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

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Kelvin J. Richards International Pacific Research Center, University of Hawaii at Manoa, Honolulu, Hawaii

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Abstract

Understanding the role of mesoscale eddies in the global ocean is fundamental to gaining insight into the factors 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 density space. Such an approach clearly demonstrates the role of layer-thickness form drag (residual effects of hydrostatic pressure perturbations), which is hidden in the classical analysis of the energetics of flows. For the first time in oceanic studies, the global distribution of layer-thickness form drag is determined. This study provides direct evidence to verify some basic characteristics of layer-thickness form drag that have often been assumed or speculated about in previous theoretical studies. The results justify most of the previous assumptions and speculations, including those associated with (i) the presence of an oceanic energy cycle explaining the relationship between layer-thickness form drag and wind forcing, (ii) the manner in which layer-thickness form drag removes the energy of vertically sheared geostrophic currents, and (iii) the reason why the work of layer-thickness form drag nearly balances the work of eddy-induced overturning circulation in each vertical column. However, the result of the analysis disagrees with speculation in previous studies that the layer-thickness form drag in the Antarctic Circumpolar Current is the agent that transfers the wind-induced momentum near the sea surface downward to the bottom layers. The authors present a new interpretation: the layer-thickness form drag reduces (and thereby cancels) the vertical shear resulting from the eddy-induced overturning circulation (rather than the vertical shear resulting from the surface wind stress). This interpretation is consistent with the results of the energy analysis conducted in this study.

Corresponding author address: Hidenori Aiki, IPRC/SOEST, University of Hawaii, 1680 East West Road, POST Bldg., 4th Floor, Honolulu, HI 96822. Email: haiki@hawaii.edu

Abstract

Understanding the role of mesoscale eddies in the global ocean is fundamental to gaining insight into the factors 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 density space. Such an approach clearly demonstrates the role of layer-thickness form drag (residual effects of hydrostatic pressure perturbations), which is hidden in the classical analysis of the energetics of flows. For the first time in oceanic studies, the global distribution of layer-thickness form drag is determined. This study provides direct evidence to verify some basic characteristics of layer-thickness form drag that have often been assumed or speculated about in previous theoretical studies. The results justify most of the previous assumptions and speculations, including those associated with (i) the presence of an oceanic energy cycle explaining the relationship between layer-thickness form drag and wind forcing, (ii) the manner in which layer-thickness form drag removes the energy of vertically sheared geostrophic currents, and (iii) the reason why the work of layer-thickness form drag nearly balances the work of eddy-induced overturning circulation in each vertical column. However, the result of the analysis disagrees with speculation in previous studies that the layer-thickness form drag in the Antarctic Circumpolar Current is the agent that transfers the wind-induced momentum near the sea surface downward to the bottom layers. The authors present a new interpretation: the layer-thickness form drag reduces (and thereby cancels) the vertical shear resulting from the eddy-induced overturning circulation (rather than the vertical shear resulting from the surface wind stress). This interpretation is consistent with the results of the energy analysis conducted in this study.

Corresponding author address: Hidenori Aiki, IPRC/SOEST, University of Hawaii, 1680 East West Road, POST Bldg., 4th Floor, Honolulu, HI 96822. Email: haiki@hawaii.edu

1. Introduction

The present study extends the theory of Bleck (1985, hereafter B85), Iwasaki (2001, hereafter Iw01), and Aiki and Yamagata (2006, hereafter AY06) to the analyses of currents and eddies simulated by a high-resolution ocean general circulation model (OGCM). This is the first attempt in oceanic studies to determine the global distribution of energy conversion done by residual effects of hydrostatic pressure perturbations (called the layer-thickness form drag, as detailed in section 2b). In the present study this drag is associated with the adiabatic effects of transient eddies. Although the layer-thickness form drag has often been assumed to vertically redistribute geostrophic momentum, it and the associated energetics have not been investigated on the basis of either the output from high-resolution numerical simulations or observations of the ocean. The effects of the Reynolds stress, diabatic processes, and the form drag caused by the bottom topography were not investigated in the present study. The context of the present study is also relevant to atmospheric dynamics in that the sum of the layer-thickness form drag and the Reynolds stress divergence corresponds to the divergence of a pressure-based Eliassen–Palm flux (Andrews 1983, hereafter A83; Lee and Leach 1996, hereafter LL96; Iw01; Tanaka et al. 2004). As an atmospheric and zonal-mean counterpart of the present study, Uno and Iwasaki (2006) have recently analyzed an output from an atmospheric general circulation model (AGCM) and shown a cascade-type energy conversion.

Despite several theoretical studies in various research areas of atmosphere and ocean dynamics (cf. Rhines 1979; A83; Johnson and Bryden 1989; Cushman-Roisin et al. 1990; Greatbatch 1998, hereafter G98) the vertical mixing of momentum by layer-thickness form drag and associated energy conversions have been little investigated (or confirmed) on the basis of output from high-resolution OGCMs. This is more or less a result of the classical energy diagram of Lorenz (1955, hereafter L55) being exclusively used in previous studies. The role of layer-thickness form drag cannot be explained by the energy diagrams of L55 and Plumb (1983); rather, it requires an adiabatic mean four-box energy diagram that was derived by B85, Iw01, and AY06. This special feature of the adiabatic mean energy diagram is attributed to its use of modified definitions of the mean and eddy kinetic energies (B85; Iw01); these are given by a thickness-weighted mean formulation of an inviscid stratified fluid in density (or isentropic) coordinates. Although it was a problem in the formulation of B85, the boundary condition of the adiabatic mean energy equations is robust and straightforward even in the presence of density surfaces intersecting (i.e., outcropping at) the top and bottom boundaries of the ocean, as recently shown by AY06 using only no-normal-flow boundary conditions of the total transport velocity (defined in section 2c) and the raw velocity.

AY06 stated that understanding the adiabatic mean energy diagram is fundamental to introducing a parameterization of layer-thickness form drag in OGCMs (McWilliams and Gent 1994; Krupitsky and Cane 1997; G98; Greatbatch and McDougall 2003; Ferreira and Marshall 2006). The energetics of layer-thickness form drag are important in a number of aspects of ocean dynamics, such as (i) the dynamics of the Antarctic Circumpolar 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 of baroclinically coupled coherent eddies (e.g., Cushman-Roisin et al. 1990; Aiki and Yamagata 2000, 2004). The present study addresses the issue of the Antarctic Circumpolar Current, and the results of our energy analysis resolve a confusion in previous studies, as explained below.

The layer-thickness form drag has received considerable attention in previous investigations of the Antarctic Circumpolar Current regarding the vertical transfer of momentum and its relationship with the wind stress applied at the sea surface. The status of modern oceanography is well represented by the debate between Olbers (1998) and Warren et al. (1996, 1998), as explained below. There has been a series of attempts to prove that the layer-thickness form drag is an agent for transferring the wind-induced momentum near the sea surface to the bottom layers (McWilliams et al. 1978; Johnson and Bryden 1989; Olbers 1998; Olbers and Ivchenko 2001). This requires the vertical flux (the form stress) of momentum associated with the layer-thickness form drag to connect the bottom of the Ekman layer (or the surface mixed layer) and the top of the deep layer interacting 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 of Olbers and Visbeck 2005). On the other hand, Warren et al. (1996, 1998) have pointed out that this view of the vertical momentum transfer may be incorrect because it appears to come from intuitions based on nonrotational fluid dynamics. Because the ocean is a rotational stratified fluid, the wind stress applied at the sea surface should induce Ekman transports (flowing perpendicular to the direction of the wind stress) such that the momentum equation is balanced in the thin Ekman layer at the sea surface, resulting in no wind-induced momentum being transferred downward.

Moreover, the layer-thickness form drag in some previous studies (Ivchenko et al. 1996; Stevens and Ivchenko 1997; Olbers and Ivchenko 2001; Marshall and Radko 2003) is misleading in that (i) this form drag is disguised as the Coriolis force associated with an eddy-induced velocity and (ii) the pressure field is not analyzed in these studies, which is a custom originating in the transformed Eulerian mean (TEM) theory and associated formulation of a velocity-based Eliassen–Palm flux (Andrews and McIntyre 1976). In contrast to the TEM theory, the layer-thickness form drag used in the present study is based on the pressure field and thus is 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 section 3, where we analyze the output from a high-resolution (0.1° × 0.1°) global OGCM based on a derivation of energy equations in density coordinates (see appendix A or AY06). The purpose of the analysis is to verify some basic characteristics of layer-thickness form drag that were often assumed or speculated about in previous studies, as explained below. Section 3a compares energy conversions caused by the layer-thickness form drag, the eddy-induced overturning circulation, the wind-induced Ekman transport, and the wind forcing. This is intended to verify the presence of an energy cycle as suggested by AY06 for the wind-driven ocean circulation, whose dependence on the time scale of a low-pass filter is investigated in section 3b. Then we investigate some characteristics of layer-thickness form drag associated with this energy cycle. To investigate how the layer-thickness form drag presents a sink of the mean kinetic energy, section 3c examines which of the vertical and horizontal redistributions of momentum by layer-thickness form drag is predominant in terms of energy conversions in the global ocean. The result of this analysis confirms the most fundamental assumption in previous studies, namely that the layer-thickness form drag redistributes momentum mainly in the vertical direction. To investigate why the work done by layer-thickness form drag nearly balances the work done by the eddy-induced overturning circulation in each vertical column, section 3d examines the presence of the geostrophic balance of the layer-thickness form drag in the global ocean. The result of this analysis justifies the frequent use of the assumption of the eddy geostrophic balance in previous studies (such as the use of the disguised form drag in the TEM theory). The adiabatic aspects of oceanic dynamics confirmed above are summarized in section 4 using a set of planetary geostrophic equations, including the energetics of layer-thickness form drag. This set of equations is simple but includes eddy effects, and it has a lot of potential to advance the understanding of the role of layer-thickness form drag in various research areas of ocean dynamics.

The primitive and energy equations in density coordinates used in sections 3 and 4 are systematically described in section 2, which serves to clarify the definitions 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, and energy equations for a continuously stratified fluid in density coordinates with inviscid, incompressible, hydrostatic, and Boussinesq approximations. Diabatic mixing, thermobaricity, friction, and boundary forcing are excluded from the present formulation for simplicity.

