# Ocean Mixed Layer Depth From Dissipation

Valentina Giunta, Brian Ward

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Valentina Giunta, Brian Ward. Ocean Mixed Layer Depth From Dissipation. Journal of Geophysical Research. Oceans, 2022, 127, 10.1029/2021jc017904. hal-05491781

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
**Key Points:**

* A new method to estimate the mixed layer depth (MLD) based on the shape of the vertical profile of dissipation is presented

* The MLD derived from dissipation offers a more realistic description of turbulent mixing than density, which tends to overestimate the MLD

* It is appropriate that dissipation-based estimates of the MLD be adopted, given that microstructure measurements are becoming increasingly available operationally

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

B. Ward,
bward@nuigalway.ie

**Citation:**

Giunta, V., & Ward, B. (2022). Ocean mixed layer depth from dissipation. *Journal of Geophysical Research: Oceans*, 127, e2021JC017904. https://doi.org/10.1029/2021JC017904

Received 17 AUG 2021
Accepted 9 MAR 2022

**Author Contributions:**

**Conceptualization:** Valentina Giunta,
Brian Ward

**Data curation:** Valentina Giunta

**Formal analysis:** Valentina Giunta

**Funding acquisition:** Brian Ward

**Methodology:** Valentina Giunta

**Software:** Valentina Giunta, Brian Ward

**Supervision:** Brian Ward

**Validation:** Valentina Giunta

**Writing – original draft:** Valentina
Giunta

**Writing – review & editing:** Brian Ward

© 2022. The Authors.

This is an open access article under
the terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

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# Ocean Mixed Layer Depth From Dissipation

Valentina Giunta¹,² ID and Brian Ward¹ ID

¹AirSea Laboratory, School of Physics and Ryan Institute, National University of Ireland, Galway, Galway, Ireland, ²Now at School for Marine Science & Technology West, University of Massachusetts Dartmouth, New Bedford, MA, USA

**Abstract** The mixed layer depth (MLD) is a widely used parameter for physical, chemical, and biological oceanography. The MLD delimits that region of the ocean, which is directly influenced by the atmosphere. There is a similar length scale referred to as the mixing layer depth, which is determined from profiles of dissipation rate of turbulent kinetic energy, but is less prevalent due to the requirement for specialized instrumentation. Here, we suggest that the MLD can be estimated using density/temperature (henceforth $h_D$) or using dissipation (henceforth $h_e$). Utilizing two data sets in the North Atlantic collected with the autonomous Air-Sea Interaction Profiler, $h_D$ and $h_e$ in the upper 100 m are compared. A new method based on the shape of the dissipation profile for estimating $h_e$ is presented. The main sources of turbulence in the upper ocean (i.e., wind, waves, and buoyancy fluxes), which tend to deepen the MLD, are more strongly correlated with $h_e$ variability for both data sets, confirming that the MLD derived from dissipation measurements represents more accurately the variability in the mixed layer. Given that dissipation measurements have become more operational due to the development of ocean microstructure technology, then it is appropriate that $h_e$ be adopted in the future for estimation of the MLD. This is additionally supported by the fact that $h_D$ typically overestimates the MLD when compared to $h_e$, thus having consequences for studies where $h_D$ is used to represent the extent of mixing in the upper ocean.

**Plain Language Summary** The mixed layer is the region of the ocean, which is directly influenced by atmospheric processes. The mixed layer plays a large role in regulating air-sea exchange and climate. The mixed layer depth (MLD) is a critical parameter for physical, chemical, and biological oceanography and is typically estimated using vertical profiles of temperature and density. However, since this layer is being actively mixed by different sources of turbulence (e.g., wind, waves, and buoyancy fluxes), a criterion based on turbulence measurements is more desirable. In this article, we explore different methods to estimate the MLD based on temperature, density, and turbulence data using two data sets in the North Atlantic collected with an autonomous vertical profiler that records these data simultaneously. Significant differences were found among the methods used in this work, showing that a method based on the shape of the turbulent profile proves to be more robust for estimating the MLD. Moreover, by looking at the sources of turbulence during these cruises, it is possible to observe that the MLD based on turbulence measurements represents more accurately the variability in the upper ocean, having consequences for studies where the MLD is used.

## 1. Introduction

Belcher et al. (2012) describe the ocean surface boundary layer (OSBL) as the region of the upper ocean where turbulent processes control the air-sea exchange of momentum, heat, and trace gases, as well as the subduction into the ocean interior. The bulk of the OSBL is referred to as the mixed layer and its associated depth, the mixed layer depth (MLD), extends from the ocean surface to the top of the pycnocline (Belcher et al., 2012). The MLD is characterized by uniform temperature (T) and density (ρ) values and it is often filled with active mixing that causes elevated values of the dissipation rate of turbulent kinetic energy (ε).

