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- Author or Editor: Rémi Tailleux x

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## Abstract

Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio *τ*, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with *τ*, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours *f* /*h ^{αc
}
* = a constant (

*α*is a near constant fixed by the strength of the stratification,

_{c}*f*is the Coriolis parameter, and

*h*is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

## Abstract

Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio *τ*, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with *τ*, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours *f* /*h ^{αc
}
* = a constant (

*α*is a near constant fixed by the strength of the stratification,

_{c}*f*is the Coriolis parameter, and

*h*is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

## Abstract

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ultimately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a nonlinear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

## Abstract

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ultimately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a nonlinear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

## Abstract

In this paper, two new quasi-neutral density variables—generalized patched potential density (GPPD) and thermodynamic neutral density *γ*
^{
T
}—are introduced, which are showed to approximate Jackett and McDougall empirical neutral density *γ*
^{
n
} significantly better than the quasi-material rational polynomial approximation *γ*
_{
a
} previously introduced by McDougall and Jackett. In contrast to *γ*
^{
n
}, *γ*
^{
T
} is easily and efficiently computed for arbitrary climatologies of temperature and salinity (both realistic and idealized), has a clear physical basis rooted in the theory of available potential energy, and does not suffer from nonmaterial effects that make *γ*
^{
n
} so difficult to use in water masses analysis. In addition, *γ*
^{
T
} is also significantly more neutral than all known quasi-material density variables, such as *σ*
_{2}, while remaining less neutral than *γ*
^{
n
}. Because unlike *γ*
^{
n
}, *γ*
^{
T
} is mathematically explicit, it can be used for theoretical as well as observational studies, as well as a generalized vertical coordinate in isopycnal models of the ocean circulation. On the downside, *γ*
^{
T
} exhibits inversions and degraded neutrality in the polar regions, where the Lorenz reference state is the furthest away from the actual state. Therefore, while *γ*
^{
T
} represents progress over previous approaches, further work is still needed to determine whether its polar deficiencies can be corrected, an essential requirement for *γ*
^{
T
} to be useful in Southern Ocean studies, for instance.

## Abstract

In this paper, two new quasi-neutral density variables—generalized patched potential density (GPPD) and thermodynamic neutral density *γ*
^{
T
}—are introduced, which are showed to approximate Jackett and McDougall empirical neutral density *γ*
^{
n
} significantly better than the quasi-material rational polynomial approximation *γ*
_{
a
} previously introduced by McDougall and Jackett. In contrast to *γ*
^{
n
}, *γ*
^{
T
} is easily and efficiently computed for arbitrary climatologies of temperature and salinity (both realistic and idealized), has a clear physical basis rooted in the theory of available potential energy, and does not suffer from nonmaterial effects that make *γ*
^{
n
} so difficult to use in water masses analysis. In addition, *γ*
^{
T
} is also significantly more neutral than all known quasi-material density variables, such as *σ*
_{2}, while remaining less neutral than *γ*
^{
n
}. Because unlike *γ*
^{
n
}, *γ*
^{
T
} is mathematically explicit, it can be used for theoretical as well as observational studies, as well as a generalized vertical coordinate in isopycnal models of the ocean circulation. On the downside, *γ*
^{
T
} exhibits inversions and degraded neutrality in the polar regions, where the Lorenz reference state is the furthest away from the actual state. Therefore, while *γ*
^{
T
} represents progress over previous approaches, further work is still needed to determine whether its polar deficiencies can be corrected, an essential requirement for *γ*
^{
T
} to be useful in Southern Ocean studies, for instance.

