• Boyer, T., S. Levitus, H. Garcia, R. A. Locarnini, C. Stephens, and J. Antonov, 2005: Objective analyses of annual, seasonal, and monthly temperature and salinity for the World Ocean on a 0.25° grid. Int. J. Climatol., 25, 931945, doi:10.1002/joc.1173.

    • Crossref
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  • Chu, P. C., and C. W. Fan, 2010: A conserved minimal adjustment scheme for stabilization of hydrographic profiles. J. Atmos. Oceanic Technol., 27, 10721083, doi:10.1175/2010JTECHO742.1.

    • Crossref
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  • de Boyer Montégut, C., G. Madec, A. S. Fischer, A. Lazar, and D. Iudicone, 2004: Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology. J. Geophys. Res., 109, C12003, doi:10.1029/2004JC002378.

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  • IOC, SCOR, and IAPSO, 2010: The International Thermodynamic Equation of Seawater—2010: Calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides 56, 220 pp. [Available online at http://www.teos-10.org/pubs/TEOS-10_Manual.pdf.]

  • Jackett, D. J., and T. J. McDougall, 1995: Minimal adjustment of hydrographic profiles to achieve static stability. J. Atmos. Oceanic Technol., 12, 381389, doi:10.1175/1520-0426(1995)012<0381:MAOHPT>2.0.CO;2.

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  • Levitus, S., 1982: Climatological Atlas of the World Ocean. NOAA Prof. Paper 13, 173 pp. and 17 microfiche.

  • Locarnini, R. A., and Coauthors, 2013: Temperature. Vol. 1, World Ocean Atlas 2013, NOAA Atlas NESDIS 73, 40 pp. [Available online at http://data.nodc.noaa.gov/woa/WOA13/DOC/woa13_vol1.pdf.]

  • McDougall, T. J., and P. M. Barker, 2011: Getting Started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox. SCOR/IAPSO Working Group 127, 28 pp.

  • McDougall, T. J., and P. M. Barker, 2014: Comment on “Buoyancy frequency profiles and internal semidiurnal tide turning depths in the oceans” by B. King et al. J. Geophys. Res. Oceans, 119, 90269032, doi:10.1002/2014JC010066.

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  • Ridgway, K. R., J. R. Dunn, and J. L. Wilkin, 2002: Ocean interpolation by four-dimensional weighted least squares—Application to the waters around Australia. J. Atmos. Oceanic Technol., 19, 13571375, doi:10.1175/1520-0426(2002)019<1357:OIBFDW>2.0.CO;2.

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  • View in gallery

    (a) Unstable cast locations in CARS2009 at which there is at least one vertically unstable bottle pair. (b) Histogram of the number of instabilities as a function of depth in CARS 2009; the N2 values have been calculated at the midlevel depths.

  • View in gallery

    (a) Conservative Temperature of the water column and (b) Absolute Salinity of the water column, which show the original profile (blue dashed line), the stabilized profile from the Chu and Fan (2010) method (red solid line), and our constrained stabilization (black line). Adjustments (original minus stabilized) using the Chu and Fan (2010) method and our constrained method for (c) the Conservative Temperature change and (d) the Absolute Salinity change.

  • View in gallery

    (a) Diagrams of for a cast from Levitus (1982) in the South Pacific Ocean (48°S, 200°E), before and after stabilization. (b) A zoomed-in view of the section of the water column that contained a vertical inversion. Term of J&McD95 is the defined in Jackett and McDougall (1995). (c) Profiles before and after stabilization. (d) Zoom-in view of the section of the water column that contained an inversion; the is defined in Jackett and McDougall (1995; gray dashed line). The colors of the lines correspond to the curves in (a) and (b). (e) Adjustments made to each bottle in the cast to achieve an N2 minimum as defined by Jackett and McDougall (1995) when scaled by and . Results from Jackett and McDougall (1995; red stars) and changes when using the constrained method of this paper (black circles). (f) Adjustments made to each bottle in the cast to achieve an minimum as defined by Jackett and McDougall (1995) when scaled by and [from Eq. (9)], which represent a smoothed version of the curve. Results from Jackett and McDougall (1995; red stars) and changes when using the constrained method of this paper (black circles).

  • View in gallery

    (a) Diagram of for a profile extracted from CARS2009 in the Pacific Ocean. (b)–(d) Zoomed-in view of the unstable sections of the cast. (e) Temperature and salinity adjustments, scaled by and , made to the each bottle in the cast to achieve an of . (f) Temperature and salinity adjustments, scaled by and of the smoothed curve, to achieve an of .

