is the reference density of seawater, f is theCoriolis parameter, and p � �z g� dz is hydrostaticpressure, with g being the acceleration caused by grav-ity. Now let the density coordinates in the present studybe spanned by a set of independent variables (x, y, s,and t), with which the primitive Eqs. (1)–(3) can berewritten respectively as (Kasahara 1974)

�t � V � �� w*�zc � 0, �4�

zstc � � � �Vzs

c��zsc � w*zc � 0, and �5�

�0�Vt � V � �V � w*Vzc � f � V� � G. �6�

The two-dimensional vector G � cp � �p g��zc is the negative of the horizontal gradient of thehydrostatic pressure, and w* � w zc

t V • �zc is thediapycnal velocity.

SEPTEMBER 2008 A I K I A N D R I C H A R D S 1847

The vertical axis s of density coordinates is set to beidentical to the potential density at depths away fromthe sea surface and bottom (i.e., the ocean interior). Asshown in Fig. 1, it is expressed as �(x, y, s, t) � s fors ∈ (�top, �btm), where �top and �btm are the densityvalues at the top and bottom edges, respectively, of awater column located at (x, y). Each surface of fixeds is an isopycnal surface. Substitution of �x� � �y� � �t� �0 into (4) yields w* � 0 for s ∈ (�top, �btm), which isessentially

w � ztc � V � �zc. �7�

The vertical velocity is given by the temporal displace-ment and advection of each isopycnal surface, as iswidely known.

The structure of density coordinates outside the den-sity range of a water column (shaded in Fig. 1) is thendetermined by considering an oceanic domain boundedby a rigid sea surface and a bottom of arbitrary depthh(�0) as follows (our procedure is very similar to that

in A83). For s ∉ (�top, �btm), each surface of fixed s iscondensed at the height of either zc � 0 or zc � h(x,y) such that �zc/�s � 0 there. Nevertheless, the potentialdensity on such a surface with a fixed s can vary: � ��top(x, y, t) or �btm(x, y, t), as illustrated by the solidvertical lines in Fig. 1. Each of �top and �btm satisfy �t �V • �� 0, which together with (4) shows w* � 0 and(7) being extensible to the outer range s ∉ (�top, �btm).This design of the density coordinates allows for thethickness based on the potential density �zc/�� to benonzero at the top and bottom boundaries (see section2d).

As a result, for all ranges of s, the momentum andthickness as described in (5) and (6) reduce to

zstc � � � �Vzs

c�� zsc � 0 and �8�

�0�Vt � V � �V � f � V� � G. �9�

Equations (7)–(9) complete the primitive equations indensity coordinates.

FIG. 1. Window of a low-pass temporal filter in density coordinates for a water columnlocated at (x, y). Solid lines represent contours of potential density � � (x, y, s, t) with aconstant interval. As defined in section 2a, � � s for s ∈ (�top, �btm), where �top and �btm arethe density values at the top and bottom edges of the water column, respectively. Shadedregions indicate s ∉ (�top, �btm), where zc

s � 0 and � � �top(x, y, t) or �btm(x, y, t). As explainedin section 2b, the minimum and maximum values of �top and �btm inside a filter window areexpressed as �min and �max (dotted lines), respectively, such that � � s and zs � 0 for s ∈ (�min,�max). As explained in section 2d, density surfaces not touching the top and bottom bound-aries are in the range s ∈ (�tmax, �bmin) where � � s holds, with s � �tmax and s � �bmin (dashedlines) being the maximum and minimum values of �top and �btm, respectively. Density surfacestouching the top and bottom boundaries are in the range of s ∉ (�tmax, �bmin) where � � s.


As explained further in section 2d, rewriting thepressure term using the Montgomery potential G ��(p � g�z) is possible only in regions where � � s;this approach is not adopted in the present study,which contrasts with A83 and LL96, who consider non-Boussinesq fluids.

b. Isopycnal mean

Surfaces of fixed s are referred to as density or iso-pycnal surfaces here, even though the density is notalways constant along each surface of fixed s (section2a). Likewise, a low-pass temporal filter operatingalong each surface of fixed s is hereafter called theisopycnal mean and is denoted by an overbar (Table 1).The height of each surface of fixed s is decomposed intothe mean height z and the deviation from it z� (z� � 0;Table 1); that is,

zc�x, y, s, t� � z�x, y, s, t� � z��x, y, s, t�. �10�

The horizontal velocity is decomposed into the thick-ness-weighted mean velocity V � zc

sV/zs and the de-viation from it V� (zc

sV� � 0; Table 1); thus,

V�x, y, s, t� � V�x, y, s, t� � V��x, y, s, t�. �11�

The thickness-weighted mean velocity V is the horizon-tal component of the total transport velocity (Table 1).The vertical component of the total transport velocity isgiven in section 2c.

Applying the thickness-weighted mean to (8) and (9)

yields the primitive equations in the mean field (B85) asfollows:

zst � � � �Vzs� � 0 and �12�

�0 �zsV�t � � � �zsVV� � f � zsV� � zsG � �0M�V�,

�13�

where M(V) � � • (zcsV�V�) is the isopycnal diver-

gence of the Reynolds stress.It is noted that the pressure term zsG � zc

sG in (13)is a thickness-weighted mean quantity, which is hereseparated into contributions from the mean and pertur-bation fields:

zsG � zsc�cp � zs

c�p � ps�zc

� zs�p � ps�z � �z�s �p� � p�s �z��

� zs�cp � �z�s �p� � p�s �z��, �14�

where p � �s g�zs ds and p� � �s g(�zcs)� ds are the

isopycnal mean of hydrostatic pressure p and the de-viation from it ( p� � 0; Table 1). The derivation of (14)is based on the horizontal gradient becoming c � � (�/�z)�z when operating on averaged quantities, suchas p and V (de Szoeke and Bennett 1993). The negativeof the horizontal gradient of the mean hydrostatic pres-sure cp and the thickness-weighted mean density� � zc

s�/zs � s (Fig. 1) are explained further in section 2d.The second term on the last line of (14) is the layer-

TABLE 1. List of symbols, where A(x, y, s, t) is an arbitrary quantity. The symbols are mostly the same as those described in AY06except for A

z, A, c, and w in the present study, corresponding to A, A

�, �H, and w in AY06, respectively.

Az

Eulerian mean at a fixed heightA Thickness-weighted mean along each surface of fixed s: A � zc

s A/ zs (zcs � �zc/�s is the thickness)

A Unweighted mean along each surface of fixed s (called the isopycnal mean for simplicity)A� Deviation from the thickness-weighted mean: A� � A A (compared at fixed s, zs

cA� � 0)A� Deviation from the isopycnal mean: A� � A A (compared at fixed s, A� � 0)c Horizontal gradient in Cartesian coordinates: c � (�/�xc, �/�yc)� Lateral gradient in density coordinates: � � (�/�x, �/�y)

(c � � (�/�zc)�zc for raw quantities and �c � � ��z��z for mean quantities)V Horizontal velocity: V � (u, �)w Vertical velocity: w � zt � V • �zc

w* Diapycnal velocity: w* � w zt V • �zc � 0V Horizontal component of the total transport velocity: V � zs

cV�zs

w Vertical component of the total transport velocity: w � zt � V � �z� Potential density: � � s in the interior, and � � �top(x, y, t) or �btm(x, y, t) at the boundaries� Mean height density: � � zs

c��zs � sp Hydrostatic pressure: p � �s g�zs

c ds (decomposed into p � �s g�zs ds and p� � �s g��zsc�� ds)

G Negative of the horizontal gradient of the hydrostatic pressure: G � (Gx, Gy) � cpGB Layer-thickness form drag: GB � G ��cp� � �z�s �p��p�s �z��zs

�hz GG

zdz Quasi-Stokes form stress

VB Horizontal component of the bolus velocity: VB � V V � z�s V��zs

wB Vertical component of the bolus velocity: wB � w w � �c � �hz V dzw

M(V) Divergence of the Reynolds stress: M�V� � � � �zscV�V��


thickness form drag, hereafter symbolized by z�s �p� �p�s �z� � zsG

B. To explain the redistribution of momen-tum, the layer-thickness form drag is often written asthe divergence of a pseudoflux (A83; Johnson and Bry-den 1989; Killworth and Nanneh 1994; LL96; Held andSchneider 1999; Smith 1999; Iw01). This is done in sec-tion 3c, where we compare the effects of the verticaland horizontal distributions of momentum in the globalocean currents. A related discussion appears in section 2d.

c. Boundary condition

If the vertical component of the total transport ve-locity is defined as w � zt � V • �z (Table 1), the totaltransport velocity (V, w) becomes three-dimensionallynondivergent (de Szoeke and Bennett 1993); that is,

�c � V � wz � � � V �z � Vz � �zt � V � �z�s�zs

� zst � � � �Vzs��zs � 0, �15�

which is an incompressibility condition in the meanfield.

