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The Center of Mass of the Ocean as an Index of the General Stratification and Its Relation to the Overturning Circulation

Benjamin Rosenthal Department of Marine Sciences, University of Gothenburg, Gothenburg, Sweden

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Fabien Roquet Department of Marine Sciences, University of Gothenburg, Gothenburg, Sweden

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https://orcid.org/0000-0003-1124-4564
Online Publication:
21 Mar 2025
Print Publication:
01 Mar 2025
DOI:
https://doi.org/10.1175/JPO-D-24-0078.1
Page(s):
277–291
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Abstract

The height anomaly, defined as the height of the ocean’s center of mass relative to the height of a fully mixed state, provides an index of the strength of the global stratification. In this paper, we propose a definition of the height anomaly accounting for compressibility effects, using the seawater Boussinesq approximation. A budget equation for the height anomaly is derived, tracking which processes control the stratification—diffusive and convective mixing and conversion into kinetic energy. To test our framework, we analyze model simulations in a rectangular-shaped basin with applied wind and buoyancy forcings. The height anomaly adjusts on time scales ranging from years to centuries due to a combination of deep convection, energy conversion, and interior mixing, reaching a value of −26 cm at equilibrium. The buoyancy forcing, which generates the ocean’s mean stratification through a balance between high-latitude convection and interior mixing, is the primary controlling factor. Wind pumping plays a secondary role here. Near equilibrium, a substantial correlation is found between the height anomaly and the meridional overturning transport, at decadal to centennial time scales. In general, a lower center of mass is associated with an increased global overturning flow, making it a useful index to study overturning circulations.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Fabien Roquet, fabien.roquet@gu.se

Abstract

The height anomaly, defined as the height of the ocean’s center of mass relative to the height of a fully mixed state, provides an index of the strength of the global stratification. In this paper, we propose a definition of the height anomaly accounting for compressibility effects, using the seawater Boussinesq approximation. A budget equation for the height anomaly is derived, tracking which processes control the stratification—diffusive and convective mixing and conversion into kinetic energy. To test our framework, we analyze model simulations in a rectangular-shaped basin with applied wind and buoyancy forcings. The height anomaly adjusts on time scales ranging from years to centuries due to a combination of deep convection, energy conversion, and interior mixing, reaching a value of −26 cm at equilibrium. The buoyancy forcing, which generates the ocean’s mean stratification through a balance between high-latitude convection and interior mixing, is the primary controlling factor. Wind pumping plays a secondary role here. Near equilibrium, a substantial correlation is found between the height anomaly and the meridional overturning transport, at decadal to centennial time scales. In general, a lower center of mass is associated with an increased global overturning flow, making it a useful index to study overturning circulations.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Fabien Roquet, fabien.roquet@gu.se
Keywords: Ocean; Deep convection; Ekman pumping/transport; Meridional overturning circulation; Energy budget/balance; General circulation models

1. Introduction

The ocean plays a crucial role in regulating global climate and supporting ecosystems by redistributing heat, carbon, and nutrients globally. The potential of the ocean to influence the climate and the atmospheric chemical composition depends largely on its storage capacity. The ocean is strongly stratified, insulating deep layers from the surface influence of the atmosphere. The general stratification of the ocean is generated by the constant renewal of deep dense waters by dense and bottom waters formed in the mid- and high latitudes, a process known as ventilation. The global overturning circulation (GOC) is the complex, large-scale, circulation pattern associated with this ventilation process. The GOC involves global transport of water masses, heat, carbon, and other properties, which regulates vertical exchanges and storage capacity between the upper- and lower-ocean domains (Cessi 2019). An important branch of the GOC is the Atlantic meridional overturning circulation (AMOC), which has a meridional volume transport of about 18 × 106 m s−1 and affects the global heat and carbon budget (Wunsch and Heimbach 2013; Frajka-Williams et al. 2019).

The question of what controls the stratification and the overturning circulation strength is the subject of much debate (Burchard et al. 2024). Early on, Sandström (1908) argued that in the absence of mechanically forced interior mixing, the deep ocean would eventually become homogeneous, filled with the densest water mass formed, and the overturning would be weak and surface trapped. This inference has been formalized by Paparella and Young (2002), using energetic arguments. This has led to the idea that the GOC cannot be powered by the buoyancy forcing and must thus be powered by winds and tides (Munk and Wunsch 1998; Wunsch and Ferrari 2004). Yet, even in the absence of mechanical mixing, molecular diffusion will provide a minimal amount of mixing to power an overturning circulation (Gayen and Griffiths 2022). Also, in the absence of buoyancy forcing, there would be no stratification at all and the circulation would be purely barotropic (Roquet 2013), so the buoyancy forcing must provide a critical control on the overturning circulation. Ekman pumping is also an obvious candidate. It is powering much of the observed surface-intensified circulation (Roquet et al. 2011), and it is thought to close the GOC in the Southern Ocean, through Ekman upwelling (Marshall and Speer 2012; Cessi 2019). However, a deep stratification can be generated even in the absence of Ekman pumping (Klocker et al. 2023), whenever two convective sites form different water masses on either side of a reentrant channel. In reality, wind, buoyancy forcings, and mixing are all needed to explain the observed circulation (Roquet 2013; Saenz et al. 2012), and a method to distinguish and quantify their relative contributions is needed.

The starting point of this work is the mere observation that, if defined carefully, the height of the center of mass is proportional to the potential energy (de Verdière 1989). Because the GOC is fundamentally a geostrophically balanced circulation, which operates at low kinetic energy, it is natural to focus on the potential energy to investigate the stratification and its controls. Wunsch and Ferrari (2004) also noted that mechanical mixing should act to constantly raise the center of mass. In fact, the center of mass is a key characteristic used in the study of any physical system, providing its “mean” position, so it is only natural to expect it to be a useful index for the ocean as well. One application is in the study of the human body, for example, for which the center of mass provides a useful signature of the posture (Farenc et al. 2003) and body movements (Minetti et al. 2011). We explore here the potential of the ocean’s center of mass to provide a simple, yet useful, index of the ocean circulation, focusing on its relation to the stratification and to the overturning circulation. Additionally, we investigate the balance of processes determining the center of mass and its variability, as a way to assess what controls the stratification.

In this manuscript, we will first develop definitions of the center of mass and its height anomaly and describe the associated budget equation (section 2). Numerical simulations and diagnostics used to provide a proof of concept are described in section 3. In section 4, we present results of the analyses, quantifying temporal variations of the center of mass at different time scales and diagnosing terms of the budget equation for the center of mass. Section 5 gives a summary and discussion of the main results. Potential applications of the center of mass are discussed including the fact that it provides a robust measure of how strong the stratification is and that it may be useful to monitor the overturning circulation on decadal time scales or longer. Our choice of a fully mixed reference state and how it differs from the Lorenz state of minimum potential energy (Lorenz 1955) generally used to define the available potential energy is also discussed.

2. Definition and equation of the center of mass

a. The center of mass in a compressible ocean

The center of mass (a.k.a. center of gravity or centroid) is the point where the mass of a given object can be considered to be concentrated. It is normally calculated by finding the mass-weighted average of the position of all the individual particles or elements that make up the object. In the case of the ocean, the true center of mass would sit very close to the center of spherical Earth, which does not tell anything useful about the ocean’s mass distribution. In the following, we will define the center of mass in a different, more meaningful, way that ignores the effects of the spherical nature of Earth. The height of the center of mass will be obtained as the weighted-mean vertical height of fluid parcels, as if the ocean surface was flat. If the ocean was homogeneous with constant density, the center of mass so defined would correspond to the average depth of the ocean, such as any deviation from this mean depth signs the presence of density variations. In fact, the center of volume is the highest position the center of mass can take for a stably stratified ocean—putting aside compressibility effects for the moment. Indeed, density must always increase with depth making the center of mass deeper. Thus, the difference between the center of mass and the center of volume should provide a measure of the strength of the general stratification, a fact that we will exploit later.

Mathematically, we define the height of the center of mass zg as the density-weighted average of the height coordinate:
zg=∭ρzdV∭ρdV=⟨ρz⟩⟨ρ⟩,
where ρ is the water density, z is the vertical height relative to a (fixed) surface level, increasing upward by convention, and the triple integral is performed on the whole volume of the ocean. The angle brackets 〈⋅〉 correspond to the global average of a quantity over the volume V of the ocean. In the following, we will refer to the center of mass to designate its height for conciseness.
Noting that the potential energy per unit volume is defined as ρgz, where g is the gravitational acceleration assumed constant, it follows that the center of mass is proportional to the ocean’s total potential energy:
zg=PgM,
where P=∭ρgzdV=Vg⟨ρz⟩ is the potential energy and M is the mass of the ocean.

