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## Abstract

The use of density surfaces in the analysis of oceanographic data and in models of the ocean circulation is widespread. The present best method of fitting these isopycnal surfaces to hydrographic data is based on a linked sequence of potential density surfaces referred to a discrete set of reference pressures. This method is both time consuming and cumbersome in its implementation. In this paper the authors introduce a new density variable, neutral density *γ*
^{n}, which is a continuous analog of these discretely referenced potential density surfaces. The level surfaces of *γ*
^{n} form neutral surfaces, which are the most appropriate surfaces within which an ocean model’s calculations should be performed or analyzed. The authors have developed a computational algorithm for evaluating *γ*
^{n} from a given hydrographic observation so that the formation of neutral density surfaces requires a simple call to a computational function. Neutral density is of necessity not only a function of the three state variables: salinity, temperature, and pressure, but also of longitude and latitude. The spatial dependence of *γ*
^{n} is achieved by accurately labeling a global hydrographic dataset with neutral density. Arbitrary hydrographic data can then be labeled with reference to this global *γ*
^{n} field. The global dataset is derived from the Levitus climatology of the world’s oceans, with minor modifications made to ensure static stability and an adequate representation of the densest seawater. An initial field of *γ*
^{n} is obtained by solving, using a combination of numerical techniques, a system of differential equations that describe the fundamental neutral surface property. This global field of *γ*
^{n} values is further iterated in the characteristic coordinate system of the neutral surfaces to reduce any errors incurred during this solution procedure and to distribute the inherent path-dependent error associated with the definition of neutral surfaces over the entire globe. Comparisons are made between neutral surfaces calculated from *γ*
^{n} and the present best isopycnal surfaces along independent sections of hydrographic data. The development of this neutral density variable increases the accuracy of the best-practice isopycnal surfaces currently in use but, more importantly, provides oceanographers with a much easier method of fitting such surfaces to hydrographic data.

## Abstract

The use of density surfaces in the analysis of oceanographic data and in models of the ocean circulation is widespread. The present best method of fitting these isopycnal surfaces to hydrographic data is based on a linked sequence of potential density surfaces referred to a discrete set of reference pressures. This method is both time consuming and cumbersome in its implementation. In this paper the authors introduce a new density variable, neutral density *γ*
^{n}, which is a continuous analog of these discretely referenced potential density surfaces. The level surfaces of *γ*
^{n} form neutral surfaces, which are the most appropriate surfaces within which an ocean model’s calculations should be performed or analyzed. The authors have developed a computational algorithm for evaluating *γ*
^{n} from a given hydrographic observation so that the formation of neutral density surfaces requires a simple call to a computational function. Neutral density is of necessity not only a function of the three state variables: salinity, temperature, and pressure, but also of longitude and latitude. The spatial dependence of *γ*
^{n} is achieved by accurately labeling a global hydrographic dataset with neutral density. Arbitrary hydrographic data can then be labeled with reference to this global *γ*
^{n} field. The global dataset is derived from the Levitus climatology of the world’s oceans, with minor modifications made to ensure static stability and an adequate representation of the densest seawater. An initial field of *γ*
^{n} is obtained by solving, using a combination of numerical techniques, a system of differential equations that describe the fundamental neutral surface property. This global field of *γ*
^{n} values is further iterated in the characteristic coordinate system of the neutral surfaces to reduce any errors incurred during this solution procedure and to distribute the inherent path-dependent error associated with the definition of neutral surfaces over the entire globe. Comparisons are made between neutral surfaces calculated from *γ*
^{n} and the present best isopycnal surfaces along independent sections of hydrographic data. The development of this neutral density variable increases the accuracy of the best-practice isopycnal surfaces currently in use but, more importantly, provides oceanographers with a much easier method of fitting such surfaces to hydrographic data.