To show that the three-dimensional velocity (V, w)satisfies a no-normal-flow boundary condition at thetop and bottom of the ocean, AY06 used a series ofintegral identities in each vertical column:

�h

0

A dz � �h

0 zscA

zszs ds � �

�max

�min

Azsc ds

� ��btm

�top

Azsc ds � �

h

0

A dzc

� �h

0

Az

dzc, �16�

where A is an arbitrary quantity, �min (�max) is the mini-mum (maximum) value of �top (�btm) in a filter window(Fig. 1), and the overbar with the superscript z denotesthe Eulerian mean at a fixed height (Table 1). Equation(16) is a generalization of the results of McDougall andMcIntosh (2001) and Killworth (2001) and was termedthe “pile-up rule” by AY06 because it explains the re-lationship between the cumulative sums of the thick-ness-weighted mean and Eulerian mean quantities inthe vertical direction.

Integrating (15) upward from the bottom boundaryyields w � c • �z

h V dz. Then the application of thepile-up rule, (16), gives the mean vertical velocity at thetop boundary as

w|z�0 � �c � �h

0

V dz � �c � �h

0

Vz

dzc

� wz|zc� 0 � 0. �17�

Equation (17) proves that mean vertical velocity w van-ishes at the top boundary, as does the Eulerian meanvertical velocity wz (Iw01). As a result, total transportvelocity (V, w) satisfies a no-normal-flow boundarycondition at the top and bottom of the ocean (cf. Mc-Dougall and McIntosh 2001).

d. Comparison with previous formulations

The design of the density coordinates in the presentstudy allows for the thickness based on the potentialdensity �zc/�� to be nonzero at the top and bottomboundaries (section 2a), whereas the density coordi-nates used in B85 and Killworth and Nanneh (1994)assume that �zc/�� vanishes at the top and bottomboundaries. In the density coordinates of the presentstudy, there are regions of �zc/�s � 0 at the top andbottom boundaries (shaded in Fig. 1) where all physicalquantities (e.g., u, �, and �) lose their dependencies ons and become functions of (x, y, t); this procedure isanalogous to the isentropic coordinates of the atmo-sphere considered in A83 with some isentropes inter-secting the ground.

One way in which our formulation differs from thatin A83 is in writing the layer-thickness form drag GB

(Table 1 and section 2b) as the divergence of a pseu-doflux. In section 3c of this paper, GB is separated intoterms representing the vertical and horizontal (not iso-pycnal) redistributions of momentum by using the pile-up rule (16), which has not been noted in previous at-mospheric and oceanic studies. On the other hand, theexpression in A83 is based on the Montgomery poten-tial, which is useful in dealing with non-Boussinesq flu-ids. As explained in section 2a, the use of the Bous-sinesq approximation in the present study makes itimpossible to derive the Montgomery potential overregions where � � s (shaded in Fig. 1). At depths suf-ficiently far from the top and bottom boundaries, p�s �g(�zc

s)� becomes gsz�s , which leads to

zsGB � z�s �p� � p�s �z� � z�s ��p� � gsz��

� z��p� � gsz��s � �g�2��z�2, �18�

where p�s � gsz� is the perturbed Montgomery poten-tial. The first term on the last line of (18) correspondsto [ p�M�x ]� in (2.21) of A83. As noted above, (18) isvalid only for density surfaces that do not touch the topand bottom boundaries in a filter window, whose den-sity range is expressed as s ∈ (�tmax, �bmin), where �tmax

and �bmin are the maximum and minimum values of �top

and �btm, respectively (Fig. 1). Equation (18) has beenused in Killworth and Nanneh (1994) and LL96.

In general, the method of determining the meanheight of each density surface is fundamental to clari-


fying the adiabatic mean formulation (cf. Nurser andLee 2004). Jacobson and Aiki (2006) considered a spe-cial case where zc is not defined over s ∉ (�top, �btm)(the shaded regions in Fig. 1) and addressed the uncer-tainty of both z� and z near the top and bottom bound-aries. In contrast to this, in the present study (section2a) we define zc over a sufficiently large range of s indensity coordinates (including the regions shaded inFig. 1) that the mean height z ranges from h to 0.Then, for fluid particles whose mean height is eitherz � 0 or h, the perturbation height becomes z� �zc z � 0. This is an important characteristic and wasoriginally described in section 7 of McDougall andMcIntosh (2001). The mean height of each density sur-face can be obtained also by rearranging all fluid par-ticles in a filter window from the bottom in order ofdecreasing density, such that the mean height z indeedranges from h to 0. It turns out that the vertical po-sition of the lightest and heaviest fluid particles in afilter window does not change (being invariably adja-cent to the boundary), resulting in the perturbationheight becoming indeed z� � zc z � 0 for fluid par-ticles whose mean height is either z � 0 or h.

Mean primitive Eqs. (12) and (13) in the presentstudy are almost the same as (1) and (2) in AY06, apartfrom our improved definition of the isopycnal mean insection 2b. For example, the density coordinates in thepresent study make it possible to show that � � s for s∈ (�tmax, �bmin) and � � s for s ∉ (�tmax, �bmin), as Fig.1 suggests. Although AY06 write the mean pressureterm as G (the isopycnal mean of G) and the meanheight density as � (the isopycnal mean of �) for sim-plicity, the formal expressions for these terms are cpand �, respectively, as derived for (14) in the presentstudy. Only at depths sufficiently far from the top andbottom boundaries does cp � �p g��z � G(the isopycnal mean of G), by using G � �p g��zc ��p gs�z for s ∈ (�tmax, �bmin). Likewise, only atdepths sufficiently far from the top and bottom bound-aries does the thickness-weighted mean density �(�zc

s�/zs � s) become identical to � (the isopycnal mean of �).The mean height density in the present study is thethickness-weighted mean density � (Table 1) ratherthan the isopycnal mean density �. The adiabatic meandensities (such as the modified mean density and thetemporal residual mean density) as defined in previousstudies can be revised accordingly (cf. de Szoeke andBennett 1993; McDougall and McIntosh 2001; Griffies2004; Jacobson and Aiki 2006).

As demonstrated by B85, de Szoeke and Bennett(1993), Greatbatch and McDougall (2003), and Jacob-son and Aiki (2006), the mean momentum Eq. (13) canbe expressed in terms of the total transport velocity (in-

stead of the isopycnal mean or Eulerian mean velocity),which was overlooked in McWilliams and Gent (1994),LL96, Wardle and Marshall (2000), and Ferreira andMarshall (2006).

e. Energy equations

Energy equations for the mean and perturbationfields are derived in appendix A. With the exceptionthat the variables are expressed in density coordinates,the context in appendix A is essentially the same as thatin AY06 in that the volume budget of the energy equa-tions is explained by using no-normal-flow boundaryconditions of the total transport velocity (V, w) and theraw velocity (V, w). Nevertheless, the explicit use ofdensity coordinates in the present study will serve toreconfirm that neither zc

� � z� � 0 nor zs � 0 is neces-sary as conditions at the top and bottom of the ocean.This improvement over the formulation of B85 is mademore transparent than in AY06 by the systematic de-scription of density coordinates in this paper.

The adiabatic mean energy diagram is derived byusing modified definitions of the mean and eddy kineticenergies as follows. The total kinetic energy (�0/2)zc

s|V|2

is separated into the mean kinetic energy (�0/2)zs|V| 2

and the eddy kinetic energy (�0/2)zcs|V�| 2, as in B85,

Røed (1997), Iw01, Jacobson and Aiki (2006), andAY06. On the other hand, the total potential energyg�zczc

s is separated into the mean potential energy g�z zs

and the eddy potential energy1 (g�zczcs g�z zs), as in

L55, Iw01, and AY06. Here we point out that the thick-ness-weighted mean velocity V (which is used to definethe mean kinetic energy) is the sum of the isopycnalmean velocity V and the so-called bolus velocity VB �z�s V�/ zs (Rhines 1982). Usually the isopycnal mean ve-locity represents the basic geostrophic currents and thewind-induced Ekman transports, with the bolus veloc-

1 As mentioned in section 2d, the biggest and most obviousadvantage of our density coordinates is having realistic distribu-tions of density and tracers at the top and bottom boundaries ofthe ocean, in contrast to traditional coordinates, which assumeunrealistic (i.e., homogeneous) distributions of density and tracersat the boundaries. However, we found it rather difficult to cat-egorize the other characteristics of our (or traditional) densitycoordinates into pros or cons. For example, with traditional den-sity coordinates where � � s everywhere, we find the expressionof the eddy potential energy becomes g�zczc

s gpz zs � gsz�z�s ,which can be shown to be a negative definite quantity gz�2/2after using the integral by parts in the vertical direction. But gen-erally we do not know whether this characteristic of the eddypotential energy (its being single signed) is a pro or con. This isbecause at least some careful theories (e.g., L55; Iw01) do not usesuch a definition of the eddy potential energy (i.e., gz�2/2). L55and Iw01 used the eddy potential energy defined as a differencebetween the total and mean potential energies, as was done in thepresent study and in AY06.


ity representing the eddy-induced overturning circula-tions (see sections 3a and 4). Note that the effect of thebolus velocity is included in the mean kinetic energy(�0/2)zs| V|2 instead of the eddy kinetic energy (�0/2)zc

s|V�|2. Because of this modified definition for themean and eddy kinetic energies, the look of the adia-batic mean energy diagram in Fig. 2 is different fromthat of the classical L55 energy diagram. Only the adia-batic mean energy diagram can describe the energyconversion done by the layer-thickness form drag�V • zsG

B), which is why this energy diagram is beingused in this study (section 3).