As indicated earlier, an ocean with homogeneous density would have its center of mass corresponding to the center of volume zυ = 〈z〉. One difficulty is that water is compressible, meaning that even a homogeneous ocean has its density increasing with depth in reality. This calls for some modification in our definition to better account for nonlinear effects of the equation of state. In practice, it would be best to use the dynamic enthalpy instead of potential energy, as it is the fraction of total potential energy that can be converted into kinetic energy (McDougall 2003; Young 2010; Nycander 2011).

Another difficulty in this definition comes from the fact that the volume of the ocean is not constant, even in the absence of mass exchanges, due to expansion and contraction of water. This makes pressure a more natural vertical coordinate to express the center of mass, as the flow is incompressible in pressure coordinates (Marshall et al. 2004). But pressure and depth are closely related in a weakly compressible fluid such as seawater. In the rest of this paper, we will use the Boussinesq approximation to exploit this fact and obtain more tractable expressions for the center of mass and its budget equation.

b. Boussinesq model with a linear equation of state

The concept of height anomaly will now be defined for a model making the Boussinesq approximation. The existence of an isomorphism between the Boussinesq and compressible models offers a natural bridge to translate back Boussinesq-based results to the compressible case (Marshall et al. 2004; Roquet 2013). Furthermore, the Boussinesq approximation is accurate to model ocean flows, and it is still used in a majority of state-of-the-art OGCMs (Roquet et al. 2015b).

To illustrate the concepts, we will first focus on the Boussinesq model with a linear EOS. The buoyancy b = g(ρ0 − ρ)/ρ0 is given here as a linear function of Absolute Salinity S and Conservative Temperature Θ:
blin(S,Θ)=gα0Θ−gβ0S,
where g is the gravity acceleration and α0 and β0 are the constants.

To define the center of mass (for Boussinesq with a linear EOS), we want to adapt the definition (1). The most natural way is to replace the mean density in the denominator by a constant ρ0 value. However, this change makes the center of mass dependent on the choice of the reference level z0 used to compute the potential energy. Is there an objective criterion to optimally define the level of reference? Consider the case of a homogeneous ocean with constant buoyancy b = b0. In this case, it is clear that the center of mass should be at the same height as the center of volume zυ. This can be achieved in practice only if the vertical coordinate is defined relative to the center of volume itself.

In other words, we choose to define the center of mass as
zgb−lin=zυ+⟨ρ(z−zυ)⟩ρ0=zυ+⟨ζb−lin⟩.
Note that a fully mixed state has its center of mass exactly at the center of volume because the EOS is linear. This provides another interpretation for the quantity zgb−lin, being the distance of the center of mass to its value if the ocean was entirely mixed. This will later offer a method to generalize this definition to the case of a nonlinear EOS. Here, the averaging is done over the global volume of the ocean. A local height anomaly is defined as ζb−lin = −blin(z − zυ)/g, averaged globally to obtain the center of mass. It is straightforward to similarly define a vertical, zonal, or meridional height anomaly by averaging ζb−lin along the appropriate coordinate.

The advantage of using the Boussinesq approximation becomes most apparent when considering the case of a homogeneous ocean being warmed. In the compressible case, the volume would increase due to water expansion resulting in a raise of the center of volume. In the Boussinesq case, the center of volume remains constant, and the raise of the center of mass is merely related to buoyancy changes.

A budget equation for the center of mass can be derived. Starting from the conservation of buoyancy,
DblinDt=b∘=gα0Θ∘−gβ0S∘,
where D/Dt is the total derivative operator and b∘ represent the local sources and sinks of buoyancy due to diffusive processes and penetrative radiation, we obtain a budget equation for the center of mass,
dzgb−lindt=⟨−b∘(z−zυ)g⟩+⟨−blinwg⟩,
where w is the vertical velocity. The first right-hand side term depends on diffusive and radiative terms and the second term is the conversion from kinetic to potential energy. This equation has been obtained by neglecting sea surface height variations, as they contribute little to the potential energy budget.

Each process acting in the system can either lower or raise the center of mass depending on the direction of the net vertical density flux it generates. A downward density flux increases the stratification, resulting in a net decrease of potential energy and a lowering of the center of mass. Conversely, an upward density flux raises the center of mass. A horizontal density flux has no effect on the center of mass. In the linear case, the rate of production of potential energy by mixing is ρ0κzN2 (Munk and Wunsch 1998), where κz is the vertical diffusivity and N2 = ∂b/∂z is the squared buoyancy frequency, so the rate of change of zgb−lin due to diffusion is 〈κzN2〉/g. In the case of sloping isopycnals, both the diapycnal and isopycnal fluxes will contribute to this vertical flux.

c. Seawater Boussinesq model

In the real ocean, additional effects complicate the energy budget because the EOS is not linear. In particular, compressibility effects must be included. We will now introduce a modified definition for the center of mass suitable for the Boussinesq approximation with a nonlinear EOS, referred to as the seawater Boussinesq approximation (McDougall 2003; Young 2010; Vallis 2017).

The key quantity in the energy budget is the Boussinesq dynamic enthalpy, defined as
hdb,z0=∫zz0b(S,Θ,z)dz.
The Boussinesq dynamic enthalpy measures the energy required to move a water parcel adiabatically from its original position up to a given reference height z0 (given relative to the surface level). Here, buoyancy is a nonlinear function of Conservative Temperature Θ, Absolute Salinity S, and height z relative to the mean sea level (Roquet et al. 2015b).

Dynamic enthalpy is usually defined relative to the ocean’s surface, i.e., with z0 = 0 (McDougall 2003; Nycander 2011; Young 2010). Here, we use the center of volume as the reference level instead, to obtain a center of mass (nearly) matching the center of volume for a mixed state as explained in the previous section. For a nonlinear EOS, the center of mass of the fully mixed state does not exactly correspond to the center of volume in general; however, the discrepancy is small in practice. This choice has also numerical advantages as the center of volume is independent of the ocean’s state and can thus be determined beforehand and because this choice minimizes the numerical range of the depth values used to compute the center of mass.

It is straightforward to compute dynamic enthalpy referenced to the center of volume from surface-referenced dynamic enthalpy:
hdb,zυ(S,Θ,z)=hdb,0(S,Θ,z)−hdb,0(S,Θ,zυ).
The function “gsw_dynamic_enthalpy” from the Thermodynamic Equation of Seawater-10 (TEOS-10) toolbox (http://www.teos-10.org/, Roquet et al. 2015b) provides the dynamic enthalpy relative to the surface, which corresponds to gz+hdb,0 and can therefore be used to compute the Boussinesq dynamic enthalpy.
By analogy with the case of a linear EOS, we define the center of mass (seawater Boussinesq, nonlinear EOS) as
zgb=zυ+⟨hdb,zυg⟩,
where dynamic enthalpy is defined using the center of volume as its reference height.
We now define the height anomaly by taking the difference with the height the center of mass would have if the ocean was fully mixed, i.e., with constant global-mean temperature and salinity:
Zb=⟨ζb⟩=zgb−zg,0b,
with ζb being the local height anomaly and zg,0b being the center of mass of the fully mixed reference state. The fully mixed state is time dependent because of the thermobaric effect changing the mean compressibility as a function of temperature. The local height anomaly is defined as ζb=[hdb,zυ(S,Θ,z)−hdb,zυ(⟨S⟩,⟨Θ⟩,z)]/g.

The height anomaly is defined here globally, but it is straightforward to define it locally, replacing the global averaging operator with a local averaging.

d. Stratification energy

The reservoir of Boussinesq dynamic enthalpy is by definition proportional to the center of mass [see Eq. (9)]:
H=ρ0∭hdb,zυdV=(zgb−zυ)ρ0gV.
We define the stratification energy as the minimum energy required to remove the stratification and completely homogenize the ocean. The stratification energy is obtained as the difference between the Boussinesq dynamic enthalpy and its value in the fully mixed state, making it proportional to the height anomaly:
S=H−H0=Zbρ0gV,
where H0=(zg,0b−zυ)ρ0gV is the energy in the mixed state.
For completeness, we define the total kinetic energy as
K=ρ02∭u⋅udV,
where only the horizontal velocity u is considered in the definition as the hydrostatic approximation was made.