  • View in gallery

    (a) Diagram of for a profile extracted from CARS2009 in the Pacific Ocean. (b) Zoomed-in view of the unstable sections of the cast. (c) Temperature and salinity adjustments, scaled by and of the smoothed curve, to achieve an of .

  • View in gallery

    (a) Diagram of for a profile extracted from CARS2009 in the Pacific Ocean. (b) Zoomed-in view of the unstable sections of the cast. (c) Temperature and salinity adjustments, scaled by and of the smoothed curve, to achieve an of .

  • View in gallery

    Histogram of the wiggliness of the stabilized casts from CARS2009. (a) The term of . (b) Zoomed-in view of (a). (c) The term of 10% of the buoyancy frequencies in observed profiles collected in the Southern Ocean, south of 50°S. (d) Zoomed-in view of (c).

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Stabilizing Hydrographic Profiles with Minimal Change to the Water Masses

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  • 1 School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia
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Abstract

Both observed and averaged oceanographic data often contain regions with density inversions. This paper presents two methods of stabilizing a water column. The first method is intended for use with observed data; it minimally adjusts Absolute Salinity while leaving the values of in situ temperature unchanged. The second method adjusts the values of both Absolute Salinity and Conservative Temperature, and these adjustments are made in such a way as to cause the least possible damage to the water-mass structure of the vertical cast.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Paul M. Barker, paul.barker@unsw.edu.au

Abstract

Both observed and averaged oceanographic data often contain regions with density inversions. This paper presents two methods of stabilizing a water column. The first method is intended for use with observed data; it minimally adjusts Absolute Salinity while leaving the values of in situ temperature unchanged. The second method adjusts the values of both Absolute Salinity and Conservative Temperature, and these adjustments are made in such a way as to cause the least possible damage to the water-mass structure of the vertical cast.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Paul M. Barker, paul.barker@unsw.edu.au

1. Introduction

Many of the profiles oceanographers use contain density inversions, which if not corrected can lead to erroneous results, particularly when considering isopycnal analyses. The vertical stability of a water column is described by the square of the buoyancy frequency . In this paper we follow the definition similar to that used in Jackett and McDougall (1995), now defined in terms of the vertical gradients of Absolute Salinity and Conservative Temperature (these being the salinity and temperature variables, respectively, of TEOS-10; IOC et al. 2010),
e1
where the vertical derivatives are taken at constant latitude and longitude with respect to absolute pressure . The gravitational acceleration is given the symbol , and is the specific volume of seawater, being the reciprocal of in situ density , and the thermal expansion and haline contraction coefficients defined with respect to Absolute Salinity and Conservative Temperature are given by
e2
There are many ways to evaluate . McDougall and Barker (2014) list six of the most commonly used definitions and show that they are all equivalent. In this paper we will also use the expression for written in terms of the vertical gradient of in situ temperature T, namely,
e3
where is the adiabatic lapse rate, and the thermal expansion and haline contraction coefficients defined with respect to and in situ temperature are given by
e4

Throughout this paper we follow the naming convention used in Jackett and McDougall (1995) so that we use the word “cast” to describe a vertical profile of either hydrographic data or data from a gridded product, and the word “bottle” to describe a data point at a particular pressure on such a cast. An instability is detected on a part of the cast where the square of the buoyancy frequency evaluated between a bottle pair is negative.

A cast usually exhibits a typical pattern; the mixed layer portion of the water column contains a minimally stable portion of the water column, while in the thermocline the water is at maximum stability, and this stability then decreases with depth below the seasonal thermocline. When isopycnal analysis is performed on oceanographic data or when an averaged dataset is used as a restoring condition for a model, the of that data must be positive, and if it is not, significant problems arise. In this paper we present numerical algorithms that minimally adjust a cast to achieve a profile of the minimum values of provided by the user.

The Commonwealth Scientific and Industrial Research Organisation (CSIRO) Atlas of Regional Seas (CARS) is a climatological atlas based on a four-dimensional least squares interpolation technique (Ridgway et al. 2002). The atlas contains a mean field along with annual and semiannual harmonics that are summed to produce seasonally varying temperature and salinity fields on a regular 1/2° grid. Calculating the buoyancy frequency of the mean field from the most recent version of the atlas (CARS2009) reveals that more than 66% of the casts contain at least one instability (Fig. 1). This is 22% higher than the number of unstable casts contained in the original Levitus (1982) atlas that Jackett and McDougall (1995) used to demonstrate their stabilization method. These density inversions arise, as the salinity and temperature values were calculated independently of each other on each of the constant depth surfaces.