Energy conversions represented by the triple lines inFig. 2—that is, V • zsG

B, caused by the layer-thicknessform drag; VB • zs

cp, caused by the bolus velocity(eddy-induced overturning circulation); and V • zs

cp,caused by the isopycnal mean velocity (Ekman trans-port)—are investigated in section 3. The other energypaths in Fig. 2 were not investigated. It is of great in-terest to directly analyze the above three paths of pres-sure-related energy conversions. According to an indi-rect analysis (scale analysis) for the global ocean(AY06), energy conversion rates caused by both thelayer-thickness form drag V • zsG

B and the bolus veloc-ity VB • zs

cp can be comparable to the input to the

mean kinetic energy by wind forcing (cf. Wunsch 1998).If this scaling is correct, then the energy conversioncaused by the isopycnal mean velocity V • zs

cp isalso important in drawing the energy cycle of the wind-driven ocean circulation (detailed in sections 3a and 4).Verifying the relationship among these three conver-sions is the first step toward understanding the ener-getics of layer-thickness form drag.

On the other hand, we generally expect (but did notexamine) that the energy conversion caused by theReynolds stress �0V • M(V) is similar to that estimatedby using the Eulerian mean formulation (e.g., L55; Ma-sina et al. 1999). There are a number of previous nu-merical investigations for the energy conversion causedby the Reynolds stress (cf. Böning and Budich 1992;Best et al. 1999; Miyazawa et al. 2004).

3. Analysis of model results

A high-resolution (0.1° � 0.1°) near-global OGCMwas integrated for a 50-yr period in the presence ofclimatological atmospheric forcing. The model codewas OFES (the OGCM for the Earth Simulator; Ma-sumoto et al. 2004), which is based on the GeophysicalFluid Dynamics Laboratory Modular Ocean Model

FIG. 2. Adiabatic mean four-box energy diagram for an inviscid hydrostatic fluid, based onthe thickness-weighted mean incompressible Boussinesq equations in density coordinates.Energy budgets were evaluated after taking the volume integral in a closed domain � basedon (A13), (A14), (A16), (A17), (A19), and (A20), as detailed in appendix A. The energypaths represented by the triple lines are investigated in section 3a (Figs. 4a–c). Although theeffects of atmospheric wind forcing are not included (for simplicity), the wind-forced energyinput to the mean kinetic energy is investigated in section 3a (Fig. 4d) by calculating V • �s,where �(x, y, s, t) � H(�top s)�(x, y, t) is the isopycnal mean of the wind stress �(x, y, t) atthe sea surface and H() is the Heaviside step function.


version 3 (Pacanowski and Griffies 2000), with substan-tial modifications for the vector-parallel hardware sys-tem of Japan’s Earth Simulator. The model domainextends from 75°S to 75°N, with the temperature andsalinity field in the southern and northern buffer zonesbeing relaxed to the monthly-mean climatological val-ues at all depths. The model has 54 depth levels, withthe discretization varying from 5 m at the surface to 330m at the maximum depth of 6065 m. The atmosphericforcing consists of wind stress, heat, and water fluxes ofmonthly-mean climatology data constructed from theNCEP–NCAR reanalysis for 1950–99 (Kalnay et al.1996). Further details and assessments of the OFESclimatological run are given in Masumoto et al. (2004),Nakamura and Kagimoto (2006a,b), and Merryfieldand Scott (2007).

A set of 3-day snapshots archived throughout the46th year of the OFES run (a total of 121 three-dimen-sional snapshots of the global ocean) were available tothe analysis. These snapshots were originally in z coor-dinates and hence were first mapped onto density co-ordinates in each vertical column. We used 80 densitylayers defined by the potential density referenced to seasurface pressure (this density was chosen in an attemptto overview the global ocean with most of the kineticenergy distributed above the main thermocline), withdensity layers being in steps of 0.02 kg m3 for s ∈(1027.2 kg m3, 1028.0 kg m3) and 0.2 kg m3 forother values of s. This mapping to density coordinateswas done in such a way as to conserve a cumulative sumof each quantity in each vertical column. In contrast tothe primitive equations of the OGCM, the hydrostaticpressure in the present analysis was calculated frompotential density referenced to the sea surface pressureinstead of in situ density, and the sea surface wastreated as a rigid rather than a free surface.

The low-pass temporal filter (overbar) in section 2was set to monthly means in the present analyses, whichwas intended to capture transient eddies while exclud-ing seasonal variability of thermocline and the surfacemixed layer. Applying the monthly means (in densitycoordinates) to the set of 3-day snapshots throughoutthe year yielded results for each month (i.e., 12 sets),but the monthly evolution and seasonal variability ofenergetics were not investigated in the present study.All results (figures and global values) presented belowrefer to the composite of the 12 sets of the monthlyresults from January to December, except that in sec-tion 3b we present brief results for cases where themonthly-mean filter was replaced by the seasonal- andannual-mean filters.

In the following, the horizontal axes (x, y) in section2 are implicitly extended to those in spherical coordi-

nates, with x and y representing the zonal and meridi-onal directions, respectively.

Figure 3 shows the mean kinetic, eddy kinetic, andeddy potential energies in each vertical column. Theglobal distributions of the mean kinetic energy (Fig. 3a)depict streamlines of major currents in the world’soceans: the Antarctic Circumpolar Current, the Agul-has Current, and the equatorial and western boundarycurrents in each basin. The eddy kinetic energy (Fig.3b) in the Gulf Stream and the Kuroshio is localized tothe regions of mean currents detaching from the west-ern boundary. The eddy kinetic and eddy potential en-ergies (Figs. 3b,c) in the Southern Ocean are isolated inregions where the mean currents are constrained bysteep topography (as in Lee and Coward 2003); theseare the Abyssal Plain (80–90°E), the area southwest ofthe New Zealand Plateau (150°E), the Scott Fracture(180°W), and the Drake Passage (60°W). Eddy kineticenergy is nearly absent over a large extent of the SouthPacific Ocean and the eastern North Pacific Ocean.

The volume integrals of the mean kinetic, eddy ki-netic, and eddy potential energies in the global oceanare 2.70, 0.85, and 1.11 EJ, respectively, where 1 EJ (anexajoule) � 1018J (Fig. 3). The fact that the eddy po-tential energy is somewhat larger than the eddy kineticenergy suggests that eddies are statistically geostrophicand on spatial scales of the internal Rossby radius. Thesum of 1.96 (� 0.85 � 1.11) EJ for the transient eddiesin the present analysis is one order of magnitudesmaller than the value of 13 EJ previously estimated byZang and Wunsch (2001) and Wunsch and Ferrari(2004) using a spectrum analysis of observations.

a. Energy conversions

We have identified three paths of energy conver-sions—represented by V • zsG

B, VB • zscp, and

V • zscp (indicated by the triple line in Fig. 2)—that

originate in the pressure effects and concern the budgetof the mean kinetic energy in (A10). AY06 speculatedthat these three terms maintain the equilibrium of themean kinetic energy with the following energy cycle(see also Fig. 2). Wind forcing provides an input to themean kinetic energy, which is then transferred to themean potential energy by the wind-induced Ekmantransport (i.e., the isopycnal mean velocity V). Never-theless, the net mean potential energy does not changebecause the eddy-induced overturning circulation (i.e.,the bolus velocity VB) extracts some of the mean po-tential energy and feeds the mean kinetic energy, whichis subsequently taken by the work of the layer-thicknessform drag GB, thereby endowing the perturbation fieldwith an energy cascade. In the following we investigatethe validity of the above energy cycle.


FIG. 3. Vertical integral ��min�max

ds of (a) mean kinetic energy (�0/2)zs| V| 2, (b) eddy kineticenergy (�0/2)zc

s|V�| 2, and (c) eddy potential energy g(�zczcs �zzs) (J m2). The net value

over the global ocean is indicated for each quantity (1 EJ � 1018 J). All the images inthe present study indicate year-round composites obtained from monthly mean analyses(section 3).