e. Budget equation of the center of mass

The budget equation of dynamic enthalpy has three terms:
Dhdb,zυDt=Gh−bw,
where Gh=hΘΘ∘+hSS∘ represents the local sources/sinks of dynamic enthalpy and w is the vertical velocity. Here, hΘ and hS are the partial derivatives of hdb,zυ w.r.t. Θ and S. The terms Θ∘ and S∘ are the diabatic source terms in the temperature and salinity equations. The budget equation for the Boussinesq center of mass becomes
dzgbdt=⟨Gh⟩g−⟨bw⟩g.
Similarly, a budget equation for the center of mass of the mixed state can be derived:
dzg,0bdt=1g(hΘ,0⟨Θ∘⟩+hS,0⟨S∘⟩),
where hΘ,0 = 〈hΘ(〈S〉, 〈Θ〉, z)〉 and hS,0 = 〈hS(〈S〉, 〈Θ〉, z)〉. This corrective term is a result of the thermobaric effect of the nonlinear EOS and depends to the leading order on the trend in global-mean temperature. It must be removed from the budget equation of zgb to obtain the budget equation of the height anomaly:
dZbdt︸trend=⟨Ghg⟩︸sources+⟨−bwg⟩︸conversion−hΘ,0⟨Θ∘⟩+hS,0⟨S∘⟩g︸thermobariccorrection.
This latter equation is the generalization of Eq. (6) for a seawater Boussinesq model.
A more detailed expression can be derived for the diabatic source term. Assuming the diabatic source terms for temperature and salinity have the forms,
Θ∘=∇⋅FΘ,
S∘=∇⋅FS,
where FΘ and Fs are the (mostly turbulent) fluxes of temperature and salinity, respectively. The temperature flux near the surface includes also irradiance associated with penetrative solar heating.
Rewriting the source term of dynamic enthalpy as
Gh=−(∇hΘ⋅FΘ+∇hS⋅FS)+∇⋅(hΘFΘ+hSFS),
the globally averaged source term becomes
⟨Gh⟩=−⟨∂hΘ∂zFΘz+∂hS∂zFSz⟩−⟨∇hhΘ⋅FΘh+∇hhS⋅FSh⟩   +1V∬Z=0(hΘFΘz+hSFSz)dA,
where superscripts z and h refers to the vertical and horizontal contributions, respectively.

The first term is the contribution due to vertical turbulent mixing, apart from the contribution from penetrative heating in the upper water column. It is the leading term, corresponding to the change of the center of mass associated with a vertical diffusive flux of density. For a linear EOS, this term simplifies to the classical 〈κzN2〉. Here, note that κz is the vertical diffusivity. In models where the diffusion operator is slanted along the isoneutral direction, κz is a combination of both isoneutral and dianeutral diffusivities with relative weight function of the local isoneutral slope.

The second term is the contribution due to horizontal turbulent mixing, related to nonlinear effects of the EOS such as cabbeling and thermobaricity (Nycander 2011). Indeed, if the EOS was linear, the horizontal gradients of dynamic enthalpy would vanish entirely. The net effect of these nonlinear effects in a steady state is to densify waters in frontal regions below the mixed layer, which is compensated by a net surface buoyancy flux (Hieronymus and Nycander 2013). On average, this tends to increase the upper stratification, therefore lowering the center of mass.

The third term is the contribution due to the net surface flux of dynamic enthalpy, the sum of contributions from surface heat and freshwater fluxes. For simplicity, we have neglected here the contribution from bottom geothermal heating, but it could readily be included.

3. Methods

a. Model setup

An idealized configuration of NEMO ocean engine (Madec et al. 2022) is used, based on Caneill et al. (2022). It is configured to simulate the most prominent large-scale ocean circulations, roughly mimicking the North Atlantic basin. It is arranged as a Northern Hemisphere 2° × 2° grid (0°–40°E, 0°–60°N), with 36 variable depth layers, an equatorial wall, a nonlinear free surface, and a flat bottom. It uses both the hydrostatic and Boussinesq approximations. Following standard practices, a large background vertical diffusivity of 10−4 m2 s−1 is used helping to produce a realistically deep stratification. The effect of eddies is parameterized, applying lateral diffusion along an approximately neutral direction and advecting tracers with a parameterized eddy-induced velocity. A convective parameterization scheme is used, applying an enhanced diffusivity coefficient of 100 m2 s−1 every time a static instability is generated in the model.

The simplified nonlinear EOS of Roquet et al. (2015a) is used,1
b(S,Θ,z)=gα0[1+λ2(Θ−Θ0)−μz](Θ−Θ0)−gβ0(S−S0),
with the parameters Θ0 = 10°C, S0 = 35 g kg−1, α0 = 1.613 × 10−4°C−1, λ = 5.952 × 10−2°C−1, μ = 1.497 × 10−4 m−1, and β0 = 7.461 (g kg−1)−1.
The Boussinesq dynamic enthalpy is obtained by vertical integration of the buoyancy keeping temperature and salinity fixed:
hdb,zυ=−b(z−zυ)−gα0μ2(Θ−Θ0)(z−zυ)2.
The surface temperature and salinity are restored to the meridional profiles of references shown in Figs. 1b and 1c. Note that the temperature restoring profile varies seasonally between −2° and 2°C. Additionally, the surface is forced by wind stress (Fig. 1a) and a seasonally varying solar heat flux is applied (Fig. 1d), consisting of a surface and a penetrative part. Because the surface radiation is already included in the relaxation heat flux, it is subtracted from the annual-mean radiative flux. More details about the surface forcing fields can be found in Caneill et al. (2022).
Fig. 1.
Fig. 1.

Numerical setup of forcings. (a) Zonal wind stress (no meridional wind stress applied). (b) Surface relaxation temperature, which varies seasonally between the upper and lower lines. (c) Surface relaxation salinity. (d) Incoming surface radiation, which varies seasonally between the upper and lower lines.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

Three modeling experiments will be considered in the following. The first experiment, EXP-REF, is our central reference run. The second experiment, EXP-NoW, experiences the same forcing fields except the wind stress is set to zero. Note that the source of turbulent kinetic energy in the mixed layer parameterization is the same for both runs, so the only difference relates to the presence or not of surface Ekman transports and of an Ekman pumping generating an interior wind-driven circulation. A third experiment, EXP-LIN, differs from EXP-REF only by having a linear EOS, i.e., by setting parameters λ and μ to zero in the simplified EOS [Eq. (22)].

The ocean simulations are started from a global-mean resting state (Θ = 10°C and S = 35 g kg−1), and they are run for a spinup period of 2000 years. The quasi-steady state is reached after 300 years. Diagnostics of the steady state are based on a 10-yr average of daily outputs for the years 2000–10. The analysis of the residual variability in the model is presented over the years 2000–2500 in section 4c, based on monthly outputs. The meridional overturning circulation (MOC) streamfunction is obtained by vertical accumulation of the zonal-integrated meridional velocity:
Ψ(ϕ,z)=∫z0∫λυ(ϕ,λ′,z′)dλ′dz′.
The meridional overturning is structured in two cells, one is the tropical MOC (TMOC) generated by the equatorial upwelling and confined to the upper 200 m south of 20°S, while the other is the deep MOC (DMOC) fed by deep-water formation in the northeastern corner of the domain and balanced by a slow upwelling across the thermocline in the subtropics (Caneill et al. 2022). We define the DMOC transport as the maximum overturning transport found north of 10°N and the TMOC transport as the maximum transport south of 10°N for simulations where a wind forcing is applied (EXP-REF and EXP-LIN).

b. Center of mass

The Boussinesq center of mass is defined using Eq. (9). The center of volume is taken constant in time, neglecting contributions from sea level variations—as if there was a rigid lid on top of the ocean.2 For the simplified nonlinear EOS defined in Eq. (22), it is obtained as
zgb=zυ−⟨b(z−zυ)⟩g−α0μ2⟨(Θ−Θ0)(z−zυ)2⟩.
The second right-hand side term is a corrective term due to thermobaric effects. Because of this term, the center of mass of a homogeneous ocean does not match the center of volume in general. However, the discrepancy remains small in practice.
The local height anomaly is defined as
ζb=−b(z−zυ)g−α0μ2(Θ−⟨Θ⟩)(z−zυ)2.
The second right-hand side term is a corrective term due to thermobaric effects, which depends on the skewness of the thermal stratification. The height anomaly is obtained as the global average of the local height anomaly Zb = 〈ζb〉.
The averaging operator is discretized as follows for any tracer T:
T=1V∑i,j,kTi,j,kVi,j,k,
where Vi,j,k is the volume of the (i, j, k) -indexed cell and V is the total volume of the basin. Hence, the center of volume is
zυ=1V∑i,j,kzi,j,kVi,j,k,
and the Boussinesq dynamic enthalpy is computed from Eq. (23). Note that, as vertical levels are irregularly spaced, the center of volume differs slightly from the actual mean depth of the ocean basin.
To compute the center of mass of the mixed state, it suffices to determine the globally averaged temperature and salinity and use them to compute the dynamic enthalpy of the fully mixed state:
zg,0b=zυ+1gV∑i,j,khdb,zυ(⟨S⟩,⟨Θ⟩,zi,j,k)Vi,j,k,
where 〈S〉 and 〈Θ〉 are the global-mean salinity and temperature, respectively. The height anomaly Zb is simply obtained as the difference between zgb and zg,0b or equivalently as the global average of the local height anomaly ζb.