## Abstract

Orthobaric density has recently been advanced as a new density variable for displaying ocean data and as a coordinate for ocean modeling. Here the extent to which orthobaric density surfaces are neutral is quantified and it is found that orthobaric density surfaces are less neutral in the World Ocean than are potential density surfaces referenced to 2000 dbar. Another property that is important for a vertical coordinate of a layered model is the quasi-material nature of the coordinate and it is shown that orthobaric density surfaces are significantly non-quasi-material. These limitations of orthobaric density arise because of its inability to accurately accommodate differences between water masses at fixed values of pressure and in situ density such as occur between the Northern and Southern Hemisphere portions of the World Ocean. It is shown that special forms of orthobaric density can be quite accurate if they are formed for an individual ocean basin and used only in that basin. While orthobaric density can be made to be approximately neutral in a single ocean basin, this is not possible in both the Northern and Southern Hemisphere portions of the Atlantic Ocean. While the helical nature of neutral trajectories (equivalently, the ill-defined nature of neutral surfaces) limits the neutrality of all types of density surface, the inability of orthobaric density surfaces to accurately accommodate more than one ocean basin is a much greater limitation.

## Abstract

Orthobaric density has recently been advanced as a new density variable for displaying ocean data and as a coordinate for ocean modeling. Here the extent to which orthobaric density surfaces are neutral is quantified and it is found that orthobaric density surfaces are less neutral in the World Ocean than are potential density surfaces referenced to 2000 dbar. Another property that is important for a vertical coordinate of a layered model is the quasi-material nature of the coordinate and it is shown that orthobaric density surfaces are significantly non-quasi-material. These limitations of orthobaric density arise because of its inability to accurately accommodate differences between water masses at fixed values of pressure and in situ density such as occur between the Northern and Southern Hemisphere portions of the World Ocean. It is shown that special forms of orthobaric density can be quite accurate if they are formed for an individual ocean basin and used only in that basin. While orthobaric density can be made to be approximately neutral in a single ocean basin, this is not possible in both the Northern and Southern Hemisphere portions of the Atlantic Ocean. While the helical nature of neutral trajectories (equivalently, the ill-defined nature of neutral surfaces) limits the neutrality of all types of density surface, the inability of orthobaric density surfaces to accurately accommodate more than one ocean basin is a much greater limitation.

## Abstract

It is shown that the ocean’s hydrography occupies little volume in the three-dimensional space defined by salinity–temperature–pressure (*S*–Θ–*p*), and the implications of this observation for the mean vertical transport across density surfaces are discussed. Although ocean data have frequently been analyzed in the two-dimensional temperature–salinity (*S*–Θ) diagram where casts of hydrographic data are often locally tight in *S*–Θ space, the relatively empty nature of the World Ocean in the three-dimensional *S*–Θ–*p* space seems not to have received attention. The World Ocean’s data lie close to a single surface in this three-dimensional space, and it is shown that this explains the known smallness of the ambiguity in defining neutral surfaces. The ill-defined nature of neutral surfaces means that lateral motion along neutral trajectories leads to mean vertical advection through density surfaces, even in the absence of small-scale mixing processes. The situation in which the ocean’s hydrography occupies a large volume in *S*–Θ–*p* space is also considered, and it is suggested that the consequent vertical diapycnal advection would be sufficiently large that the ocean would not be steady.

## Abstract

It is shown that the ocean’s hydrography occupies little volume in the three-dimensional space defined by salinity–temperature–pressure (*S*–Θ–*p*), and the implications of this observation for the mean vertical transport across density surfaces are discussed. Although ocean data have frequently been analyzed in the two-dimensional temperature–salinity (*S*–Θ) diagram where casts of hydrographic data are often locally tight in *S*–Θ space, the relatively empty nature of the World Ocean in the three-dimensional *S*–Θ–*p* space seems not to have received attention. The World Ocean’s data lie close to a single surface in this three-dimensional space, and it is shown that this explains the known smallness of the ambiguity in defining neutral surfaces. The ill-defined nature of neutral surfaces means that lateral motion along neutral trajectories leads to mean vertical advection through density surfaces, even in the absence of small-scale mixing processes. The situation in which the ocean’s hydrography occupies a large volume in *S*–Θ–*p* space is also considered, and it is suggested that the consequent vertical diapycnal advection would be sufficiently large that the ocean would not be steady.