One of the earliest studies with combined temperature, density, and dissipation measurements was carried out by Brainerd and Gregg (1995). They set out to establish the difference between the mixed layer and the mixing layer (mixing layer depth (XLD). The former was defined as the layer formed by the history of mixing, whereas the latter was defined as the depth to which material entering the ocean at the surface is actively mixed. The fundamental difference is that the mixed and mixing layers refer to different timescales over which mixing occurs. They are also determined using different methods: The MLD is calculated from density or temperature profiles and the XLD from dissipation. Figure 1.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Schematic showing profiles of density, temperature, and dissipation with the corresponding estimates of $h_D$ and $h_e$. These profiles were generated from data taken with Air-Sea Interaction Profiler (ASIP) (see Section 2) and were highly averaged and modified for clarity. A consistent characteristic is that $\epsilon$ profiles are much more variable than $T$ or $\rho$, which reflects the turbulent nature of the upper ocean.</figcaption></figure>

Brainerd and Gregg (1995) also discussed the application of the concept of mixed layer to a particular process. For modeling sea surface temperature (SST), the diurnal mixing layer is appropriate. For the study of phytoplankton blooms, where populations can double in less than a single day, they suggest the diurnal mixed layer. For gas exchange between the ocean and atmosphere, they suggest it is necessary to consider the diurnal mixing cycle. In the case of air-sea momentum fluxes, the seasonal mixed layer is appropriate. Here, the terms mixing and mixed layers are used interchangeably when considering processes over the diurnal time scales.

Compared to the availability of $T$ and $\rho$ data, there is a paucity of $\epsilon$ data as it requires specialized instruments. However, recent developments in technology have led to possibility of $\epsilon$ data becoming more operational. We refer to the MLD determined from density (or temperature) and dissipation as $h_D$ and $h_e$, respectively.

There exist many different methods to determine the MLD. The gradient method defines the $h_D$ as the depth at which the gradient of $\rho$ (or $T$) exceeds a specific value (e.g., Banyte et al., 2018; Brainerd & Gregg, 1995; Monterey & Levitus, 1997). The difference method uses a threshold to find the depth at which the deviation between $\rho$ (or $T$) and a value at a near-surface reference depth exceeds a specific value (e.g., Brainerd & Gregg, 1995; de Boyer Montégut et al., 2004; Kara et al., 2000; Monterey & Levitus, 1997; Thomson & Fine, 2003). The choice of threshold and reference depth is crucial in this method as the resulting $h_D$ is strongly dependent on these values. de Boyer Montégut et al. (2004) estimated $h_D$ using the difference criterion but with a variable potential density threshold to compensate for nonlinearity in thermal expansion of seawater. There are other criteria based on integrated methods (e.g., Chu & Fan, 2011; Huang et al., 2018, 2019). Yet another approach was taken by Holte and Talley (2009), who developed a hybrid method, which combines the threshold criterion, the shape of each profile, and the gradient criterion using potential density and temperature for sub-Antarctic and Antarctic profiles. Estimates of $h_e$ are determined by finding the depth at which $\epsilon$ drops below a specific value (e.g., Brainerd & Gregg, 1995; Dewey & Moum, 1990; Fer & Sundfjord, 2007; Lozovatsky et al., 2006; Lozovatsky & Shapovalov, 2012).

The aim of this paper is to compare different estimates of the MLD using measurements of density, temperature, and dissipation rates of turbulent kinetic energy to study the variability of $h_D$ and $h_e$ under the main sources of turbulence: wind, waves and buoyancy fluxes. The data used in this paper were derived from the Air-Sea Interaction Profiler—ASIP (Ward et al., 2014) for two different cruises in the North and Subtropical Atlantic. ASIP is an autonomous upwardly rising instrument, which is programmed to profile from below the maximal MLD estimate to the surface, acquiring data within the water column at high resolution. It was designed to study small-scale high-frequency events in the mixed layer and is therefore ideally suited for this study. In Section 2, we describe the field data and conditions under which the data were acquired. In Section 3, we describe the various methods for estimating the MLD with an evaluation of the performance of $h_D$ and $h_e$ using a quality index. The main results with accompanying discussion are provided in Section 4 and our conclusions from this study are presented in Section 5.

## 2. Observations

The data used in this study were obtained from two different research cruises. The first took place from late June to mid-July 2011 in the North Atlantic aboard the R/V Knorr (henceforth referred to as Knorr). The overall objectives of the Knorr campaign were focused on gas exchange (e.g., Esters et al., 2017; Scanlon & Ward, 2013; Sutherland, Christensen, & Ward, 2014). A full complement of meteorological parameters was recorded continuously onboard. The surface net heat flux ($Q$) was computed as the combination between the incoming shortwave, net incoming and emitted longwave, sensible heat, and latent heat. The surface buoyancy flux ($B_0$) is estimated as $B_0 = -gQ_\rho$, with $Q_\rho = \rho(\alpha FT + \beta FS)$, where $Q_\rho$ is the density flux, $g$ is the gravity acceleration, and $\alpha$ and $\beta$ are the coefficients of thermal expansion of sea water and of haline contraction, respectively. Here, $FT = -Q/\rho C_p$ and $FS = (E - P)S/(1 - SSS/1000)$, where $C_p$ is the specific heat of sea water, $E$ and $P$ are the evaporation and
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) The positions for the Knorr and Sub-Tropical Atlantic Surface Salinity Experiment (STRASSE) cruises. The numbers in parentheses indicate the number of Air-Sea Interaction Profiler profiles for each campaign. (b) The location of deployments for the Knorr cruise. Every three profiles were averaged resulting in the number of profiles per deployment. (c) The deployments for the STRASSE cruise indicating the number of averaged profiles per deployment.</figcaption></figure>

precipitation rates in m/s, respectively, and SSS is the sea surface salinity in practical salinity scale. The ocean surface becomes less buoyant (i.e., more dense) when there is a positive buoyancy flux out of the ocean surface (upward) due to the minus sign in the $B_0$ equation. This means that positive values of $B_0$ (into the ocean) occur when the density flux is negative and may force convection. On the contrary, negative values of $B_0$ occur under stabilizing conditions (stratification).