A multiplication of height anomalies by the constant factor ρ0gV yields the dynamic enthalpies H and H0 and the stratification energy S [see Eq. (11)]. Steady-state estimates of these quantities are given in Table 1.

Table 1.

Total energy reservoirs in exajoules (EJ; 1018 J). Here, P is the total potential energy referenced to the surface, K is the kinetic energy, H is the dynamic enthalpy, H0 is obtained for the fully mixed state, and their difference S corresponds to the stratification energy. Mass center heights are given in centimeters. Maximum values of the MOC streamfunction are given for the deep and tropical cells (DMOC and TMOC, respectively) in Sverdrups (1 Sv ≡ 106 m3 s−1). An average of modeled years 2000–10 is used to compute the different quantities.
Table 1.

c. Diagnostic of zg-tendency terms

A diagnostic of the zg-tendency terms is carried out on model outputs based on Eq. (17). For the simplified EOS, the partial derivatives of dynamic enthalpy needed to compute Gh are obtained as follows:
hΘ=−gα0[1+λ(Θ−Θ0)−μ2(z+zυ)](z+zυ),
hS=−gβ0(z−zυ).
The tendency term due to changes in the reference mixed state simplifies because 〈hΘ,0〉 = gα0μ〈(z − zυ)2〉/2 is a constant and 〈hΘ,0〉 = 0. We confirm here that this term is caused by thermobaricity, as it is proportional to the thermobaric coefficient μ. It accounts for the fact that cold water is more compressible than warm water, so it creates a stronger background stratification which lowers the center of mass of the mixed state.

Outputs of tendency terms in the equations of temperature and salinity are obtained from the NEMO ocean model. The conversion term bw is computed online which stems from the kinetic energy diagnostics but can additionally be reproduced afterward with good agreement. Second, h and its derivatives with respect to Θ and S are required. They are calculated from analytical functions derived for the respective EOS used. All analytical functions and general computations on model data are implemented in a Python framework that builds on functions of the Python XGCM package, extended with implementations required for processing results of the center of mass presented here.

The simulations that we analyze have a coarse resolution, and thus, they incorporate an eddy mixing parameterization. In this parameterization, a residual velocity, the sum of model and bolus velocities, is estimated and used to advect tracers. Advection by the bolus velocity acts as a net sink of potential energy, used to generate eddy kinetic energy in the real ocean (von Storch et al. 2012). In practice, this introduces a modified conversion term to the Eq. (15), −〈bwbolus〉/g, with wbolus being the vertical bolus velocity.

To distinguish the contribution from different mixing sources, we separate the source term into two main contributions. The model uses a nonpenetrative convective scheme, drastically increasing vertical diffusivity in case of static instability. The contribution from convection [to Eq. (20)] is computed as
Ghconv=−[(∂hΘ∂z)FΘz+(∂hS∂z)FSz]H(κ−κ0)nκ,
with the Heaviside function H(κ−κ0)=1 if κ is greater than κ0 = 1 m2 s−1 and =0 otherwise. The nκ is a fractional scaling based on the percentage of time steps the convective scheme was activated in the averaging period. The term in the square brackets corresponds to the model’s vertical diffusion term, which can be reconstructed directly from the vertical fluxes of temperature and salinity (respectively, FΘz and FSz). We are aware that this way of separating convection from other mixing processes is not perfect. The model is using an implicit scheme for computing vertical diffusion term, making it difficult to achieve a clean separation. We argue that this simplified method correctly captures the bulk of convection.

The residual Gdiff=Gh−Ghconv is attributed to turbulent diffusion. For simplicity, we included penetrative heating into the diffusion term, although it is not a diffusive process; however, it is confined to the top 20 m of the water column, and it is thus not affecting budgets in the interior.

While the global average of the conversion term is unambiguously defined, there is no unique way to define local values of conversion. Following Roquet (2013), we consider the horizontal anomaly of buoyancy to compute local rates of conversion:
C*=b*w=(b−b¯)w.
This definition does not affect the global integrals because the horizontal mean of the vertical velocity has to be zero to satisfy volume continuity. It produces, however, local values that are arguably more meaningful. The best way to see why is to consider a barotropic case, where potential density is homogeneous. In that case, local conversion values would be proportional to the vertical velocity; however, C*=0 everywhere as expected from a flow that has no stratification. This definition will be retained to compute local values of conversion in Fig. 6.

4. Results

a. Simulation characteristics

Global estimates of the energetic content highlight the vast difference in magnitude between the largely inert potential energy P and the more physically relevant dynamic enthalpy H (Table 1). Referencing dynamic enthalpy at the center of volume and comparing it to the mixed-state potential energy helps isolating only the energy stored in the stratification. Note that in EXP-LIN, the EOS is linear so the mixed-state potential energy is exactly zero by construction. In the two other cases, the nonlinear thermobaric term in the EOS introduces a nonzero, time-varying, mixed-state potential energy.

The stratification energy S is negative due to the presence of a mean deep-reaching stratification (Fig. 2). Small differences in stratification are observed between the three simulations. The simulation EXP-REF with wind is more stratified than EXP-NoW without wind. The stratification of the case EXP-LIN with a linear EOS is also less stratified. Kinetic energy is a much smaller reservoir than the stratification energy, five orders of magnitude lower, which is expected from a noneddying flow in near geostrophic balance. Note that the presence of a wind forcing in EXP-REF increases the reservoir of kinetic energy by a factor of 3 as compared to in EXP-NoW, but it increases the stratification energy by about 10% only, a first indication that the mean stratification is controlled to the leading order by the buoyancy forcing alone. Our simulations are based on a coarse-resolution model where eddies are not resolved. In the real ocean, the eddy kinetic energy (EKE) reservoir would be much larger than the mean kinetic energy, by an order of magnitude or more. This would still be smaller than the stratification energy reservoir.

Fig. 2.
Fig. 2.

Vertical profiles of (a) horizontal-mean density anomalies δρ(S, Θ, z) = ρ − ρ(Θ, S, z) and (b) the differences of EXP-NoW and EXP-LIN with EXP-REF computed for steady states (average of years 2000–10). (c) Vertical profiles of horizontally averaged local height anomaly and (d) their differences. Note that the global height anomaly is the vertical average of the horizontally averaged local height anomaly shown in (c) and (d).

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

When converted into a height anomaly, we find that the stratification energy corresponds to a lowering of about 26 cm of the center of mass in the EXP-REF run compared to the fully mixed state. One might be puzzled by the smallness of the variations of the center of mass. This can be traced back to how little density varies in the ocean and shows how inefficient surface forcings are in producing a deep-reaching ocean circulation. The lowering of the center of mass is of 24 cm in the EXP-NoW run, 2 cm less than in the EXP-REF experiment with wind, showing that the wind pumping contributes to strengthening the stratification by about 10% in our domain. The EXP-LIN run with a linear EOS has a 24-cm height reduction as well, indicating that nonlinear effects of the EOS are about as significant as the wind forcing in shaping the mean stratification.

The MOC in EXP-REF shows two distinct maxima, the TMOC at ≈5°N and DMOC at ≈35°N. Both feature a mean density stratification with a shallow pycnocline in the subtropics. In the absence of wind forcing, the MOC consists of a single hemispheric DMOC cell (Fig. 3b; EXP-NoW). The MOC structure is more complicated near the surface in the EXP-REF case, reflecting the action of Ekman pumping on the upper thermocline. As a result of the wind, surface waters are lighter and the thermocline is deeper in the subtropics. Interestingly, the influence of the wind on the mean stratification and on the deep overturning structure appears rather secondary despite its large influence on local horizontal circulation patterns.

Fig. 3.
Fig. 3.

Steady-state distributions of density anomaly and MOC transport in (a) EXP-REF and (b) EXP-NoW. The color contour shows the zonal average density anomaly (as defined in Fig. 2), and the line contour indicates the MOC transport (Sv). (c) Height anomaly averaged in depth and longitude, providing an index for the mean stratification as a function of latitude in the two runs. The global height anomaly is indicated with horizontal lines for both simulations.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

More details can be derived on the differences in stratification between the simulations by looking at the local density anomaly (defined as the density difference with the fully mixed state) and the local height anomaly. Horizontal means of these two quantities are shown in Fig. 2. The simulation with no wind has a shallower center of mass compared to EXP-REF primarily because it has a shallower pycnocline and lighter bottom waters. This creates a significantly larger height anomaly in the 100–300-m depth range. The meridional distribution of the height anomaly in Fig. 3c shows that the wind present in EXP-REF produces a local minimum (stronger stratification) centered at 10°. Figure 4 provides the spatial distribution of this increased height anomaly, indicating that this difference arises primarily from the subtropics where Ekman pumping present in EXP-REF produces a convergence of light waters and deepens the thermocline along the western side of the basin.