## Abstract

Hydrographic data, be it raw or highly averaged observational data, contain substantial regions having vertical density inversions. An algorithm is described that minimally modifies such data so that the resulting hydrographic casts have vertical buoyancy frequency profiles larger than a specified lower bound. The method underlying the algorithm is based on the solution of a constrained weighted least-squares problem and maximizes the smoothness of the resulting salinity-potential temperature diagram. Examples are provided that demonstrate the effectiveness of the technique in minimally altering hydrographic data only in the immediate vicinity of the data that do not already satisfy the buoyancy frequency constraint.

A modified equation of state, identical in form to the international equation of state of seawater but written in terms of potential rather than in situ temperature, is also provided, enabling rapid computation of the thermal expansion and saline contraction coefficients.

## Abstract

Hydrographic data, be it raw or highly averaged observational data, contain substantial regions having vertical density inversions. An algorithm is described that minimally modifies such data so that the resulting hydrographic casts have vertical buoyancy frequency profiles larger than a specified lower bound. The method underlying the algorithm is based on the solution of a constrained weighted least-squares problem and maximizes the smoothness of the resulting salinity-potential temperature diagram. Examples are provided that demonstrate the effectiveness of the technique in minimally altering hydrographic data only in the immediate vicinity of the data that do not already satisfy the buoyancy frequency constraint.

A modified equation of state, identical in form to the international equation of state of seawater but written in terms of potential rather than in situ temperature, is also provided, enabling rapid computation of the thermal expansion and saline contraction coefficients.

## Abstract

A comparison is made of the meridional overturning circulation in a coarse-resolution World Ocean model when the integration is performed along (i) level, (ii) potential density, and (iii) neutral density surfaces. In the level-surface calculation, all the usual cells are evident, including the Atlantic “conveyor,” the Deacon cell, and the direct Antarctic cell. In the potential or neutral density calculations, all cells remain present; however, the Deacon cell is greatly reduced in strength (to just a few Sverdrups). An analysis of the thermodynamics underlying the dianeutral motion is conducted. Most dianeutral motion results from fluxes associated with the vertical diffusivity and the (unphysical) horizontal diffusivity. Caballing is not important, despite the inclusion of isopycnal diffusivity. The mechanism of the residual Deacon cell involves densification near 40°5 resulting from fluxes associated with the horizontal diffusivity. Horizontal diffusivity results in substantial dianeutral motion in several other parts of the ocean. Most significant is motion toward lesser density in the far Southern Ocean, which integrates zonally and between 67°S and 57°S to give a transport of about 25 Sv across density surfaces. This transport dominates other dianeutral transports at high density in the ocean interior and indicates serious distortion of the solution by the horizontal diffusivity.

A second model run is conducted where the horizontal diffusivity is reduced to near the (experimentally determined) limit for the numerical integrity of water properties on the large scale. Dianeutral transports associated with horizontal diffusivity generally decline modestly. In neutral density coordinates, the Deacon cell now vanishes almost completely. The Deacon cell of the level-surface integration results mainly from large-scale isopycnal motions occurring on sloping density surfaces, which superpose to yield a cell upon zonal integration at constant depth. Finally, it is apparent that the neutral density coordinate provides a clearer picture of the ocean circulation than do potential density coordinates, because of inherent ambiguity in choosing the reference pressure of potential density.

## Abstract

A comparison is made of the meridional overturning circulation in a coarse-resolution World Ocean model when the integration is performed along (i) level, (ii) potential density, and (iii) neutral density surfaces. In the level-surface calculation, all the usual cells are evident, including the Atlantic “conveyor,” the Deacon cell, and the direct Antarctic cell. In the potential or neutral density calculations, all cells remain present; however, the Deacon cell is greatly reduced in strength (to just a few Sverdrups). An analysis of the thermodynamics underlying the dianeutral motion is conducted. Most dianeutral motion results from fluxes associated with the vertical diffusivity and the (unphysical) horizontal diffusivity. Caballing is not important, despite the inclusion of isopycnal diffusivity. The mechanism of the residual Deacon cell involves densification near 40°5 resulting from fluxes associated with the horizontal diffusivity. Horizontal diffusivity results in substantial dianeutral motion in several other parts of the ocean. Most significant is motion toward lesser density in the far Southern Ocean, which integrates zonally and between 67°S and 57°S to give a transport of about 25 Sv across density surfaces. This transport dominates other dianeutral transports at high density in the ocean interior and indicates serious distortion of the solution by the horizontal diffusivity.

