`mld` documentation

The `mld` function calculates the mixed layer depth of a water column based on profiles of temperature, salinity, and/or potential density.

The oceanic mixed layer refers to the well-mixed upper layer of the ocean that is turbulently mixed and characterized by constant temperature and salinity values. While the concept of a mixed layer is relatively straightforward, a concensus quantitative definition for how to specifically define it is lacking. Numerous algorithms exist, ranging from simple threshold algorithms to complex shape-fitting tools, and there is a large body of literature debating the relative merits of each.

For this particular toolbox function, we chose to adapt the algorithms of Holte and Talley, 2009. This paper details a multi-algorithm approach to calculating the mixed layer for oceanic Argo profiles. Those algorithms encompass the most commonly-used metrics, and therefore serves as a nice starting point for this function that also strives to offer multiple approaches to mixed layer depth. The examples below detail some of the most common approaches to this calculation. Always keep in mind that mixed layer depth calculation is not a one-size-fits-all calculation, and you may need to experiment with many different options to determine which one is most applicable to your data.

Note that this function currently includes some quirks (for example, salinity profiles are treated differently than temperature and density profiles under certain algorithms) that are specific to the Holte & Talley paper and are tailored to open ocean, float-measured profiles. These may or may not be appropriate to all datasets. We preserve them here, along with default parameter choices, for consistency with the original Holte & Talley code on which this function is based.

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## Contents

## Syntax

pmld = mld(pres, temp) pmld = mld(pres, temp, salt) pmld = mld(..., Name, Value, ...) [pmld, mldtbl, pth, h] = mld(...)

## Description

`pmld = mld(pres, temp)` calculates pressure corresonding to the mixed layer depth of a water column defined by pressure vector `pres` (in db) and temperature vector `temp` (in deg C). In the temperature-only calculations, the pressure values are used only as a vertical axis, and therefore depth values (in m) can be substituted.

`pmld = mld(pres, temp)` calculates the mixed layer depth pressure of a water column defined by pressure vector `pres` (in db), temperature vector `temp` (in deg C), and salinity vector `salt`. Note that in this instance pressure values will be used to calculate potential density, and therefore users should be cautious about substituting depth values (in m) in place of pressure.

`pmld = mld(..., Name, Value, ...)` allows users to modify algorithm parameters via the Name-Value parameters listed in the next section.

`[pmld, mldtbl, pth, h] = mld(...)` returns additional information about the internal algorithms used. The `mldtbl` table holds all mixed layer depth pressure candidate values used if the "best" algorithm is specified. The `pth` value returns the algorithm pathway used if the "best" algorithm is used; this is primarily for debugging purposes and was designed to match the analysis_t, analysis_s, and analysis_d variables in the original Holte&Talley 2009 supplementary online code (note that this code is not identical to that online supplement, but reproduces the same pathways in most instances). Finally, if the plot input option is specified, graphics handles will be returned in the handle structure `h`.

## Name-Value pair arguments

**metric** `'threshold'`, `'gradient'`, `'fit'`, `'extrema'`, `'subsurface'`, `'best'` (default)

This function allows for 6 different methods of defining the mixed layer depth:

`threshold`: The mixed layer depth is the first depth, measured from the reference depth downward, where profile value differs from the reference depth value by more than a threshold amount. Applicable to temperature and density profiles only.`gradient`: first depth, measured from reference depth downward, where profile gradient exceeds an indicated threshold amount. If the gradient never exceeds the threshold, the depth of highest absolute gradient is used. For salinity profiles, simply looks for maximum gradient.`fit`: A line is fit to the top, vertical-ish part of the profile and to the highest-gradient, horizontal-ish part of the profile. The mixed layer depth is set to the pressure where these lines intersect, or 0 if they don't intersect.`extrema`: pressure of the extreme values. For temperature, the maximum value is found, and for salinity and density, the minimum. Note that while these values may be applicable to most oceanic profiles, there are circumstances (for example, temperature inversions in salinity-dominated under-ice environments) where this simple metric will definitely fail.`subsurface`: This method attempts to find subsurface anomalies at the base of the mixed layer that are the result of surface cooling and mixing events. The profiles are characterized by a temperature gradient maximum in close proximity to the temperature maximum (or minima, for salinity). If found, the mld value reflects the shallower of the two values; if not found, it is set to 0. Applicable to temperature and salinity profiles only.`'best'`: All applicable potential mld values are calculated from the above options, and the Holte & Talley 2009 algorithm is used to choose which is the best estimate for a given profile.