Fig. 4.
Fig. 4.

Spatial distribution of the vertically averaged local height anomaly ∫zζbdz/H in (a) EXP-REF and (b) EXP-NoW simulations, providing a local index of stratification. Associated zonal mean curves are shown in Fig. 3c. The purple contour marks where the local height anomaly equals the global height anomaly, i.e., where the local stratification matches the global one.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

The spatial distribution of the vertically averaged local height anomaly is shown in Fig. 4. Both simulations show deepest height anomaly in the tropics (maximum stratification), decreasing toward the north. The effect of the wind in EXP-REF is evidenced by a deeper subtropical height anomaly and more zonal variations in the tropics related to the equatorial upwelling and to the western intensification in the subtropics. The location where the vertical height anomaly crosses the global height anomaly value (shown as the purple contour in Fig. 4) is not changed significantly, however.

b. Ocean adjustment

During the spinup phase, it is instructive to track how the center of mass changes over time (Fig. 5b) and to see how it relates to changes in the DMOC strength (Fig. 5a). The DMOC strength and height anomalies are plotted using a logarithmic time scale to better highlight the different phases of the adjustment process. A rolling mean of 1 year periods is applied to the center of mass to filter the contribution of the seasonal cycle. The overturning strength is smoothed with a lowess filter to highlight the evolution of the long-term trend.

Fig. 5.
Fig. 5.

Time series of (a) DMOC transport, (b) heights of the center of mass, and (c) tendency terms in the height anomaly budget in the EXP-REF and EXP-NoW simulations. Solid lines correspond to the reference simulation EXP-REF, and dashed lines correspond to the no wind simulation EXP-NoW. The adjustment is divided into four periods separated at 5, 20, and 300 simulated years. Note that the time axis is logarithmic.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

The adjustment is divided into four main phases, with time scales similar to de Verdière (1988). During the first 5 years, intense deep convection is triggered, reaching the bottom in the northern half of the basin. During this phase, the center of mass drops slightly by about 8 cm while the DMOC transport takes unrealistically large values around 70 Sv (1 Sv ≡ 106 m3 s−1). The simulations experience substantial changes in their mean properties and energetics. The global-mean temperature abruptly drops from the initial 10°–3.3°C, and the global-mean salinity increases by 1 g kg−1. Salinity is not conserved as we are using a restoring term at the surface.

In the second phase (5–20 years), the transport in the MOC has reached its maximum and remains relatively stable, although with large variability at all time scales. This is during this phase that the center of mass drops the most rapidly, reaching its lowest value within a couple of decades. This indicates that the mean density stratification is generated mostly during this period, with cold waters in the north and a strong thermocline in the south. The geostrophic circulation begins to settle during this period.

In the third phase, lasting approximately 300 years, the DMOC reduces gradually, while the center of mass rises back slightly, mostly as a result of an 11-cm reduction of the reference height for the corresponding mixed state. This indicates that heat is slowly penetrating at depth, which modulates the thermobaric effect. This points to the importance of slow diffusive processes in the interior during this phase. It is also during this phase that differences between the simulations with and without wind pumping becomes more apparent.

In the last phase, the overturning circulation and the center of mass have nearly stabilized and a quasi-steady state has been reached. The DMOC stabilizes to 6.2 Sv (EXP-REF) and 7.3 Sv (EXP-NoW). The center of mass stabilizes as well to values of −26 cm (EXP-REF) and −24 cm (EXP-NoW).

Interestingly, the center of mass adjusts much faster than the MOC as it reaches a value within 10%–20% of its equilibrium value within a few decades only, while it takes several hundred years for the MOC to equilibrate. This points to the importance of fast processes such as convection in controlling global stratification, although the subsequent adjustments rely on slower processes such as interior diffusion and geostrophic advection.

While we do not find a straightforward correlation between MOC strength and height anomalies during the spinup period, we observe a relation between these two quantities in the form of a tendency to have a stronger MOC whenever the center of mass is deeper. Note that this connection is hidden in zgb alone, which continues to decrease even when the MOC weakens, showing the importance of defining the height anomaly relative to the mixed state.

c. Tendencies on the center of mass and power sources

A budget of the tendency terms in the height anomaly [Eq. (10)] is used here to assess which processes are controlling the height anomaly and its variability. Figure 5c presents the temporal evolution of each tendency term for the simulations EXP-REF and EXP-NoW, with EXP-REF values tabulated in Table 2. Contributions are decomposed into convection, diffusion, conversion, and thermobaric correction.

Table 2.

Global Zb tendencies for the different processes in the EXP-REF simulation, averaged over four different time periods (10−9 m s−1). For reference, one unit 10−9 m s−1 is equivalent to 3 cm yr−1.
Table 2.

Consistent with the height anomaly variations described in the previous section, four separate phases can be observed in the time evolution, with substantial shifts in the balance of processes. Consistent with the argument of Sandström, the source of energy seen as an upward tendency on the center of mass comes from diffusion, and it is positive at all time. However, it has little temporal variations, decreasing smoothly from a large initial value of 4.4 cm yr−1 to a mere 0.1 cm yr−1 at equilibrium.

Both convection and conversion remove potential energy from the system, lowering the center of mass. In the first 5 years of the simulation, deep convection reduces it at a rapid −5.8 cm yr−1 rate. As the overturning circulation settles in, the conversion term becomes a dominant sink of potential energy soon after. Both terms become similar in magnitude after 10 years of simulation and begin to decrease in amplitude as the steady state is approached.

In the third phase of the adjustment, the thermobaric correction becomes nonnegligible, tending to raise the height anomaly as the global-mean temperature gradually decreases. The steady state consists of a balance between diffusion raising the center, while convection and conversion contribute approximately at a ratio of 2:1 to lower it.

Figure 6 presents zonally integrated sections of these three processes once the steady state has been reached, illustrating the large differences in the spatial distribution of the different processes. Convection acts mainly in the northern part of the basin and reaches deep in regions associated with deep mixed layers. In contrast, diffusion is maximal in the shallow thermocline south of 40°N. The overturning is powered by the conversion from potential to kinetic energy, which features a complex pattern, with overall positive conversion rates south of 30°N and negative rates north of it.

Fig. 6.
Fig. 6.

Zonal integrals of the main tendency terms in the height anomaly budget for (a)–(c) EXP-REF and (d)–(f) EXP-NoW simulations: (a),(d) convection, (b),(e) vertical diffusion, and (c),(f) conversion.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

Interestingly, the inclusion or not of a wind forcing does not modify the balance of processes and their distribution much, except in the upper 300 m of the water column where conversion rates are markedly enhanced and alternating sign, following the pattern of Ekman pumping. Overall, the effect of wind on the global stratification and the overturning appears secondary, albeit nonnegligible. The net effect of the wind in this configuration is to deepen the center of mass, i.e., to increase the global stratification.

d. Correlations during near equilibrium state

The long-term internal variability of Zb and DMOC, during the near equilibrium state, is investigated focusing on the years 2000–2500 (Fig. 7). The DMOC strength and zgb show a Pearson correlation coefficient of −0.72 for the 10-yr smoothing and −0.83 for the 100-yr window is used. A linear regression analysis indicates that the DMOC increases by about 0.1 Sv for every 2-mm variation of the center of mass. This sensitivity only reflects the highly simplified configuration presented here, and they cannot be expected to match the real ocean sensitivity. However, this confirms a relationship between the center of mass and the overturning circulation.

Fig. 7.
Fig. 7.

Comparison of the DMOC transport and height anomaly Zb time series in (a),(b) EXP-REF and (c),(d) EXP-NoW, respectively. Plots include yearly data (gray), decadal smoothed data (10-yr interval lowess filter; colored), and centennial smoothed data (100-yr interval lowess filter; dashed colored).

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

The center of mass has a large seasonal variability linked to density variations in the upper pycnocline. The DMOC time series also has a substantial interannual variability which does not compare well with the fast variability of the center of mass. This shows that these two indices have different adjustment times and that the center of mass is more sensitive to the surface variability than the DMOC. It may imply that in a realistic case with large natural variability in the surface forcing, the correlation between the DMOC and zgb may be lower than found here.