An ultrasonic altimeter was mounted on the bow to obtain wave measurements due to nonlocally generated swell and local wind waves. Data collected while the ship was cruising were discarded due to contamination of the ship's wake. To correct for ship motion, the altimeter was combined with an accelerometer. Using the full 1D-wave spectra, it is possible to calculate $H_s$ and mean wave periods (for a detailed description, see Christensen et al., 2013).

The second research campaign was the Sub-Tropical Atlantic Surface Salinity Experiment (STRASSE), which took place from 21 August to 9 September 2012, as part of the Salinity Processes in the Upper-ocean Regional Study (SPURS) campaign. The main objective of this cruise was to study the North Atlantic salinity maximum (NASM), which is located in the North Atlantic subtropical gyre (see Reverdin et al., 2015; Sutherland, Reverdin, et al., 2014; Sutherland et al., 2016 for further details). Radiative fluxes and wind speed measurements were recorded onboard during the STRASSE campaign. Details for both cruises are shown in Figure 2.

The buoyancy flux $B_0$ was calculated using the TOGA COARE 3.0 algorithm Fairall et al. (2003) (same methodology as in the Knorr11 campaign). The main difference with the Knorr11 campaign is the fact that no wave measurements were carried out onboard. Consequently, data obtained using wave model data from the European Centre for Medium-Range Weather Forecasts Ocean Wave Model (ECWAM) are only considered.

Vertical profiles of temperature, density, and dissipation rates of turbulent kinetic energy were obtained with ASIP (Ward et al., 2014). Once deployed, ASIP can capture repeated microstructure profiles in the upper ocean. Using three thrusters, ASIP dives up to 100 m and then ascends using its own buoyancy at a rise velocity of 0.5 $ms^{-1}$, acquiring data until the sensors reach the air-sea interface. At the surface, ASIP transmits a short
<table><caption>**Table 1**

Methods for Determination of the Mixed Layer Depth Used in This Study</caption><thead><tr><th>Citation</th><th>Key</th><th>MLD</th><th>Description</th></tr></thead><tbody><tr><td>Brainerd and Gregg (1995)</td><td>BG95<sub>D</sub></td><td>h<sub>D</sub></td><td>V<sub>σθ</sub>=0.05 kg m<sup>-3</sup> m<sup>-1</sup></td></tr><tr><td>Monterey and Levitus (1997)</td><td>ML97</td><td>h<sub>D</sub></td><td>ΔT = 0.5°C; z<sub>ref</sub> = 0 m</td></tr><tr><td>Kara et al. (2000)</td><td>K00</td><td>h<sub>D</sub></td><td>ΔT = 0.8°C; z<sub>ref</sub> = 10 m</td></tr><tr><td>de Boyer Montégut et al. (2004)</td><td>BM04</td><td>h<sub>D</sub></td><td>Δσ<sub>θ</sub> = σ<sub>θ</sub>(T<sub>10</sub> - 0.2°C, S<sub>10</sub>, P<sub>0</sub>) - σ<sub>θ</sub>(T<sub>10</sub>, S<sub>10</sub>, P<sub>0</sub>)</td></tr><tr><td>Holte and Talley (2009)</td><td>HT09</td><td>h<sub>D</sub></td><td>Uses shape of the profiles and a set of possible MLD values</td></tr><tr><td>Dewey and Moum (1990)</td><td>DW90</td><td>h<sub>e</sub></td><td>Δ<sub>e</sub> = 5 × 10<sup>-9</sup> m<sup>2</sup>s<sup>-3</sup></td></tr><tr><td>Brainerd and Gregg (1995)</td><td>BG95<sub>e</sub></td><td>h<sub>e</sub></td><td>Δ<sub>e</sub> = 10<sup>-9</sup> m<sup>2</sup>s<sup>-3</sup></td></tr><tr><td>Lozovatsky et al. (2006)</td><td>L06</td><td>h<sub>e</sub></td><td>Δ<sub>e</sub> = 10<sup>-8</sup> m<sup>2</sup>s<sup>-3</sup></td></tr><tr><td>Fer and Sundfjord (2007)</td><td>FS07</td><td>h<sub>e</sub></td><td>Δ<sub>e</sub> = 3 × 10<sup>-8</sup> m<sup>2</sup>s<sup>-3</sup></td></tr></tbody></table>

Note. $V_{\sigma\theta}$ is the threshold for the gradient criterion using profiles of $\sigma_{\theta}$, $\Delta T$ and $\Delta\sigma_{\theta}$ are the thresholds for the difference criterion regarding a depth of reference ($z_{ref}$) for the profiles of temperature and potential density, respectively. Finally, $\Delta_{e}$ is the threshold for the $\epsilon$ profile.

message after the completion of the profile and the repeats the same process. One of the chief advantages that ASIP has against previous studies of turbulence measurements is that it is not necessary to discard the measurements taken in the first 10 m of the water column (Brainerd & Gregg, 1995; Oakey & Elliott, 1982) as it is an upwardly rising profiler.