Fig 3 live 4/C

The quantity V • zsGB is the work done by the layer-

thickness form drag, whose vertical integral in each ver-tical column is shown in Fig. 4a, where negative (posi-tive) values indicate a decrease (increase) in the meankinetic energy. Significant work values leading to anenergy cascade to the perturbation field are present inthe Southern Ocean (the aforementioned eddy-activeregions), the Agulhas Current, the Gulf Stream, andthe Kuroshio, whereas no significant work is evidentover the equatorial regions. The global work of thelayer-thickness form drag is 0.50 TW (1 TW � 1012

W), which would be sufficient to exhaust the net meankinetic energy in the global ocean (Fig. 3a) within80 days [i.e., (2.7 EJ/0.50 TW)/(86 400 s day1)] if therewere no source of mean kinetic energy. The work ofthe layer-thickness form drag is examined further insection 3c.

The quantity VB • zscp is the work done by the

bolus velocity, whose vertical integral in each verticalcolumn is shown in Fig. 4b, where positive (negative)values indicate transformation from mean potential (ki-netic) energy to mean kinetic (potential) energy (Fig.2). The work of the bolus velocity is clearly positiveover each region of high eddy activity in the global

ocean (Figs. 3b,c): some of the mean potential energy isextracted to provide an input to the mean kinetic en-ergy when the eddy-induced overturning circulation re-laxes the slope of isopycnal surfaces. The global workof the bolus velocity, which is �0.46 TW in Fig. 4b,nearly balances the work of the layer-thickness formdrag presented above.

The quantity V • zscp is the work of the isopycnal

mean velocity, whose vertical integral in each verticalcolumn is shown in Fig. 4c. Although significant posi-tive and negative signs are evident in Fig. 4c, the globalwork is negative (0.30 TW in Fig. 4c), which reducesthe mean kinetic energy and increases the mean poten-tial energy (see Fig. 2). The work of the isopycnal meanvelocity reflects the wind-induced Ekman transportsthat steepen the slope of isopycnal surfaces over theglobal ocean (as detailed in section 4). Strictly speaking,the work of the isopycnal mean velocity also includesthe effects of standing eddies (and transient eddies withfrequencies lower than 2� (30day)1 because we usethe monthly-mean filter), which is indicated by positivevalues in Fig. 4c (e.g., around the Drake passage) ifstanding eddies convert some of the mean potentialenergy into mean kinetic energy.

FIG. 4. Vertical integral ��min�max

ds of the work (energy conversion rate) caused by (a) layer-thickness form drag V • zsGB, (b) bolus

velocity VB • zscp, (c) isopycnal mean velocity V • zs

cp, and (d) wind stress V • �s (W m2). The sign is relative to the budget ofthe mean kinetic energy. The net value over the global ocean is calculated for each quantity (1 TW � 1012 W). The pressure-relatedenergy conversions [(a)–(c)] are plotted with horizontal Gaussian smoothing with a radius of 1.5°.


Fig 4 live 4/C

In addition to the pressure-related energy conver-sions described above, there is an input to the meankinetic energy caused by wind forcing (cf. Wunsch1998). We quantified the work of the wind stress (Fig.4d) by calculating ��min

�maxV • �s ds , where �(x, y, s, t) �

H(�top s)�(x, y, t) is the isopycnal mean of the windstress � � (�x, �y) at the sea surface and H() is theHeaviside step function. The global work of wind forc-ing in the OFES output is �1.04 TW, of which �0.63TW comes from mid- and high latitudes (i.e., out of the20°S–20°N band).

The budget of the mean kinetic energy in mid- andhigh latitudes sums to �0.29 TW, comprising �0.63 TWfor wind forcing, 0.30 TW for isopycnal mean veloc-ity, �0.46 TW for bolus velocity, and 0.50 TW forlayer-thickness form drag. Therefore, the processes ex-cluded in the present analysis, such as the boundaryfriction, the Reynolds stress, the eddy viscosity, diabaticprocesses, and the difference between the potentialdensity and in situ density (in calculating the horizontalgradient of hydrostatic pressure) must conjointly con-tribute 0.29 TW.

The above results have generally confirmed the pres-ence of the energy cycle suggested by AY06.

b. Temporal filters other than the monthly mean

All of the above results (and the results in sections 3cand 3d) were derived using the monthly-mean filter tofocus on transient eddies while excluding seasonal vari-ability of the thermocline and the surface mixed layer.It is of interest to assess the effects of changing the timescale of the low-pass temporal filter on the value of theenergy and work terms. Table 2 compares the globalenergy and work terms in cases where the same 3-daysnapshots obtained from the OFES climatological runare analyzed with seasonal (3-month)-mean and an-nual-mean filters, where the seasonal-mean (the second

column of Table 2) represents the composites of thefour sets of analysis results for December–February,March–May, June–August, and September–November.

The values in Table 2 can be explained as follows.The inclusion of the intraseasonal and intra-annualvariabilities in the perturbation field produces leadingorder decreases (increases) in the mean (eddy) energyin the global ocean; the annual-mean kinetic energy(1.21 EJ) is about half the monthly-mean kinetic energy(2.75 EJ), whereas the annual eddy kinetic energy (2.37EJ) is about 3 times larger than the monthly eddy ki-netic energy (0.88 EJ), which may be caused by a sea-sonal shift (or reversal) of the current pathways. Theannual eddy potential energy (4.90 EJ) turns out to beabout 4 times larger than the monthly eddy potentialenergy (1.15 EJ), which is caused by the seasonal cycleof the thermocline and mixed layer depth.

In contrast to the energy terms, the work terms of thepressure effects and wind forcing are not greatly alteredby changes in the filtering, except that the work of theisopycnal mean velocity in the annual-mean analysisbecomes 0.66 TW, which is close to the work of theform drag and the bolus velocity (0.60 and �0.80 TW,respectively). Note that the effects of slowly oscillatingtransient eddies are included in the work of the bolus(isopycnal mean) velocity in the annual (monthly)-mean analysis. Moreover, in the annual-mean analysisthe work of the isopycnal mean velocity (0.66 TW)better balances the work of wind forcing in mid- andhigh latitudes (�0.61 TW). We conclude that the oce-anic energy cycle discussed in section 3a (Fig. 4) is veryrobust in the annual-mean analysis.

In the next subsections we investigate the character-istics of layer-thickness form drag that are associatedwith the presence of the energy cycle confirmed above.Generally speaking, section 3c investigates how thelayer-thickness form drag presents a sink of the mean

TABLE 2. The global energies (in EJ) and works (in TW) in cases where the low-pass temporal filter (overbar in section 2) is set tobe the monthly mean, the seasonal (3 month) mean, and the annual mean, in analyzing the 3-day snapshots obtained from the OFESclimatological run (throughout the 46th year). Values for the monthly mean (first column) are adapted from Figs. 3–5. Values for theseasonal mean (second column) are the composites of the four sets of analysis results for December–February, March–May, June–August, and September–November. Bracketed values of the work of the form drag represent the contribution from the quasi-Stokesform stress (explained in Fig. 5a and section 3c). Bracketed values of the work of the wind stress represent the contribution of the mid-and high latitudes (i.e., outside the 20°S–20°N band).

Composite Monthly Mean Composite Seasonal Mean Annual Mean

Mean kinetic energy 2.75 EJ 1.91 EJ 1.24 EJEddy kinetic energy 0.88 EJ 1.71 EJ 2.37 EJEddy potential energy 1.15 EJ 2.90 EJ 4.90 EJWork of form drag 0.50 TW (0.46 TW) 0.58 TW (0.54 TW) 0.60 TW (0.56 TW)Work of bolus velocity �0.46 TW �0.64 TW �0.80 TWWork of isopycnal mean velocity 0.30 TW 0.49 TW 0.66 TWWork of wind stress �1.04 TW (�0.63 TW) �1.01 TW (�0.63 TW) �0.94 TW (�0.61 TW)


kinetic energy, and section 3d investigates why thework done by the layer-thickness form drag (Fig. 4a)nearly balances the work done by the eddy-inducedoverturning circulation (Fig. 4b) in each vertical col-umn. [Note that all numerical results (figures and val-ues) in sections 3c and 3d are shown based on themonthly-mean analysis (as in section 3a), with the in-tent of providing useful metrics to be used in futurenumerical studies (see appendix B for details).]

c. Momentum redistributions

The most fundamental assumption that has been fre-quently used in previous studies for layer-thicknessform drag is that it allows the vertical transfer of mo-mentum (cf. Johnson and Bryden 1989; G98), whichseems to be consistent if waves and eddies in baroclinicinstability tend to break the thermal wind balance ofbasic currents. Strictly speaking, such momentum trans-fer is possible in both the vertical and horizontal (orisopycnal) directions. Distributions of the momentumfluxes caused by layer-thickness form drag have notbeen determined to any significant extent in previousoceanic studies, except for LL96, who analyzed an ide-alized jet in a rectangular channel. The purpose of thissubsection is to extend the analysis of LL96 to the out-put of a high-resolution global ocean model and inves-tigate the energetics. In particular, we compare the ef-fects of the vertical and horizontal transfers of momen-tum by layer-thickness form drag.

