## 1. Introduction

The global kinetic energy (KE) budget plays a key role in ocean energetics, for it is often the natural starting point for discussing how the ocean circulation is forced and dissipated. In its standard form, the kinetic energy budget reveals that kinetic energy is primarily controlled by 1) the power input due to the wind forcing and tidal forcing, 2) viscous dissipation, and 3) the net conversion between potential energy (PE) and kinetic energy (Gregory and Tailleux 2011). Of these three terms, only the first one is well understood, and quantifiable from observations, as discussed in Roquet et al. (2011) and Zhai et al. (2012), among others. Although local turbulent viscous dissipation rates are routinely observed as part of campaigns to measure turbulent mixing in the oceans, these are so variable spatially and temporally that it is difficult to infer what the volume-integrated viscous dissipation is. Oort et al. (1994) originally assumed that in the ocean there is a net conversion of potential energy into kinetic energy. This was then disputed by Toggweiler and Samuels (1998) and in subsequent modeling studies using ocean-only models (Gnanadesikan et al. 2005) and coupled climate models (Gregory and Tailleux 2011), in which net conversion from kinetic energy to potential energy has been found, dominated by wind power input through Ekman pumping in the Antarctic circumpolar region.

The standard kinetic energy budget, however, does not directly reveal the role and importance of surface buoyancy fluxes and interior mixing processes of heat and salt in forcing and dissipating the ocean circulation. These processes control, at least in part, the sign and magnitude of the net conversion between potential and kinetic energy in the oceans. Linking the mechanical energy (gravitational potential plus kinetic energy) budget to surface buoyancy fluxes and interior mixing processes has been a controversial topic and has been overlooked in recent reviews on ocean energetics by Wunsch and Ferrari (2004), Kuhlbrodt et al. (2007), and Ferrari and Wunsch (2009). In these reviews, the mechanism by which surface buoyancy forcing by heat and freshwater fluxes provides mechanical energy to the ocean circulation has been related to the mechanical work of expansion and contraction produced by a heat engine, which in classical thermodynamics can be done by combining energy conservation with the entropy budget. For a stratified fluid, however, the available potential energy (APE) budget introduced by Lorenz (1955) is better suited for representing the mechanical power input by buoyancy forces that results from surface heat and freshwater fluxes. Also, available potential energy decomposes the net conversion between potential and kinetic energy as the difference between a production term by surface buoyancy fluxes minus a dissipation term by interior mixing (Hughes et al. 2009; Tailleux 2010, 2012).

Quantifying available potential energy in the framework of Lorenz (1955) relies on the definition of a globally static and stably stratified reference state obtained from the actual state by means of an adiabatic rearrangement of the fluid parcels conserving mass and salt. Defining the reference state for a fluid with a linear equation of state (EOS) in a simple domain with no sills or enclosed basins is made straightforward by, for instance, sorting water parcels according to density and filling the ocean volume level by level with the sorted density field (e.g., Winters et al. 1995). This sorting formulation has been used recently to investigate the mechanical energy budget of ocean circulation in a number of idealized situations (Hughes et al. 2009; Saenz et al. 2012; Hogg et al. 2013; Dijkstra et al. 2014).

Generalizing the concept of a reference state for the ocean under realistic conditions has remained problematic because of the presence of topographic sills and because of the nonlinearities of the equation of state for seawater. Topographic barriers block heavy waters from flowing between ocean basins, and it has been unclear how to represent this effect on the APE budget. Stewart et al. (2014) show that energy fluxes in the APE budget are largely insensitive to how the effects of topographic barriers are represented. On the other hand, the compressibility and the nonlinear dependence that density has on pressure leads to difficulties when sorting water parcels according to density, as the density needs to be recalculated at every level that is being filled. This makes calculating the reference state computationally expensive and causes the position of water parcels in the reference state to depend on the number of levels used and on whether the ocean volume is being sorted by increasing or decreasing density (e.g., Ilicak et al. 2012). Ad hoc sorting methods have nevertheless been devised to overcome these difficulties and have been used to investigate, for example, available potential energy in the global oceans (Huang 2005) and mixing in fluids with a nonlinear EOS (Ilicak et al. 2012; Petersen et al. 2015; Butler et al. 2013), but they provide no physical insight into the effects of compressibility and nonlinearities on the resulting reference density profile and remain computationally expensive. The lack of a physically tractable and well understood method to construct the Lorenz reference state has led to skepticism about the applicability of the APE framework to the ocean with a nonlinear equation of state. However, the physical basis for such skepticism has been lacking.