Fig. 1.
Fig. 1.

(a) Unstable cast locations in CARS2009 at which there is at least one vertically unstable bottle pair. (b) Histogram of the number of instabilities as a function of depth in CARS 2009; the N2 values have been calculated at the midlevel depths.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

Hydrographic observational data also contain instabilities, with some of these being real while others being due to instrumental difficulties, such as salinity spiking.

Historical correction methods

Jackett and McDougall (1995) proposed a method for correcting instabilities that involved a constrained least squares technique that maximized the smoothness of the temperature–salinity profile. However, this method required a predefined independent weighting of salinity and temperature to reduce unrealistic water-mass changes in the resulting diagram. The software provided with Jackett and McDougall (1995) required access to a mathematical Fortran library that is not widely used in oceanography.

The World Ocean Atlas 2005 (WOA; Boyer et al. 2005) adopted a method that is based on the Jackett and McDougall (1995) method. In WOA either salinity or temperature is adjusted based on the signs of the vertical gradients of Practical Salinity and temperature of a bottle pair: When the temperature gradient was positive, the temperature was adjusted; when the Practical Salinity gradient was negative, the salinity was adjusted; and when the Practical Salinity gradient was negative and the temperature gradient was positive, both the temperature and Practical Salinity were adjusted (Locarnini et al. 2013).

Another method is that of Chu and Fan (2010), which is an iterative Newton method for correcting instabilities. This method conserves both salt and heat while adjusting the static stability using predetermined temperature and salinity ratios. This iterative method starts from the bottom of the cast and works upward to the ocean surface, sequentially adjusting temperature and salinity. We note that when there is a spike or outlier near the bottom of the cast, applying this method has the undesired effect of significantly changing the structure of the whole water column (Fig. 2).

Fig. 2.
Fig. 2.

(a) Conservative Temperature of the water column and (b) Absolute Salinity of the water column, which show the original profile (blue dashed line), the stabilized profile from the Chu and Fan (2010) method (red solid line), and our constrained stabilization (black line). Adjustments (original minus stabilized) using the Chu and Fan (2010) method and our constrained method for (c) the Conservative Temperature change and (d) the Absolute Salinity change.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

We have developed two methods to remove instabilities from a water column (see the appendix). The first involves adjusting only and is intended for use with observational data. This method of stabilizing a water column is motivated by the idea that the measured in situ temperature is error free and that the Absolute Salinity contains salinity spiking errors or anomalies due to biological material passing through the conductivity sensor. In the second method, we adjust both SA and and this second method is intended for use with gridded data, such as that from hydrographic atlases or model output. This second method is designed to minimally change the water-mass structure of the diagram. Both methods will stabilize a cast to be at least as stable as a vertical profile of minimum that the user supplies.

The variables and equation of state used in this paper are based on the International Thermodynamic Equation of Seawater—2010 (TEOS-10), which are what has been recommended by the Intergovernmental Oceanographic Commission (IOC), the Scientific Committee on Oceanic Research (SCOR), and the International Association for the Physical Sciences of the Oceans (IAPSO) for use in physical oceanography. However, the concepts of how and why the adjustments are made do hold when they are applied with the International Equation of State of Seawater (EOS-80).

2. Stabilization by adjusting only Absolute Salinity, keeping in situ temperature unchanged

A single cast of observed data contains values of Absolute Salinity, in situ temperature, and pressure p, for n bottles, that is , j = 1, 2, …, n, and the first step is to evaluate for each bottle pair. We want to adjust only the Absolute Salinity of the bottles while keeping the bottles’ in situ temperatures unchanged. The values of down the cast can be compared to the lower limit that we specify, , as follows
e5
where are the differences between the absolute pressures between adjacent bottles (deepest minus shallowest). The values of b for each bottle pair will be positive if the water column is more stable than our lower limit at this depth interval, and if the value of b is negative for a bottle pair, then the absolute salinities of one or other (or both) of the bottles of the pair will need to be altered in order to increase the value of .
The motivation for the form of Eq. (5) comes from Eq. (3), realizing that we want to minimize the perturbations of Absolute Salinity (i.e., we minimize the sum of the square of all j salinity perturbations) while obeying the inequality constraint
e6
with the matrix being a sparse bidiagonal matrix of values of +1 and −1 representing the differences of the perturbation Absolute Salinities for each bottle pair, and the values of for each bottle pair being evaluated as in Eq. (5). Equation (3) suggests that if both the in situ temperatures and the values of the adiabatic lapse rate remain unchanged, then Eqs. (5) and (6) should give an accurate estimate of the required changes in Absolute Salinity for each bottle.

