## 1. Introduction

*T*, namely,where

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

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.

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

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).

We have developed two methods to remove instabilities from a water column (see the appendix). The first involves adjusting only *S*_{A} and

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

*p*, for

*n*bottles, that is

*j*= 1, 2, …,

*n*, and the first step is to evaluate

*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

*j*salinity perturbations) while obeying the inequality constraintwith the matrix

## 3. Stabilization by adjusting both Absolute Salinity and Conservative Temperature

*p*recorded for

*n*bottles—that is,

*j*= 1, 2, …,

*n*—and we minimally adjust both the Absolute Salinity by

## 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.

Figures 3a and 3b depict a *N*^{2}) 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

The salinity-only and the unconstrained

Plots of the temperature changes scaled by

Figures 4–6 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 *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 *WOA* has relied on changing either the temperature or the salinity.

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

*j*= 1, 2, …,

*n*, we consider the set of difference vectors (

*u*

_{j},

*j*= 1, 2, …,

*n−*1) that are defined byand the set of changes in angle is given byand the wiggliness is defined asHere

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

## 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

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

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

Once any spikes that were present have been removed, the vertical ^{−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

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 *gsw_Nsquared_lowerlimit*. In future releases of GSW, it is intended that this function will be expanded to include an

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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