## Abstract

have proposed a new theoretical framework for studying ocean heat uptake in potential temperature coordinates. One important step in their derivations requires understanding the temporal changes of the volume of water *V* with temperature greater than some value, which they write as the sum of two terms. The first one is due to the surface freshwater fluxes and is well defined, but the second one—attributed to the volume fluxes through the lower boundary of the domain—is given no explicit expression. What the authors mean exactly is unclear, however, because in the incompressible Boussinesq approximation, the use of a divergenceless velocity field implies that the sum of the volume fluxes through any kind of control volume must integrate to zero at all times. In this comment, we provide two alternative explicit mathematical expressions linking the volume change of to the diabatic sources and sinks of heat that clarify their result. By contrasting approach with that for a fully compressible ocean, it is concluded that the volume considered by is best interpreted as a proxy for the Boussinesq mass *M*
_{0} = *ρ*
_{0}
*V*, where *ρ*
_{0} is the reference Boussinesq density. If *V* were truly meant to represent volume rather than a proxy for the Boussinesq mass, the Boussinesq expression for *dV*/*dt* would have to be regarded as inaccurate because of its neglect of the volume changes resulting from mean density changes.

## Abstract

have proposed a new theoretical framework for studying ocean heat uptake in potential temperature coordinates. One important step in their derivations requires understanding the temporal changes of the volume of water *V* with temperature greater than some value, which they write as the sum of two terms. The first one is due to the surface freshwater fluxes and is well defined, but the second one—attributed to the volume fluxes through the lower boundary of the domain—is given no explicit expression. What the authors mean exactly is unclear, however, because in the incompressible Boussinesq approximation, the use of a divergenceless velocity field implies that the sum of the volume fluxes through any kind of control volume must integrate to zero at all times. In this comment, we provide two alternative explicit mathematical expressions linking the volume change of to the diabatic sources and sinks of heat that clarify their result. By contrasting approach with that for a fully compressible ocean, it is concluded that the volume considered by is best interpreted as a proxy for the Boussinesq mass *M*
_{0} = *ρ*
_{0}
*V*, where *ρ*
_{0} is the reference Boussinesq density. If *V* were truly meant to represent volume rather than a proxy for the Boussinesq mass, the Boussinesq expression for *dV*/*dt* would have to be regarded as inaccurate because of its neglect of the volume changes resulting from mean density changes.

## Abstract

The influences of topography on the propagation, spatial patterns, and amplitude variations of long baroclinic Rossby waves are investigated with a wind-forced, two-layer model above a midocean ridge. With steep topography the evolution equation for the baroclinic mode is shown to differ from that for a flat bottom in several ways: 1) The phase speed is systematically faster by the factor *H*/*H*
_{2}, where *H* is the total ocean depth and *H*
_{2} is the lower layer thickness, though the propagation remains westward and nearly nondispersive; 2) an effectively dissipative transfer to the barotropic mode occurs whenever the baroclinic mode is locally parallel to *f*/*H* contours, where *f* is the Coriolis frequency; and 3) the wind-forced response is amplified in proportion to the topographic steepness, (*f*/*H*)(*dH*/*dx*)/(*df*/*dy*), for a longitudinally varying topography, which can be a large factor, but the amplification is only by the modest factor *H*/*H*
_{2} for a latitudinally varying topography. Effects 2 and 3 are the result of energy exchanges to and from the barotropic mode, respectively. Effect 3 causes freely propagating, baroclinic Rossby waves to be generated west of the ridge. These effects collectively cause distortions of the baroclinic wave pattern as it traverses the ridge. These effects account qualitatively for several features seen in altimetric measurements in the vicinity of major topographic features: an increase in variance of baroclinic signals on the west side, an enhanced phase speed overall (compared to flat-bottom waves), and an abrupt change in the phase speed at midocean ridges.

## Abstract

The influences of topography on the propagation, spatial patterns, and amplitude variations of long baroclinic Rossby waves are investigated with a wind-forced, two-layer model above a midocean ridge. With steep topography the evolution equation for the baroclinic mode is shown to differ from that for a flat bottom in several ways: 1) The phase speed is systematically faster by the factor *H*/*H*
_{2}, where *H* is the total ocean depth and *H*
_{2} is the lower layer thickness, though the propagation remains westward and nearly nondispersive; 2) an effectively dissipative transfer to the barotropic mode occurs whenever the baroclinic mode is locally parallel to *f*/*H* contours, where *f* is the Coriolis frequency; and 3) the wind-forced response is amplified in proportion to the topographic steepness, (*f*/*H*)(*dH*/*dx*)/(*df*/*dy*), for a longitudinally varying topography, which can be a large factor, but the amplification is only by the modest factor *H*/*H*
_{2} for a latitudinally varying topography. Effects 2 and 3 are the result of energy exchanges to and from the barotropic mode, respectively. Effect 3 causes freely propagating, baroclinic Rossby waves to be generated west of the ridge. These effects collectively cause distortions of the baroclinic wave pattern as it traverses the ridge. These effects account qualitatively for several features seen in altimetric measurements in the vicinity of major topographic features: an increase in variance of baroclinic signals on the west side, an enhanced phase speed overall (compared to flat-bottom waves), and an abrupt change in the phase speed at midocean ridges.