ASIP is equipped with an FP07 thermistor and SBE7 conductivity sensor, which provides high-resolution temperature and salinity from which density is calculated. These sensors are calibrated against a reference CT sensor to maintain accuracy. To estimate $\epsilon$, ASIP has two orthogonally mounted airfoil shear probes. Details about the different cruises and $\epsilon$ estimation can be found in Sutherland, Christensen, and Ward (2014) and Esters et al. (2017). Profiles of temperature, salinity, density, and dissipation were estimated using a vertical resolution of about 0.5 m and averaged over three successive profiles to obtain mean values approximately every hour for Knorr and every 1.5 hr for STRASSE.

### 3. Estimating the Mixed Layer Depth

Table 1 provides details of the MLD estimates using vertical profiles of T, potential density ($\sigma_{\theta}$), and $\epsilon$. Most of those methods were used to compute MLD climatologies (de Boyer Montégut et al., 2004; Holte et al., 2017; Kara et al., 2003; Monterey & Levitus, 1997) and therefore can be used in any region of the globe.

In addition to the MLD estimates provided in Table 1, an estimate was also determined through a visual inspection of the profile. Visual estimates have been used on several occasions to evaluate the efficacy of different MLD methodologies and to choose thresholds, which allow MLD determination. For example, Thomson and Fine (2003) and de Boyer Montégut et al. (2004) used the visual inspection to choose possible threshold values for estimating the MLD. Castro-Morales and Kaiser (2012) used MLD values obtained from the visual inspection of $O_2$ profiles to evaluate the performance of the new MLD method introduced. Huang et al. (2018) used the same approach. Lorbacher et al. (2006) and Holte and Talley (2009) selected a number of random profiles and performed a visual inspection to assess which MLD method was closer to the visual estimate. González-Pola et al. (2007) discussed the importance of having MLD visual estimates to obtain a better comparison of MLD method and recommended to perform a visual inspection after any MLD method is applied. Lozovatsky et al. (2006) used the visual inspection of $\epsilon$ profiles to determine a threshold for estimating $h_e$. The visual method (VIS<sub>D</sub>) involves identification of the pycnocline and a quasi-homogeneous profile of density to the surface. If the pycnocline cannot be identified easily, the thermocline (and possibly the halocline) is used. In the case that $h_D$ is identified in the upper 10 m of the profile, this is rejected as it is attributed to diurnal warming (de Boyer Montégut et al., 2004).

As estimates of $h_e$ rely solely on thresholds of $\epsilon$, we developed a new methodology that uses a combination of the shape of the profiles and a threshold (Figure 3). The $\epsilon$ data are initially filtered to remove the high-frequency components of the signal using a ten-point moving average filter. Then, it is necessary to locate the
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Example showing two different dissipation profiles and their respective $h_e$ estimate. The black line is the raw $\epsilon$ profile and the green line is the filtered $\epsilon$. Relative extremes are shown using the red circles, and red-dashed lines illustrate the interpolated lines between them. Black-dashed lines show two log limit values for $log(\epsilon)$: -8.5 and -9. Blue-dashed lines indicate the estimation of $h_e$. The shear probe raw data are shown in gray. (Left Panel) Example of $\epsilon$ profile, which reaches the lower limit ($log(\epsilon) = -9$). (Right Panel) example of $\epsilon$ profile, which does not reach the lower limit, but it contains values between the lower and the upper limits.</figcaption></figure>

profile-relative extremes on the smoothed data. A profile is then reconstructed by linear interpolation between the relative extremes in order to avoid any irregularities that may affect the determination of $h_e$. Using limits of -8.5 to -9 for the $log(\epsilon)$, the location of the maximum negative slope for the interpolated profile between these limits was selected as the $h_e$ value. Any data where $\epsilon < 10^{-9} m^2 s^{-3}$ were ignored. The election of these thresholds was based on the visual inspection of $\epsilon$ profiles and those used in previous publications in Table 1.

To visually estimate the MLD using $\epsilon$ profiles (VIS$_{\epsilon}$), it is necessary to first identify the section of the profile at which the $\epsilon$ values drop rapidly. In the case of finding more than one section, the shear profiles were used as an indicator of the depth at which the active turbulence occurs. Once the section at which the dissipation rate values fall abruptly was identified, $h_e$ is selected as the maximum decreasing depth.

### 3.1. MLD Performance Evaluation

Lorbacher et al. (2006) developed a quality index to evaluate the performance of the MLD estimate. The quality index for $h_D$ is given by

$$QI_D = 1 - \frac{\sigma(T(z) - \langle T \rangle)|_{(z_0, h_D)}}{\sigma(T(z) - \langle T \rangle)|_{(z_0, 1.5 \times h_D)}} \quad (1)$$

This index assumes that above the MLD exists a quasi-homogeneous layer in which the standard deviation ($\sigma$) between the physical properties ($T$, $S$, and $\rho$) and their vertical mean (denoted using $\langle \rangle$) is almost zero. Below the MLD, the density (temperature) increases (decreases) with depth resulting in a gradient and therefore increasing their value of $\sigma$. The index estimates the ratio of the standard deviation of the physical property between the surface ($z_0$) and $h_D$ and between the surface and $1.5 \times h_D$. A quality index close to one indicates a higher performance in estimating the MLD. As the criteria used in this work do not use the same physical properties, each $QI_D$ was estimated using the respective property with which $h_D$ was estimated. Based on the work of Lorbacher et al. (2006), the $QI_D$ of $h_D$ could be divided into three categories: C1 represents a well-defined $h_D$ with $QI_D > 0.8$; C2 represents acceptable $h_D$ estimates with $0.8 > QI_D > 0.5$; and finally C3 with $QI_D < 0.5$ represents rejected $h_D$ values.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Example showing $QI_e$. The gray line is the raw $\epsilon$ profile and the black line is the filtered profile. The estimated $h_e$ is shown using a black circle and the boxes represent the two sections of the profile at which the standard deviation ($\sigma$) is estimated. For this profile, $h_e = 24.6$ m and $QI_e = 0.95$.</figcaption></figure>