3. Stabilization by adjusting both Absolute Salinity and Conservative Temperature

The second method of correcting for instabilities is more complex and involves minimally adjusting both Absolute Salinity and Conservative Temperature. This time a single cast contains Absolute Salinity , and pressure p recorded for n bottles—that is, , j = 1, 2, …, n—and we minimally adjust both the Absolute Salinity by and Conservative Temperature by to achieve a stable cast that has greater than a predefined value between each pair of bottles. In this case the unknown vector in the inequality constraint involves the perturbations of both Absolute Salinity and Conservative Temperature, and the matrix consists of four diagonal bands, with elements of , , and , while is defined as
e7
The constrained least squares technique minimizes the salinity and temperature perturbations,
e8
Note that we have weighted the Conservative Temperature perturbations using the ratio to arrive at the same units as Absolute Salinity.
As described by Jackett and McDougall (1995), the abovementioned constrained least squares problem achieves a stable vertical water column but oftentimes at the expense of making unphysical changes to the water-mass structure of the water column; that is, the resulting curve of the water column can be perturbed in a way that does not seem realistic. Here we address this issue by adding a requirement that and of each bottle must exactly obey the linear constraint
e9
where the differences of Conservative Temperature and Absolute Salinity— and , respectively—are chosen so that the ratio is representative of a smoothed version of the curve of the water column.

4. Results/examples

Figure 3 depicts a Southern Ocean profile from Levitus (1982). This profile was used in Jackett and McDougall (1995) to show the effectiveness of their minimal adjustment to achieve a stable water column. We have included their adjusted profile for comparison, shown as the red line.

Fig. 3.
Fig. 3.

(a) Diagrams of for a cast from Levitus (1982) in the South Pacific Ocean (48°S, 200°E), before and after stabilization. (b) A zoomed-in view of the section of the water column that contained a vertical inversion. Term of J&McD95 is the defined in Jackett and McDougall (1995). (c) Profiles before and after stabilization. (d) Zoom-in view of the section of the water column that contained an inversion; the is defined in Jackett and McDougall (1995; gray dashed line). The colors of the lines correspond to the curves in (a) and (b). (e) Adjustments made to each bottle in the cast to achieve an N2 minimum as defined by Jackett and McDougall (1995) when scaled by and . Results from Jackett and McDougall (1995; red stars) and changes when using the constrained method of this paper (black circles). (f) Adjustments made to each bottle in the cast to achieve an minimum as defined by Jackett and McDougall (1995) when scaled by and [from Eq. (9)], which represent a smoothed version of the curve. Results from Jackett and McDougall (1995; red stars) and changes when using the constrained method of this paper (black circles).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

Figures 3a and 3b depict a profile before and after stabilization; the original data contained one inversion. Figure 3c shows the stability of the profile; note that log(N2) is plotted and there is a gap in the original profile (blue), since negative values cannot be plotted. We have included the results from the salinity-only correction and the adjustments to show the effect of not including the constraint of Eq. (9). All of the methods eliminated the instability and increased the minimum to be greater than . Of the five curves shown, four have defined by Jackett and McDougall (1995; gray dashed line in Fig. 3b) and the other has an of .

The salinity-only and the unconstrained and Θ adjustment resulted in unrealistic large changes in the resulting profile, which appear as a water-mass intrusion, and are features also observed by Jackett and McDougall (1995) when they applied only local values of and . The constrained and Θ adjustment with a low of detailed in this paper also produce a single sharp water-mass-like intrusion that was not present in the results from the version that used the from Jackett and McDougall (1995) and the output from Jackett and McDougall (1995).

Plots of the temperature changes scaled by versus the salinity changes scaled by (Fig. 3e) for the constrained and Θ method and for Jackett and McDougall (1995) show that both methods do not equally weight and adjust and Θ. Whereas the unconstrained and Θ method has the two properties equally weighted, which can be seen as the changes lie along a line angled at −45°, shown as the black dashed line, but as mentioned earlier the resulting profile contains unrealistic intrusions. This and scaled ratio of the changes (Fig. 3e) does not indicate how much damage is done to the water masses by applying the adjustments. A graphical method that does show how much damage or change is done to the structure of the water column is to scale the changes by the slopes of a smoothed profile (Fig. 3f). In this plot the least possible damage is done to the water column when the changes occur along a line sloping at 45°, shown in Fig. 3f as the dashed line. The unconstrained and Θ method makes fewer total changes to both temperature and salinity, but to do this the adjustments are at almost normal to the profile.