The explicit use of density coordinates in the present study is intended to demonstrate how density surfaces are handled near the top and bottom of the ocean. Previous formulations of density coordinates (B85; Killworth and Nanneh 1994) assume that unused density surfaces are condensed at the top and bottom of the ocean with infinitesimal thickness (i.e., the buoyancy frequency is infinite there). This problem is fixed in the present study by adopting a formulation similar to that in A83 to achieve realistic (i.e., nonhomogeneous) distributions of density and other tracers at the top and bottom boundaries (section 2a). Elaborating the structure of density coordinates near the top and bottom boundaries of the ocean is fundamental also to clarifying the definition of the isopycnal (unweighted) mean in the presence of density surfaces that intersect (outcrop at) the top and bottom boundaries of the ocean (section 2b).

The other aspects of the derivation of the adiabatic mean equations are consistent with those in AY06, including the no-normal-flow boundary condition of the total transport velocity explained by using an integral identity (section 2c) and the use of energy equations based on modified definitions of the mean and eddy kinetic energies. Comparisons with the formulations used in previous studies are given in section 2d. Characteristics of the energy equations and the adiabatic mean four-box energy diagram are briefly explained in section 2e.

a. Density coordinates

Let (u, υ, w) be the three-dimensional velocity in Cartesian coordinates (often called z coordinates in oceanic studies) spanned by a set of independent variables (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 ρ, incompressibility, and the horizontal velocity V ≡ (u, υ) are
i1520-0485-38-9-1845-e1
i1520-0485-38-9-1845-e2
i1520-0485-38-9-1845-e3
where ∇c = (∂/∂xc, ∂/∂yc) is the horizontal gradient operator, ρ0 is the reference density of seawater, f is the Coriolis parameter, and p ≡ ∫z gρ dz is hydrostatic pressure, with g being the acceleration caused by gravity. Now let the density coordinates in the present study be spanned by a set of independent variables (x, y, s, and t), with which the primitive Eqs. (1)(3) can be rewritten respectively as (Kasahara 1974)
i1520-0485-38-9-1845-e4
i1520-0485-38-9-1845-e5
i1520-0485-38-9-1845-e6
The two-dimensional vector G ≡ −∇cp = −pzc is the negative of the horizontal gradient of the hydrostatic pressure, and w* ≡ wzctV · zc is the diapycnal velocity.
The vertical axis s of density coordinates is set to be identical to the potential density at depths away from the sea surface and bottom (i.e., the ocean interior). As shown in Fig. 1, it is expressed as ρ(x, y, s, t) ≡ s for s ∈ (ρtop, ρbtm), where ρtop and ρbtm are the density values at the top and bottom edges, respectively, of a water column located at (x, y). Each surface of fixed s is an isopycnal surface. Substitution of ∂xρ = ∂yρ = ∂tρ = 0 into (4) yields w* = 0 for s ∈ (ρtop, ρbtm), which is essentially
i1520-0485-38-9-1845-e7
The vertical velocity is given by the temporal displacement and advection of each isopycnal surface, as is widely known.

The structure of density coordinates outside the density range of a water column (shaded in Fig. 1) is then determined by considering an oceanic domain bounded by a rigid sea surface and a bottom of arbitrary depth h(>0) as follows (our procedure is very similar to that in A83). For s ∉ (ρtop, ρbtm), each surface of fixed s is condensed at the height of either zc = 0 or zc = −h(x, y) such that ∂zc/∂s = 0 there. Nevertheless, the potential density on such a surface with a fixed s can vary: ρρtop(x, y, t) or ρbtm(x, y, t), as illustrated by the solid vertical lines in Fig. 1. Each of ρtop and ρbtm satisfy ρt + V · ρ = 0, which together with (4) shows w* = 0 and (7) being extensible to the outer range s ∉ (ρtop, ρbtm). This design of the density coordinates allows for the thickness based on the potential density ∂zc/∂ρ to be nonzero at the top and bottom boundaries (see section 2d).

As a result, for all ranges of s, the momentum and thickness as described in (5) and (6) reduce to
i1520-0485-38-9-1845-e8
i1520-0485-38-9-1845-e9
Equations (7)(9) complete the primitive equations in density coordinates.

As explained further in section 2d, rewriting the pressure term using the Montgomery potential G = −(p + gρz) is possible only in regions where ρ = 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 isopycnal surfaces here, even though the density is not always constant along each surface of fixed s (section 2a). Likewise, a low-pass temporal filter operating along each surface of fixed s is hereafter called the isopycnal mean and is denoted by an overbar (Table 1). The height of each surface of fixed s is decomposed into the mean height z and the deviation from it z′′′ (> ≡ 0; Table 1); that is,
i1520-0485-38-9-1845-e10
The horizontal velocity is decomposed into the thickness-weighted mean velocity /zs and the deviation from it V″ ( ≡ 0; Table 1); thus,
i1520-0485-38-9-1845-e11
The thickness-weighted mean velocity is the horizontal component of the total transport velocity (Table 1). The vertical component of the total transport velocity is given in section 2c.
Applying the thickness-weighted mean to (8) and (9) yields the primitive equations in the mean field (B85) as follows:
i1520-0485-38-9-1845-e12
i1520-0485-38-9-1845-e13
where M(V) ≡ − · () is the isopycnal divergence of the Reynolds stress.
It is noted that the pressure term zsĜ in (13) is a thickness-weighted mean quantity, which is here separated into contributions from the mean and perturbation fields:
i1520-0485-38-9-1845-e14
where p ≡ ∫s gρ̂zs ds and p′′′ ≡ ∫s g(ρzcs)′′′ ds are the isopycnal mean of hydrostatic pressure p and the deviation from it ( ≡ 0; Table 1). The derivation of (14) is based on the horizontal gradient becoming ∇c = − (∂/∂z)z when operating on averaged quantities, such as p and (de Szoeke and Bennett 1993). The negative of the horizontal gradient of the mean hydrostatic pressure −∇cp and the thickness-weighted mean density ρ̂ ≡ /zs = s (Fig. 1) are explained further in section 2d.

The second term on the last line of (14) is the layer-thickness form drag, hereafter symbolized by zsGB. To explain the redistribution of momentum, the layer-thickness form drag is often written as the divergence of a pseudoflux (A83; Johnson and Bryden 1989; Killworth and Nanneh 1994; LL96; Held and Schneider 1999; Smith 1999; Iw01). This is done in section 3c, where we compare the effects of the vertical and horizontal distributions of momentum in the global ocean currents. A related discussion appears in section 2d.

c. Boundary condition

If the vertical component of the total transport velocity is defined as zt + · z (Table 1), the total transport velocity (, ) becomes three-dimensionally nondivergent (de Szoeke and Bennett 1993); that is,
i1520-0485-38-9-1845-e15
which is an incompressibility condition in the mean field.
To show that the three-dimensional velocity (, ) satisfies a no-normal-flow boundary condition at the top and bottom of the ocean, AY06 used a series of integral identities in each vertical column:
i1520-0485-38-9-1845-e16
where A is an arbitrary quantity, ρmin (ρmax) is the minimum (maximum) value of ρtop (ρbtm) in a filter window (Fig. 1), and the overbar with the superscript z denotes the Eulerian mean at a fixed height (Table 1). Equation (16) is a generalization of the results of McDougall and McIntosh (2001) and Killworth (2001) and was termed the “pile-up rule” by AY06 because it explains the relationship between the cumulative sums of the thickness-weighted mean and Eulerian mean quantities in the vertical direction.
Integrating (15) upward from the bottom boundary yields = −∇c · ∫zh dz. Then the application of the pile-up rule, (16), gives the mean vertical velocity at the top boundary as
i1520-0485-38-9-1845-e17
Equation (17) proves that mean vertical velocity vanishes at the top boundary, as does the Eulerian mean vertical velocity wz (Iw01). As a result, total transport velocity (, ) satisfies a no-normal-flow boundary condition at the top and bottom of the ocean (cf. McDougall and McIntosh 2001).

d. Comparison with previous formulations

The design of the density coordinates in the present study allows for the thickness based on the potential density ∂zc/∂ρ to be nonzero at the top and bottom boundaries (section 2a), whereas the density coordinates used in B85 and Killworth and Nanneh (1994) assume that ∂zc/∂ρ vanishes at the top and bottom boundaries. In the density coordinates of the present study, there are regions of ∂zc/∂s = 0 at the top and bottom boundaries (shaded in Fig. 1) where all physical quantities (e.g., u, υ, and ρ) lose their dependencies on s and become functions of (x, y, t); this procedure is analogous to the isentropic coordinates of the atmosphere considered in A83 with some isentropes intersecting the ground.

One way in which our formulation differs from that in A83 is in writing the layer-thickness form drag GB (Table 1 and section 2b) as the divergence of a pseudoflux. In section 3c of this paper, GB is separated into terms representing the vertical and horizontal (not isopycnal) redistributions of momentum by using the pile-up rule (16), which has not been noted in previous atmospheric and oceanic studies. On the other hand, the expression in A83 is based on the Montgomery potential, which is useful in dealing with non-Boussinesq fluids. As explained in section 2a, the use of the Boussinesq approximation in the present study makes it impossible to derive the Montgomery potential over regions where ρs (shaded in Fig. 1). At depths sufficiently far from the top and bottom boundaries, p′′′s ≡ −g(ρzcs)′′′ becomes −gsz′′′s, which leads to
i1520-0485-38-9-1845-e18
where p′′′s + gsz′′′ is the perturbed Montgomery potential. The first term on the last line of (18) corresponds to []θ in (2.21) of A83. As noted above, (18) is valid only for density surfaces that do not touch the top and bottom boundaries in a filter window, whose density range is expressed as s ∈ (ρtmax, ρbmin), where ρtmax and ρbmin are the maximum and minimum values of ρtop and ρbtm, respectively (Fig. 1). Equation (18) has been used in Killworth and Nanneh (1994) and LL96.

In general, the method of determining the mean height of each density surface is fundamental to clarifying the adiabatic mean formulation (cf. Nurser and Lee 2004). Jacobson and Aiki (2006) considered a special case where zc is not defined over s ∉ (ρtop, ρbtm) (the shaded regions in Fig. 1) and addressed the uncertainty of both z′′′ and z near the top and bottom boundaries. In contrast to this, in the present study (section 2a) we define zc over a sufficiently large range of s in density coordinates (including the regions shaded in Fig. 1) that the mean height z ranges from −h to 0. Then, for fluid particles whose mean height is either z = 0 or −h, the perturbation height becomes z′′′zcz = 0. This is an important characteristic and was originally described in section 7 of McDougall and McIntosh (2001). The mean height of each density surface can be obtained also by rearranging all fluid particles in a filter window from the bottom in order of decreasing density, such that the mean height z indeed ranges from −h to 0. It turns out that the vertical position of the lightest and heaviest fluid particles in a filter window does not change (being invariably adjacent to the boundary), resulting in the perturbation height becoming indeed z′′′zcz = 0 for fluid particles whose mean height is either z = 0 or −h.