To evaluate the efficacy of $h_e$, we developed a quality index based on Lorbacher et al. (2006) and Abdulla et al. (2016):

$$QI_e = 1 - \frac{\sigma(\epsilon(z) - \langle\epsilon\rangle)_{(h_e+5m, h_e+15m)}}{\sigma(\epsilon(z) - \langle\epsilon\rangle)_{(z_0, h_e)}}. \quad (2)$$

where $\sigma$ is calculated from two sections of the $\epsilon$ profile: (a) $z_0$ to $h_e$ and (b) $h_e+5$ m to $h_e+15$ m. For profile section (i), it is assumed that turbulence levels drop rapidly with increasing $\sigma$. Below $h_e$, the levels of turbulence are nearly constant and diminished compared to $\epsilon$ above $h_e$, represented by section (ii) of the profile (Figure 4). A filtered version of the $\epsilon$ profile using a ten-point moving average filter to remove excessive variability is used to calculate $QI_e$ (Figure 4). The same categorization as for Equation 1 is applied here: C1 with a well-defined $h_e$ for $QI_e > 0.8$; C2 with acceptable $h_e$ values for $0.8 > QI_e > 0.5$; and C3 with rejected $h_e$ estimates for $QI_e < 0.5$.

## 4. Results and Discussion

The MLD was estimated using the 9 different methods described in Table 1 and also using a visual inspection of T and $\epsilon$ profiles ($VIS_D$ and $VIS_e$ respectively). Table 2 provides an overview of the success of each algorithm in estimating the MLD. This includes the number of profiles at which each criterion was able to find an MLD value for each cruise and the quality index values. Figure 5 presents $h_D$ for Knorr and STRASSE cruises separately.

Some of the criteria did not return an $h_D$ value for some of the profiles as their thresholds were not reached. For example, $BG95_D$, a gradient criterion, requires a threshold of $0.05 \text{ kg m}^{-3} \text{ m}^{-1}$, but this was found in only 61 and 167 profiles for Knorr and STRASSE, respectively. Regarding the difference criterion, ML97 and K00 were able to return an $h_D$ value in almost all profiles (with the exception of two in Knorr cruise). This may be related to the fact that the difference criterion is more stable than the gradient criterion (Brainerd & Gregg, 1995; Thomson & Fine, 2003).

For the Knorr cruise, $VIS_D$ has the highest number of $h_D$ estimates with a $QI_D > 0.8$ for all profiles with only four estimates defined as not acceptable. For STRASSE, the $VIS_D$ performance was acceptable with 174 profiles that have a $QI_D > 0.5$. BM04 and HT09 have the highest number of $h_D$ values with a $QI_D > 0.8$ for the STRASSE cruise although 24 $h_D$ estimates for HT09 were rejected.

It is important to note that STRASSE cruise presents a significant number of profiles of T and $\sigma_\theta$ where the mixed layer is not well delimited and the stratification is weak and not stable (see profiles in Supporting Information S1). Moreover, the different nature of the profiles found among STRASSE cruise creates a dispersion among the $h_D$ results using different methods. This is evident for the $BG95_D$ method, which is closer to $VIS_D$ for the Knorr cruise but seems to overestimate $h_D$ for the STRASSE cruise since it shows the deepest values among all methods (Figure 5). The performance of the $QI_D$ for $BG95_D$ is higher for Knorr than for STRASSE, since $C3_S = 41$, which explains the deeper values of MLD found for this cruise. Consequently, the threshold used for the gradient criterion overestimates the actual $h_D$ value in the case of profiles with weak stratification.

Among the methods that uses a difference criterion, ML97 was effective in finding an $h_D$ value in almost every profile. It uses the temperature profile and a threshold of $0.5^\circ C$, which has better agreement with $VIS_D$ for Knorr than for STRASSE (Figure 5). Despite the high correlation with $VIS_D$ during Knorr cruise, ML97 has a significant number of poor MLD estimates for

