## 1. Introduction

The Southern Ocean (SO) plays a crucial role in transforming and transporting ocean water masses. The Atlantic, Pacific, and Indian Oceans are connected through the SO, and no description of the global ocean circulation is complete without a full understanding of this region. One wishes to understand the Antarctic Circumpolar Current (ACC) system, the polar gyres, and the meridional overturning circulation (MOC), which are linked as they represent branches of the three-dimensional pathways of ocean water masses. Here we use a synthesis of observations, a numerical model, and theory to investigate the force balance of the SO limb of the MOC.

Standard scaling analysis for the large-scale ocean circulation assumes a small Rossby number, leading to the thermocline equations based on the linearized planetary geostrophic equations (Robinson and Stommel 1959; Welander 1959; Phillips 1963; Pedlosky 1987). But in the Drake Passage latitude band of the SO, at depths where there are no lateral topographic boundaries to support a zonal pressure or buoyancy gradient, nonlinear eddy terms become important: one can show that below the surface Ekman and diabatic layers and above any bottom boundary layers, the planetary geostrophic equations, ignoring eddy fluxes of buoyancy and momentum, would imply constant vertical velocities, *f* is the Coriolis parameter, *w* is the vertical velocity, *τ ^{x}* is the zonal wind stress, and

*b*is the buoyancy. We denote zonal and temporal means with overbars; fluctuations about them will be denoted by primes. Therefore, the unblocked latitudes of the SO would be unstratified, and, for

^{1}This is contrary to observations, which show significant stratification in the unblocked latitudes and demand a poleward heat transport that cannot be achieved with surface waters flowing equatorward (Mazloff et al. 2010).

*υ*,

_{e}*w*) = (−∂

_{e}_{z}, ∂

_{y})

*ψ*. Thus eddy fluxes contribute to the mean transport, and models that do not resolve these eddies must parameterize them (Treguier et al. 1997).

_{e}Marshall and Radko (2003) used Eqs. (1)–(4) to construct a model of the overturning circulation of the SO. Their model offers useful insights into the dynamics of unblocked zonal flows. Questions remain, however, whether this and similar models are quantitatively accurate because the solution is determined by the boundary conditions at the surface, where the quasigeostrophic approximations used to derive the planetary geostrophic equations for the residual circulation do not apply. Held and Schneider (1999), Schneider et al. (2003), and Schneider (2005) showed that nonquasigeostrophic effects at the boundaries (specifically, relatively large isopycnal slopes) modify the overall residual circulation of the atmosphere. Similar issues may arise in the ocean.

Plumb and Ferrari (2005) extended the planetary geostrophic system in (1)–(4) to account for nonquasigeostrophic effects. However, their momentum and buoyancy equations involve terms that are difficult to diagnose from observations or numerical models. For example, their equations involve terms like

A second limitation of the reentrant channel model described by (1)–(4) is that it ignores the remote forcing of the ACC by the subtropical and polar gyres abutting it. Gill (1968) showed that, even though the ACC may be governed by channel dynamics, a full theory of its transport requires specification of the northern and southern inflow and outflow conditions. A complete understanding of the ACC circulation cannot be attained solely with either Sverdrupian models of gyre circulations or channel models, but requires a merging of the two approaches, as attempted in some recent theories (Nadeau and Straub 2009; LaCasce and Isachsen 2010; Nikurashin and Vallis 2011; Nadeau and Straub 2012; Nikurashin and Vallis 2012).

In this study, we address these issues by diagnosing the momentum and buoyancy budgets of the SO from a model that is fit to observations and in which eddies and their associated transports are explicitly represented. Our primary analysis tool is a Southern Ocean state estimate (SOSE) for the years 2005 and 2006 (Mazloff et al. 2010).^{2} The ocean state is estimated with a general circulation model run at ⅙° resolution and optimized to match observations in a weighted least squares sense. Convergence to the state estimate is achieved by systematically adjusting the atmospheric driving and initial conditions using an adjoint model. A cost function compares the model state to in situ observations (Argo float profiles, CTD synoptic sections, seal-mounted SEaOS instrument profiles, and XBTs), altimetric observations [*Envisat*, *Geosat*, Jason, Ocean Topography Experiment (TOPEX)/Poseidon)], and other datasets (e.g., sea surface temperatures inferred from infrared and microwave radiometers). In contrast to other data assimilation approaches, no spurious nudging terms are introduced in the dynamical equations, and therefore the state estimate represents a physically sound solution that satisfies the discretized equations of motion of the model. Mazloff (2008) describes in detail the assimilation procedure and the observations used. SOSE is a good resource for quantifying the dynamical balances in question because it has eddy kinetic energy on par with that observed, it is largely consistent with individual observations, and it is consistent with integrated fluxes inferred from previous static inverse models (Mazloff et al. 2010).