## Abstract

In layered models of the ocean, the assumption of a deep resting layer is often made, motivated by the surface intensification of many phenomena. The propagation speed of first-mode, baroclinic Rossby waves in such models is always faster than in models with all the layers active. The assumption of a deep-resting layer is not crucial for the phase-speed enhancement since the same result holds if the bottom pressure fluctuations are uncorrelated from the overlying wave dynamics.

*C*

_{fast}is the enhanced phase speed,

*C*

_{standard}the standard phase speed,

^{′}

_{1}(

*z*)

*H*

_{0}is the reference depth serving to define it. In the case WKB theory is applicable in the vertical direction, the above formula reduces to

*N*is the deep Brunt–Väisälä frequency and

_{b}*N*

The amplification factor is computed from a global hydrographic climatology. The comparison with observational estimates shows a reasonable degree of consistency, although with appreciable scatter. The theory appears to do as well as the previously published mean-flow theories of Killworth et al. and others. The link between the faster mode and the surface-intensified modes occurring over steep topography previously discussed in the literature is also established.

## Abstract

In layered models of the ocean, the assumption of a deep resting layer is often made, motivated by the surface intensification of many phenomena. The propagation speed of first-mode, baroclinic Rossby waves in such models is always faster than in models with all the layers active. The assumption of a deep-resting layer is not crucial for the phase-speed enhancement since the same result holds if the bottom pressure fluctuations are uncorrelated from the overlying wave dynamics.

*C*

_{fast}is the enhanced phase speed,

*C*

_{standard}the standard phase speed,

^{′}

_{1}(

*z*)

*H*

_{0}is the reference depth serving to define it. In the case WKB theory is applicable in the vertical direction, the above formula reduces to

*N*is the deep Brunt–Väisälä frequency and

_{b}*N*

The amplification factor is computed from a global hydrographic climatology. The comparison with observational estimates shows a reasonable degree of consistency, although with appreciable scatter. The theory appears to do as well as the previously published mean-flow theories of Killworth et al. and others. The link between the faster mode and the surface-intensified modes occurring over steep topography previously discussed in the literature is also established.

## Abstract

Subducted temperature anomalies have been invoked as a possible way for midlatitudes to alter the climate variability of equatorial regions through the so-called thermocline bridge, both in the Pacific and Atlantic Oceans. To have a significant impact on the equatorial heat balance, however, temperature anomalies must reach the equatorial regions sufficiently undamped. In the oceans, the amplitude of propagating temperature (and salinity) anomalies can be altered both by diabatic (nonconservative) and adiabatic (conservative) effects. The importance of adiabatic alterations depends on whether the anomalies are controlled by wave dynamics or by passive advection associated with density compensation. Waves being relatively well understood, this paper seeks to understand the amplitude variations of density-compensated temperature and salinity anomalies caused by adiabatic effects, for which no general methodology is available. The main assumption is that these can be computed independent of amplitude variations caused by diabatic effects. Because density compensation requires the equality *T*′/*S*′ = *β _{S}
*/

*α*to hold along mean trajectories, the ratio

*T*′/

*S*′ may potentially undergo large amplitude variations if the ratio

*β*/

_{S}*α*does, where

*α*and

*β*are the thermal expansion and haline contraction coefficients, respectively. In the oceans, the ratio