In this paper, we investigate the reference state for the ocean with a nonlinear equation of state for seawater by generalizing the approach proposed by Tseng and Ferziger (2001), based on the volume frequency distribution of water masses in temperature–salinity space. We demonstrate that, for all practical purposes, the oceanic reference state can be regarded as well defined. In doing so, we provide a framework by which the reference state can be obtained and characterized systematically in a physically tractable and computationally efficient manner. The work by Stewart et al. (2014) and the work we present in this paper together lay the groundwork for using available potential energy to quantify the mechanical energy budget in ocean circulation under realistic conditions.

In section 2, we summarize basic concepts of the local formulation of APE theory for a Boussinesq ocean that will be used in this paper. Then, in section 3, we present a new approach based on the volume frequency distribution of water masses to link the reference state to the thermophysical properties of the ocean. In section 4, we discuss the reference position of the ocean parcels obtained using our approach, and in section 5, we discuss the links of our approach with sorting-based methods. Last, in section 6, we briefly discuss the implications for ocean APE estimates, and we summarize and discuss the results in section 7.

## 2. A review of available potential energy for a Boussinesq fluid

To set the context and to introduce concepts that will be used in this paper, in this section we present a brief summary of the general theoretical framework in Tailleux (2013) and present it in a form that is applicable to the ocean.

### a. Model assumptions and governing equations

*w*is the vertical velocity;

*p*is the pressure;

*ρ*is density;

*g*is the acceleration of gravity;

*f*is the Coriolis parameter;

*S*is salinity; and

*S*, respectively. The equation of state

### b. Available potential energy: Work by buoyancy forces

*z*

_{r}to its actual position

*z*. We refer to

*z*

_{r}of a fluid parcel in the reference state corresponds to the depth at which its density is equal to the reference density:which is referred to as the level of neutral buoyancy (LNB) equation. Equation (8) defines

*t*,

### c. Interpretation of APE density and its relation to global APE

### d. Local description of the ocean energy cycle

*p*), which leads towhere

*α*and

*β*are the thermal expansion and haline contraction coefficients defined relative to the

### e. APE production and dissipation

*Q*is the surface heat flux into the ocean (with

*α*and

*β*applicable to each parcel at the surface. Since

## 3. Linking the reference state **ρ**_{r}(*z*) to the stratification and thermophysical properties of the ocean

**ρ**

In this section, we outline a new approach to constructing the Lorenz reference state *N* of target depths *N* target depths *z* if needed. Once *S*, so that we can write

From a fundamental viewpoint, the above method divorces the task of constructing the reference density profile *ρ*_{r}(*z*) from the task of computing the reference position *z*_{r}(**x**) for each parcel of water. This is an important departure from current sorting-based approaches, which accomplish the two tasks simultaneously. As we justify below, such a divorce is not only feasible but also essential for elucidating the nature of the difficulties associated with the nonlinearity of the equation of state and its binary character through the dependence on temperature and salinity. From a computational viewpoint, such a divorce also proves essential for understanding how to design algorithms that are computationally more efficient and parallelizable than sorting-based approaches.

In this section, we describe the construction of the reference density profile. We start in section 3a by deriving a general equation of the form *f*[*ρ*_{r}(*z*)] = 0, which can be easily solved numerically for the reference density profile *ρ*_{r}(*z*). In section 3b, we discuss how compressibility may lead to a special situation that may arise when solving *f*[*ρ*_{r}(*z*)] = 0 for *ρ*_{r}(*z*). We then describe the two choices that can be made for arriving at a solution of this equation in section 3c.

### a. An implicit equation for ρ_{r}*(*z*)* based on linking volume in physical space and thermohaline space

We now set out to define *ρ*_{r}(*z*) by linking the volumetric properties of the ocean stratification to the thermophysical properties of seawater in temperature–salinity (thermohaline) space. This is done for a Boussinesq ocean at some instant in time *t*, with potential or Conservative Temperature *x*, latitude *y*, and depth *z* increasing upward, with the ocean surface being located at *z* = 0. For brevity, we omit the time dependence in the following. In this paper, the density of a fluid parcel is estimated using the nonlinear equation of state of Jackett and McDougall (1995), which is defined in terms of potential temperature, practical salinity, and pressure.