Mean primitive Eqs. (12) and (13) in the present study are almost the same as (1) and (2) in AY06, apart from our improved definition of the isopycnal mean in section 2b. For example, the density coordinates in the present study make it possible to show that ρ = s for s ∈ (ρtmax, ρbmin) and ρs for s ∉ (ρtmax, ρbmin), as Fig. 1 suggests. Although AY06 write the mean pressure term as (the isopycnal mean of G) and the mean height density as ρ̃ (the isopycnal mean of ρ) for simplicity, the formal expressions for these terms are −∇cp and ρ̂, respectively, as derived for (14) in the present study. Only at depths sufficiently far from the top and bottom boundaries does −∇cp ≡ −pgρ̂z = G (the isopycnal mean of G), by using G ≡ −pg = −pgsz for s ∈ (ρtmax, ρbmin). Likewise, only at depths sufficiently far from the top and bottom boundaries does the thickness-weighted mean density ρ̂(≡/zs = s) become identical to ρ (the isopycnal mean of ρ). The mean height density in the present study is the thickness-weighted mean density ρ̂ (Table 1) rather than the isopycnal mean density ρ. The adiabatic mean densities (such as the modified mean density and the temporal residual mean density) as defined in previous studies can be revised accordingly (cf. de Szoeke and Bennett 1993; McDougall and McIntosh 2001; Griffies 2004; Jacobson and Aiki 2006).

As demonstrated by B85, de Szoeke and Bennett (1993), Greatbatch and McDougall (2003), and Jacobson and Aiki (2006), the mean momentum Eq. (13) can be expressed in terms of the total transport velocity (instead of the isopycnal mean or Eulerian mean velocity), which was overlooked in McWilliams and Gent (1994), LL96, Wardle and Marshall (2000), and Ferreira and Marshall (2006).

e. Energy equations

Energy equations for the mean and perturbation fields are derived in appendix A. With the exception that the variables are expressed in density coordinates, the context in appendix A is essentially the same as that in AY06 in that the volume budget of the energy equations is explained by using no-normal-flow boundary conditions of the total transport velocity (, ) and the raw velocity (V, w). Nevertheless, the explicit use of density coordinates in the present study will serve to reconfirm that neither zcρ = zρ = 0 nor zs = 0 is necessary as conditions at the top and bottom of the ocean. This improvement over the formulation of B85 is made more transparent than in AY06 by the systematic description of density coordinates in this paper.

The adiabatic mean energy diagram is derived by using modified definitions of the mean and eddy kinetic energies as follows. The total kinetic energy (ρ0/2) is separated into the mean kinetic energy (ρ0/2)zs||2 and the eddy kinetic energy (ρ0/2), as in B85, Røed (1997), Iw01, Jacobson and Aiki (2006), and AY06. On the other hand, the total potential energy g is separated into the mean potential energy gρ̂z zs and the eddy potential energy1 (ggρ̂z zs), as in L55, Iw01, and AY06. Here we point out that the thickness-weighted mean velocity (which is used to define the mean kinetic energy) is the sum of the isopycnal mean velocity V and the so-called bolus velocity VB/zs (Rhines 1982). Usually the isopycnal mean velocity represents the basic geostrophic currents and the wind-induced Ekman transports, with the bolus velocity representing the eddy-induced overturning circulations (see sections 3a and 4). Note that the effect of the bolus velocity is included in the mean kinetic energy (ρ0/2)zs||2 instead of the eddy kinetic energy (ρ0/2). Because of this modified definition for the mean and eddy kinetic energies, the look of the adiabatic mean energy diagram in Fig. 2 is different from that of the classical L55 energy diagram. Only the adiabatic mean energy diagram can describe the energy conversion done by the layer-thickness form drag ( · zsGB), which is why this energy diagram is being used in this study (section 3).

Energy conversions represented by the triple lines in Fig. 2—that is, · zsGB, caused by the layer-thickness form drag; −VB · zscp, caused by the bolus velocity (eddy-induced overturning circulation); and −V · zscp, caused by the isopycnal mean velocity (Ekman transport)—are investigated in section 3. The other energy paths in Fig. 2 were not investigated. It is of great interest to directly analyze the above three paths of pressure-related energy conversions. According to an indirect analysis (scale analysis) for the global ocean (AY06), energy conversion rates caused by both the layer-thickness form drag · zsGB and the bolus velocity −VB · zscp 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 conversion caused by the isopycnal mean velocity −V · zscp is also important in drawing the energy cycle of the wind-driven ocean circulation (detailed in sections 3a and 4). Verifying the relationship among these three conversions is the first step toward understanding the energetics of layer-thickness form drag.

On the other hand, we generally expect (but did not examine) that the energy conversion caused by the Reynolds stress ρ0 · M(V) is similar to that estimated by using the Eulerian mean formulation (e.g., L55; Masina et al. 1999). There are a number of previous numerical investigations for the energy conversion caused by 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 OGCM was integrated for a 50-yr period in the presence of climatological atmospheric forcing. The model code was OFES (the OGCM for the Earth Simulator; Masumoto et al. 2004), which is based on the Geophysical Fluid Dynamics Laboratory Modular Ocean Model version 3 (Pacanowski and Griffies 2000), with substantial modifications for the vector-parallel hardware system of Japan’s Earth Simulator. The model domain extends from 75°S to 75°N, with the temperature and salinity field in the southern and northern buffer zones being relaxed to the monthly-mean climatological values at all depths. The model has 54 depth levels, with the discretization varying from 5 m at the surface to 330 m at the maximum depth of 6065 m. The atmospheric forcing consists of wind stress, heat, and water fluxes of monthly-mean climatology data constructed from the NCEP–NCAR reanalysis for 1950–99 (Kalnay et al. 1996). Further details and assessments of the OFES climatological run are given in Masumoto et al. (2004), Nakamura and Kagimoto (2006a, b), and Merryfield and Scott (2007).

A set of 3-day snapshots archived throughout the 46th year of the OFES run (a total of 121 three-dimensional snapshots of the global ocean) were available to the analysis. These snapshots were originally in z coordinates and hence were first mapped onto density coordinates in each vertical column. We used 80 density layers defined by the potential density referenced to sea surface pressure (this density was chosen in an attempt to overview the global ocean with most of the kinetic energy distributed above the main thermocline), with density layers being in steps of 0.02 kg m−3 for s ∈ (1027.2 kg m−3, 1028.0 kg m−3) and 0.2 kg m−3 for other values of s. This mapping to density coordinates was done in such a way as to conserve a cumulative sum of each quantity in each vertical column. In contrast to the primitive equations of the OGCM, the hydrostatic pressure in the present analysis was calculated from potential density referenced to the sea surface pressure instead of in situ density, and the sea surface was treated as a rigid rather than a free surface.

The low-pass temporal filter (overbar) in section 2 was set to monthly means in the present analyses, which was intended to capture transient eddies while excluding seasonal variability of thermocline and the surface mixed layer. Applying the monthly means (in density coordinates) to the set of 3-day snapshots throughout the year yielded results for each month (i.e., 12 sets), but the monthly evolution and seasonal variability of energetics were not investigated in the present study. All results (figures and global values) presented below refer to the composite of the 12 sets of the monthly results from January to December, except that in section 3b we present brief results for cases where the monthly-mean filter was replaced by the seasonal- and annual-mean filters.

In the following, the horizontal axes (x, y) in section 2 are implicitly extended to those in spherical coordinates, with x and y representing the zonal and meridional directions, respectively.

Figure 3 shows the mean kinetic, eddy kinetic, and eddy potential energies in each vertical column. The global distributions of the mean kinetic energy (Fig. 3a) depict streamlines of major currents in the world’s oceans: the Antarctic Circumpolar Current, the Agulhas Current, and the equatorial and western boundary currents in each basin. The eddy kinetic energy (Fig. 3b) in the Gulf Stream and the Kuroshio is localized to the regions of mean currents detaching from the western boundary. The eddy kinetic and eddy potential energies (Figs. 3b,c) in the Southern Ocean are isolated in regions where the mean currents are constrained by steep topography (as in Lee and Coward 2003); these are the Abyssal Plain (80–90°E), the area southwest of the New Zealand Plateau (150°E), the Scott Fracture (180°W), and the Drake Passage (60°W). Eddy kinetic energy is nearly absent over a large extent of the South Pacific Ocean and the eastern North Pacific Ocean.

The volume integrals of the mean kinetic, eddy kinetic, and eddy potential energies in the global ocean are 2.70, 0.85, and 1.11 EJ, respectively, where 1 EJ (an exajoule) = 1018J (Fig. 3). The fact that the eddy potential energy is somewhat larger than the eddy kinetic energy suggests that eddies are statistically geostrophic and on spatial scales of the internal Rossby radius. The sum of 1.96 (= 0.85 + 1.11) EJ for the transient eddies in the present analysis is one order of magnitude smaller than the value of 13 EJ previously estimated by Zang and Wunsch (2001) and Wunsch and Ferrari (2004) using a spectrum analysis of observations.

a. Energy conversions

We have identified three paths of energy conversions—represented by · zsGB, −VB · zscp, and −V · zscp (indicated by the triple line in Fig. 2)—that originate in the pressure effects and concern the budget of the mean kinetic energy in (A10). AY06 speculated that these three terms maintain the equilibrium of the mean kinetic energy with the following energy cycle (see also Fig. 2). Wind forcing provides an input to the mean kinetic energy, which is then transferred to the mean potential energy by the wind-induced Ekman transport (i.e., the isopycnal mean velocity V). Nevertheless, the net mean potential energy does not change because the eddy-induced overturning circulation (i.e., the bolus velocity VB) extracts some of the mean potential energy and feeds the mean kinetic energy, which is subsequently taken by the work of the layer-thickness form drag GB, thereby endowing the perturbation field with an energy cascade. In the following we investigate the validity of the above energy cycle.