The primary focus of this paper is on the thermally indirect SO MOC cell that upwells deep waters in the Drake Passage latitude band and returns intermediate waters to ventilate the thermoclines of the subtropical gyres. This is often referred to as the upper SO MOC cell, but because our study region extends into the subtropics where there is an even shallower thermocline, with an additional MOC cell, we here refer to this as the middle MOC cell. The overall goal is to assess the relative importance of eddies and mechanical wind forcing in driving the circulations and to understand how the MOC in the ACC latitude band merges with the gyroscopic flows north and south of it.

The paper is organized as follows. In section 2, the mass and zonal momentum budgets are formulated in isopycnal coordinates to illuminate the different factors contributing to the dynamical balance of the MOC. SOSE output is used to diagnose the MOC in section 3 and the terms in the zonal momentum budget in section 4. A synthesis of the dominant dynamical balance of the MOC is presented in section 5, and implications for models of the MOC are discussed in section 6.

## 2. Mass and zonal momentum budget in isopycnal coordinates

In this section, the mass and zonal momentum budgets are formulated in neutral density coordinates, assuming that the ocean is stably stratified (see Vallis 2006, section 3.9 for a derivation). Neutral density *γ*, which largely removes the effects of compressibility from in situ density (Jackett and McDougall 1997), is used as the vertical coordinate. Surfaces of constant *γ* will be referred to as isopycnals.

*h*=

*ρ*

_{0}

*∂*is the “thickness” of isopycnal layers (

_{γ}z*z*is their height),

*ρ*

_{0}is a constant reference density, (

*u*,

*υ*) is the horizontal velocity, and

*Q*=

*Dγ*/

*Dt*the material density tendency (diapycnal “velocity”). Horizontal and time derivatives are understood as taken at constant neutral density. Taking an average over long times and along a latitude circle giveswhere the averaging convention used is to set the argument to zero when isopycnals vanish. We denote thickness-weighted averages by

The meridional residual velocity *z* coordinates of the thickness-weighted meridional velocity

*γ*can be written aswhere

*ζ*≡

*υ*−

_{x}*u*is the relative vorticity of the fluid and

_{y}*F*accounts for all mechanical forcing and viscous effects. The pressure gradient in isopycnal coordinates is

^{x}= ∂

_{x}

*p*+

*gρ*∂

_{x}

*z*, with pressure

*p*, gravitational acceleration

*g*, and in situ density

*ρ*.

^{3}In the Boussinesq approximation, this reduces to the

*x*derivative of the Montgomery potential,

_{x}≈ ∂

_{x}

_{x}(

*p*+

*gρ*

_{0}

*z*). Here we keep the general form that accounts for compressibility effects.

*q*≡ (

*f*+

*ζ*)/

*h*is the Rossby–Ertel potential vorticity (PV).

### a. Mean potential vorticity at outcrops

The thickness-weighted mean potential vorticity *h* = 0 (at outcrops). Koh and Plumb (2004) suggest to interrupt the zonal averaging in those regions without adjusting the normalization of the averages by still dividing by the full length of the latitude circle and duration of the averaging period, which corresponds to setting *h* = 0. This approach is followed here, and the reader is referred to Schneider (2005) and Jansen and Ferrari (2013, manuscript submitted to *J. Atmos. Sci.*) for a discussion of alternative approaches.

### b. Mean pressure gradient at outcrops

*z*and

*γ*coordinates. In

*z*coordinates, the zonal average reduces to the sum of pressure differences across topographic features. In

*γ*coordinates, in the Boussinesq approximation, the zonal average reduces to the sum of the Montgomery potential difference across both topographic features and density outcrops. Consider the Montgomery potential integrated over a layer bounded by the free surface on top and by a submerged isopycnal

*γ*at the bottom. In the Boussinesq approximation, we havewhere

_{n}*γ*

_{surf}is the density at the free surface of the ocean and we substituted

*z*=

*η*and

*p*= 0 at this free surface. Taking the zonal average of (11) and assuming there are no lateral boundaries, we obtainwhere

*γ*

_{surf}is proportional to the correlations between the Montgomery potential and isopycnal slope at the surface. It is analogous to the form drag in

*z*coordinates that results from correlations between pressure and topographic slope (Andrews 1983).