The layer-thickness form drag GB (Table 1 and sec-tion 2b) can be written as the divergence of a pseudo-flux in the forms of several expressions. Before consid-ering expression (19), used in the present analysis, webriefly explain three other expressions suggested in pre-vious studies. The first expression is (18), as alreadygiven in section 2d (A83; LL96), which cannot be de-rived near the top and bottom boundaries because ofthe Boussinesq approximation. The second expression,zsG

B � (z��p�)s � �(z�p�s ), is applicable to the com-plete range of s, where the vertical flux z��p� vanishesat the top and bottom boundaries (because z� � 0 atz � 0 and h; see section 2d), representing the verticalredistribution of momentum. The third expression,zsG

B � (p��z�)s �(p�z�s ), is discussed in Rhines andHolland (1979), Johnson and Bryden (1989), andIvchenko et al. (1996). The most plausible in terms ofthe boundary condition is the second expression, but itsuse in a numerical analysis will result in pressure gra-dient errors (cf. Haney 1991; Mellor et al. 1994; Shche-petkin and McWilliams 2003) near the top and bottomboundaries where � � s (Fig. 1).

A solution to elucidate the pseudoflux of momentumassociated with the layer-thickness form drag is to write

the divergence in Cartesian coordinates. Rememberingthat GB � G � cp (Table 1 and section 2b), it can beshown that

GB � �G Gz� � �G

z� �cp�

��

�z ��h

z

�G Gz� dz� �c�

z

g��z �� dz.

�19�

The first term on the last line of (19) is essentially G G

zand has no barotropic effects in each vertical col-

umn because of the pile-up rule, (16). The vertical fluxof momentum �z

h(G Gz) dz is hereafter called the

quasi-Stokes form stress, following the terminology andconcept of the quasi-Stokes velocity developed by Mc-Dougall (1998) and McDougall and McIntosh (2001).The second term on the last line of (19) originates fromthe difference between the Eulerian mean density �z

and the mean height density �. The difference between�z and � has received considerable attention in recentstudies (Killworth 2001; McDougall and McIntosh2001; Greatbatch and McDougall 2003), but it has notbeen subject to realistic numerical investigations andhence was worth analyzing in the present study.

Figure 5 compares the distributions of the work doneby the first (the quasi-Stokes form stress) and second(the density residual) terms on the last line of (19),whose sum becomes the quantity shown in Fig. 4a. Thework of the quasi-Stokes form stress (Fig. 5a) is nega-tive in most regions of the global ocean, suggesting thatthe vertical redistribution of momentum reduces themean kinetic energy and leads to an energy cascade tothe perturbation fields, which is as expected if the meancurrents in the global ocean are close to being in ther-mal wind balance. On the other hand, the work done bythe density residual (Fig. 5b) has both positive andnegative values, and its global integral of 0.04 TW isone order of magnitude smaller than that of the quasi-Stokes form stress (estimated at 0.46 TW in Fig. 5a).We conclude that the work of the layer-thickness formdrag results mostly from the quasi-Stokes form stress(the vertical redistribution of momentum) and not fromthe density residual (the horizontal redistribution ofmomentum).

Figure 6 shows the meridional–vertical views of thezonal component of the thickness-weighted mean ve-locity, the quasi-Stokes form stress, and the wind stress,obtained by taking a zonal average in density coordi-nates over the global ocean. The velocity field (Fig. 6a)comprises (i) the Antarctic Circumpolar Current in theSouthern Ocean, whose velocity profiles (flowing east-ward) reach depths of 1000–2000 m; (ii) the extensions


of the Gulf Stream and the Kuroshio at midlatitudes ofthe Northern Hemisphere, whose velocity profiles(flowing eastward) reach depths of 100–500 m; and (iii)equatorial currents, whose velocity profiles (flowingeastward and westward) are near the sea surface. Incontrast to the zonal component of velocity (Fig. 6a),the quasi-Stokes form stress �s

�max(Gx Gx)zs ds (Fig.

6b) is significant only in the Southern Ocean at densitysurfaces of s ∈ (1027.7 kg m3, 1027.85 kg m3) anddepths of 1000–2000 m. The positive sign for the quasi-Stokes form stress indicates the downward transfer ofthe eastward momentum (as suggested by Johnson andBryden 1989), with the maximum value of �0.12 N m2

being of the same order of magnitude as the wind stressapplied at the sea surface (Fig. 6c; discussed further in

section 4). The quasi-Stokes form stress is maximal at alatitude of 58°S, corresponding to the location of theDrake Passage. Figure 6b also indicates that the quasi-Stokes form stress appears to be absent outside theSouthern Ocean, despite significant work values of thequasi-Stokes form stress appearing in the Gulf Streamand the Kuroshio (Fig. 5a).

It is of interest to examine the horizontal distributionof the quasi-Stokes form stress. Figure 7a shows thequasi-Stokes form stress (for the zonal momentum) inthe Antarctic Circumpolar Current (around the DrakePassage in the Southern Ocean) on a density surfaces � 1027.82 kg m3, which is where the zonal average ofthe quasi-Stokes form stress is maximal in Fig. 6b. Theform stress in Fig. 7a is clearly positive, suggesting

FIG. 5. As in Fig. 4a, but separated into contributions from (a) the quasi-Stokes form stressV • zs(G G

z) and (b) the density residual V • zs

c�z g(�z �)dz (W m2).


Fig 5 live 4/C

downward transfers of the eastward momentum atdepths of 2000–3000 m. Its magnitude is as high as �3N m2 in Fig. 7a, which is around 30 times larger thanthe typical value of the wind stress acting at the sea

surface (�0.1 N m2; Fig. 6c). Figure 7b shows thequasi-Stokes form stress (for the zonal momentum) inthe Kuroshio Extension (in the western North PacificOcean) on a density surface of 1026.0 kg m3. The form

FIG. 6. Meridional–vertical view of (a) thickness-weighted mean velocity u (cm s1), (b)quasi-Stokes form stress �s

�max(Gx Gxz

)zs ds (N m2), and (c) wind stress �x (N m2) (all arethe zonal component), obtained by taking a zonal average � dx/L in density coordinates overthe global ocean, where L is the zonal length of the global ocean measured at the sea surface.Contours are the mean height of each isopycnal shown by the zonal average in densitycoordinates �z dx/L (m). In (b), positive (negative) values of the quasi-Stokes form stressindicate downward (upward) transfer of the eastward momentum.


Fig 6 live 4/C

stress in Fig. 7b is largely positive, again suggesting thatthe eastward momentum is transferred downward atdepths of 100–300 m. Its magnitude of about �1 N m2

in the Kuroshio Extension is somewhat smaller than—but of the same order of magnitude as—the quasi-Stokes form stress around the Drake Passage in Fig. 7a.The quasi-Stokes form stress is detectable in western

boundary currents, which is consistent with the work ofthe quasi-Stokes form stress being significant in theKuroshio and the Gulf Stream (Fig. 5a).

d. Eddy geostrophic balance

The layer-thickness form drag is often assumed to bein a geostrophic balance (i.e., �0f � VB � GB, hereafter

FIG. 7. Quasi-Stokes form stress for zonal momentum �s�max

(Gx Gxz)zs ds (N m2) (a)

around the Drake Passage at a density surface of s � 1027.82 kg m3 and (b) over the westernNorth Pacific Ocean at a density surface of s � 1026.0 kg m3, where positive (negative)values indicate the downward (upward) transfer of the eastward momentum. Contours are themean height z (m) of the corresponding density surface.


Fig 7 live 4/C

called the eddy geostrophic balance) in theoreticalstudies of mesoscale eddy parameterization (McWil-liams and Gent 1994; Gent et al. 1995, hereafter GW95;G98; Smith 1999; Aiki et al. 2004) comparing the char-acteristics of the tracer approach (using the eddy-induced extra advection) and the momentum approach(using the eddy form drag); see appendix B for detailsof these two approaches. Also, many studies of the Ant-arctic Circumpolar Current have implicitly used theeddy geostrophic balance to justify their analysis of thevelocity-based Eliassen–Palm flux (Ivchenko et al.1996; Stevens and Ivchenko 1997; Olbers and Ivchenko2001; Marshall and Radko 2003). Despite the frequentuse of this assumption of eddy geostrophic balance inprevious studies, its presence has not been confirmedbased on outputs from high-resolution OGCMs. Onlyalong 60°S (the Southern Ocean) have Killworth andNanneh (1994) confirmed the eddy geostrophic balanceof standing eddies (i.e., not transient eddies) by analyz-ing output of the Fine Resolution Antarctic Model. Thepresent study extends the analysis of Killworth andNanneh (1994) to transient eddies over the globalocean. Verifying the eddy geostrophic balance is alsoimportant for explaining why the work done by thelayer-thickness form drag (Fig. 4a) nearly balancesthe work done by the eddy-induced overturning circu-lation (Fig. 4b) in each vertical column (see sections 3aand 4).