A second model run is conducted where the horizontal diffusivity is reduced to near the (experimentally determined) limit for the numerical integrity of water properties on the large scale. Dianeutral transports associated with horizontal diffusivity generally decline modestly. In neutral density coordinates, the Deacon cell now vanishes almost completely. The Deacon cell of the level-surface integration results mainly from large-scale isopycnal motions occurring on sloping density surfaces, which superpose to yield a cell upon zonal integration at constant depth. Finally, it is apparent that the neutral density coordinate provides a clearer picture of the ocean circulation than do potential density coordinates, because of inherent ambiguity in choosing the reference pressure of potential density.

## Abstract

Warming of the atmosphere as a result of an increased concentration of greenhouse gases is expected to lead to a significant rise is global sea level. We present estimates of the component of this sea level rise caused by thermal expansion of the ocean. These estimates are based on the idea that the upper layers of the main gyres of the ocean are ventilated by the subduction of water at higher latitudes and its subsequent equatorward and downward flow into the main thermocline along surfaces of constant “density”. In this mechanism, heat enters the ocean by an advection process rather than by vertical diffusion, as in previous estimates of the component of sea level rise that is caused by thermal expansion. After the heat initially enters the subtropical gyres by subduction, it is then redistributed to preserve gradients of the depth-integrated pressure field, by an adjustment involving low vertical-mode baroclinic waves. Estimates of historical sea level rise based on this simple ventilation scheme, when combined with estimates of nonpolar glacial melt, are about equal to the observed sea level rise. For a global mean 3.0°C (1.5°C, 4.5°C) temperature rise by 2050 (and with the spatial distribution predicted by three climate models), we estimate the component of sea level rise that is caused by thermal expansion to be about 0.2 to 0.3 m (0.1 m, 0.4 m) by 2050. Low-mode internal Rossby and Kelvin waves appear to be quite efficient at distributing the sea level rise evenly over the earth without major distortions to the thermocline. A delayed warming, as suggested by transient coupled ocean-atmosphere models, can be simulated by using a smaller temperature rise, say 1.5°C rather than 3.0°C, by 2050. Changes in sea level arising from variations in the wind field could be estimated, but so far our calculations are based on the assumption that the wind stress field does not change from its present value. We estimate the maximum rate of sea level rise caused by changes in deep water formation is 0.1 meter per century. Contributions from the cryosphere reported in the literature range from near zero to about 0.35 m. When added to the thermal expansion components, our total sea level rise scenario for 2050 for a temperature rise of 3.0°C (1.5°C to 4.5°C) is about 0.35 m (0.15 and 0.70 m).

## Abstract

Warming of the atmosphere as a result of an increased concentration of greenhouse gases is expected to lead to a significant rise is global sea level. We present estimates of the component of this sea level rise caused by thermal expansion of the ocean. These estimates are based on the idea that the upper layers of the main gyres of the ocean are ventilated by the subduction of water at higher latitudes and its subsequent equatorward and downward flow into the main thermocline along surfaces of constant “density”. In this mechanism, heat enters the ocean by an advection process rather than by vertical diffusion, as in previous estimates of the component of sea level rise that is caused by thermal expansion. After the heat initially enters the subtropical gyres by subduction, it is then redistributed to preserve gradients of the depth-integrated pressure field, by an adjustment involving low vertical-mode baroclinic waves. Estimates of historical sea level rise based on this simple ventilation scheme, when combined with estimates of nonpolar glacial melt, are about equal to the observed sea level rise. For a global mean 3.0°C (1.5°C, 4.5°C) temperature rise by 2050 (and with the spatial distribution predicted by three climate models), we estimate the component of sea level rise that is caused by thermal expansion to be about 0.2 to 0.3 m (0.1 m, 0.4 m) by 2050. Low-mode internal Rossby and Kelvin waves appear to be quite efficient at distributing the sea level rise evenly over the earth without major distortions to the thermocline. A delayed warming, as suggested by transient coupled ocean-atmosphere models, can be simulated by using a smaller temperature rise, say 1.5°C rather than 3.0°C, by 2050. Changes in sea level arising from variations in the wind field could be estimated, but so far our calculations are based on the assumption that the wind stress field does not change from its present value. We estimate the maximum rate of sea level rise caused by changes in deep water formation is 0.1 meter per century. Contributions from the cryosphere reported in the literature range from near zero to about 0.35 m. When added to the thermal expansion components, our total sea level rise scenario for 2050 for a temperature rise of 3.0°C (1.5°C to 4.5°C) is about 0.35 m (0.15 and 0.70 m).