*z*, which can be obtained from the knowledge of the ocean bathymetry by defining depth

*ρ*at constant pressure

*p*is best viewed as the locus in

*z*only, the salinity curve of equation

*z*alone, which is indicated by the tilde over

*S*. As a result, a mathematical expression for the volume

*σ*as per Eq. (31) is one way to avoid the ambiguity and the double counting of water masses and allows the bottom-up and top-down approaches discussed below to be connected with the bottom-up and top-down sorting strategies discussed later in the text.

We construct a normalized volume frequency distribution function *World Ocean Atlas* (*WOA09*; Locarnini et al. 2010; Antonov et al. 2010). Data from the *WOA09* were classified on a discretized *d*Θ = 0.005°C and *dS* = 0.002 practical salinity units (psu). The *WOA09* dataset is defined on a vertical grid consisting of 33 discrete levels separated by *dz* varying from 10 m at the surface to 500 m at the bottom of the ocean. We use the *WOA09* vertical grid but with higher resolution in the first 10 m, where we defined levels every 1 m, resulting in a vertical grid with 42 levels. Values for thermodynamic constants were obtained from McDougall and Barker (2011), and we used gravity *g* = 9.81 m s^{−2} and reference density *ρ*_{0} = 1027.0 kg m^{−3}. The left-hand side of Eq. (29) was calculated using topography from the gridded global relief model of Earth’s surface in Amante and Eakins (2009), assuming a sphere radius of *R* = 6.4 × 10^{6} m. We solve Eq. (29) at each of the 42 levels of our vertical grid by integrating from the surface toward the bottom of the ocean. The reference density profile obtained this way is shown in Fig. 2. In the remainder of this paper, we will discuss the physical properties of this profile.

### b. Effects of compressibility: Intersection of salinity curves

For a given target depth

*WOA09*annual-mean climatology, Fig. 4 shows that a large volume of the ocean is concentrated in a very small region in

Mathematically,

### c. Top-down versus bottom-up approaches

There are two ways to construct

### d. Topographic barriers

Topographic enclosures may trap dense fluids and isolate them from flowing into other ocean basins. The effects that topographic barriers may have on APE or the reference state are ignored here in the sense that all parcels are treated as if they were all in the same ocean basin. The reader is referred to Stewart et al. (2014) for a more detailed discussion of the effects of topographic barriers.

## 4. Position of a water parcel in the reference state

*S*and

Two distinct regions can be identified in the reference state profile *WOA09* annual-mean climatology (Fig. 5, lower panel). In the upper 1 km of the reference state profile, the adiabatic change in in situ density with depth of a water parcel occupying a given level is significantly smaller than the rate at which the reference density increases with depth: *z* < −1 km on the other hand,

The condition

The existence of the overlap region and two stable solutions explains why, in general, the bottom-up and top-down algorithms discussed previously yield two distinct reference density profiles. This is because the top-down approach naturally selects the stable solution with the largest

Based on our analysis of the *WOA09* annual-mean climatology, we find that the volume of water masses found in the overlap region represents a negligible fraction of the overall ocean volume. It follows that the reference position for a large majority of the fluid parcels is unique and stable. The volume of the water parcels inside the overlap region is 0.0023% of the global ocean volume in the *WOA09* dataset. The vast majority of these water parcels are Antarctic surface waters, in the Southern Ocean south of 60°S, with the remaining volume located in the Arctic Ocean (Figs. 7, 8). All parcels with three possible reference levels are within the upper 600 m of the surface of the ocean, most of them above 200 m (Fig. 7). In other words, their actual location is at much shallower depths than their shallow reference level

Cabbeling, the densification that occurs when water masses with different

By systematically implementing the procedure to obtain the reference level described so far, we are able to calculate *WOA09* annual-mean climatology. The results of these calculations have a remarkable resemblance to the calculations by Saenz et al. (2012) for an idealized ocean basin with realistic forcing. Water parcels with the deepest reference level (lowest

## 5. The reference configuration obtained by sorting water parcels

The procedures to calculate *k* at level *k* after being “flattened” out to occupy the volume of the ocean at that level. This pressure is used to calculate the density that the unsorted parcels would have if placed at level *m*, is assigned to level *WOA09* annual-mean climatology we analyzed; however, parcels with the same *m* divided by the area of the ocean at level *k*,

The special case (extremely unlikely, however, as we have found with the *WOA09* data) in which more than one water parcel has the same density at level

By assigning a parcel to a given level *k*, an adiabatic sorting scheme is equivalent to solving Eq. (37) for

As a result, when the above restrictions are enforced, the reference density curve obtained by integrating Eq. (29) with *WOA09* annual-mean climatology by sorting from the surface toward the bottom is −8.394 82 × 10^{25} J, which is 1.4 × 10^{−4} % higher than