Because the main component of the layer-thicknessform drag is the vertical divergence of the quasi-Stokesform stress (Fig. 5; section 3c), we investigated the eddygeostrophic balance by using the following slight modi-fications. The zonal component of the eddy geostrophicbalance becomes �0 f�B � GxB � (Gx Gxz

), and therelationship �B � (Gx Gxz

)/(f�0) is then integratedupward from the bottom and subsequently integratedin the zonal direction over the global ocean; that is,

��h

z

�B dz dxc �1�0 f ��h

z

�Gx Gxz� dz dxc, �20�

where both sides of the approximate equality are in theunit of volume transport (m3 s1). We now examine thevalidity of (20).

The left-hand side of (20) represents the eddy-induced meridional transport in the global ocean asshown in Fig. 8a, with positive (negative) values repre-senting anticlockwise (clockwise) rotation in the me-ridional–vertical plane. The zonal integral � dxc in (20)is taken in Cartesian coordinates so Fig. 8a can be com-pared with the parameterized eddy-induced overturn-ing circulation that is typically used in coarse-resolution

OGCMs, such as that shown in Figs. 6 and 7 of GW95.The maximum overturning rate in Fig. 8a is 16 � 106

m3 s1 in the Southern Ocean at depths of 2000–3000 mwith deeper (upper) waters flowing toward the equator(pole), which is in good agreement with the parameter-ized circulation reported in GW95. In low latitudes(20°S and 20°N) in Fig. 8a, there is a cell in each hemi-sphere with reversed sign, which tends to lower theupwelling thermocline at rates of about 10 � 106 and3 � 106 m3 s1 above and below a depth of 100 m,respectively. These antisymmetric cells tend to cancel(or reduce) the tropical cells (Hazeleger et al. 2001). Inthe midlatitudes of the Northern Hemisphere (Fig. 8a),there appears to be a rather weak negative (clockwise)cell. Figure 8a also indicates that the bolus velocity in-herently includes slight barotropic components (cf. Mc-Dougall and McIntosh 2001; AY06; Lee et al. 2007).

The right-hand side of (20), hereafter called the re-scaled quasi-Stokes form stress, is shown in Fig. 8b; itsdistribution in the Southern Ocean (with a maximum of16 � 106 m s1 at a depth of 2000–3000 m) stronglyresembles that of the meridional bolus transports inFig. 8a. Note that the quantity plotted in Fig. 8b is thesame as that in Fig. 6b except for differences in scalingand mapping. Although the quasi-Stokes form stress inFig. 6b is invisible outside the Southern Ocean, Fig. 8bshows that the rescaled quasi-Stokes form stress is an-tisymmetric across the equator: each hemisphere con-tains both positive and negative cells, suggesting down-welling (upwelling) circulation at upper (lower) depths.The negative (positive) cell in the Northern (Southern)Hemisphere extends poleward with increasing depth.We conclude that the eddy geostrophic balance islargely valid at depths below 100 m in mid and highlatitudes, which is useful for various theoretical analy-ses, such as idealizing oceanic dynamics (section 4) andparameterizing unresolved eddies in OGCMs (McWil-liams and Gent 1994; G98).

In contrast to the present study directly investigatingpressure-based fields (Figs. 6b and 8b), many previousstudies have employed indirect analyses of the layer-thickness form drag. The quantity investigated inIvchenko et al. (1996), Stevens and Ivchenko (1997),Held and Schneider (1999), and Olbers and Ivchenko(2001) is essentially an eddy-induced meridional trans-port corresponding to Fig. 8a of the present study. It isan open question why these previous studies haveavoided direct analyses of the pressure field derivedfrom the outputs of OGCMs and AGCMs.

When the above analyses in sections 3a–3d were re-peated using the potential density referenced to thepressure at 2000-m depth (which replaces the potential


density referenced to the sea surface pressure, as origi-nally used), we found no qualitative difference from theresults presented above (not shown). The global ratesof the pressure-related energy conversions changed byonly �0.1 TW, and the vertical profiles of the formstress were largely the same as the ones presentedabove.

4. Planetary geostrophic equations for the globalocean

The results of the analyses in section 3 elucidate theadiabatic aspects of oceanic dynamics in terms of theequilibrium of energy and momentum, as explained be-low. Here we summarize momentum and energy bal-ances in mid and high latitudes (where the presence ofthe eddy geostrophic balance was confirmed in section

3d) using a minimum set of equations. As shown below,these equations are simple yet include eddy effects, andthey could be fundamental to applying the energetics oflayer-thickness form drag in various research areas ofocean dynamics (excluding at least equatorial dynam-ics) in future studies, such as (i) the dynamics of theAntarctic Circumpolar Current, (ii) the ventilated ther-mocline theory, (iii) mesoscale eddy parameterizationin coarse-resolution OGCMs, and (iv) the dynamics ofbaroclinically coupled coherent eddies.

For simplicity and for convenience of explanation ofthe boundary condition at the end, we begin by rewrit-ing (A9) and (A10) in Cartesian coordinates. By using(A11), Eq. (A9) for the mean potential energy becomes

�

�tc ��gz� � �c � V�g�z � p�� w�g�z � p��z � V � �cp,

�21�

FIG. 8. Meridional–vertical view of (a) the eddy-induced meridional transport ��zh �B dz

dxc (Sv: 1 Sv � 106 m3 s1), and (b) the rescaled quasi-Stokes form stress for the zonalmomentum ��z

h(Gx Gxz)dzdxc/(f�0) (Sv), where � dxc is the zonal integral in Cartesian

coordinates over the global ocean.


Fig 8 live 4/C

which includes the pressure-flux divergence in themean field, (A15). In situations where the mean poten-tial energy is locally in equilibrium, the temporal de-rivative in (21) can be dropped, yielding

�c � V�g�z � p�� w�g�z � p��z � V � �cp. �22�

On the other hand, by using (A11) Eq. (A10) for themean kinetic energy becomes

�

�tc ��0

2|V|2� � �c � �V

�0

2|V|2� � �w

�0

2|V|2�

z�

V � �cp � V � GB ��0

zsV � M�V� � V � �z, �23�

which includes an introduced wind stress term �z � �s/zs

(section 3a). In situations where the work of the pres-sure effect and the wind forcing are locally balanced (aswe show in sections 3a and 3b; see also Fig. 4), theleft-hand side and the Reynolds stress term in (23) canbe dropped; thus,

0 � V � �cp � V � GB � V � �z. �24�

Equations (22) and (24) represent the local balances ofthe mean potential and mean kinetic energies, respec-tively, in the planetary geostrophic limit. For example,if we assume a combined geostrophic and Ekman bal-ance �0 f � V � cp � GB � �z, substitution of thisinto (24) results in the right-hand side of (24) vanishing,which is consistent with the left-hand side.

The momentum balance is investigated in detail byseparating the isopycnal mean velocity as V � Vekm �Vgeo. The wind-induced Ekman velocity Vekm, the geo-strophic velocity Vgeo, and the eddy-induced (bolus)velocity VB are characterized by a set of balanced equa-tions, namely,

�0f � Vekm � �z, �25�

�0f � Vgeo � �cp, and �26�

�0f � VB � GB, �27�

where the last equation is the eddy geostrophic balanceinvestigated in section 3d. We now classify oceanic dy-namics into wind- and eddy-driven regimes. Equations(28)–(31) can be applied to both the zonally periodiccurrents (i.e., Antarctic Circumpolar Current) and theclosed gyre-scale circulations in each basin of the globalocean.

a. Wind-driven regime

The dynamics of the wind-driven regime apply tonear the sea surface where the momentum balance isdescribed by (25), (26), and VB � 0, such that

�0f � �Vekm � Vgeo� � �cp � �z. �28�

The surface wind stress �z tends to enhance the verticalshear of the basic currents, which is blocked by theCoriolis force associated with the Ekman transportVekm. In other words, the Ekman transport tends toreduce the vertical shear of the basic currents, which isconsistent with this transport reducing the mean kineticenergy, as explained below.

The local balance of the mean kinetic energy in (24)becomes

0 � Vekm � �cp � Vgeo � �z, �29�

which can be verified by comparing Figs. 4c and 4d.Over the bulk of the global ocean, the wind-inducedmean kinetic energy is transformed into the mean po-tential energy when the wind-induced Ekman transportsteepens the slope of isopycnal surfaces near the seasurface. This explanation is consistent with the com-ments of Warren et al. (1996, 1998) and with classicalviews of oceanic dynamics (cf. Kuhlbrodt et al. 2007).

b. Eddy-driven regime

The dynamics of the eddy-driven regime hold belowthe surface Ekman layer and in regions where eddiesare more important than the wind stress. The momen-tum balance is given by (26), (27), and Vekm � 0, suchthat

�0f � �Vgeo � VB� � �cp � GB, �30�

where layer-thickness form drag GB tends to reduce thevertical shear of the basic currents, which is blocked bythe Coriolis force associated with eddy-induced over-turning circulation VB. In other words, the eddy-induced overturning circulation is accompanied by thedeflective (i.e., Coriolis) force, the latter of which en-hances the vertical shear of the basic currents, as in thecase of baroclinic geostrophic adjustment establishingthermal wind balance. Here we find that it is the verti-cal shear resulting from the eddy-induced overturningcirculation rather than the surface wind stress that can-cels the vertical redistribution of momentum by thelayer-thickness form drag (or the quasi-Stokes formstress; section 3c).