<table><caption>**Table 2**

Knorr and Sub-Tropical Atlantic Surface Salinity Experiment (STRASSE)
Statistical Characteristics: N is the Number of Profiles for Which an Mixed Layer Depth (MLD) Could be Determined (the Total Number of Profiles is 94 for Knorr and 176 for STRASSE)</caption><thead><tr><th>Key</th><th>N<sub>K</sub></th><th>N<sub>S</sub></th><th>C1<sub>K</sub></th><th>C1<sub>S</sub></th><th>C2<sub>K</sub></th><th>C2<sub>S</sub></th><th>C3<sub>K</sub></th><th>C3<sub>S</sub></th></tr></thead><tbody><tr><td>BG95<sub>D</sub></td><td>61</td><td>167</td><td>35</td><td>26</td><td>23</td><td>98</td><td>3</td><td>41</td></tr><tr><td>ML97</td><td>92</td><td>176</td><td>62</td><td>81</td><td>25</td><td>68</td><td>5</td><td>27</td></tr><tr><td>K00</td><td>92</td><td>176</td><td>42</td><td>36</td><td>45</td><td>136</td><td>5</td><td>4</td></tr><tr><td>BM04</td><td>94</td><td>176</td><td>50</td><td>112</td><td>33</td><td>61</td><td>11</td><td>3</td></tr><tr><td>HT09</td><td>94</td><td>176</td><td>54</td><td>105</td><td>37</td><td>47</td><td>3</td><td>24</td></tr><tr><td>VIS<sub>D</sub></td><td>94</td><td>176</td><td>73</td><td>94</td><td>17</td><td>80</td><td>4</td><td>2</td></tr><tr><td>DW90</td><td>94</td><td>176</td><td>15</td><td>57</td><td>35</td><td>85</td><td>44</td><td>34</td></tr><tr><td>BG95<sub>e</sub></td><td>94</td><td>176</td><td>23</td><td>91</td><td>44</td><td>74</td><td>25</td><td>11</td></tr><tr><td>L05</td><td>94</td><td>176</td><td>12</td><td>32</td><td>30</td><td>88</td><td>52</td><td>56</td></tr><tr><td>FS07</td><td>94</td><td>176</td><td>5</td><td>13</td><td>27</td><td>76</td><td>62</td><td>87</td></tr><tr><td>G20</td><td>94</td><td>176</td><td>23</td><td>97</td><td>49</td><td>70</td><td>22</td><td>9</td></tr><tr><td>VIS<sub>e</sub></td><td>94</td><td>176</td><td>22</td><td>99</td><td>45</td><td>65</td><td>27</td><td>12</td></tr></tbody></table>

Note. The quality index ($QI_D$ and $QI_e$) is indicated as follows: C1 (quality index >0.8) corresponds to a well-defined MLD; C2 (0.5 < quality index <0.8) represents an acceptable MLD value; and C3 (quality index <0.5) indicates a rejected MLD
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Comparison of the different criteria used in this work to estimate $h_D$. Upper plot (a) corresponds to Knorr data and lower plot (b) to Sub-Tropical Atlantic Surface Salinity Experiment. The x label represents the profile number taken for each deployment.</figcaption></figure>

STRASSE associated with shallower values of $h_D$ and bigger differences with VIS$_{T,rho}$ during the first deployment (Figure 5b). The difference criterion K00, with a threshold of 0.8°C, has a better performance regarding quality index results and proximity to VIS$_D$ (Figure 5). Most of the $h_D$ values found by K00 have a $QI_D$ with values between 0.5 and 1 with less than 10 profiles with a $QI_D$ < 0.5 in total for both cruises. K00 also performs very well for STRASSE since it is closer to VIS$_D$ (Figure 5b) and it has a high number of values with a $QI_D$ > 0.5.

The only MLD criteria that use methods other than the difference of threshold are BM04 and HT09. BM04 uses a variable threshold and HT09 uses a hybrid method. Both have similar performance in estimating $h_D$ and are closer to VIS$_D$ for most of the deployments in Knorr cruise (Figure 5a). The high values for C1 and C2 for both cruises show an accurate $h_D$ estimation, particularly for STRASSE, which has more $h_D$ estimates in C1 than the VIS$_D$. However, HT09 does have over 20 profiles in STRASSE where the $h_D$ values were rejected. BM04 has a small number of poor $h_D$ estimates for STRASSE but the highest amount of not acceptable $h_D$ estimates (11 values) of all methods for the Knorr cruise (Table 1). Taking into consideration that BM04 is based on a difference criterion with a variable threshold, it seems that the difference criterion has a better performance in estimating $h_D$ over the gradient and hybrid methods. Moreover, the simplicity of the implementation of the difference criterion makes them much more accessible for $h_D$ estimation.

To understand why BM04 and HT09 differ from VIS$_D$ and other criteria, even though have good results according to the $QI_D$, it is necessary to investigate their performance for individual profiles. One of the reasons of this difference could be that the $QI_D$ can show values close to 1 for unacceptable $h_D$ estimates in profiles where a short range gradient is found within the mixed layer (Abdulla et al., 2016). This can be observed in Figure 6 during Knorr and STRASSE cruises. On the third deployment of Knorr cruise (Figure 6a), BM04 and HT09 returned shallower values for the MLD in contrast to the other two criteria and VIS$_D$. In this case, those methods are not able to find the mixing that extends below their $h_D$ values, and they identified a mixed layer caused by weak and shallow stratification and do not capture the deepening of $h_D$ that happens during nighttime. For the STRASSE cruise, the biggest differences with VIS$_D$ are located in the first and second deployments. For the first deployment (Figure 6b), BM04 and HT09 found MLD values between 10 and 20 m and VIS$_{h_D}$ between 30 and 40 m. In this case, it is possible to observe that the nature of the profiles makes it difficult for all criteria to identify the same $h_D$. If we take into consideration the seven deployments of STRASSE cruise, BM04 is closer to VIS$_D$ than HT09, since the last one shows the shallowest values in comparison with the rest of the methods.
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Temperature profiles with the corresponding $h_D$ estimates using the methods described in Table 1 separated by an offset of 1°C for (a) deployment 3 of Knorr cruise and (b) deployment 1 of Sub-Tropical Atlantic Surface Salinity Experiment cruise.</figcaption></figure>