**refpres** numeric scalar, default = 10

reference pressure (db). Calculations are based on this point of the profile downward; points above are ignored, with the intent to eliminate any diurnal heating artifacts at the surface.

**tthresh** numeric scalar, default = 0.2

threshold value used for temperature threshold calculation (deg C)

**tgradthresh** numeric scalar, default = 0.005

threshold value used for temperature gradient calculation (deg C m^-1)

**dthresh** numeric scalar, default = 0.03

threshold value used for density threshold calculation (kg m^-3)

**dgradthresh** numeric scalar, default = 0.0005

threshold value used for density gradient calculation (kg m^-4)

**errortol** numeric scalar, default = 1e-10

error tolerance used to choose which points to include in the surface-verticalish-line in the fit algorithm. Cutoff is set to the deepest point where the normalized sum squared error of the fit is less than this tolerance.

**range** numeric scalar, default = 25

range parameter used to identify clusters of mld values during the choose-the-best algorithm, i.e. r in Holte & Talley, 2009, section 3b. (db)

**deltad** numeric scalar, default = 100

maximum distance allowed between maximim[minimum] value and gradient in temperature[salinity] subsurface calculation. (db)

**tcutoffu** numeric scalar, default = 0.5

upper value of temperature change cutoff when classifying a temperature profile as winter-like during the choose-the-best algorithm. (deg C)

**tcutoffl** numeric scalar, default = -0.25

lower value of temperature change cutoff when classifying a temperature profile as winter-like during the choose-the-best algorithm. (deg C)

**dcutoff** numeric scalar, default = -0.06

potential density change cutoff above which a profile is classified as winter-like during the choose-the-best algorithm. (kg m^-3)

**plot** logical scalar, default = False

true to create a figure with the profiles, mixed layer depth points, and some reference detail.

**tblformat** `'table'` (default), `'array'`

The `mldtbl` output consists of a 3 x 5 array, with rows corresponding to the 5 algorithms (excluding best), and columns corresponding to temperature, salinity, and density, respectively. The default table format is more self-describing, since the columns and rows are clearly labeled, but the creation of a table array can add significant time to the function execution; array output is recommended if calling this function many times. (Note that if this function is called with only one output, this becomes irrelevant).

## Example 1: The threshold approach

The threshold definition is by far the most commonly-used mixed layer depth metric, and underlies several global mixed layer depth products, such as Monterey and Levitus (1997) and de Boyer Montegut et al. (2004).

To replicate this type of calculation, we start with example data from an Argo float. We visualize this data using the `transect` function to turn the 34 profiles into a transect plot and set colormaps with `cmocean`.

A = load('example_argo.mat'); nprof = length(A.date); ax(1) = subplot(3,1,1); transect(A.date, A.PRES, A.TEMP,'markersize',2); hold on; cb(1) = colorbar('eastoutside'); ylabel(cb(1), 'Temperature'); datetick; cmocean('thermal'); ax(2) = subplot(3,1,2); transect(A.date, A.PRES, A.PSAL,'markersize',2); hold on; cb(2) = colorbar('eastoutside'); ylabel(cb(2), 'Salinity'); datetick; cmocean('haline'); ax(3) = subplot(3,1,3); transect(A.date, A.PRES, A.pden,'markersize',2); hold on; cb(3) = colorbar('eastoutside'); ylabel(cb(3), 'Density'); datetick; cmocean('dense'); set(ax, 'YLim', [0 500]); set(ax(3), 'clim', [25 30]);