In EXP-NoW (Fig. 8), the decadal correlation is nearly null; however, a substantial correlation coefficient of −0.49 is found when the time series are temporally shifted, with the DMOC lagging behind the center of mass by 16 years. A similar correlation coefficient (−0.41) is found at a centennial time scale. Comparing correlations for EXP-REF and EXP-NoW, we find that the wind forcing increases substantially at both decadal and centennial time scales pointing to a coupling between wind-driven and buoyancy-driven processes. We do not know the origin of this coupling but speculate it may be linked to the modulation of Rossby wave properties by the mean flow (Ferjani et al. 2014). This is also consistent with the finding that wind stress played a larger role on centennial time scales, while convection and conversion played a larger role in decadal adjustments.

Fig. 8.
Fig. 8.

Correlation analysis of DMOC vs Zb time series of (a) EXP-REF and (b) EXP-NoW, highlighting the synchronization in variability. Plots include unfiltered data points (colored), linear regression for decadal smoothed data (10-yr interval lowess filter; black solid line), and centennial smoothed data (100-yr interval lowess filter; black dashed line) and Pearson coefficients for the respective regressions. EXP-NoW is extended with an analysis for 16-yr time-shifted data (DMOC prior to Zb; gray solid and dashed lines). (c) The decadal Pearson coefficient over timeshift, with a clear peak for EXP-NoW for a 16-yr timeshift.

Citation: Journal of Physical Oceanography 55, 3; 10.1175/JPO-D-24-0078.1

5. Conclusions and discussion

This paper explores the use of the center of mass as a global indicator of ocean circulation and its variability. Here, the center of mass must be understood as the density-weighted mean vertical depth of the ocean and not as the actual center of mass of the ocean on spherical Earth. As such, the center of mass is found close to the average depth of the ocean (center of volume). The center of mass defined in this way relates directly to the potential energy reservoir, as noted early on by de Verdière (1989), and it provides an intuitive way to think about the integrated effect of various processes on the stratification. Despite being briefly discussed in Wunsch and Ferrari (2004), it has never been diagnosed before to the best of our knowledge. The center of mass provides information about the stratification, with a lower center of mass corresponding to a larger stratification. Processes can then be distinguished based on their overall stratifying or destratifying action, by comparing their effect on the center of mass.

We provide for the first time different methods to compute the center of mass from knowledge of the temperature and salinity distributions. The suggested definitions deviate from the true center of mass, preferring close quantities that are arguably more useful dynamically. In particular, to better account for compressibility effects, we replace potential energy by dynamic enthalpy in the definition of the center of mass (Nycander 2011; Young 2010). We also focus on the seawater Boussinesq approximation which has a simplified energy conservation compared to the compressible case, with energy present in two forms only, Boussinesq dynamic enthalpy and kinetic energy. The effect of volume expansion on the total volume of the ocean is also naturally filtered out, as volume is conserved in the Boussinesq approximation. An equivalent expression could be derived for the case of a compressible model, following the methodology of Marshall and Schott (1999), where the “center of mass” would be defined in pressure coordinate.

The center of mass is compared to its value in a fully mixed state, leading us to define the height anomaly [Eq. (10)], which corresponds to their differences. For a linear EOS, this corresponds simply to the distance to the center of volume. However, a correction term must be subtracted in the case of nonlinear EOS, induced by thermobaricity, i.e., due to the fact that cold water is more compressible than warm water. Note that this state of reference is time dependent, as the ocean exchanges net amounts of heat and freshwater with its environment. The height anomaly is a robust index of the strength of the global stratification because it differs from zero only when density varies as a result of temperature and salinity variations. Hence, a uniform warming or a sea level rise caused by glacier melting would both raise the center of mass without affecting the height anomaly. For a realistic stratification, it has a magnitude of a few tens of centimeters. This is no coincidence that it scales as the standard deviation of the free surface elevation, as most of the large-scale flow is in thermal wind balance. In the idealized configurations we presented here, we found height anomalies of about 25 cm. In the real ocean, the height anomaly would be larger, around 70 cm, reflecting the presence of a deeper thermocline enabled by the presence of an open channel across the Southern Ocean (Marshall and Speer 2012; Cessi 2019; Klocker et al. 2023).

The height anomaly is a direct measure of how strong the mean stratification is, being proportional to the minimal amount of work that would be needed to wipe out the stratification [that we coined here the stratification energy, see Eq. (12)]. However, other metrics exist. Stratification and its variations are often measured using the buoyancy frequency averaged over a certain depth (Li et al. 2020; Cheng et al. 2024). This metric is useful; however, it is an integrated metric insensitive to the density distribution between the bounds of integration. Indeed, the buoyancy stratification is proportional to the vertical gradient of potential density; hence, its vertical average is proportional to the density difference. When averaged over a large depth thickness (typically 2000 m), it ends up being solely correlated with the surface density variation. The height anomaly, in contrast, is sensitive to the vertical distribution of stratification. Approximations of the height anomaly can be derived using potential density—similar to how N2 can be approximated using potential density. For example, the height anomaly could be approximated for the volume of ocean above H = 2000 m taken in places that are deeper than 2000 m as follows:
ZH=1ρ0AHH∬x,y∫z=−H0σθ(z+H2)dzdxdy,
where σθ is the potential density anomaly defined relative to the surface and AH is the area of the ocean deeper than H. Ocean heat content is a better, more stable, indicator of climate change than sea surface temperature (Li et al. 2020). For similar reasons, the height anomaly should be a more stable indicator of climate variability.

A relation between the height anomaly and the strength of the overturning circulation is expected from theory because the mean stratification depends to the leading order on the depth of the thermocline (δ) and the density contrast across it (Δρ). Simple theories of the thermocline suggest that the depth of the thermocline is a balance between processes involving an overturning flow crossing density contours (e.g., Gnanadesikan 1999). The strength of the overturning flow must be related to the density contrast through the thermal wind balance, acting to generate the returning flow in the form of a western boundary flow. It is in fact not fortuitous that the height anomaly would be of similar amplitude to the free surface variations because both scale as the product δ × Δρ/ρ0. Here, we show that a relation exists between the height anomaly and the overturning strength both during transitory regimes (Fig. 5) and near the equilibrium states (Fig. 6) at decadal and longer time scales. This suggests a potential of tracking the center of mass to monitor the overturning circulation. Interestingly, our run with wind (EXP-REF) has a deeper center of mass than our run without wind (EXP-NoW), which should mean that it has a stronger overturning. Yet, its deep MOC is weaker. This apparent contradiction may be resolved when considering that the run with wind has a strong tropical MOC, indicating that the center of mass provides a more integrated measure of global overturning than MOC streamfunction maxima. We leave to future studies to explore this relation in more realistic ocean cases.

This study also presents a budget of the tendency equation for the center of mass, used to identify the main balance of processes generating the stratification [Eq. (17)]. The method is applied on model outputs of a square basin configuration, as a proof of concept. Several conclusions can be drawn already: 1) diffusion is the main source of potential energy continuously lifting the center of mass, at a rate that increases the farther the ocean is from being mixed; 2) convection and conversion represent two sinks of potential energy of comparable magnitude, pushing the center of mass away from its equilibrium height; and 3) the addition of a wind forcing modulates the contribution of these processes but does not change the main balance to leading order. We must emphasize here that some of these results will be strongly dependent on the basin geometry, as well as the realism of mixing parameterizations and the type of surface forcings. It is already very encouraging to obtain a clear picture of what is the main balance controlling the stratification, and by extension the overturning circulation, from a comparatively simple scalar diagnostic.

To conclude, we have used here a fully mixed state as reference to quantify the useful fraction of potential energy present in the stratification. Another state of interest which is often considered is the Lorenz state, i.e., the state of minimum potential energy (and therefore lowest center of mass) that could be obtained by adiabatic rearrangement (Lorenz 1955). The difference in potential energy with this state defines the available potential energy (APE). The exact definition of APE can be complicated in practice by the nonlinear nature of the EOS, but satisfactory methods exist (Saenz et al. 2015). Interestingly, the Lorenz state and the fully mixed state provide lower and upper bounds to the center of mass at constant global-mean temperature and salinity and for a statically stable state. In fact, simple scaling arguments show that the APE reservoir should typically be around 10%–20% of the stratification energy, meaning that there is substantially more energy stored in the mean stratification than in sloping isopycnals.

The overturning circulation is a fundamentally diabatic circulation, being powered by mechanical mixing (Burchard et al. 2024). The ocean would naturally tend toward a fully mixed state, even if it might take several millennia to achieve in practice with current rates of turbulent mixing. The stratification energy provides a direct measure of how much energy would be required to mix up the stratification. Perhaps counterintuitively, potential energy must be reduced to create the conditions of a circulation. This goes against the usual expectation that higher potential energy produces stronger kinetic energy once converted—as in a pendulum. The correlation between the center of mass and the overturning strength indicates that a lower center of mass (less potential energy) generates a stronger flow. The reason is that a stronger stratification (vertical density gradient) has a greater ability to produce more available potential energy (horizontal density gradient).