There are four published methods to estimate $h_e$ described in Table 1 as well as the novel method, which is here-
with referred to as G20 (this publication). The number of profiles for which $h_e$ was found by each criterion (N)
and the $QI_e$ are shown in Table 2 for Knorr and STRASSE separately. Figure 7 presents a comparison of each of
the $h_e$ methods (including a visual estimate $VIS_e$) for Knorr and STRASSE profiles.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Comparison of the different criteria used in this work to estimate $h_e$. Upper plot (a) corresponds to Knorr data and lower plot (b) to Sub-Tropical Atlantic Surface Salinity Experiment. The x label represents the profile number taken for each deployment.</figcaption></figure>
All five methods proved to be effective in estimating $h_e$ for both cruises in terms of number profiles with $h_e$ estimates. Along with BG95$_e$ and G20, VIS$_e$ has the highest number of profiles with $QI_e > 0.8$ and the lowest with a $QI_e < 0.5$ for both cruises. The performance was even better for the STRASSE cruise rather than Knorr, contrary to what was found for $h_D$. The same result was also found among the other $h_e$ methods.

DW90 uses a threshold of $5 \times 10^{-9} m^2 s^{-3}$ and it is possible to observe that this method follows the VIS$_e$ values (Figure 7) with the exception of some shallow values (Figure 7a, deployment 2). Nonetheless, DW90 presents a significant amount of rejected $h_e$ values ($QI_e < 0.5$) for both cruises with nearly half of the Knorr $h_e$ estimates categorized as not well defined. BG95$_e$ uses the smallest threshold for $\epsilon$ among all criteria ($1 \times 10^{-9} m^2 s^{-3}$). This criterion produces deeper values for $h_e$ in comparison with VIS$_e$ during Knorr cruise (Figure 7a, deployment 1), but there is good agreement for STRASSE (Figure 7b). The $QI_e$ found for BG95$_e$ is similar to the one estimated for VIS$_e$, having a low number of rejected $h_e$ estimates for both cruises.

Among the bigger thresholds for $\epsilon$, L05 uses a value of $1 \times 10^{-8} m^2 s^{-3}$ and presents some disagreements with VIS$_e$ (Figure 7). This criterion tends to underestimate $h_e$ compared to VIS$_e$ (Figure 7a, deployments 2 and 3) and presents a higher number of poor $h_e$ estimates for both cruises. The method that deviates most from VIS$_e$ is FS07. This may be caused by the fact that this method applies the biggest threshold of all ($3 \times 10^{-8} m^2 s^{-3}$). The shallowest values of $h_e$ estimated by FS07 can be observed in Figure 7 during both cruises. One of the reasons that caused this differences is that FS07 uses a threshold that was originally designed for open ocean and ice-covered conditions, which may not be suitable for all regions. This disagreement is also evident in its $QI_e$ performance, which has the lowest values of well-defined $h_e$ estimates and the biggest number of rejected $h_e$ estimates.

Finally, the G20 method is the closest to VIS$_e$. This advantage is due to the fact that it is a dynamic method, which combines the characteristics of the shape of the $\epsilon$ profile and thresholds that are adequate for profiles with strong variability due to the nature of $\epsilon$ profiles, a method mostly based on the shape of profile is more appropriate than fixed thresholds that sometimes lead to misleading results due to the spikes in $\epsilon$, resulting in shallower $h_e$ values when in fact there is enough turbulent mixing below it. Figure 7 shows good agreement between G20 and VIS$_e$, particularly for the STRASSE cruise with a few overestimates for Knorr. G20 is not only well correlated with VIS$_e$ but also has a good performance in terms of $QI_e$. It has the lowest number of rejected $h_e$ estimates for both cruises.

It is also possible to understand the difference among $h_e$ criteria by looking at Figure 8. During deployment 3 for the Knorr cruise, L05 and FS07 did not detect high values of dissipation below their $h_e$ estimates, which are captured by the other methods (Figures 7a and 8a). However, in those cases where $\epsilon$ profiles show a sharp decline in $\epsilon$ values below the surface, all the methods find nearly the same $h_e$. For deployment 1 of STRASSE cruise, it is possible to observe bigger differences among the methods with the shallower values found by FS07 (Figure 8b). The variability of $h_e$ during that deployment is mostly caused by the diurnal cycle since that deployment started in the afternoon, deepening $h_e$ during nighttime and shallowing $h_e$ during the diurnal heating.

## 4.1. Relationship Between $h_D$ and $h_e$ and the Sources of Turbulence

To understand how accurate $h_D$ and $h_e$ represent the depth of the OSBL, it is necessary to study their variability from surface forcing. There are different sources of turbulence, which drive the mixing in the upper ocean: wind, waves, and buoyancy fluxes (Belcher et al., 2012). The influence of the wind component in the upper portion of the ocean is analyzed through the energy flux. Following Yang et al. (2014), the wind energy flux at a height of 10 m ($E_{10}$) is given by

$$E_{10} = \rho_a C_D U_{10}, \quad (3)$$

where $\rho_a$ is the density of the air, $C_D$ is the drag coefficient, and $U_{10}$ is the wind speed at 10 m. The influence of each of the sources of turbulence (through $E_{10}$, the surface buoyancy flux ($B_0$), and the significant wave height ($H_s$)) during the Knorr and STRASSE cruises are shown in Figure 9.