Figures 46 are three typical examples of the thousands of unstable casts that we have looked at from the CARS2009 climatology. Applying the Chu and Fan (2010) method, the method applied in WOA, and the constrained adjustment to a cast in the Pacific Ocean from CARS2009 (Fig. 4a) that contains three separate density inversions (Figs. 4b–d), we see that all three methods successfully remove the instability, but all methods produce slightly different results. The changes when scaled by and (Fig. 4e) reveal that none of the methods, except for the unconstrained , apply equal amounts of change to salinity and temperature. WOA corrects the density inversions by adjusting either temperature or salinity. The Chu and Fan (2010) method applies a uniform but unequal weighting for temperature and salinity, if they were equally weighted the changes would be aligned along a line at −45° as indicated by the dashed line in Fig. 4e. Our constrained method does not have a favored direction of change for the adjustments. When we apply this method to the CARS2009 Pacific Ocean cast, it shows that we do not create artificial water masses (Fig. 4f) and minimal damage to the water column occurs when the changes correspond to the dashed line. The Chu and Fan (2010) method opted to make the smallest possible changes, but in order to do so the bottles move normal to the curve. As with Fig. 4e, we can see that the WOA has relied on changing either the temperature or the salinity.

Fig. 4.
Fig. 4.

(a) Diagram of for a profile extracted from CARS2009 in the Pacific Ocean. (b)–(d) Zoomed-in view of the unstable sections of the cast. (e) Temperature and salinity adjustments, scaled by and , made to the each bottle in the cast to achieve an of . (f) Temperature and salinity adjustments, scaled by and of the smoothed curve, to achieve an of .

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

Fig. 5.
Fig. 5.

(a) Diagram of for a profile extracted from CARS2009 in the Pacific Ocean. (b) Zoomed-in view of the unstable sections of the cast. (c) Temperature and salinity adjustments, scaled by and of the smoothed curve, to achieve an of .

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

Fig. 6.
Fig. 6.

(a) Diagram of for a profile extracted from CARS2009 in the Pacific Ocean. (b) Zoomed-in view of the unstable sections of the cast. (c) Temperature and salinity adjustments, scaled by and of the smoothed curve, to achieve an of .

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

These comments related to the profile depicted in Fig. 4 also apply to Figs. 5 and 6. We have not included the figures showing the and scaled adjustments for these casts, as they do not provide a clear measure of how much change is made to the original water column.

The realism of a stabilized profile can be measured by the smoothness of the profile. A realistic result will typically be smooth and not contain large water-mass-like intrusions or wiggles. We define the amount of wiggliness as the integral of the change in angle on the curve between consecutive bottles multiplied by the geometric mean of the distance between the bottles . For the cast given by , j = 1, 2, …, n, we consider the set of difference vectors (uj, j = 1, 2, …, n−1) that are defined by
e10
and the set of changes in angle is given by
e11
and the wiggliness is defined as
e12
Here is the average value for values found in the oceanographic range.

The histogram of this wiggliness measure for the constrained and unconstrained methods when applied to the global CARS2009 data (Fig. 7) reveals that the constrained method (black line) produces profiles with approximately half the number of large deviations (fewer wiggles). Increasing the minimum stability from (Figs. 7a and 7b) to 10% of the buoyancy frequencies in observed profiles collected in the Southern Ocean, south of 50°S (Figs. 7c and 7d), produced a similar result. However, if one chooses to adopt the resulting cast with the smaller wiggliness of the two, then the histogram curve (blue line) is less than for either of the two individual methods.

Fig. 7.
Fig. 7.

Histogram of the wiggliness of the stabilized casts from CARS2009. (a) The term of . (b) Zoomed-in view of (a). (c) The term of 10% of the buoyancy frequencies in observed profiles collected in the Southern Ocean, south of 50°S. (d) Zoomed-in view of (c).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-16-0111.1

5. Summary

In this paper we have described methods for minimally adjusting a vertical cast of hydrographic data to achieve a stable cast. We have presented two different methods of correcting for instabilities. The first method is intended for use with an observed hydrographic profile that involves adjusting Absolute Salinity only. The second method is intended for adjusting either averaged data, such as a climatology, or model data. This second method involves adjusting both methods and Absolute Salinity so as to cause the least possible damage to the water masses contained within the profile.