The quantity · zsGB is the work done by the layer-thickness form drag, whose vertical integral in each vertical column is shown in Fig. 4a, where negative (positive) values indicate a decrease (increase) in the mean kinetic energy. Significant work values leading to an energy cascade to the perturbation field are present in the Southern Ocean (the aforementioned eddy-active regions), the Agulhas Current, the Gulf Stream, and the Kuroshio, whereas no significant work is evident over the equatorial regions. The global work of the layer-thickness form drag is −0.50 TW (1 TW = 1012 W), which would be sufficient to exhaust the net mean kinetic energy in the global ocean (Fig. 3a) within 80 days [i.e., (2.7 EJ/0.50 TW)/(86 400 s day−1)] if there were no source of mean kinetic energy. The work of the layer-thickness form drag is examined further in section 3c.

The quantity −VB · zscp is the work done by the bolus velocity, whose vertical integral in each vertical column is shown in Fig. 4b, where positive (negative) values indicate transformation from mean potential (kinetic) energy to mean kinetic (potential) energy (Fig. 2). The work of the bolus velocity is clearly positive over each region of high eddy activity in the global ocean (Figs. 3b,c): some of the mean potential energy is extracted to provide an input to the mean kinetic energy when the eddy-induced overturning circulation relaxes the slope of isopycnal surfaces. The global work of the bolus velocity, which is +0.46 TW in Fig. 4b, nearly balances the work of the layer-thickness form drag presented above.

The quantity −V · zscp is the work of the isopycnal mean velocity, whose vertical integral in each vertical column is shown in Fig. 4c. Although significant positive and negative signs are evident in Fig. 4c, the global work is negative (−0.30 TW in Fig. 4c), which reduces the mean kinetic energy and increases the mean potential energy (see Fig. 2). The work of the isopycnal mean velocity reflects the wind-induced Ekman transports that steepen the slope of isopycnal surfaces over the global ocean (as detailed in section 4). Strictly speaking, the work of the isopycnal mean velocity also includes the effects of standing eddies (and transient eddies with frequencies lower than 2π (30day)−1 because we use the monthly-mean filter), which is indicated by positive values in Fig. 4c (e.g., around the Drake passage) if standing eddies convert some of the mean potential energy into mean kinetic energy.

In addition to the pressure-related energy conversions described above, there is an input to the mean kinetic energy caused by wind forcing (cf. Wunsch 1998). We quantified the work of the wind stress (Fig. 4d) by calculating ∫ρminρmax · τs ds , where τ(x, y, s, t) ≡ is the isopycnal mean of the wind stress τ ≡ (τx, τy) at the sea surface and H() is the Heaviside step function. The global work of wind forcing in the OFES output is +1.04 TW, of which +0.63 TW comes from mid- and high latitudes (i.e., out of the 20°S–20°N band).

The budget of the mean kinetic energy in mid- and high latitudes sums to +0.29 TW, comprising +0.63 TW for wind forcing, −0.30 TW for isopycnal mean velocity, +0.46 TW for bolus velocity, and −0.50 TW for layer-thickness form drag. Therefore, the processes excluded in the present analysis, such as the boundary friction, the Reynolds stress, the eddy viscosity, diabatic processes, and the difference between the potential density and in situ density (in calculating the horizontal gradient of hydrostatic pressure) must conjointly contribute −0.29 TW.

The above results have generally confirmed the presence 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 3c and 3d) were derived using the monthly-mean filter to focus on transient eddies while excluding seasonal variability of the thermocline and the surface mixed layer. It is of interest to assess the effects of changing the time scale of the low-pass temporal filter on the value of the energy and work terms. Table 2 compares the global energy and work terms in cases where the same 3-day snapshots obtained from the OFES climatological run are analyzed with seasonal (3-month)-mean and annual-mean filters, where the seasonal-mean (the second column of Table 2) represents the composites of the four 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-annual variabilities in the perturbation field produces leading order decreases (increases) in the mean (eddy) energy in 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 (2.37 EJ) is about 3 times larger than the monthly eddy kinetic energy (0.88 EJ), which may be caused by a seasonal shift (or reversal) of the current pathways. The annual eddy potential energy (4.90 EJ) turns out to be about 4 times larger than the monthly eddy potential energy (1.15 EJ), which is caused by the seasonal cycle of the thermocline and mixed layer depth.

In contrast to the energy terms, the work terms of the pressure effects and wind forcing are not greatly altered by changes in the filtering, except that the work of the isopycnal mean velocity in the annual-mean analysis becomes −0.66 TW, which is close to the work of the form drag and the bolus velocity (−0.60 and +0.80 TW, respectively). Note that the effects of slowly oscillating transient eddies are included in the work of the bolus (isopycnal mean) velocity in the annual (monthly)-mean analysis. Moreover, in the annual-mean analysis the work of the isopycnal mean velocity (−0.66 TW) better balances the work of wind forcing in mid- and high latitudes (+0.61 TW). We conclude that the oceanic energy cycle discussed in section 3a (Fig. 4) is very robust in the annual-mean analysis.

In the next subsections we investigate the characteristics of layer-thickness form drag that are associated with the presence of the energy cycle confirmed above. Generally speaking, section 3c investigates how the layer-thickness form drag presents a sink of the mean kinetic energy, and section 3d investigates why the work done by the layer-thickness form drag (Fig. 4a) nearly balances the work done by the eddy-induced overturning circulation (Fig. 4b) in each vertical column. [Note that all numerical results (figures and values) in sections 3c and 3d are shown based on the monthly-mean analysis (as in section 3a), with the intent of providing useful metrics to be used in future numerical studies (see appendix B for details).]

c. Momentum redistributions

The most fundamental assumption that has been frequently used in previous studies for layer-thickness form drag is that it allows the vertical transfer of momentum (cf. Johnson and Bryden 1989; G98), which seems to be consistent if waves and eddies in baroclinic instability tend to break the thermal wind balance of basic currents. Strictly speaking, such momentum transfer is possible in both the vertical and horizontal (or isopycnal) directions. Distributions of the momentum fluxes caused by layer-thickness form drag have not been determined to any significant extent in previous oceanic studies, except for LL96, who analyzed an idealized jet in a rectangular channel. The purpose of this subsection is to extend the analysis of LL96 to the output of a high-resolution global ocean model and investigate the energetics. In particular, we compare the effects of the vertical and horizontal transfers of momentum by layer-thickness form drag.

The layer-thickness form drag GB (Table 1 and section 2b) can be written as the divergence of a pseudoflux in the forms of several expressions. Before considering expression (19), used in the present analysis, we briefly explain three other expressions suggested in previous studies. The first expression is (18), as already given in section 2d (A83; LL96), which cannot be derived near the top and bottom boundaries because of the Boussinesq approximation. The second expression, zsGB = −()s + (), is applicable to the complete range of s, where the vertical flux vanishes at the top and bottom boundaries (because z′′′ = 0 at z = 0 and −h; see section 2d), representing the vertical redistribution of momentum. The third expression, zsGB = ()s(), is discussed in Rhines and Holland (1979), Johnson and Bryden (1989), and Ivchenko et al. (1996). The most plausible in terms of the boundary condition is the second expression, but its use in a numerical analysis will result in pressure gradient errors (cf. Haney 1991; Mellor et al. 1994; Shchepetkin and McWilliams 2003) near the top and bottom boundaries where ρs (Fig. 1).

A solution to elucidate the pseudoflux of momentum associated with the layer-thickness form drag is to write the divergence in Cartesian coordinates. Remembering that GBĜ + ∇cp (Table 1 and section 2b), it can be shown that
i1520-0485-38-9-1845-e19
The first term on the last line of (19) is essentially ĜGz and has no barotropic effects in each vertical column because of the pile-up rule, (16). The vertical flux of momentum ∫zh(ĜGz) dz is hereafter called the quasi-Stokes form stress, following the terminology and concept of the quasi-Stokes velocity developed by McDougall (1998) and McDougall and McIntosh (2001). The second term on the last line of (19) originates from the difference between the Eulerian mean density ρz and the mean height density ρ̂. The difference between ρz and ρ̂ has received considerable attention in recent studies (Killworth 2001; McDougall and McIntosh 2001; Greatbatch and McDougall 2003), but it has not been subject to realistic numerical investigations and hence was worth analyzing in the present study.

Figure 5 compares the distributions of the work done by 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. The work of the quasi-Stokes form stress (Fig. 5a) is negative in most regions of the global ocean, suggesting that the vertical redistribution of momentum reduces the mean kinetic energy and leads to an energy cascade to the perturbation fields, which is as expected if the mean currents in the global ocean are close to being in thermal wind balance. On the other hand, the work done by the density residual (Fig. 5b) has both positive and negative values, and its global integral of −0.04 TW is one 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 form drag results mostly from the quasi-Stokes form stress (the vertical redistribution of momentum) and not from the density residual (the horizontal redistribution of momentum).

Figure 6 shows the meridional–vertical views of the zonal component of the thickness-weighted mean velocity, the quasi-Stokes form stress, and the wind stress, obtained by taking a zonal average in density coordinates over the global ocean. The velocity field (Fig. 6a) comprises (i) the Antarctic Circumpolar Current in the Southern Ocean, whose velocity profiles (flowing eastward) reach depths of 1000–2000 m; (ii) the extensions of the Gulf Stream and the Kuroshio at midlatitudes of the Northern Hemisphere, whose velocity profiles (flowing eastward) reach depths of 100–500 m; and (iii) equatorial currents, whose velocity profiles (flowing eastward and westward) are near the sea surface. In contrast to the zonal component of velocity (Fig. 6a), the quasi-Stokes form stress ∫sρmax()zs ds (Fig. 6b) is significant only in the Southern Ocean at density surfaces of s ∈ (1027.7 kg m−3, 1027.85 kg m−3) and depths of 1000–2000 m. The positive sign for the quasi-Stokes form stress indicates the downward transfer of the eastward momentum (as suggested by Johnson and Bryden 1989), with the maximum value of +0.12 N m−2 being of the same order of magnitude as the wind stress applied at the sea surface (Fig. 6c; discussed further in section 4). The quasi-Stokes form stress is maximal at a latitude of 58°S, corresponding to the location of the Drake Passage. Figure 6b also indicates that the quasi-Stokes form stress appears to be absent outside the Southern Ocean, despite significant work values of the quasi-Stokes form stress appearing in the Gulf Stream and the Kuroshio (Fig. 5a).