If the top-to-bottom sorting scheme and the volume integration procedure solving Eq. (29) each use the highest possible vertical resolution, given by the volume of each individual water parcel grid cell divided by the area of the ocean at each level, the difference between the two methods should be accounted for by the differences induced by not including parcels in the overlap region more than once in the sorting scheme. This is confirmed in Fig. 2, which shows the reference density profile obtained using the sorting scheme at the highest resolution possible, along with the four other reference density profiles obtained as follows: with the volume frequency distribution method by integrating from the bottom toward the top onto 42 levels and with the volume frequency distribution method by integrating from the top toward the bottom onto 42 levels and onto levels spaced at 10 and 0.1 m. Since the overlap region obtained using the *WOA09* data is so small, the difference between the reference density profiles obtained with high vertical resolution with each method is very small, as shown in the bottom panel of Fig. 2. The error induced by coarser-resolution calculations is also small.

Existing sorting schemes are computationally very expensive. For example, the most accurate sorting-based approach one can conceive of uses as many target depths *i*th iteration is *k*. Once

## 6. Consequence for estimates of the ocean APE

By considering a fully compressible ocean, including possible volume changes upon adiabatic rearrangement, Huang (2005) previously estimated the oceanic average APE to be 624.2 J m^{−3}. Since the total ocean volume is approximately ^{20} J. In contrast, by computing the APE density and estimating its volume integral, we estimate that the total APE is 1.08 × 10^{21} J, which yields an average APE value of 834 J m^{−3} (assuming the same value for *WOA09* database). From a numerical viewpoint, the estimate for the total APE in Huang (2005) is based on computing the difference in potential energy between the actual state and the reference state, that is, between two extremely large numbers, which is potentially ill conditioned and prone to numerical errors [see also the relevant discussion by Winters and Barkan (2013)]. In contrast, our approach is based on integrating only relatively small positive numbers and is therefore expected to be better conditioned numerically. Whether this is sufficient to account for the differences between our estimates for APE and that of Huang (2005) is unclear and warrants further study. This is beyond the scope of this work, as the computational issues associated with deriving accurate estimates of APE require a full separate study of a more technical nature.

## 7. Summary and conclusions

We have discussed the effects that compressibility and nonlinearities of the equation of state of seawater have on the reference state and have illustrated these effects using annual climatological data for temperature and salinity in the ocean. Most of the water parcels in the ocean have a single, well-defined reference level in the reference state. Variations in compressibility with temperature and salinity cause a very small volume fraction of the ocean to have up to two stable positions in the reference state profile. We argue that because these volumes are located at high latitudes and shallow depths, one reference level, the shallowest, is energetically more accessible.

Our formulation, through Eqs. (29) and (32), allows us to strongly constrain (to a desired level of precision) the range of temperature and salinity properties that a water parcel at a given level must have in the reference state. Because we know either the overall minimum (surface) or maximum (bottom) density in the reference state ocean, we can proceed to efficiently construct the reference density profile by progressively filling the ocean volume adiabatically either downward from the surface or upward from the bottom.

We show that the adiabatic sorting schemes are equivalent to classifying water masses using the volume frequency distribution in temperature and salinity space, provided that the latter accounts for the water parcels that have been assigned a location in the reference profile. Because of the multiple equilibrium positions that some water parcels may have, different methods can yield different reference states. The differences between these profiles are so small that, within the uncertainty of available data from simulations and from climatologies, the reference state profile can be thought of as well defined for all practical purposes. Uniqueness can be enforced if one chooses to define the reference state as that obtained by filling the ocean volume downward from the surface, which we have argued is more energetically accessible than the one resulting from filling the ocean upward from the bottom of the ocean.

Our work here, along with the results in Stewart et al. (2014), provides the foundation for the investigation of the available potential energy budget of the ocean circulation under realistic conditions.

## Acknowledgments

Discussions with R. W. Griffiths, A. M. Hogg, Y. Dossmann, S. Downes, and T. McDougall are gratefully acknowledged. Comments by three anonymous reviewers have led to significant improvements on the quality of this paper. JAS was supported by Australian Research Council Grants DP1094542 and DP120102744 for this work. RT was supported by NERC Grant NE/K016083/1, “Improving simple climate models through a traceable and process-based analysis of ocean heat uptake (INSPECT).” GOH was supported by Australian Research Council Future Fellowship FT100100869.

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