_{S}*β*/

_{S}*α*may decrease by an order-1 factor between the extratropical and tropical latitudes, but such large variations are in general associated with diapycnal rather than isopycnal motion and hence are likely to be superimposed in practice with diabatically induced variations. To understand the individual variations of

*T*′ and

*S*′ along the mean streamlines, two distinct theories are constructed that respectively use density/salinity and density/spiciness as prognostic variables. If the coupling between the prognostic variables is neglected, as is usually done, both theories predict at leading order that temperature (salinity) anomalies should be systematically and significantly attenuated (conserved or amplified), on average, when propagating from extratropical to tropical latitudes. Along particular trajectories following isopycnals, however, both attenuation and amplification appear to be locally possible. Assuming that the density/spiciness formulation is the most accurate, which is supported by a theoretical assessment of higher-order effects, the present results provide an amplification mechanism for subducted salinity anomalies propagating equatorward, by which the latter could potentially affect decadal equatorial climate variability through their slow modulation of the equatorial mixed layer, perhaps more easily than their attenuated temperature counterparts. This could be by affecting, for instance, barrier layers by which salinity is known to strongly affect local heat fluxes and heat content.

## Abstract

Subducted temperature anomalies have been invoked as a possible way for midlatitudes to alter the climate variability of equatorial regions through the so-called thermocline bridge, both in the Pacific and Atlantic Oceans. To have a significant impact on the equatorial heat balance, however, temperature anomalies must reach the equatorial regions sufficiently undamped. In the oceans, the amplitude of propagating temperature (and salinity) anomalies can be altered both by diabatic (nonconservative) and adiabatic (conservative) effects. The importance of adiabatic alterations depends on whether the anomalies are controlled by wave dynamics or by passive advection associated with density compensation. Waves being relatively well understood, this paper seeks to understand the amplitude variations of density-compensated temperature and salinity anomalies caused by adiabatic effects, for which no general methodology is available. The main assumption is that these can be computed independent of amplitude variations caused by diabatic effects. Because density compensation requires the equality *T*′/*S*′ = *β _{S}
*/

*α*to hold along mean trajectories, the ratio

*T*′/

*S*′ may potentially undergo large amplitude variations if the ratio

*β*/

_{S}*α*does, where

*α*and

*β*are the thermal expansion and haline contraction coefficients, respectively. In the oceans, the ratio

_{S}*β*/

_{S}*α*may decrease by an order-1 factor between the extratropical and tropical latitudes, but such large variations are in general associated with diapycnal rather than isopycnal motion and hence are likely to be superimposed in practice with diabatically induced variations. To understand the individual variations of

*T*′ and

*S*′ along the mean streamlines, two distinct theories are constructed that respectively use density/salinity and density/spiciness as prognostic variables. If the coupling between the prognostic variables is neglected, as is usually done, both theories predict at leading order that temperature (salinity) anomalies should be systematically and significantly attenuated (conserved or amplified), on average, when propagating from extratropical to tropical latitudes. Along particular trajectories following isopycnals, however, both attenuation and amplification appear to be locally possible. Assuming that the density/spiciness formulation is the most accurate, which is supported by a theoretical assessment of higher-order effects, the present results provide an amplification mechanism for subducted salinity anomalies propagating equatorward, by which the latter could potentially affect decadal equatorial climate variability through their slow modulation of the equatorial mixed layer, perhaps more easily than their attenuated temperature counterparts. This could be by affecting, for instance, barrier layers by which salinity is known to strongly affect local heat fluxes and heat content.

## Abstract

A key idea in the study of the Atlantic meridional overturning circulation (AMOC) is that its strength is proportional to the meridional density gradient or, more precisely, to the strength of the meridional pressure gradient. A physical basis that would indicate how to estimate the relevant meridional pressure gradient locally from the density distribution in numerical ocean models to test such an idea has been lacking however. Recently, studies of ocean energetics have suggested that the AMOC is driven by the release of available potential energy (APE) into kinetic energy (KE) and that such a conversion takes place primarily in the deep western boundary currents. In this paper, the authors develop an analytical description linking the western boundary current circulation below the interface separating the North Atlantic Deep Water (NADW) and Antarctic Intermediate Water (AAIW) to the shape of this interface. The simple analytical model also shows how available potential energy is converted into kinetic energy at each location and that the strength of the transport within the western boundary current is proportional to the local meridional pressure gradient at low latitudes. The present results suggest, therefore, that the conversion rate of potential energy may provide the necessary physical basis for linking the strength of the AMOC to the meridional pressure gradient and that this could be achieved by a detailed study of the APE to KE conversion in the western boundary current.