It is of interest to examine the horizontal distribution of the quasi-Stokes form stress. Figure 7a shows the quasi-Stokes form stress (for the zonal momentum) in the Antarctic Circumpolar Current (around the Drake Passage in the Southern Ocean) on a density surface s = 1027.82 kg m−3, which is where the zonal average of the quasi-Stokes form stress is maximal in Fig. 6b. The form stress in Fig. 7a is clearly positive, suggesting downward transfers of the eastward momentum at depths of 2000–3000 m. Its magnitude is as high as +3 N m−2 in Fig. 7a, which is around 30 times larger than the typical value of the wind stress acting at the sea surface (∼0.1 N m−2; Fig. 6c). Figure 7b shows the quasi-Stokes form stress (for the zonal momentum) in the Kuroshio Extension (in the western North Pacific Ocean) on a density surface of 1026.0 kg m−3. The form stress in Fig. 7b is largely positive, again suggesting that the eastward momentum is transferred downward at depths of 100–300 m. Its magnitude of about +1 N m−2 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 of the quasi-Stokes form stress being significant in the Kuroshio and the Gulf Stream (Fig. 5a).

d. Eddy geostrophic balance

The layer-thickness form drag is often assumed to be in a geostrophic balance (i.e., ρ0f × VBGB, hereafter called the eddy geostrophic balance) in theoretical studies of mesoscale eddy parameterization (McWilliams and Gent 1994; Gent et al. 1995, hereafter GW95; G98; Smith 1999; Aiki et al. 2004) comparing the characteristics of the tracer approach (using the eddy-induced extra advection) and the momentum approach (using the eddy form drag); see appendix B for details of these two approaches. Also, many studies of the Antarctic Circumpolar Current have implicitly used the eddy geostrophic balance to justify their analysis of the velocity-based Eliassen–Palm flux (Ivchenko et al. 1996; Stevens and Ivchenko 1997; Olbers and Ivchenko 2001; Marshall and Radko 2003). Despite the frequent use of this assumption of eddy geostrophic balance in previous studies, its presence has not been confirmed based on outputs from high-resolution OGCMs. Only along 60°S (the Southern Ocean) have Killworth and Nanneh (1994) confirmed the eddy geostrophic balance of standing eddies (i.e., not transient eddies) by analyzing output of the Fine Resolution Antarctic Model. The present study extends the analysis of Killworth and Nanneh (1994) to transient eddies over the global ocean. Verifying the eddy geostrophic balance is also important for explaining why the work done by the layer-thickness form drag (Fig. 4a) nearly balances the work done by the eddy-induced overturning circulation (Fig. 4b) in each vertical column (see sections 3a and 4).

Because the main component of the layer-thickness form drag is the vertical divergence of the quasi-Stokes form stress (Fig. 5; section 3c), we investigated the eddy geostrophic balance by using the following slight modifications. The zonal component of the eddy geostrophic balance becomes −ρ0f υBGxB ≃ (z), and the relationship υB ≃ (z)/(−fρ0) is then integrated upward from the bottom and subsequently integrated in the zonal direction over the global ocean; that is,
i1520-0485-38-9-1845-e20
where both sides of the approximate equality are in the unit of volume transport (m3 s−1). We now examine the validity of (20).

The left-hand side of (20) represents the eddy-induced meridional transport in the global ocean as shown in Fig. 8a, with positive (negative) values representing anticlockwise (clockwise) rotation in the meridional–vertical plane. The zonal integral ∫ dxc in (20) is taken in Cartesian coordinates so Fig. 8a can be compared with the parameterized eddy-induced overturning 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 s−1 in the Southern Ocean at depths of 2000–3000 m with deeper (upper) waters flowing toward the equator (pole), which is in good agreement with the parameterized circulation reported in GW95. In low latitudes (20°S and 20°N) in Fig. 8a, there is a cell in each hemisphere with reversed sign, which tends to lower the upwelling thermocline at rates of about 10 × 106 and 3 × 106 m3 s−1 above and below a depth of 100 m, respectively. These antisymmetric cells tend to cancel (or reduce) the tropical cells (Hazeleger et al. 2001). In the 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 inherently includes slight barotropic components (cf. McDougall and McIntosh 2001; AY06; Lee et al. 2007).

The right-hand side of (20), hereafter called the rescaled quasi-Stokes form stress, is shown in Fig. 8b; its distribution in the Southern Ocean (with a maximum of 16 × 106 m s−1 at a depth of 2000–3000 m) strongly resembles that of the meridional bolus transports in Fig. 8a. Note that the quantity plotted in Fig. 8b is the same as that in Fig. 6b except for differences in scaling and mapping. Although the quasi-Stokes form stress in Fig. 6b is invisible outside the Southern Ocean, Fig. 8b shows that the rescaled quasi-Stokes form stress is antisymmetric across the equator: each hemisphere contains both positive and negative cells, suggesting downwelling (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 is largely valid at depths below 100 m in mid and high latitudes, which is useful for various theoretical analyses, such as idealizing oceanic dynamics (section 4) and parameterizing unresolved eddies in OGCMs (McWilliams and Gent 1994; G98).

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

When the above analyses in sections 3a3d were repeated using the potential density referenced to the pressure at 2000-m depth (which replaces the potential density referenced to the sea surface pressure, as originally used), we found no qualitative difference from the results presented above (not shown). The global rates of the pressure-related energy conversions changed by only ±0.1 TW, and the vertical profiles of the form stress were largely the same as the ones presented above.

4. Planetary geostrophic equations for the global ocean

The results of the analyses in section 3 elucidate the adiabatic aspects of oceanic dynamics in terms of the equilibrium of energy and momentum, as explained below. Here we summarize momentum and energy balances in mid and high latitudes (where the presence of the 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, and they could be fundamental to applying the energetics of layer-thickness form drag in various research areas of ocean dynamics (excluding at least equatorial dynamics) in future studies, such as (i) the dynamics of the Antarctic Circumpolar Current, (ii) the ventilated thermocline theory, (iii) mesoscale eddy parameterization in coarse-resolution OGCMs, and (iv) the dynamics of baroclinically coupled coherent eddies.

For simplicity and for convenience of explanation of the boundary condition at the end, we begin by rewriting (A9) and (A10) in Cartesian coordinates. By using (A11), Eq. (A9) for the mean potential energy becomes
i1520-0485-38-9-1845-e21
which includes the pressure-flux divergence in the mean field, (A15). In situations where the mean potential energy is locally in equilibrium, the temporal derivative in (21) can be dropped, yielding
i1520-0485-38-9-1845-e22
On the other hand, by using (A11) Eq. (A10) for the mean kinetic energy becomes
i1520-0485-38-9-1845-e23
which includes an introduced wind stress term τz = τs/zs (section 3a). In situations where the work of the pressure effect and the wind forcing are locally balanced (as we show in sections 3a and 3b; see also Fig. 4), the left-hand side and the Reynolds stress term in (23) can be dropped; thus,
i1520-0485-38-9-1845-e24
Equations (22) and (24) represent the local balances of the mean potential and mean kinetic energies, respectively, in the planetary geostrophic limit. For example, if we assume a combined geostrophic and Ekman balance ρ0f × ≃ −∇cp + GB + τz, substitution of this into (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 by separating the isopycnal mean velocity as V = Vekm + Vgeo. The wind-induced Ekman velocity Vekm, the geostrophic velocity Vgeo, and the eddy-induced (bolus) velocity VB are characterized by a set of balanced equations, namely,
i1520-0485-38-9-1845-e25
i1520-0485-38-9-1845-e26
i1520-0485-38-9-1845-e27
where the last equation is the eddy geostrophic balance investigated in section 3d. We now classify oceanic dynamics into wind- and eddy-driven regimes. Equations (28)(31) can be applied to both the zonally periodic currents (i.e., Antarctic Circumpolar Current) and the closed gyre-scale circulations in each basin of the global ocean.

a. Wind-driven regime

The dynamics of the wind-driven regime apply to near the sea surface where the momentum balance is described by (25), (26), and VB ≃ 0, such that
i1520-0485-38-9-1845-e28
The surface wind stress τz tends to enhance the vertical shear of the basic currents, which is blocked by the Coriolis force associated with the Ekman transport Vekm. In other words, the Ekman transport tends to reduce the vertical shear of the basic currents, which is consistent with this transport reducing the mean kinetic energy, as explained below.
The local balance of the mean kinetic energy in (24) becomes
i1520-0485-38-9-1845-e29
which can be verified by comparing Figs. 4c and 4d. Over the bulk of the global ocean, the wind-induced mean kinetic energy is transformed into the mean potential energy when the wind-induced Ekman transport steepens the slope of isopycnal surfaces near the sea surface. This explanation is consistent with the comments of Warren et al. (1996, 1998) and with classical views of oceanic dynamics (cf. Kuhlbrodt et al. 2007).

b. Eddy-driven regime

The dynamics of the eddy-driven regime hold below the surface Ekman layer and in regions where eddies are more important than the wind stress. The momentum balance is given by (26), (27), and Vekm ≃ 0, such that
i1520-0485-38-9-1845-e30
where layer-thickness form drag GB tends to reduce the vertical shear of the basic currents, which is blocked by the Coriolis force associated with eddy-induced overturning circulation VB. In other words, the eddy-induced overturning circulation is accompanied by the deflective (i.e., Coriolis) force, the latter of which enhances the vertical shear of the basic currents, as in the case of baroclinic geostrophic adjustment establishing thermal wind balance. Here we find that it is the vertical shear resulting from the eddy-induced overturning circulation rather than the surface wind stress that cancels the vertical redistribution of momentum by the layer-thickness form drag (or the quasi-Stokes form stress; section 3c).

We thereby conclude that the form stress is not the agent that transfers the wind-induced momentum at the sea surface to the bottom layers, in sharp contrast to the traditional theories of McWilliams et al. (1978), Johnson and Bryden (1989), Olbers (1998), and Olbers and Visbeck (2005). Thus, there is no need to assume that the form stress or the associated momentum flux has a constant (i.e., divergence free) vertical profile between the bottom of the Ekman layer and the top of the bottom layer (see section 1). Realistic vertical profiles of the form stress are shown in Figs. 6b and 8b.