We thereby conclude that the form stress is not theagent that transfers the wind-induced momentum at thesea surface to the bottom layers, in sharp contrast tothe traditional theories of McWilliams et al. (1978),Johnson and Bryden (1989), Olbers (1998), and Olbersand Visbeck (2005). Thus, there is no need to assumethat the form stress or the associated momentum flux


has a constant (i.e., divergence free) vertical profile be-tween the bottom of the Ekman layer and the top of thebottom layer (see section 1). Realistic vertical profilesof the form stress are shown in Figs. 6b and 8b.

The enhancement of the vertical shear of the basiccurrents by eddy-induced overturning circulation isconsistent with this circulation providing an input to themean kinetic energy, as explained below. The local bal-ance of the mean kinetic energy in (24) becomes

0 � VB � �cp � Vgeo � GB, �31�

which can be verified by comparing Figs. 4a and 4b.Over several regions and depths of high eddy activity inmid and high latitudes of the global ocean (i.e., somelocalized regions over the Southern Ocean and over thewestern boundary currents of each basin; see Figs.3b,c), eddies formed by baroclinic instability relax theslope of isopycnal surfaces, transforming some of themean potential energy back to the mean kinetic energy,which is then taken by the layer-thickness form drag.

The two types of regions in the wind- and eddy-driven regimes mentioned above are connectedthrough the mean potential energy as given in (22),whose right-hand side also becomes Vekm • cp �VB • cp. It is the three-dimensional divergence of thecombined potential energy and hydrostatic pressureflux [V(g�z � p), w(g�z � p)] on the left-hand side of(22) that accounts for both the horizontal and verticaltransfer of energy between regions in the wind- andeddy-driven regimes. The boundary condition of theabove three-dimensional flux is straightforward be-cause the total transport velocity (V, w) satisfies a no-normal-flow boundary condition (section 2c). It wouldbe interesting to investigate this three-dimensional fluxin a future study.

5. Summary

To examine some basic characteristics of layer-thick-ness form drag that were often assumed or speculatedabout in previous theoretical studies, here we have de-scribed the first oceanic study to apply the adiabaticmean four-box energy diagram of B85, Iw01, and AY06to the analysis of currents and eddies in the globalocean. This analysis was performed by using the thick-ness-weighted temporal-averaged mean primitive andenergy equations in density coordinates.

We have determined the global distribution of en-ergy conversion done by layer-thickness form drag. Itwas found that the layer-thickness form drag yields anenergy cascade to the perturbation field, whose rate isas intense as the work values of the eddy-induced over-turning circulation and the wind-induced Ekman trans-

port in the global ocean (sections 3a and 3b). This resultconfirms the presence of an oceanic energy cycle in-volving an equilibrium of the mean kinetic energy assuggested by AY06, which is drawn by sequentiallyconnecting the roles of the layer-thickness form drag,the eddy-induced overturning circulation, the wind-induced Ekman transport, and the direct wind forcing.

The presence of the above energy cycle stems from atleast two characteristics of layer-thickness form drag, asfollows. The first characteristic—the layer-thicknessform drag allowing the vertical (rather than the hori-zontal) transfer of momentum—is the most fundamen-tal assumption that has been frequently used in previ-ous studies. Here we found that the work associatedwith the horizontal redistribution of momentum by thelayer-thickness form drag is one order of magnitudesmaller than that associated with the vertical redistri-bution of momentum (section 3c). This result essen-tially explains how the layer-thickness form drag pre-sents a sink of the mean kinetic energy: dumping thevertical shear of the basic currents in thermal wind bal-ance is an effective way for the layer-thickness formdrag to take the mean kinetic energy and provide anenergy cascade to the perturbation field. The secondcharacteristic of the layer-thickness form drag—beingin an eddy geostrophic balance—is another importantassumption in previous studies and has been confirmedin the present study for transient eddies in the globalocean currents: both the zonally periodic currents andthe western boundary currents in mid and high latitudes(see section 3d). This result essentially explains why thework done by the layer-thickness form drag nearly bal-ances the work done by the eddy-induced overturningcirculation in each vertical column (as detailed in sec-tion 4).

Based on the results of the above analyses, we havepresented a set of planetary geostrophic equations thatelucidate the adiabatic aspects of oceanic dynamics inequilibrium of energy and momentum in the mid andhigh latitudes (section 4). This set of equations is simpleyet includes eddy effects, and it has a lot of potential toadvance the understanding of the role of layer-thick-ness form drag in various research areas of ocean dy-namics. For example, we can resolve a previous debateon the dynamics of the Antarctic Circumpolar Currentby the finding that rather than the vertical shear result-ing from the surface wind stress, in fact the layer-thickness form drag reduces (and thereby cancels) thevertical shear resulting from the eddy-induced over-turning circulation. This undermines Johnson and Bry-den (1989) and Olbers’s (1998) arguments that the formstress is the agent that transfers the wind-induced mo-mentum at the sea surface to the bottom layers.


These results confirm the self-consistency and utilityof the adiabatic mean four-box energy diagram as aformulation to identify the role of layer-thickness formdrag in a rotating stratified fluid.

Acknowledgments. Comments from Peter Gent andan anonymous reviewer improved the presentationquality of this paper. The authors are grateful to ToshioYamagata, Hiro Sakuma, and Taroh Matsuno for pro-viding insightful comments, T. Iwasaki and K. Miyazakifor stimulating discussions at Tohoku University, S.Griffies, G. Holloway, P. Rhines, R. Furue, and K.Takaya for their interest and encouragement, and H.Sasaki for helping to access the OFES output. This pa-per is dedicated to the late Hajime Miyoshi, who di-rected the development of the Earth Simulator. HA issupported by the Postdoctoral Fellowship for ResearchAbroad from the Japan Society for the Promotion ofScience. KJR is supported by the Japan Agency forMarine-Earth Science and Technology through its spon-sorship of the International Pacific Research Center.

APPENDIX A

Derivation of Energy Equations

Evolution of the potential and kinetic energies in athickness zc

s can be derived from the primitive Eqs.(7)–(9) as

�

�t�g�zczs

c� � � � �Vg�zczsc� � g�wzs

c, �A1�

�

�t ��0

2zs

c|V|2� � � � �V�0

2zs

c|V|2� � V � zscG. �A2�

We consider the volume integral of these equations in aclosed domain � with solid boundaries. Note that theshape of the domain changes in density coordinates(��/�t � 0), resulting in the domain integral �� and thetemporal derivative in density coordinates �/�t not com-mutating. To avoid this complexity, the left-hand sidesof (A1) and (A2) are expressed using

�Azsc�t � � � �VAzs

c� � �At � V � �A�zsc � A zst

c � � � �Vzsc��

� �Atc � V � �cA � wAzc�zsc � A��c � V � wzc�zs

c

� Atc � �c � �VA� � �wA�zc�zsc, �A3�

where A is an arbitrary quantity. The volume integral of (A3) becomes

�

�Azsc�t � � � �VAzs

c� ds d2x � �

Atczsc ds d2x �

d

dtc �

Azsc ds d2x, �A4�

where a no-normal-flow boundary condition of thethree-dimensional velocity (V, w) is applied and d2x �dxdy � dxcdyc. Equation (A4) shows that the temporalderivative in Cartesian coordinates �/�tc can be movedout of the integral operator �� because the shape of thedomain does not change in Cartesian coordinates (��/�tc � 0).

Applying (A4) to (A1) and (A2) yields the volumebudgets of the potential and kinetic energies:

d

dtc �

g�zczsc ds d2x � �

g�wzsc ds d2x, �A5�

d

dtc �

�0

2zs

c|V|2 ds d2x � �

V � zscG ds d2x. �A6�

The evolution of the potential energy is given by fluidparticles moving in the vertical direction, and the evo-lution of the kinetic energy is given by fluid particlesmoving in the direction of the horizontal gradient ofhydrostatic pressure, as is widely known.

The interaction between the potential and kinetic en-ergies is determined by the pressure-flux divergence

g�wzsc � V � zs

cG � �wpzc � V � �cp�zsc

� �wp�zc � �c � �Vp��zsc, �A7�

which includes the incompressibility condition, (2). Be-cause there is no flow crossing the boundary of domain�, the volume integral of (A7) becomes

�

g�wzsc ds d2x � �

V � zscG ds d2x � 0. �A8�

Equations (A5), (A6), and (A8) show that the sum of thedomain integrals of the potential and kinetic energies isconstant in the absence of boundary forcing and friction.