Much remains to be explored, including how the height anomaly index may help to better quantify current changes in the global stratification or compare different ocean model products. The link with the general circulation is another potential application of this index. Just as one studies the motion of a body by analyzing separately the motions of each limbs and how they move relative to each other, it would be interesting to compute the height anomaly separately for different ocean basins, to possibly gain more insight into the global connections between the different circulation pathways and their variability. Finally, budgets of the height anomaly in more realistic model simulations may help to settle the long-standing debate on the relative importance of the different controls of the global overturning circulation.

1

The equation of state is usually given as a function of pressure. We use here the conversion Z (m) ↔ −p (dbar).

2

In a realistically stratified ocean, the potential energy associated with free surface variations is negligible compared to the potential energy stored in the stratification and can thus be safely neglected.

Acknowledgments.

We want to thank Romain Caneill for his initial help in setting up the model configuration and developing numerical diagnostics. Jonas Nycander, Rainer Feistel, and two anonymous reviewers made useful comments that helped improve the manuscript.

Data availability statement.

All model data are available upon request. The diagnostics were computed offline using a Python-based package available on GitHub at https://github.com/BenniSchmiedel/ECO.

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  • Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7, 157–167, https://doi.org/10.3402/tellusa.v7i2.8796.

    • Search Google Scholar
    • Export Citation

  • Madec, G., and Coauthors, 2022: NEMO ocean engine. Scientific Notes of IPSL Climate Modelling Center 27, 401 pp., https://epic.awi.de/id/eprint/39698/1/NEMO_book_v6039.pdf.

  • Marshall, J., and F. Schott, 1999: Open-ocean convection: Observations, theory and models. Rev. Geophys., 37 (1), 1–64, https://doi.org/10.1029/98RG02739.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171–180, https://doi.org/10.1038/ngeo1391.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., A. Adcroft, J.-M. Campin, C. Hill, and A. White, 2004: Atmosphere–ocean modeling exploiting fluid isomorphisms. Mon. Wea. Rev., 132, 2882–2894, https://doi.org/10.1175/MWR2835.1.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat content and heat fluxes. J. Phys. Oceanogr., 33, 945–963, https://doi.org/10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2.

    • Search Google Scholar
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  • Minetti, A. E., C. Cisotti, and O. S. Mian, 2011: The mathematical description of the body centre of mass 3D path in human and animal locomotion. J. Biomech., 44, 1471–1477, https://doi.org/10.1016/j.jbiomech.2011.03.014.

    • Search Google Scholar
    • Export Citation

  • Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.

    • Search Google Scholar
    • Export Citation

  • Nycander, J., 2011: Energy conversion, mixing energy, and neutral surfaces with a nonlinear equation of state. J. Phys. Oceanogr., 41, 28–41, https://doi.org/10.1175/2010JPO4250.1.

    • Search Google Scholar
    • Export Citation

  • Paparella, F., and W. R. Young, 2002: Horizontal convection is non-turbulent. J. Fluid Mech., 466, 205–214, https://doi.org/10.1017/S0022112002001313.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., 2013: Dynamical potential energy: A new approach to ocean energetics. J. Phys. Oceanogr., 43, 457–476, https://doi.org/10.1175/JPO-D-12-098.1.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., C. Wunsch, and G. Madec, 2011: On the patterns of wind-power input to the ocean circulation. J. Phys. Oceanogr., 41, 2328–2342, https://doi.org/10.1175/JPO-D-11-024.1.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., G. Madec, L. Brodeau, and J. Nycander, 2015a: Defining a simplified yet “realistic” equation of state for seawater. J. Phys. Oceanogr., 45, 2564–2579, https://doi.org/10.1175/JPO-D-15-0080.1.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., G. Madec, T. J. McDougall, and P. M. Barker, 2015b: Accurate polynomial expressions for the density and specific volume of seawater using the TEOS-10 standard. Ocean Modell., 90, 29–43, https://doi.org/10.1016/j.ocemod.2015.04.002.

    • Search Google Scholar
    • Export Citation

  • Saenz, J. A., A. M. Hogg, G. O. Hughes, and R. W. Griffiths, 2012: Mechanical power input from buoyancy and wind to the circulation in an ocean model. Geophys. Res. Lett., 39, L13605, https://doi.org/10.1029/2012GL052035.

    • Search Google Scholar
    • Export Citation

  • Saenz, J. A., R. Tailleux, E. D. Butler, G. O. Hughes, and K. I. C. Oliver, 2015: Estimating Lorenz’s reference state in an ocean with a nonlinear equation of state for seawater. J. Phys. Oceanogr., 45, 1242–1257, https://doi.org/10.1175/JPO-D-14-0105.1.

    • Search Google Scholar
    • Export Citation

  • Sandström, J. W., 1908: Dynamicsche versuche mit meerwasser. Ann. Hydrogr. Martimen Meteor., 36, 6–23.

  • Vallis, G. K., 2017: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. 2nd ed. Cambridge University Press, 946 pp., https://doi.org/10.1017/9781107588417.

  • von Storch, J.-S., C. Eden, I. Fast, H. Haak, D. Hernández-Deckers, E. Maier-Reimer, J. Marotzke, and D. Stammer, 2012: An estimate of the Lorenz energy cycle for the World Ocean based on the 1/10° STORM/NCEP simulation. J. Phys. Oceanogr., 42, 2185–2205, https://doi.org/10.1175/JPO-D-12-079.1.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and R. Ferrari, 2004: Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36, 281–314, https://doi.org/10.1146/annurev.fluid.36.050802.122121.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and P. Heimbach, 2013: Two decades of the Atlantic meridional overturning circulation: Anatomy, variations, extremes, prediction, and overcoming its limitations. J. Climate, 26, 7167–7186, https://doi.org/10.1175/JCLI-D-12-00478.1.

    • Search Google Scholar
    • Export Citation

  • Young, W. R., 2010: Dynamic enthalpy, conservative temperature, and the seawater Boussinesq approximation. J. Phys. Oceanogr., 40, 394–400, https://doi.org/10.1175/2009JPO4294.1.

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    • Export Citation
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  • Klocker, A., D. Munday, B. Gayen, F. Roquet, and J. H. La Casce, 2023: Deep-reaching global ocean overturning circulation generated by surface buoyancy forcing. Tellus, 75A, 392–409, https://doi.org/10.16993/tellusa.3231.

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  • Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7, 157–167, https://doi.org/10.3402/tellusa.v7i2.8796.

    • Search Google Scholar
    • Export Citation

  • Madec, G., and Coauthors, 2022: NEMO ocean engine. Scientific Notes of IPSL Climate Modelling Center 27, 401 pp., https://epic.awi.de/id/eprint/39698/1/NEMO_book_v6039.pdf.

  • Marshall, J., and F. Schott, 1999: Open-ocean convection: Observations, theory and models. Rev. Geophys., 37 (1), 1–64, https://doi.org/10.1029/98RG02739.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171–180, https://doi.org/10.1038/ngeo1391.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., A. Adcroft, J.-M. Campin, C. Hill, and A. White, 2004: Atmosphere–ocean modeling exploiting fluid isomorphisms. Mon. Wea. Rev., 132, 2882–2894, https://doi.org/10.1175/MWR2835.1.

    • Search Google Scholar
    • Export Citation

  • McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat content and heat fluxes. J. Phys. Oceanogr., 33, 945–963, https://doi.org/10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Minetti, A. E., C. Cisotti, and O. S. Mian, 2011: The mathematical description of the body centre of mass 3D path in human and animal locomotion. J. Biomech., 44, 1471–1477, https://doi.org/10.1016/j.jbiomech.2011.03.014.

    • Search Google Scholar
    • Export Citation

  • Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.

    • Search Google Scholar
    • Export Citation

  • Nycander, J., 2011: Energy conversion, mixing energy, and neutral surfaces with a nonlinear equation of state. J. Phys. Oceanogr., 41, 28–41, https://doi.org/10.1175/2010JPO4250.1.

    • Search Google Scholar
    • Export Citation

  • Paparella, F., and W. R. Young, 2002: Horizontal convection is non-turbulent. J. Fluid Mech., 466, 205–214, https://doi.org/10.1017/S0022112002001313.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., 2013: Dynamical potential energy: A new approach to ocean energetics. J. Phys. Oceanogr., 43, 457–476, https://doi.org/10.1175/JPO-D-12-098.1.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., C. Wunsch, and G. Madec, 2011: On the patterns of wind-power input to the ocean circulation. J. Phys. Oceanogr., 41, 2328–2342, https://doi.org/10.1175/JPO-D-11-024.1.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., G. Madec, L. Brodeau, and J. Nycander, 2015a: Defining a simplified yet “realistic” equation of state for seawater. J. Phys. Oceanogr., 45, 2564–2579, https://doi.org/10.1175/JPO-D-15-0080.1.