The enhancement of the vertical shear of the basic currents by eddy-induced overturning circulation is consistent with this circulation providing an input to the mean kinetic energy, as explained below. The local balance of the mean kinetic energy in (24) becomes
i1520-0485-38-9-1845-e31
which can be verified by comparing Figs. 4a and 4b. Over several regions and depths of high eddy activity in mid and high latitudes of the global ocean (i.e., some localized regions over the Southern Ocean and over the western boundary currents of each basin; see Figs. 3b,c), eddies formed by baroclinic instability relax the slope of isopycnal surfaces, transforming some of the mean 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 connected through 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 the combined potential energy and hydrostatic pressure flux [(gρ̂z + p), (gρ̂z + p)] on the left-hand side of (22) that accounts for both the horizontal and vertical transfer of energy between regions in the wind- and eddy-driven regimes. The boundary condition of the above three-dimensional flux is straightforward because the total transport velocity (, ) satisfies a no-normal-flow boundary condition (section 2c). It would be interesting to investigate this three-dimensional flux in a future study.

5. Summary

To examine some basic characteristics of layer-thickness form drag that were often assumed or speculated about in previous theoretical studies, here we have described the first oceanic study to apply the adiabatic mean four-box energy diagram of B85, Iw01, and AY06 to the analysis of currents and eddies in the global ocean. This analysis was performed by using the thickness-weighted temporal-averaged mean primitive and energy equations in density coordinates.

We have determined the global distribution of energy conversion done by layer-thickness form drag. It was found that the layer-thickness form drag yields an energy cascade to the perturbation field, whose rate is as intense as the work values of the eddy-induced overturning circulation and the wind-induced Ekman transport in the global ocean (sections 3a and 3b). This result confirms the presence of an oceanic energy cycle involving an equilibrium of the mean kinetic energy as suggested by AY06, which is drawn by sequentially connecting 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 at least two characteristics of layer-thickness form drag, as follows. The first characteristic—the layer-thickness form drag allowing the vertical (rather than the horizontal) transfer of momentum—is the most fundamental assumption that has been frequently used in previous studies. Here we found that the work associated with the horizontal redistribution of momentum by the layer-thickness form drag is one order of magnitude smaller than that associated with the vertical redistribution of momentum (section 3c). This result essentially explains how the layer-thickness form drag presents a sink of the mean kinetic energy: dumping the vertical shear of the basic currents in thermal wind balance is an effective way for the layer-thickness form drag to take the mean kinetic energy and provide an energy cascade to the perturbation field. The second characteristic of the layer-thickness form drag—being in an eddy geostrophic balance—is another important assumption in previous studies and has been confirmed in the present study for transient eddies in the global ocean currents: both the zonally periodic currents and the western boundary currents in mid and high latitudes (see section 3d). This result essentially explains why the work done by the layer-thickness form drag nearly balances the work done by the eddy-induced overturning circulation in each vertical column (as detailed in section 4).

Based on the results of the above analyses, we have presented a set of planetary geostrophic equations that elucidate the adiabatic aspects of oceanic dynamics in equilibrium of energy and momentum in the mid and high latitudes (section 4). This set of equations is simple yet includes eddy effects, and it has a lot of potential to advance the understanding of the role of layer-thickness form drag in various research areas of ocean dynamics. For example, we can resolve a previous debate on the dynamics of the Antarctic Circumpolar Current by the finding that rather than the vertical shear resulting from the surface wind stress, in fact the layer-thickness form drag reduces (and thereby cancels) the vertical shear resulting from the eddy-induced overturning circulation. This undermines Johnson and Bryden (1989) and Olbers’s (1998) arguments that the form stress is the agent that transfers the wind-induced momentum at the sea surface to the bottom layers.

These results confirm the self-consistency and utility of the adiabatic mean four-box energy diagram as a formulation to identify the role of layer-thickness form drag in a rotating stratified fluid.

Acknowledgments

Comments from Peter Gent and an anonymous reviewer improved the presentation quality of this paper. The authors are grateful to Toshio Yamagata, Hiro Sakuma, and Taroh Matsuno for providing insightful comments, T. Iwasaki and K. Miyazaki for 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 paper is dedicated to the late Hajime Miyoshi, who directed the development of the Earth Simulator. HA is supported by the Postdoctoral Fellowship for Research Abroad from the Japan Society for the Promotion of Science. KJR is supported by the Japan Agency for Marine-Earth Science and Technology through its sponsorship of the International Pacific Research Center.

REFERENCES

  • Aiki, H., and T. Yamagata, 2000: Successive formation of planetary lenses in an intermediate layer. Geophys. Astrophys. Fluid Dyn., 92 , 129.

    • Search Google Scholar
    • Export Citation
  • Aiki, H., and T. Yamagata, 2004: A numerical study on the successive formation of Meddy-like lenses. J. Geophys. Res., 109 .C06020, doi:10.1029/2003JC001952.

    • Search Google Scholar
    • Export Citation
  • Aiki, H., and T. Yamagata, 2006: Energetics of the layer-thickness form drag based on an integral identity. Ocean Sci., 2 , 161171.

  • Aiki, H., T. Jacobson, and T. Yamagata, 2004: Parameterizing ocean eddy transports from surface to bottom. Geophys. Res. Lett., 31 .L19302, doi:10.1029/2004GL020703.

    • Search Google Scholar
    • Export Citation
  • Andrews, D. G., 1983: A finite-amplitude Eliassen–Palm theorem in isentropic coordinates. J. Atmos. Sci., 40 , 18771883.

  • Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33 , 20312048.

    • Search Google Scholar
    • Export Citation
  • Best, S. E., V. O. Ivchenko, K. J. Richards, R. D. Smith, and R. C. Malone, 1999: Eddies in numerical models of the Antarctic Circumpolar Current and their influence on the mean flow. J. Phys. Oceanogr., 29 , 328350.

    • Search Google Scholar
    • Export Citation
  • Bleck, R., 1985: On the conversion between mean and eddy components of potential and kinetic energy in isentropic and isopycnic coordinates. Dyn. Atmos. Oceans, 9 , 1737.

    • Search Google Scholar
    • Export Citation
  • Böning, C. W., and R. G. Budich, 1992: Eddy dynamics in a primitive equation model: Sensitivity to horizontal resolution and friction. J. Phys. Oceanogr., 22 , 361381.

    • Search Google Scholar
    • Export Citation
  • Cushman-Roisin, B., E. P. Chassignet, and B. Tang, 1990: Westward motion of mesoscale eddies. J. Phys. Oceanogr., 20 , 758768.

  • De Szoeke, R. A., and A. F. Bennett, 1993: Microstructure fluxes across density surfaces. J. Phys. Oceanogr., 23 , 22542264.

  • Ferreira, D., and J. Marshall, 2006: Formulation and implementation of a “residual-mean” ocean circulation model. Ocean Modell., 13 , 86107.

    • Search Google Scholar
    • Export Citation
  • Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20 , 150155.

  • Gent, P. R., J. Willebrand, T. J. McDougall, and J. C. McWilliams, 1995: Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr., 25 , 463474.

    • Search Google Scholar
    • Export Citation
  • Greatbatch, R. J., 1998: Exploring the relationship between eddy-induced transport velocity, vertical momentum transfer, and the isopycnal flux of potential vorticity. J. Phys. Oceanogr., 28 , 422432.

    • Search Google Scholar
    • Export Citation
  • Greatbatch, R. J., and K. G. Lamb, 1990: On parameterizing vertical mixing of momentum in non-eddy-resolving ocean models. J. Phys. Oceanogr., 20 , 16341637.

    • Search Google Scholar
    • Export Citation
  • Greatbatch, R. J., and T. J. McDougall, 2003: The non-Boussinesq temporal residual mean. J. Phys. Oceanogr., 33 , 12311239.

  • Griffies, S. M., 2004: Fundamentals of Ocean Climate Models. Princeton University Press, 518 pp.

  • Haney, R. L., 1991: On the pressure gradient force over steep topography in sigma coordinate ocean models. J. Phys. Oceanogr., 21 , 610619.

    • Search Google Scholar
    • Export Citation
  • Hazeleger, W., P. de Vries, and G. J. van Oldenborgh, 2001: Do tropical cells ventilate the Indo-Pacific equatorial thermocline? Geophys. Res. Lett., 28 , 17631766.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56 , 16881697.

    • Search Google Scholar
    • Export Citation
  • Ivchenko, V. O., K. J. Richards, and D. P. Stevens, 1996: The dynamics of the Antarctic Circumpolar Current. J. Phys. Oceanogr., 26 , 753774.

    • Search Google Scholar
    • Export Citation
  • Iwasaki, T., 2001: Atmospheric energy cycle viewed from wave–mean-flow interaction and Lagrangian mean circulation. J. Atmos. Sci., 58 , 30363052.

    • Search Google Scholar
    • Export Citation
  • Jacobson, T., and H. Aiki, 2006: An exact energy for TRM theory. J. Phys. Oceanogr., 36 , 558564.

  • Johnson, G. C., and H. L. Bryden, 1989: On the size of the Antarctic Circumpolar Current. Deep-Sea Res., 36 , 3953.

  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77 , 437471.

  • Kasahara, A., 1974: Various vertical coordinate system used for numerical weather prediction. Mon. Wea. Rev., 102 , 509522.

  • Killworth, P. D., 2001: Boundary conditions on quasi-Stokes velocities in parameterizations. J. Phys. Oceanogr., 31 , 11321155.

  • Killworth, P. D., and M. M. Nanneh, 1994: Isopycnal momentum budget of the Antarctic Circumpolar Current in the Fine Resolution Antarctic Model. J. Phys. Oceanogr., 24 , 12011223.

    • Search Google Scholar
    • Export Citation
  • Krupitsky, A., and M. A. Cane, 1997: A two-layer wind-driven ocean model in a multiply connected domain with bottom topography. J. Phys. Oceanogr., 27 , 23952404.

    • Search Google Scholar
    • Export Citation
  • Kuhlbrodt, T., A. Griesel, M. Montoya, A. Levermann, M. Hofmann, and S. Rahmstorf, 2007: On the driving processes of the Atlantic meridional overturning circulation. Rev. Geophys., 45 .RG2001, doi:10.1029/2004RG000166.

    • Search Google Scholar
    • Export Citation
  • Lee, M-M., and H. Leach, 1996: Eliassen–Palm flux and eddy potential vorticity flux for a nonquasigeostrophic time-mean flow. J. Phys. Oceanogr., 26 , 13041319.

    • Search Google Scholar
    • Export Citation
  • Lee, M-M., and A. Coward, 2003: Eddy mass transport for the Southern Ocean in an eddy-permitting global ocean model. Ocean Modell., 5 , 249266.