Monterey and Levitus (1997) use a temperature threshold of 0.5 deg C and a density threshold of 0.125 kg/m^3. This is a typical default choice for MLD calculations.

pmld = zeros(nprof,3); for ix = 1:nprof pmld(ix,:) = mld(A.PRES{ix}, A.TEMP{ix}, A.PSAL{ix}, 'metric', 'threshold', ... 'tthresh', 0.05, 'dthresh', 0.125); end plot(ax(1), A.date, pmld(:,1), 'b'); plot(ax(3), A.date, pmld(:,3), 'r');

de Boyer Montegut et al. (2004) argue that the Monterey and Levitus (1997) thresholds work for gridded and averaged profiles, but are less suitable to direct profile measurements. They instead use temperature and density thresholds of 0.2 deg C and 0.03 kg/m^3, respectively. Let's see how this compares to the previous estimates for this set of profiles:

pmld = zeros(nprof,3); for ix = 1:nprof pmld(ix,:) = mld(A.PRES{ix}, A.TEMP{ix}, A.PSAL{ix}, 'metric', 'threshold', ... 'tthresh', 0.02, 'dthresh', 0.03); end plot(ax(1), A.date, pmld(:,1), '--b'); plot(ax(3), A.date, pmld(:,3), '--r');

## Example 2: The fit approach

Holte & Talley, 2009's fit approach is often a bit more robust at identifying deep winter mixed layer depths, and can be applied to all three profile types. This function works by fitting two lines to the profile, one near-vertical one representing the mixed part of the profile, and a diagonal one representing the transition part at the bottom of the mixed layer. Note that this is the most time-consuming option of the five metrics.

pmld = zeros(nprof,3); for ix = 1:nprof pmld(ix,:) = mld(A.PRES{ix}, A.TEMP{ix}, A.PSAL{ix}, 'metric', 'fit'); end plot(ax(1), A.date, pmld(:,1), ':b'); plot(ax(2), A.date, pmld(:,2), ':r'); plot(ax(3), A.date, pmld(:,3), ':r');

## Example 3: Alternate methods

The remaining methods of MLD calculation presented in this function are less common, and have for the most part been developed to deal with situations where a simple threshold calculation fails to identify the desired MLD value. We refer you to the full Holte & Talley, 2009 paper for a detailed description of these algorithms, and the motivation behind their development.

The automatic plotting included with this function provides a quick summary of the possibilities. In this figure, T = temperature, S = salinity, D = density (sigma-theta), and P = pressure. The dashed lines correspond to the fit-method best-fit lines, and the solid line is the final "best" MLD. All those individual candidate MLD values can be found in the `mldtbl` output:

```
[pmld, mldtbl, pth, h] = mld(A.PRES{22}, A.TEMP{22}, A.PSAL{22}, 'plot', true);
```

mldtbl

mldtbl = 3×5 table threshold gradient fit extrema subsurface _________ ________ ______ _______ __________ temp 85.875 81 100.18 45 9 salt NaN 99 83 45 45 pdens 15.885 15 94.6 9 NaN

The most appropriate algorithm varies based on the properties of a given water mass. Keep in mind that the "best" method was specifically designed for Argo float profiles (and predominately tested on Southern Ocean profiles); we provide it as the default simply due to the history of this function (designed to mimic the Holte & Talley, 2009 supplementary code but with more flexibility).

## References

de Boyer Montégut C, Madec G, Fischer AS, Lazar A, Iudicone D (2004) Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology. J Geophys Res C Ocean 109:1–20

Holte J, Talley L (2009) A new algorithm for finding mixed layer depths with applications to argo data and subantarctic mode water formation. J Atmos Ocean Technol 26:1920–1939 doi:10.1175/2009JTECHO543.1

Monterey, G. and Levitus, S., 1997: Seasonal Variability of Mixed Layer Depth for the World Ocean. NOAA Atlas NESDIS 14, U.S. Gov. Printing Office, Wash., D.C., 96 pp.

## Author Info

This function and supporting documentation were written by Kelly A. Kearney for the Climate Data Toolbox for Matlab, 2019.