*υ*=

_{g}*f*

^{−1}

*g*∂

_{x}

*η*:The surface form drag is proportional to the geostrophic meridional density flux, as derived in the atmospheric context by Andrews (1983) and Schneider (2005). Thus, even if there are no lateral boundaries, a mean Montgomery potential gradient can exist in the surface layer owing to correlations between velocity and density fluctuations. An equivalent bottom form drag results from interactions of isopycnals with the ocean bottom.

## 3. Meridional overturning circulation of the Southern Ocean

*γ*:with

*γ*the neutral density at the ocean bottom. Temporally averaging the streamfunction over the two years of the state estimate, Fig. 1, shows the MOC to be characterized by three primary cells, each with a transport of about 10 Sv (Sv ≡ 10

_{b}^{6}m

^{3}s

^{−1}). The upper overturning cell, which is part of the horizontal subtropical gyres, rotates counterclockwise and is confined to the lightest density classes, all of which outcrop somewhere along the latitude circle. This cell consists of a poleward surface flow compensated by an equatorward interior return flow of Antarctic Intermediate Water (AAIW) and Antarctic Mode Water (AMW). The middle cell rotates clockwise and consists of Upper Circumpolar Deep Water (UCDW) flowing into the SO at depth and a return flow of AAIW. The abyssal cell rotates counterclockwise and consists of inflowing Lower Circumpolar Deep Water (LCDW) and an abyssal outflow of Antarctic Bottom Water (AABW).

The MOC undergoes a substantial seasonal cycle in the SO, as also found by Wunsch and Heimbach (2009). In Fig. 2, the MOC is shown separately for each season, that is, time averaging for two years over three-month windows: December–February (austral summer), March–May (fall), June–August (winter), and September–November (spring). The intermediate cell exhibits large seasonal shifts, with its transport becoming quite weak across ~40°S in summer and fall and increasing to over 20 Sv in the spring and winter. The greatest seasonal change occurs in the upper cell, which is strong (~10 Sv) in spring and winter and almost nonexistent poleward of 40°S in summer and fall.

The strong seasonal variability in the MOC raises the question of whether *ψ** can be interpreted as a streamfunction tracking the pathways of water masses: *ψ** is a streamfunction only if the mass budget is approximately in a statistically steady state over the time interval considered. As given by (6), the temporal and zonal mean of the continuity equation is *γ* = 27.8 kg m^{−3}, and most significantly in fall. What appears to be a closed abyssal cell in the circulation is actually a change in the volume of the isopycnal layer. This is likely a model drift that would be eliminated by averaging over a longer time period. The upper two MOC cells are in statistically steady state, however, as the thickness tendency *ψ** can be interpreted as a streamfunction in the upper cells.

The primary focus in this paper is on the middle cell, as this cell dominates the buoyancy budget in the ACC latitudes (Fig. 3). There is, however, a small compensating poleward transport of buoyancy from the upper cell especially in spring and winter. The middle MOC cell vanishes near the subtropical front in fall and summer because the subduction of waters into the ocean interior is stalled as surface waters become very buoyant. The upper cell is part of the subtropical gyres and dominates the total poleward buoyancy transport in the subtropics from spring to fall, with a middle cell contribution in winter. In the polar regions, the abyssal MOC cell controls the relatively small poleward buoyancy transport, as it is the only overturning cell at those latitudes. Its contribution to buoyancy transport becomes insignificant north of the polar front because the difference in buoyancy between LCDW and AABW is small. In summary, the overall buoyancy transport in the SO is dominated by the upper and middle MOC cells, with the middle cell accounting for most the buoyancy transport across the ACC. We now diagnose the forces balancing this transport.

## 4. Force balance of the Southern Ocean meridional overturning circulation

Thus the volume transport in each layer,

### a. Mean wind stress

The volume transport driven by mean wind stresses is given by

The cumulative magnitude of this mean wind transport is stronger than the ~30 Sv of mean Ekman flow previously reported in *z* coordinate calculations (e.g., Mazloff et al. 2010). This difference is due to the *γ* coordinate calculation; there are strong *γ* fluctuations at the surface causing the wind transport to be partitioned between the zonally averaged frictional forcing and the eddy PV flux as shown in the next section.