## Abstract

An equation of state for seawater is presented that contains 25 terms and is an excellent fit to the Feistel and Hagen equation of state. It is written in terms of potential temperature (rather than in situ temperature), as required for efficient ocean model integrations. The maximum density error of the fit is 3 × 10^{–3} kg m^{–3} in the oceanographic ranges of temperature, salinity, and pressure. The corresponding maximum error in the thermal expansion coefficient is 4 × 10^{–7} °C^{–1}, which is a factor of 12 less than the corresponding maximum difference between the Feistel and Hagen equation of state and the widely used but less accurate international equation of state.

A method is presented to convert between potential temperature and in situ temperature using specific entropy based on the Gibbs function of Feistel and Hagen. The resulting values of potential temperature are substantially more accurate than those based on the lapse rate derived from the international equation of state.

## Abstract

An equation of state for seawater is presented that contains 25 terms and is an excellent fit to the Feistel and Hagen equation of state. It is written in terms of potential temperature (rather than in situ temperature), as required for efficient ocean model integrations. The maximum density error of the fit is 3 × 10^{–3} kg m^{–3} in the oceanographic ranges of temperature, salinity, and pressure. The corresponding maximum error in the thermal expansion coefficient is 4 × 10^{–7} °C^{–1}, which is a factor of 12 less than the corresponding maximum difference between the Feistel and Hagen equation of state and the widely used but less accurate international equation of state.

A method is presented to convert between potential temperature and in situ temperature using specific entropy based on the Gibbs function of Feistel and Hagen. The resulting values of potential temperature are substantially more accurate than those based on the lapse rate derived from the international equation of state.

## Abstract

Algorithms are presented for density, potential temperature, conservative temperature, and the freezing temperature of seawater. The algorithms for potential temperature and density (in terms of potential temperature) are updates to routines recently published by McDougall et al., while the algorithms involving conservative temperature and the freezing temperatures of seawater are new. The McDougall et al. algorithms were based on the thermodynamic potential of Feistel and Hagen; the algorithms in this study are all based on the “new extended Gibbs thermodynamic potential of seawater” of Feistel. The algorithm for the computation of density in terms of salinity, pressure, and conservative temperature produces errors in density and in the corresponding thermal expansion coefficient of the same order as errors for the density equation using potential temperature, both being twice as accurate as the International Equation of State when compared with Feistel’s new equation of state. An inverse function relating potential temperature to conservative temperature is also provided. The difference between practical salinity and absolute salinity is discussed, and it is shown that the present practice of essentially ignoring the difference between these two different salinities is unlikely to cause significant errors in ocean models.

## Abstract

Algorithms are presented for density, potential temperature, conservative temperature, and the freezing temperature of seawater. The algorithms for potential temperature and density (in terms of potential temperature) are updates to routines recently published by McDougall et al., while the algorithms involving conservative temperature and the freezing temperatures of seawater are new. The McDougall et al. algorithms were based on the thermodynamic potential of Feistel and Hagen; the algorithms in this study are all based on the “new extended Gibbs thermodynamic potential of seawater” of Feistel. The algorithm for the computation of density in terms of salinity, pressure, and conservative temperature produces errors in density and in the corresponding thermal expansion coefficient of the same order as errors for the density equation using potential temperature, both being twice as accurate as the International Equation of State when compared with Feistel’s new equation of state. An inverse function relating potential temperature to conservative temperature is also provided. The difference between practical salinity and absolute salinity is discussed, and it is shown that the present practice of essentially ignoring the difference between these two different salinities is unlikely to cause significant errors in ocean models.