    • Search Google Scholar
    • Export Citation
  • Lee, M-M., A. J. G. Nurser, A. C. Coward, and B. A. de Cuevas, 2007: Eddy advective and diffusive transports of heat and salt in the Southern Ocean. J. Phys. Oceanogr., 37 , 13761393.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7 , 157167.

  • Luyten, J. R., J. Pedlosky, and H. Stommel, 1983: The ventilated thermocline. J. Phys. Oceanogr., 13 , 292309.

  • Marshall, J., and T. Radko, 2003: Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. J. Phys. Oceanogr., 33 , 23412354.

    • Search Google Scholar
    • Export Citation
  • Masina, S., S. G. H. Philander, and A. B. G. Bush, 1999: An analysis of tropical instability waves in a numerical model of the Pacific Ocean. 2. Generation and energetics of the waves. J. Geophys. Res., 104 , 2963729661.

    • Search Google Scholar
    • Export Citation
  • Masumoto, Y., and Coauthors, 2004: A fifty-year eddy-resolving simulation of the world ocean—Preliminary outcomes of OFES (OGCM for the Earth Simulator). J. Earth Simulator, 1 , 3556.

    • Search Google Scholar
    • Export Citation
  • McDougall, T. J., 1998: Three-dimensional residual-mean theory. Ocean Modelling and Parameterization, E. P. Chassignet and J. Verron, Eds., Kluwer Academic, 269–302.

    • Search Google Scholar
    • Export Citation
  • McDougall, T. J., and P. C. McIntosh, 2001: The temporal-residual-mean velocity. Part II: Isopycnal interpretation and the tracer and momentum equations. J. Phys. Oceanogr., 31 , 12221246.

    • Search Google Scholar
    • Export Citation
  • McWilliams, J. C., and P. R. Gent, 1994: The wind-driven ocean circulation with an isopycnal-thickness mixing parameterization. J. Phys. Oceanogr., 24 , 4665.

    • Search Google Scholar
    • Export Citation
  • McWilliams, J. C., W. R. Holland, and J. H. S. Chow, 1978: A description of numerical Antarctic Circumpolar Currents. Dyn. Atmos. Oceans, 2 , 213291.

    • Search Google Scholar
    • Export Citation
  • Mellor, G., T. Ezer, and L-Y. Oey, 1994: The pressure gradient conundrum of sigma coordinate ocean models. J. Atmos. Oceanic Technol., 11 , 11261134.

    • Search Google Scholar
    • Export Citation
  • Merryfield, W. J., and R. B. Scott, 2007: Bathymetric influence on mean currents in two high-resolution near-global ocean models. Ocean Modell., 16 , 7694.

    • Search Google Scholar
    • Export Citation
  • Miyazawa, Y., X. Guo, and T. Yamagata, 2004: Roles of mesoscale eddies in the Kuroshio paths. J. Phys. Oceanogr., 34 , 22032222.

  • Nakamura, M., and T. Kagimoto, 2006a: Potential vorticity and eddy potential enstrophy in the North Atlantic Ocean simulated by a global eddy-resolving model. Dyn. Atmos. Oceans, 41 , 2859.

    • Search Google Scholar
    • Export Citation
  • Nakamura, M., and T. Kagimoto, 2006b: Transient wave activity and its fluxes in the North Atlantic Ocean simulated by a global eddy-resolving model. Dyn. Atmos. Oceans, 41 , 6084.

    • Search Google Scholar
    • Export Citation
  • Nurser, A. J. G., and M-M. Lee, 2004: Isopycnal averaging at constant height. Part I: The formulation and a case study. J. Phys. Oceanogr., 34 , 27212739.

    • Search Google Scholar
    • Export Citation
  • Olbers, D., 1998: Comments on “On the obscurantist physics of ‘form drag’ in theorizing about the Circumpolar Current”. J. Phys. Oceanogr., 28 , 16471654.

    • Search Google Scholar
    • Export Citation
  • Olbers, D., and V. O. Ivchenko, 2001: On the meridional circulation and balance of momentum in the Southern Ocean of POP. Ocean Dyn., 52 , 7993.

    • Search Google Scholar
    • Export Citation
  • Olbers, D., and M. Visbeck, 2005: A model of the zonally averaged stratification and overturning in the Southern Ocean. J. Phys. Oceanogr., 35 , 11901205.

    • Search Google Scholar
    • Export Citation
  • Pacanowski, R. C., and S. M. Griffies, 2000: MOM 3.0 Manual. NOAA/Geophysical Fluid Dynamics Laboratory, 680 pp.

  • Plumb, R. A., 1983: A new look at the energy cycle. J. Atmos. Sci., 40 , 16691688.

  • Rhines, P., 1979: Geostrophic turbulence. Annu. Rev. Fluid Mech., 11 , 404441.

  • Rhines, P., 1982: Basic dynamics of the large-scale geostrophic circulation. 1982 Summer Study Program in Geophysical Fluid Dynamics: Particle Motions in Fluids, Woods Hole Oceanographic Institution, 1–47.

    • Search Google Scholar
    • Export Citation
  • Rhines, P., and W. R. Holland, 1979: A theoretical discussion of eddy-driven mean flows. Dyn. Atmos. Oceans, 3 , 289325.

  • Røed, L. P., 1997: Energy diagnostics in a 1½-layer, nonisopycnic model. J. Phys. Oceanogr., 27 , 14721476.

  • Shchepetkin, A. F., and J. C. McWilliams, 2003: A method for computing horizontal pressure gradient force in an oceanic model with a nonaligned vertical coordinate. J. Geophys. Res., 108 .3090, doi:10.1029/2001JC001047.

    • Search Google Scholar
    • Export Citation
  • Smith, R. D., 1999: The primitive equations in the stochastic theory of adiabatic stratified turbulence. J. Phys. Oceanogr., 29 , 18651880.

    • Search Google Scholar
    • Export Citation
  • Stevens, D. P., and V. O. Ivchenko, 1997: The zonal momentum balance in an eddy-resolving global-circulation model of the Southern Ocean. Quart. J. Roy. Meteor. Soc., 123 , 929951.

    • Search Google Scholar
    • Export Citation
  • Tanaka, D., T. Iwasaki, S. Uno, M. Ujiie, and K. Miyazaki, 2004: Eliassen–Palm flux diagnosis based on isentropic representation. J. Atmos. Sci., 61 , 23702383.

    • Search Google Scholar
    • Export Citation
  • Uno, S., and T. Iwasaki, 2006: A cascade-type global energy conversion diagram based on wave–mean flow interactions. J. Atmos. Sci., 63 , 32773295.

    • Search Google Scholar
    • Export Citation
  • Wardle, R., and J. Marshall, 2000: Representation of eddies in primitive equation models by a PV flux. J. Phys. Oceanogr., 30 , 24812503.

    • Search Google Scholar
    • Export Citation
  • Warren, B. A., J. H. LaCasce, and P. E. Robbins, 1996: On the obscurantist physics of “form drag” in theorizing about the Circumpolar Current. J. Phys. Oceanogr., 26 , 22972301.

    • Search Google Scholar
    • Export Citation
  • Warren, B. A., J. H. LaCasce, and P. E. Robbins, 1998: Reply. J. Phys. Oceanogr., 28 , 16551658.

  • Wunsch, C., 1998: The work done by the wind on the oceanic general circulation. J. Phys. Oceanogr., 28 , 23322340.

  • Wunsch, C., and R. Ferrari, 2004: Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36 , 281314.

    • Search Google Scholar
    • Export Citation
  • Zang, X., and C. Wunsch, 2001: Spectral description of low-frequency oceanic variability. J. Phys. Oceanogr., 31 , 30733095.

APPENDIX A

Derivation of Energy Equations

Evolution of the potential and kinetic energies in a thickness zcs can be derived from the primitive Eqs. (7)(9) as
i1520-0485-38-9-1845-ea1
i1520-0485-38-9-1845-ea2
We consider the volume integral of these equations in a closed domain Ω with solid boundaries. Note that the shape of the domain changes in density coordinates (∂Ω/∂t ≠ 0), resulting in the domain integral ∫Ω and the temporal derivative in density coordinates ∂/∂t not commutating. To avoid this complexity, the left-hand sides of (A1) and (A2) are expressed using
i1520-0485-38-9-1845-ea3
where A is an arbitrary quantity. The volume integral of (A3) becomes
i1520-0485-38-9-1845-ea4
where a no-normal-flow boundary condition of the three-dimensional velocity (V, w) is applied and d2xdxdy = dxcdyc. Equation (A4) shows that the temporal derivative in Cartesian coordinates ∂/∂tc can be moved out of the integral operator ∫Ω because the shape of the domain does not change in Cartesian coordinates (∂Ω/∂tc = 0).
Applying (A4) to (A1) and (A2) yields the volume budgets of the potential and kinetic energies:
i1520-0485-38-9-1845-ea5
i1520-0485-38-9-1845-ea6
The evolution of the potential energy is given by fluid particles moving in the vertical direction, and the evolution of the kinetic energy is given by fluid particles moving in the direction of the horizontal gradient of hydrostatic pressure, as is widely known.
The interaction between the potential and kinetic energies is determined by the pressure-flux divergence
i1520-0485-38-9-1845-ea7
which includes the incompressibility condition, (2). Because there is no flow crossing the boundary of domain Ω, the volume integral of (A7) becomes
i1520-0485-38-9-1845-ea8
Equations (A5), (A6), and (A8) show that the sum of the domain integrals of the potential and kinetic energies is constant in the absence of boundary forcing and friction.

Mean field

Equations for the mean potential and mean kinetic energies can be derived from the primitive Eqs. (12) and (13) in the mean field as
i1520-0485-38-9-1845-ea9
i1520-0485-38-9-1845-ea10
where ρ̂ = s and zt + · 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
i1520-0485-38-9-1845-ea11
where  is a low-pass filtered quantity. Remembering that total transport velocity (, ) has no component crossing the boundaries (section 2c), the volume integral of (A11) becomes
i1520-0485-38-9-1845-ea12
which indicates that the temporal derivative in Cartesian coordinates ∂/∂tc can be moved out of the integral operator ∫Ω and thus allows the volume budgets of the mean potential and mean kinetic energies in (A9) and (A10) to be determined as
i1520-0485-38-9-1845-ea13
i1520-0485-38-9-1845-ea14
The second integral on the right-hand side of (A14) allows the transfer of energy between the mean and perturbation fields, with the connection between them provided by both the layer-thickness form drag zsGB and the Reynolds stress divergence M(V).
The interaction between the potential and kinetic energies in the mean field is determined by the pressure-flux divergence in the mean field
i1520-0485-38-9-1845-ea15
which includes the incompressibility condition in the mean field, (15). Because there is no mean flow (, ) crossing the boundaries of the domain (section 2c), the volume integral of (A15) becomes
i1520-0485-38-9-1845-ea16
As illustrated in Fig. 2, this energy conversion is closed in the mean field, with no energy connection between the mean and eddy potential energies. In Fig. 2, the thickness-weighted mean velocity has been separated into the isopycnal mean velocity V and the bolus velocity VB (Table 1). The vertical component of the bolus velocity is given by wB = −w − ∇c · ∫zh dz from the three-dimensional nondivergence of the total transport velocity, (15). Both the isopycnal mean velocity and the bolus velocity are three-dimensionally divergent (McDougall 1998).