### b. Ageostrophic eddy transport

*υ*=

_{a}*υ*−

*υ*. There are ageostrophic velocities in the densest layers; however, the ageostrophic eddy PV flux

_{g}*υ*such that

_{a}*q*≈

*f*/

*h*in the term

### c. Mean geostrophic transport

The mean geostrophic volume transport associated with the mean pressure gradient is given in (14) by

In these outcropping layers the relationships derived in section 2b imply that the mean geostrophic volume transport is related to both standing and transient geostrophic eddy density fluxes at the ocean surface and bottom. For the surface layer in the unblocked latitudes we find that standing eddy density fluxes are typically a factor 5 stronger than transient eddy fluxes. Thus the surface layer mean geostrophic volume transport in this region is dominated by a time-mean standing eddy component.

### d. Geostrophic eddy transport

*γ*, and then taking the time and zonal averages to compute the fluxes in

*γ*coordinates. Five-day averages are used because that is the frequency at which the SOSE output was saved; we do not expect the results to change if a higher frequency output were used. It is important to map properties onto

*γ*surfaces first, before averaging, so as not to underestimate eddy contributions. (This ordering of the calculation has not been consistently carried out in previous studies.)

*γ*surfaces, the transient and standing eddy fluxes are defined aswhere

In contrast to the surface fluxes in the unblocked latitudes, the transient geostrophic eddy PV flux is about an order of magnitude stronger than the standing eddy component. The volume transport associated with this transient flux,

The volume transport associated with the standing geostrophic eddy PV flux,

## 5. Synthesis of the MOC force balance

### a. Subtropical and polar gyres

The subtropical gyre regime extends from the northern edge of the domain to ~40°S where the upper overturning cell ends (Fig. 2) approximately in correspondence with the subtropical front. There are three overturning cells in the subtropical gyre region, and in all of them the mean geostrophic volume flux dominates the circulation. In addition, there is a strong mechanically forced convergence of the most buoyant waters. Transient eddy PV fluxes are significant. They partially compensate the mean geostrophic transport and, thus, they reduce the strength of the overturning circulation that would exist without them. The dynamical balance of the subtropical upper overturning cell that emerges from this analysis is consistent with Sverdrup theory. The mean pressure gradients drive the interior flow equatorward, in accordance with vorticity conservation, while geostrophic western boundary currents and surface wind-driven flows close the mass and momentum budgets.

Mechanical forcing is weak poleward of Drake Passage in the Ross and Weddell Polar Gyres. The overturning in these latitudes consists primarily of the SO abyssal cell, though at these latitudes the “abyssal cell” spans all depths. With the exception of the weaker wind-driven flow, the balance in this region is much like in the subtropical gyre, consisting of a mean geostrophic volume transport with significant compensation by transient eddies.

### b. ACC and Drake Passage latitudes

Strong winds drive surface waters equatorward in the subpolar ACC region. With the exception of the dominance of this forcing on the most buoyant waters, the circulation in this region is balanced in the same way as in the polar and subtropical regions: a mean geostrophic mass transport is partially compensated by transient eddy PV fluxes. Comparing Fig. 6 with Figs. 8a,d shows that the overturning structure below the mechanically forced layer is primarily a mean geostrophic transport.

There is a gap in the continental boundaries between ~65°S and ~55°S, yet the dynamical balance in this Drake Passage region is similar to that of the northern ACC latitudes, which reach to about ~40°S. In particular, the zonal pressure gradient averaged along isopycnals does not vanish (Figs. 8a,d) because isopycnal surfaces are blocked along these latitude circles, and thus the zonally averaged zonal pressure gradient is nonzero. The isopycnal surfaces either run into bathymetry, predominantly at the Macquarie Ridge and the Kerguelen Plateau, or they outcrop at the surface, predominantly in the Weddell Sea (Fig. 9). It has become common in theorizing about the ACC (e.g., Olbers et al. 2004) to call pressure gradients against topography bottom form drag, and analogously we call pressure gradients resulting from surface outcrops surface form drag.