    • Search Google Scholar
    • Export Citation

  • Roquet, F., G. Madec, T. J. McDougall, and P. M. Barker, 2015b: Accurate polynomial expressions for the density and specific volume of seawater using the TEOS-10 standard. Ocean Modell., 90, 29–43, https://doi.org/10.1016/j.ocemod.2015.04.002.

    • Search Google Scholar
    • Export Citation

  • Saenz, J. A., A. M. Hogg, G. O. Hughes, and R. W. Griffiths, 2012: Mechanical power input from buoyancy and wind to the circulation in an ocean model. Geophys. Res. Lett., 39, L13605, https://doi.org/10.1029/2012GL052035.

    • Search Google Scholar
    • Export Citation

  • Saenz, J. A., R. Tailleux, E. D. Butler, G. O. Hughes, and K. I. C. Oliver, 2015: Estimating Lorenz’s reference state in an ocean with a nonlinear equation of state for seawater. J. Phys. Oceanogr., 45, 1242–1257, https://doi.org/10.1175/JPO-D-14-0105.1.

    • Search Google Scholar
    • Export Citation

  • Sandström, J. W., 1908: Dynamicsche versuche mit meerwasser. Ann. Hydrogr. Martimen Meteor., 36, 6–23.

  • Vallis, G. K., 2017: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. 2nd ed. Cambridge University Press, 946 pp., https://doi.org/10.1017/9781107588417.

  • von Storch, J.-S., C. Eden, I. Fast, H. Haak, D. Hernández-Deckers, E. Maier-Reimer, J. Marotzke, and D. Stammer, 2012: An estimate of the Lorenz energy cycle for the World Ocean based on the 1/10° STORM/NCEP simulation. J. Phys. Oceanogr., 42, 2185–2205, https://doi.org/10.1175/JPO-D-12-079.1.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and R. Ferrari, 2004: Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36, 281–314, https://doi.org/10.1146/annurev.fluid.36.050802.122121.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and P. Heimbach, 2013: Two decades of the Atlantic meridional overturning circulation: Anatomy, variations, extremes, prediction, and overcoming its limitations. J. Climate, 26, 7167–7186, https://doi.org/10.1175/JCLI-D-12-00478.1.

    • Search Google Scholar
    • Export Citation

  • Young, W. R., 2010: Dynamic enthalpy, conservative temperature, and the seawater Boussinesq approximation. J. Phys. Oceanogr., 40, 394–400, https://doi.org/10.1175/2009JPO4294.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Numerical setup of forcings. (a) Zonal wind stress (no meridional wind stress applied). (b) Surface relaxation temperature, which varies seasonally between the upper and lower lines. (c) Surface relaxation salinity. (d) Incoming surface radiation, which varies seasonally between the upper and lower lines.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Vertical profiles of (a) horizontal-mean density anomalies δρ(S, Θ, z) = ρ − ρ(Θ, S, z) and (b) the differences of EXP-NoW and EXP-LIN with EXP-REF computed for steady states (average of years 2000–10). (c) Vertical profiles of horizontally averaged local height anomaly and (d) their differences. Note that the global height anomaly is the vertical average of the horizontally averaged local height anomaly shown in (c) and (d).

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Steady-state distributions of density anomaly and MOC transport in (a) EXP-REF and (b) EXP-NoW. The color contour shows the zonal average density anomaly (as defined in Fig. 2), and the line contour indicates the MOC transport (Sv). (c) Height anomaly averaged in depth and longitude, providing an index for the mean stratification as a function of latitude in the two runs. The global height anomaly is indicated with horizontal lines for both simulations.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Spatial distribution of the vertically averaged local height anomaly ∫zζbdz/H in (a) EXP-REF and (b) EXP-NoW simulations, providing a local index of stratification. Associated zonal mean curves are shown in Fig. 3c. The purple contour marks where the local height anomaly equals the global height anomaly, i.e., where the local stratification matches the global one.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Time series of (a) DMOC transport, (b) heights of the center of mass, and (c) tendency terms in the height anomaly budget in the EXP-REF and EXP-NoW simulations. Solid lines correspond to the reference simulation EXP-REF, and dashed lines correspond to the no wind simulation EXP-NoW. The adjustment is divided into four periods separated at 5, 20, and 300 simulated years. Note that the time axis is logarithmic.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Zonal integrals of the main tendency terms in the height anomaly budget for (a)–(c) EXP-REF and (d)–(f) EXP-NoW simulations: (a),(d) convection, (b),(e) vertical diffusion, and (c),(f) conversion.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Comparison of the DMOC transport and height anomaly Zb time series in (a),(b) EXP-REF and (c),(d) EXP-NoW, respectively. Plots include yearly data (gray), decadal smoothed data (10-yr interval lowess filter; colored), and centennial smoothed data (100-yr interval lowess filter; dashed colored).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Correlation analysis of DMOC vs Zb time series of (a) EXP-REF and (b) EXP-NoW, highlighting the synchronization in variability. Plots include unfiltered data points (colored), linear regression for decadal smoothed data (10-yr interval lowess filter; black solid line), and centennial smoothed data (100-yr interval lowess filter; black dashed line) and Pearson coefficients for the respective regressions. EXP-NoW is extended with an analysis for 16-yr time-shifted data (DMOC prior to Zb; gray solid and dashed lines). (c) The decadal Pearson coefficient over timeshift, with a clear peak for EXP-NoW for a 16-yr timeshift.

Fig. 1.

Numerical setup of forcings. (a) Zonal wind stress (no meridional wind stress applied). (b) Surface relaxation temperature, which varies seasonally between the upper and lower lines. (c) Surface relaxation salinity. (d) Incoming surface radiation, which varies seasonally between the upper and lower lines.
Fig. 2.

Vertical profiles of (a) horizontal-mean density anomalies δρ(S, Θ, z) = ρ − ρ(Θ, S, z) and (b) the differences of EXP-NoW and EXP-LIN with EXP-REF computed for steady states (average of years 2000–10). (c) Vertical profiles of horizontally averaged local height anomaly and (d) their differences. Note that the global height anomaly is the vertical average of the horizontally averaged local height anomaly shown in (c) and (d).
Fig. 3.

Steady-state distributions of density anomaly and MOC transport in (a) EXP-REF and (b) EXP-NoW. The color contour shows the zonal average density anomaly (as defined in Fig. 2), and the line contour indicates the MOC transport (Sv). (c) Height anomaly averaged in depth and longitude, providing an index for the mean stratification as a function of latitude in the two runs. The global height anomaly is indicated with horizontal lines for both simulations.
Fig. 4.

Spatial distribution of the vertically averaged local height anomaly ∫zζbdz/H in (a) EXP-REF and (b) EXP-NoW simulations, providing a local index of stratification. Associated zonal mean curves are shown in Fig. 3c. The purple contour marks where the local height anomaly equals the global height anomaly, i.e., where the local stratification matches the global one.
Fig. 5.

Time series of (a) DMOC transport, (b) heights of the center of mass, and (c) tendency terms in the height anomaly budget in the EXP-REF and EXP-NoW simulations. Solid lines correspond to the reference simulation EXP-REF, and dashed lines correspond to the no wind simulation EXP-NoW. The adjustment is divided into four periods separated at 5, 20, and 300 simulated years. Note that the time axis is logarithmic.
Fig. 6.

Zonal integrals of the main tendency terms in the height anomaly budget for (a)–(c) EXP-REF and (d)–(f) EXP-NoW simulations: (a),(d) convection, (b),(e) vertical diffusion, and (c),(f) conversion.
Fig. 7.

Comparison of the DMOC transport and height anomaly Zb time series in (a),(b) EXP-REF and (c),(d) EXP-NoW, respectively. Plots include yearly data (gray), decadal smoothed data (10-yr interval lowess filter; colored), and centennial smoothed data (100-yr interval lowess filter; dashed colored).
Fig. 8.

Correlation analysis of DMOC vs Zb time series of (a) EXP-REF and (b) EXP-NoW, highlighting the synchronization in variability. Plots include unfiltered data points (colored), linear regression for decadal smoothed data (10-yr interval lowess filter; black solid line), and centennial smoothed data (100-yr interval lowess filter; black dashed line) and Pearson coefficients for the respective regressions. EXP-NoW is extended with an analysis for 16-yr time-shifted data (DMOC prior to Zb; gray solid and dashed lines). (c) The decadal Pearson coefficient over timeshift, with a clear peak for EXP-NoW for a 16-yr timeshift.
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