Mean field

Equations for the mean potential and mean kineticenergies can be derived from the primitive Eqs. (12)and (13) in the mean field as


�

�t�g�z zs� � � � �Vg�z zs� � g�wzs, �A9�

�

�t ��0

2zs|V|2� � � � �V

�0

2zs|V|2� � V � �zs�

cp � zsGB� � �0V � M�V�, �A10�

where � � s and w � zt � V • �z (Table 1).To show the volume integral of the mean potential and mean kinetic energies, the left-hand sides of (A9) and

(A10) are expressed using

�Azs�t � � � �VAzs� � �At � V � �A�zs � A zst � � � �Vzs��

� �Atc � V � �cA � wAz�zs � A��c � V � wz�zs

� Atc � �c � �VA� � �wA�z�zs, �A11�

where A is a low-pass filtered quantity. Remembering that total transport velocity (V, w) has no componentcrossing the boundaries (section 2c), the volume integral of (A11) becomes

�

�Azs�t � � � �VAzs� ds d2x � �

Atczs ds d2x �d

dtc�

Azs ds d2x, �A12�

which indicates that the temporal derivative in Cartesian coordinates �/�tc can be moved out of the integraloperator �� and thus allows the volume budgets of the mean potential and mean kinetic energies in (A9) and(A10) to be determined as

d

dtc�

g�z zs ds d2x � �

g�wzs ds d2x, �A13�

d

dtc�

�0

2zs|V|2 ds d2x � �

V � zs�cp ds d2x � �

V � zsGB � �0M�V�� ds d2x. �A14�

The second integral on the right-hand side of (A14)allows the transfer of energy between the mean andperturbation fields, with the connection between themprovided by both the layer-thickness form drag zsG

B

and the Reynolds stress divergence M(V).The interaction between the potential and kinetic en-

ergies in the mean field is determined by the pressure-flux divergence in the mean field

g�wzs V � zs�cp � zs�wpz � V � �cp�

� zs �wp�z � �c � �Vp��, �A15�

which includes the incompressibility condition in themean field, (15). Because there is no mean flow (V, w)crossing the boundaries of the domain (section 2c), thevolume integral of (A15) becomes

�

g�wzs ds d2x �

V � zs�cp ds d2x � 0. �A16�

As illustrated in Fig. 2, this energy conversion is closedin the mean field, with no energy connection betweenthe mean and eddy potential energies. In Fig. 2, thethickness-weighted mean velocity V has been separatedinto the isopycnal mean velocity V and the bolus ve-locity VB (Table 1). The vertical component of the bo-lus velocity is given by wB � w c • �z

h Vdz fromthe three-dimensional nondivergence of the total trans-port velocity, (15). Both the isopycnal mean velocityand the bolus velocity are three-dimensionally diver-gent (McDougall 1998).

Perturbation field

We now derive the eddy energy as the differencebetween (the residual of) the total and mean energies.

An expression for the eddy potential energy is ob-tained by subtracting (A9) from a low-pass filtered ver-sion of (A1), and the volume budget of the eddy po-tential energy is obtained by subtracting (A13) from alow-pass filtered version of (A5); thus,

d

dtc�

g��zczsc �z zs� ds d2x � �

g��wzsc �wzs� ds d2x. �A17�


An expression for the eddy kinetic energy is obtained by subtracting (A10) from a low-pass filtered version of(A2). We first show the volume budget of the total kinetic energy by low-pass filtering (A6); that is,

d

dtc�0

2 �

�zs|V|2 � zsc|V�|2� ds d2x � �

zsV � G � zscV� � G� ds d2x, �A18�

where G� � G G is the deviation from the thickness-weighted mean (zcsG� � 0). Subtracting (A14) from the

above equation yields

d

dtc�

�0

2zs

c|V�|2 ds d2x � �

zscV� � G� �0V � M�V�� ds d2x, �A19�

which is the volume budget of the eddy kinetic en-ergy.

An expression for the pressure-flux divergence in theperturbation field is obtained by subtracting (A16)

from a low-pass filtered version of (A7), and the vol-ume budget of the pressure-flux divergence in the per-turbation field is obtained by subtracting (A14) from alow-pass filtered version of (A8); that is,

�

g��wzsc �wzs� ds d2x � �

�V � zsGB � zs

cV� � G�� ds d2x � 0, �A20�

which relates the source and sink terms of the meankinetic, eddy potential, and eddy kinetic energies in(A14), (A17), and (A19).

We now have a complete set of equations for themean and eddy energies, leading to the four-box energydiagram shown in Fig. 2, which is consistent with theresults of AY06. The budget of each energy box is givenrespectively by (A13), (A14), (A17), and (A19), andthe connections between the four boxes are given by(A16) and (A20).

APPENDIX B

Discussion of Mesoscale Eddy Parameterization

Although the paper by GW95 brought a major ad-vance in the field of ocean modeling and has been citedby hundreds of papers, it appears that not many peopleinterpret the context of this paper in the same manneras it was originally written. We note that GW95 pre-sented a debate on the two types of approaches sug-gested at the time for mesoscale eddy parameterizationto be used in coarse-resolution OGCMs (cf. McWil-liams and Gent 1994).

The first type (the so-called tracer approach) followsGent and McWilliams (1990), who parameterized theeddy-induced advection of tracers in such a way as todecrease the mean potential energy based on the clas-sical L55 energy diagram. The second type (the so-called momentum approach) follows Greatbatch andLamb (1990), who parameterized the layer-thicknessform drag in the modified momentum equations of de

Szoeke and Bennett (1993) in such a way as to redis-tribute geostrophic momentum in the vertical direction(cf. G98). GW95 commented that nothing was wrongwith the modified momentum equations of de Szoekeand Bennett (1993) except that they were not familiarwith the characteristics of layer-thickness form drag(however, we note the energetics of B85). This appearsto be the reason why GW95 proceeded with the tracerapproach (rather than momentum), which has been fol-lowed by most OGCMs currently maintained at worldclimate centers. The momentum approach has not beenused in modern OGCMs except for a recent study byFerreira and Marshall (2006).

Comparing the two approaches further will serve toadvance understanding of eddy effects in ocean dynam-ics and modeling. GW95 discussed the likelihood that ifthe eddy geostrophic balance (section 3d) holds, thentwo types of OGCM simulations adopting the tracerand momentum approaches could present similar re-sults (cf. AY06). This hypothesis of the eddy geo-strophic balance was examined for the first time by thepresent study and confirmed based on the output of ahigh-resolution global ocean model. In addition, ourresult of comparing the vertical and horizontal redistri-butions of momentum by the layer-thickness form drag(section 3c) justifies the previous assumption that layer-thickness form drag is parameterized in the form of thevertical (rather than horizontal) diffusion of momen-tum (cf. Greatbatch and Lamb 1990).

The numerical results presented in the present studycan be regarded as useful metrics for tuning coarse-


resolution OGCM simulations in future studies. Notethat OGCMs are supposed to resolve variability on sea-sonal and longer time scales. This is why we have pre-sented results as a composite from monthly-meananalyses; the mean (perturbation) field in the presentstudy includes (excludes) seasonal and intra-annualvariability.

For example, Figs. 5a and 8b will be useful as metricsfor tuning OGCMs adopting the momentum approach.The fact that the work V • GB is mostly negative in eachvertical column (Fig. 5a) suggests that this relation canbe used as a principle for parameterizing the quasi-Stokes form stress.

Although the L55 energy diagram was not used in thepresent study, Figs. 4b and 8a will be useful as metricsfor tuning OGCMs adopting the tracer approach. Thefact that the work VB • cp is mostly positive in eachvertical column (Fig. 4b) supports GW95, who used thisrelation as a principle to parameterize the eddy-induced velocity. Nevertheless, the Gent and McWil-liams (1990) scheme is not the only solution that canmake the sign of VB • cp positive in each verticalcolumn, as recently shown by Aiki et al. (2004). Asexplained in section 3d, the maximum overturning ratein Fig. 8a is 16 � 106 m3 s1 in the Southern Ocean,which is nearly the same as that in Figs. 6 and 7 ofGW95. This result corroborates, at least in the South-ern Ocean, the use of the standard horizontal diffusivity� � 1000 m2 s1 in the parameterization of Gent andMcWilliams (1990).

Finally, we touch on the small effect of the work doneby the density residual (Fig. 5b), which originates in thesecond term on the last line of (19). McDougall andMcIntosh (2001) and Greatbatch and McDougall(2003) argued that this term (which is missing also inOGCMs adopting the tracer approach) would be lessimportant than the other terms in the momentum equa-tions, such as the Reynolds stress. The result of ouranalysis justifies their argument (section 3c).

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