Perturbation field

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

An expression for the eddy potential energy is obtained by subtracting (A9) from a low-pass filtered version of (A1), and the volume budget of the eddy potential energy is obtained by subtracting (A13) from a low-pass filtered version of (A5); thus,
i1520-0485-38-9-1845-ea17
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,
i1520-0485-38-9-1845-ea18
where GĜ is the deviation from the thickness-weighted mean ( ≡ 0). Subtracting (A14) from the above equation yields
i1520-0485-38-9-1845-ea19
which is the volume budget of the eddy kinetic energy.
An expression for the pressure-flux divergence in the perturbation field is obtained by subtracting (A16) from a low-pass filtered version of (A7), and the volume budget of the pressure-flux divergence in the perturbation field is obtained by subtracting (A14) from a low-pass filtered version of (A8); that is,
i1520-0485-38-9-1845-ea20
which relates the source and sink terms of the mean kinetic, eddy potential, and eddy kinetic energies in (A14), (A17), and (A19).

We now have a complete set of equations for the mean and eddy energies, leading to the four-box energy diagram shown in Fig. 2, which is consistent with the results of AY06. The budget of each energy box is given respectively by (A13), (A14), (A17), and (A19), and the 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 advance in the field of ocean modeling and has been cited by hundreds of papers, it appears that not many people interpret the context of this paper in the same manner as it was originally written. We note that GW95 presented a debate on the two types of approaches suggested at the time for mesoscale eddy parameterization to be used in coarse-resolution OGCMs (cf. McWilliams and Gent 1994).

The first type (the so-called tracer approach) follows Gent and McWilliams (1990), who parameterized the eddy-induced advection of tracers in such a way as to decrease the mean potential energy based on the classical L55 energy diagram. The second type (the so-called momentum approach) follows Greatbatch and Lamb (1990), who parameterized the layer-thickness form drag in the modified momentum equations of de Szoeke and Bennett (1993) in such a way as to redistribute geostrophic momentum in the vertical direction (cf. G98). GW95 commented that nothing was wrong with the modified momentum equations of de Szoeke and Bennett (1993) except that they were not familiar with the characteristics of layer-thickness form drag (however, we note the energetics of B85). This appears to be the reason why GW95 proceeded with the tracer approach (rather than momentum), which has been followed by most OGCMs currently maintained at world climate centers. The momentum approach has not been used in modern OGCMs except for a recent study by Ferreira and Marshall (2006).

Comparing the two approaches further will serve to advance understanding of eddy effects in ocean dynamics and modeling. GW95 discussed the likelihood that if the eddy geostrophic balance (section 3d) holds, then two types of OGCM simulations adopting the tracer and momentum approaches could present similar results (cf. AY06). This hypothesis of the eddy geostrophic balance was examined for the first time by the present study and confirmed based on the output of a high-resolution global ocean model. In addition, our result of comparing the vertical and horizontal redistributions 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 the vertical (rather than horizontal) diffusion of momentum (cf. Greatbatch and Lamb 1990).

The numerical results presented in the present study can be regarded as useful metrics for tuning coarse-resolution OGCM simulations in future studies. Note that OGCMs are supposed to resolve variability on seasonal and longer time scales. This is why we have presented results as a composite from monthly-mean analyses; the mean (perturbation) field in the present study includes (excludes) seasonal and intra-annual variability.

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

Although the L55 energy diagram was not used in the present study, Figs. 4b and 8a will be useful as metrics for tuning OGCMs adopting the tracer approach. The fact that the work −VB · ∇cp is mostly positive in each vertical column (Fig. 4b) supports GW95, who used this relation as a principle to parameterize the eddy-induced velocity. Nevertheless, the Gent and McWilliams (1990) scheme is not the only solution that can make the sign of −VB · ∇cp positive in each vertical column, as recently shown by Aiki et al. (2004). As explained in section 3d, the maximum overturning rate in Fig. 8a is 16 × 106 m3 s−1 in the Southern Ocean, which is nearly the same as that in Figs. 6 and 7 of GW95. This result corroborates, at least in the Southern Ocean, the use of the standard horizontal diffusivity κ = 1000 m2 s−1 in the parameterization of Gent and McWilliams (1990).

Finally, we touch on the small effect of the work done by the density residual (Fig. 5b), which originates in the second term on the last line of (19). McDougall and McIntosh (2001) and Greatbatch and McDougall (2003) argued that this term (which is missing also in OGCMs adopting the tracer approach) would be less important than the other terms in the momentum equations, such as the Reynolds stress. The result of our analysis justifies their argument (section 3c).

Fig. 1.
Fig. 1.

Window of a low-pass temporal filter in density coordinates for a water column located at (x, y). Solid lines represent contours of potential density ρ = (x, y, s, t) with a constant interval. As defined in section 2a, ρs for s ∈ (ρtop, ρbtm), where ρtop and ρbtm are the density values at the top and bottom edges of the water column, respectively. Shaded regions indicate s ∉ (ρtop, ρbtm), where zcs = 0 and ρρtop(x, y, t) or ρbtm(x, y, t). As explained in section 2b, the minimum and maximum values of ρtop and ρbtm inside a filter window are expressed as ρmin and ρmax (dotted lines), respectively, such that ρ̂ = s and zs ≠ 0 for s ∈ (ρmin, ρmax). As explained in section 2d, density surfaces not touching the top and bottom boundaries are in the range s ∈ (ρtmax, ρbmin) where ρ = s holds, with s = ρtmax and s = ρbmin (dashed lines) being the maximum and minimum values of ρtop and ρbtm, respectively. Density surfaces touching the top and bottom boundaries are in the range of s ∉ (ρtmax, ρbmin) where ρs.

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 2.
Fig. 2.

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

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 3.
Fig. 3.

Vertical integral ∫ρminρmax ds of (a) mean kinetic energy (ρ0/2)zs||2, (b) eddy kinetic energy (ρ0/2), and (c) eddy potential energy g( − ρ̂z zs) (J m−2). The net value over the global ocean is indicated for each quantity (1 EJ = 1018 J). All the images in the present study indicate year-round composites obtained from monthly mean analyses (section 3).

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 4.
Fig. 4.

Vertical integral ∫ρminρmax ds of the work (energy conversion rate) caused by (a) layer-thickness form drag · zsGB, (b) bolus velocity −VB · zscp, (c) isopycnal mean velocity −V · zscp, and (d) wind stress · τs (W m−2). The sign is relative to the budget of the mean kinetic energy. The net value over the global ocean is calculated for each quantity (1 TW = 1012 W). The pressure-related energy conversions [(a)–(c)] are plotted with horizontal Gaussian smoothing with a radius of 1.5°.

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 5.
Fig. 5.

As in Fig. 4a, but separated into contributions from (a) the quasi-Stokes form stress · zs(ĜGz) and (b) the density residual − · zscz g(ρz − ρ̂)dz (W m−2).

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 6.
Fig. 6.

Meridional–vertical view of (a) thickness-weighted mean velocity û (cm s−1), (b) quasi-Stokes form stress ∫sρmax(z)zs ds (N m−2), and (c) wind stress τx (N m−2) (all are the zonal component), obtained by taking a zonal average ∫ dx/L in density coordinates over the 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 density coordinates ∫z dx/L (m). In (b), positive (negative) values of the quasi-Stokes form stress indicate downward (upward) transfer of the eastward momentum.

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 7.
Fig. 7.

Quasi-Stokes form stress for zonal momentum ∫sρmax (z)zs ds (N m−2) (a) around the Drake Passage at a density surface of s = 1027.82 kg m−3 and (b) over the western North Pacific Ocean at a density surface of s = 1026.0 kg m−3, where positive (negative) values indicate the downward (upward) transfer of the eastward momentum. Contours are the mean height z (m) of the corresponding density surface.

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

Fig. 8.
Fig. 8.

Meridional–vertical view of (a) the eddy-induced meridional transport ∫∫zh υB dz dxc (Sv: 1 Sv ≡ 106 m3 s−1), and (b) the rescaled quasi-Stokes form stress for the zonal momentum ∫∫zh(z)dzdxc/( fρ0) (Sv), where ∫ dxc is the zonal integral in Cartesian coordinates over the global ocean.

Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3820.1

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 AY06 except for Az, A, ∇c, and in the present study, corresponding to A,Aρ, H, and ŵ in AY06, respectively.

Table 1.
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 to be the monthly mean, the seasonal (3 month) mean, and the annual mean, in analyzing the 3-day snapshots obtained from the OFES climatological run (throughout the 46th year). Values for the monthly mean (first column) are adapted from Figs. 3 –5. Values for the seasonal 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-Stokes form 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).

Table 2.

1

As mentioned in section 2d, the biggest and most obvious advantage of our density coordinates is having realistic distributions of density and tracers at the top and bottom boundaries of the ocean, in contrast to traditional coordinates, which assume unrealistic (i.e., homogeneous) distributions of density and tracers at the boundaries. However, we found it rather difficult to categorize the other characteristics of our (or traditional) density coordinates into pros or cons. For example, with traditional density coordinates where ρ = s everywhere, we find the expression of the eddy potential energy becomes ĝpz zs = gs, which can be shown to be a negative definite quantity g/2 after using the integral by parts in the vertical direction. But generally we do not know whether this characteristic of the eddy potential energy (its being single signed) is a pro or con. This is because at least some careful theories (e.g., L55; Iw01) do not use such a definition of the eddy potential energy (i.e., −/2). L55 and Iw01 used the eddy potential energy defined as a difference between the total and mean potential energies, as was done in the present study and in AY06.

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