## Abstract

A key idea in the study of the Atlantic meridional overturning circulation (AMOC) is that its strength is proportional to the meridional density gradient or, more precisely, to the strength of the meridional pressure gradient. A physical basis that would indicate how to estimate the relevant meridional pressure gradient locally from the density distribution in numerical ocean models to test such an idea has been lacking however. Recently, studies of ocean energetics have suggested that the AMOC is driven by the release of available potential energy (APE) into kinetic energy (KE) and that such a conversion takes place primarily in the deep western boundary currents. In this paper, the authors develop an analytical description linking the western boundary current circulation below the interface separating the North Atlantic Deep Water (NADW) and Antarctic Intermediate Water (AAIW) to the shape of this interface. The simple analytical model also shows how available potential energy is converted into kinetic energy at each location and that the strength of the transport within the western boundary current is proportional to the local meridional pressure gradient at low latitudes. The present results suggest, therefore, that the conversion rate of potential energy may provide the necessary physical basis for linking the strength of the AMOC to the meridional pressure gradient and that this could be achieved by a detailed study of the APE to KE conversion in the western boundary current.

## Abstract

The study of the mechanical energy budget of the oceans using the Lorenz available potential energy (APE) theory is based on knowledge of the adiabatically rearranged Lorenz reference state of minimum potential energy. The compressible and nonlinear character of the equation of state for seawater has been thought to cause the reference state to be ill defined, casting doubt on the usefulness of APE theory for investigating ocean energetics under realistic conditions. Using a method based on the volume frequency distribution of parcels as a function of temperature and salinity in the context of the seawater Boussinesq approximation, which is illustrated using climatological data, the authors show that compressibility effects are in fact minor. The reference state can be regarded as a well-defined one-dimensional function of depth, which forms a surface in temperature, salinity, and density space between the surface and the bottom of the ocean. For a very small proportion of water masses, this surface can be multivalued and water parcels can have up to two statically stable levels in the reference density profile, of which the shallowest is energetically more accessible. Classifying parcels from the surface to the bottom gives a different reference density profile than classifying in the opposite direction. However, this difference is negligible. This study shows that the reference state obtained by standard sorting methods is equivalent to, though computationally more expensive than, the volume frequency distribution approach. The approach that is presented can be applied systematically and in a computationally efficient manner to investigate the APE budget of the ocean circulation using models or climatological data.

## Abstract

The study of the mechanical energy budget of the oceans using the Lorenz available potential energy (APE) theory is based on knowledge of the adiabatically rearranged Lorenz reference state of minimum potential energy. The compressible and nonlinear character of the equation of state for seawater has been thought to cause the reference state to be ill defined, casting doubt on the usefulness of APE theory for investigating ocean energetics under realistic conditions. Using a method based on the volume frequency distribution of parcels as a function of temperature and salinity in the context of the seawater Boussinesq approximation, which is illustrated using climatological data, the authors show that compressibility effects are in fact minor. The reference state can be regarded as a well-defined one-dimensional function of depth, which forms a surface in temperature, salinity, and density space between the surface and the bottom of the ocean. For a very small proportion of water masses, this surface can be multivalued and water parcels can have up to two statically stable levels in the reference density profile, of which the shallowest is energetically more accessible. Classifying parcels from the surface to the bottom gives a different reference density profile than classifying in the opposite direction. However, this difference is negligible. This study shows that the reference state obtained by standard sorting methods is equivalent to, though computationally more expensive than, the volume frequency distribution approach. The approach that is presented can be applied systematically and in a computationally efficient manner to investigate the APE budget of the ocean circulation using models or climatological data.