Applying our salinity-only method corrects all of the unstable profiles in CARS2009, while applying the second method where both salinity and temperature are adjusted, 94% of the cast were able to be corrected with constraints with the remaining 6% corrected without constraints. Of the profiles, 6% consisted of profiles where the cast contained a very small density range or the cast was in the polar regions, where the temperatures were situated along the freezing line. The polar regions proved to be a very difficult region to produce stable realistic profiles; the upper section of the water column typically contains a freshwater layer with temperatures hovering just above the freezing point.

If the instability is caused by a real physical process, such as the Kelvin–Helmhotz instability, then the correction method that should be applied is the constrained and Θ adjustment; however, if the instability is caused by a bad value in either salinity or temperature, then the unconstrained solution is the better correction method, because in this case we do not want to try to keep the error in the profile.

Acknowledgments

The authors thank P.C. Chu and C. W. Fan for providing a copy of their software, and T. Boyer for the stabilized data from the WOA routines. Also thanks must go to the anonymous reviewers, whose comments improved this manuscript. We gratefully acknowledge the Australian Research Council for its support through Grant FL150100090.

APPENDIX

Description of the GSW Code for Stabilization

The stabilization methods detailed in this paper are included as functions in the Gibbs Seawater (GSW) Oceanographic Toolbox of TEOS-10 (McDougall and Barker 2011). The toolbox is a collection of programs to compute the TEOS-10 properties and is available online (www.TEOS-10.org).

The algorithm to stabilize the water column by adjusting Absolute Salinity while keeping in situ temperature unchanged is available with the function gsw_stabilise_SA_const_t. This method is intended for use with observed data where salinity is the least reliable measurement. In the standard form of this function, we set for each bottle pair to be , which is approximately one-fifth of the square of Earth’s rotation rate. If desired, a vector of values of can be supplied, one for each bottle pair, and in this case, the code will ensure that the resulting stabilized cast is at least as stable as this specified vertical profile.

We developed a multistep procedure to implement the second method, where both Absolute Salinity and Conservative Temperature are adjusted, in the GSW code gsw_stabilise_SA_CT.

Initially the profile is despiked, salinity first, then temperature. A spike is defined when the gradient, either temperature or salinity, between consecutive bottles is greater than three standard deviations of gradients between all of the bottle pairs on the cast and the stability is less than . When a spike is detected in Absolute Salinity, the bottles above and below the spike are replaced with bottles stabilized by adjusting Absolute Salinity while keeping Conservative Temperature unchanged. However, when a spike is detected in Conservative Temperature, we replace both the Absolute Salinity and Conservative Temperature values for the bottles above and below using the unconstrained and Θ adjustment. We do this because a small change in salinity, particularly in cold water, can eliminate the need for a very large change in temperature.

Once any spikes that were present have been removed, the vertical profile is stabilized by adjusting and without the linear constraint of Eq. (9) to have an of . This very small lower limit enables the calculation of the mixed-layer pressure (MLP) as described in de Boyer Montégut et al. (2004). A background cast is constructed to have bottles regularly spaced in vertically integrated with a resolution of about 0.1 kg m−3; this is achieved by using the data below the MLP and vertically averaging over a range of ~0.2 kg m−3, such that each average has an overlap of 0.05 kg m−3. This sparse, smoothed cast is then stabilized without the individual bottle constraints [Eq. (9)], such that the minimum stability is greater than the user-defined . The resulting cast is used to calculate and of Eq. (9). The full cast is then stabilized using the individual bottle constraints of Eq. (9).

When the constrained solution has been obtained, we then compare it with the unconstrained solution and return the solution that contains the smaller wiggliness [Eq. (12)].

Also included is a polynomial that is based on the smallest 10% of the buoyancy frequencies (as a function of pressure) in observed profiles collected in the Southern Ocean, south of 50°S, where the lowest values are typically found. This code is called gsw_Nsquared_lowerlimit. In future releases of GSW, it is intended that this function will be expanded to include an polynomial that is a function of pressure, latitude, and longitude.

In the code gsw_stabilise_SA_CT, we have included the option to conserve heat or salt. We have set the default to conserve neither heat nor salt in our calculations, as we found that sometimes stabilizing the water column at one height caused changes far above or below this height, as was observed when applying the Chu and Fan (2010) method (Figs. 2a–d). In the case of observed data, correcting for spikes in the data and conserving properties can cause the profile to shift away from the observed profile shape.

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