When zonally integrated at constant depth, the zonal pressure gradient vanishes by periodicity in Drake Passage. However, when integrating along neutral density surfaces, significant mean geostrophic flows can occur supported by surface form drag. A neutral density zonal section at 58°S shows there is a large-scale equatorward geostrophic transport of relatively buoyant waters in the south Indian and Pacific regions above the sill depth (Fig. 10). In the South Atlantic region, there is a poleward flow of relatively dense water at these same depths. These flows compensate such that the zonally integrated geostrophic volume transport at a fixed depth is negligible. However, a mean geostrophic volume transport exists when averaged along a density surface that outcrops, and thus an overturning circulation occurs in density coordinates, even in the unblocked latitudes. This is analogous to the overturning in the subtropical gyres resulting from the basin interior transport occurring at a different density than the western boundary current return flow. In the ACC, this phenomenon occurs at all length scales (note the wiggles in the mean density and sea surface height contours in Fig. 11). While this effect may be subtle, especially for small-scale meanders, it sums to a significant effect in the zonal integral.

We have found that standing meanders in the presence of isopycnal outcrops result in the mean geostrophic transport being a major contribution to the MOC in the ACC latitude band. This is different from the force balance diagnosed in idealized channel models (e.g., Abernathey et al. 2011) and often assumed in theorizing about the ACC (e.g., Marshall and Radko 2003). Typically a balance is expected between the Ekman volume fluxes and those associated with the transient eddy PV fluxes, with a minor contribution from surface geostrophic buoyancy fluxes (e.g., Marshall and Radko 2003). Instead, we find that the Ekman volume flux has the same sign as that associated with the transient eddy PV fluxes, and both are opposed by the mean geostrophic volume transport. However, the fact that this geostrophic transport in the ACC latitude band is largely due to standing meanders associated with density outcrops makes this discrepancy less puzzling. It is indeed eddies that balance the Ekman volume flux, but primarily standing rather than transient eddies and through their transport of surface buoyancy rather than PV.

It is possible that some of the differences between our results and previous work are due to the fact that we take averages along latitude circles rather than along streamlines. Starting with Marshall et al. (1993), it has been argued that the decomposition of the mean and eddy contributions to the Southern Ocean overturning is better achieved by averaging along the mean transport streamlines to follow the ACC standing meanders, rather than taking zonal averages. We found that rotating the momentum equations into along- and across-stream directions does not result in a significant reduction of the mean geostrophic transport. Apparently the structure of the ACC departs from equivalent barotropic, at least where there are significant standing meanders, to an extent that it is impossible to find “mean streamlines” that remove the geostrophic transport at all depths. Furthermore, the ACC meanders are very sharp in the SOSE solution and cumbersome curvature terms would need to be included in the momentum equations if one insists on following them accurately.

## 6. Summary and conclusions

We analyzed the Southern Ocean limb of the meridional overturning circulation using an eddy-permitting state estimate run for the years 2005 and 2006. The circulation was diagnosed as a function of neutral density, a natural coordinate system for the ocean where motions flow along these surfaces outside boundary layers. We found that the global overturning *z* coordinates. (The transport value on each isopycnal is mapped to the zonal mean depth of that isopycnal and then integrated to determine the streamfunction.)

The wind-driven transport *ψ*_{Ek} is significant at all latitudes near the surface. In the ACC latitude band, it is responsible for a sizable equatorward transport of buoyant waters. The mean geostrophic transport *ψ _{g}* is supported by mean zonal pressure gradients that arise from isopycnal outcrops into either bottom topography or the surface. In the Drake passage latitude band, the primary blockage in isopycnal coordinates comes from bottom outcrops at the Kerguelan Plateau (~70°E) and Macquarie Ridge (~160°E) and surface outcrops in the eastern Weddell Sea (~0°). The Campbell Plateau at 155°E and the Pacific–Antarctic Ridge to its east are also notable constriction points. The surface outcrops span many more neutral density classes than do the bottom outcrops (Fig. 9).

The transport associated with transient eddy PV fluxes *ψ _{e}* has the same pattern and opposite sign of the mean geostrophic transport at most latitudes and depths (Figs. 8 and 12). The transient eddy PV fluxes are, for the most part, oriented down the mean PV gradient along isopycnals. In

*z*coordinates, one primarily focuses on the PV gradients between the well-stratified pycnocline and the thick isopycnal layers in the interior. The interior PV generally decreases southward, and we do find a southward transient eddy PV flux, which is associated with poleward volume transport (Figs. 8b,e). Isopycnal coordinates also make manifest the contribution of the surface layer where isopycnals outcrop. There, PV, with the averaging convention we adopted, generally decreases northward. Correspondingly, the eddy PV flux is directed northward and is associated with equatorward volume transport. The isopycnal analysis shows that, by transporting volume toward the pycnocline (equatorward along isopycnals), the transient eddy PV fluxes and Ekman transport together tend to oppose the mean geostrophic transport.

The transport balance that we diagnose in the ACC latitude band differs from the prevailing view, which posits that the Ekman transport is opposed by a transient eddy transport associated with adiabatic PV fluxes (e.g., Johnson and Bryden 1989; Olbers et al. 2004). However, the two balances are not as different upon closer inspection. In both cases there is a balance between an Ekman transport and an eddy transport, but in our analysis the eddy transport is dominated by horizontal diabatic geostrophic buoyancy fluxes at the surface rather than by adiabatic PV fluxes. The isopycnal averaging used in this paper, as opposed to the averaging at fixed *z* used in previous analyses of the momentum budget, shows that upon circumpolar integration almost all isopycnals pass through a surface or bottom boundary layer at some point in the seasonal average (Fig. 9). One expects diabatic processes to be influential in these boundary layers, and thus it is not surprising that diabatic fluxes dominate the transport. Indeed, a large geostrophic flow across density outcrops has been previously identified in a suite of idealized studies (e.g., Treguier et al. 1997; Marshall and Radko 2003; Kuo et al. 2005) and has been shown to have observable consequences for watermass transport and subduction (Sallée et al. 2010). Our analysis suggests that this mechanism is responsible for a large fraction of the total volume transport in the ACC region.

Consistent with previous model diagnoses (Stevens and Ivchenko 1997; Lee and Coward 2003; Dufour et al. 2012), we have found that standing eddy transports play a significant role in the Southern Ocean MOC. The geostrophic buoyancy flux across a latitude circle is achieved by transporting negative buoyancy anomalies when the ACC veers north and positive ones when it veers south. Significant meanders can be seen in the lee of major topographic features, but there are also many smaller ones. The dominance of standing eddies raises a challenge for parameterizations of eddy transport in the ACC. Transient eddy fluxes of buoyancy can be parameterized as a downgradient buoyancy flux (Treguier et al. 1997), while standing eddies are supported by seasonal-mean outcrops and there is no guiding principle for their parameterization.

Our analysis shows that isopycnal outcropping allows mean pressure gradients to oppose the Ekman transport in the ACC latitudes. If the equatorward Ekman transport (i.e., the zonal wind stress) increases, it is likely that more deep water will be brought to the surface, resulting in more outcrops, which will then be able to support a stronger mean geostrophic circulation. This balance is, to some degree, observable. In winter when the mean zonal wind stress is greatest (Fig. 7c), zonal outcropping is most significant (Fig. 9b), and the mean geostrophic transport is strongest (Fig. 7d). More outcroppings also result in more significant air–sea gas exchange, with implications for biogeochemical climate predictions.

## Acknowledgments

We acknowledge the National Science Foundation (NSF) for support of this research through Grants OCE-1233832, OCE-1234473, and OPP-0961218. SOSE was produced using the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant MCA06N007.

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^{1}

Throughout this paper, we use the south to north plotting convention, such that a clockwise overturning cell consists of northward flow of relatively buoyant waters and southward flow of less buoyant waters. In the SO, a clockwise overturning is thermally indirect, as it transports buoyancy upgradient toward the equator.

^{2}

As part of the Antarctic Treaty, the International Hydrographic Organization has defined the Southern Ocean to extend from 60°S to Antarctica. The region of study in this work is the oceans south of 25°S, which, for ease, will be referred to collectively as the Southern Ocean.

^{3}

When converting from *z* coordinates to *γ* coordinates, the pressure gradient term becomes ∂_{x}*p*|_{z} = ∂_{x}*p*|_{γ} − ∂_{x}*z*|_{γ} ∂_{x}*p*|_{γ} ∂_{z}*p* = ∂_{x}*p*|_{γ} + *gρ*∂_{x}*p*|_{γ}, where the hydrostatic relation, ∂_{x}*p* = −*gρ*, has been used.