Regarding the Knorr cruise, winds were light on deployments 2 and 4 and moderate on deployment 1. The strongest winds are presented in deployment 3, at the beginning and at the end, with a peak of $E_{10} \sim 6 W m^{-2}$ ($U_{10} > 16 m s^{-1}$) at profile number 18 (Figure 9c). The peak is coincident with the maximum values of $\epsilon$ found from the surface down to $h_e$ (Figure 9a), which implies that the energy introduced by the wind-generated
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Dissipation rate of turbulent kinetic energy profiles (smoothed for a better visualization) with the corresponding $h_c$ estimates using the methods described in Table 1 separated by an offset of 1 m$^2$ s$^{-3}$ for (a) deployment three of Knorr cruise and (b) deployment one of Sub-Tropical Atlantic Surface Salinity Experiment cruise.</figcaption></figure>

turbulence and increased the mixing in the upper ocean, which deepened $h_c$. There is a correlation between both parameters for deployment 3, but not with the rest of the deployments. As a consequence, the variability of the turbulence in those deployments could be regulated by other mixing processes and/or be a combination of them.

<figure><img src="image_11.png" /><figcaption>**Figure 9.** (a) Vertical profiles of $log_{10}(\epsilon)$ from all of the stations for the four deployments. The yellow line represents $h_c$ and the aquamarine line $h_D$. (b) Same as for the Sub-Tropical Atlantic Surface Salinity Experiment (STRASSE) cruise for its seven deployments. (c) $E_{10}$, (d) $B_0$, and (e) $H_s$ for all stations for Knorr (orange solid line) and for STRASSE (green solid line) cruises. The x label represents the profile number taken for each deployment and vertical dashed lines in (a) and (b) delimit the deployments.</figcaption></figure>
The surface buoyancy flux $B_0$ (Figure 9d) is nearly zero for deployment 3; therefore, it has almost no influence on the mixing. However, it has a local minimum on the other deployments. It is important to note that negative values of $B_0$ are favorable for stratification, and positive values represent losses from the ocean making favorable conditions for convection. In the case of deployment 2, $B_0$ is positive at the beginning, which allows vertical mixing. As a result, both $h_D$ and $h_e$ estimates are nearly equal, since the upper portion of the water column is well mixed and the convective turbulence is dominating the mixing (Figure 9a). At station number 20, the $B_0$ has turned negative, thereby inhibiting the turbulent mixing and creating a shallower $h_e$. However, $h_D$ does not change considerably since it is not calculated through $\epsilon$ vertical profiles.

The significant wave height $H_s$ is shown in Figure 9d. The average value for $H_s$ for the four deployments is around 2 m. However, at the end of deployment 2, it reaches almost half of that value, decreasing the turbulence and producing shallower $h_e$ (Figure 9a). There is no well-pronounced local maximum of $H_s$ in deployment 1 or in deployment 3 and its effect is combined with the other sources of turbulences.

For the STRASSE cruise, the main source of mixing regulating the variability of the OSBL is the buoyancy flux (Figure 9d). During the first deployment, the wind energy has large values (nearly 2 W m⁻²). However, for the other deployments, $E_{10}$ is weaker (lower than 1 W m⁻²). The buoyancy flux shows a cycle associated with the diurnal heating. In the majority of the deployments, it is possible to identify a deep $h_e$ caused by convective turbulence during nighttime and the opposite behavior when there is an intensification of the stratification during diurnal heating. It is possible to observe this correlation between $h_e$ and $B_0$ but not with $h_D$. Regarding $H_s$, the maximum value does not exceed 2 m and it could create a deeper $h_e$ when it is combined with a significant wind energy (Figure 9b, deployment 1). However, $h_e$ is mostly influenced by the action of the buoyancy fluxes, which are stronger than the other turbulence sources. $h_D$ does not follow any evident pattern that is related to the three turbulence sources.

For both cruises, the variability of $h_e$ is mainly regulated by the three main sources of turbulence: wind, waves, and buoyancy fluxes. This is evident when one source of turbulence dominated over the other two (e.g., deployments 2 and 3 of Knorr cruise). Nonetheless, this is not evident if $h_D$ is used instead. It is not possible to predict the variability of $h_D$ by only looking at the main sources of turbulence and other physical processes must be taken into account. The rapid changes of $h_e$ forced by wind, waves, and buoyancy fluxes variability set a precedent toward obtaining a scaling of this parameter using the surface forcing, which will help to improve the parametrizations used to study the OSBL.

## 5. Conclusions

We present an analysis of different methods to estimate the MLD using vertical profiles of temperature, density, and dissipation rates of turbulent kinetic energy. These methods were analyzed using data from two research cruises in the North Atlantic—Knorr and STRASSE. The data used were acquired with the ASIP autonomous profiler. ASIP was designed to study the surface mixed layer and provides estimates of dissipation and ocean state variables up to 100 m, making it an ideal instrument for the study presented here.

The MLD estimated using temperature and density, $h_D$, has been widely used due to the fact that $T$ and $\rho$ are widely available from oceanographic cruises. To evaluate the performance of each of the methods for estimating $h_D$, a quality index ($QI_D$) was calculated. In addition, $h_D$ was estimated using a visual inspection, which has the highest values of $QI_D$ for both cruises. This is evidenced by the fact that Knorr cruise was stratified with a well-defined mixed layer for most of the cases, whereas the STRASSE cruise profiles were characterized by weak and unstable stratification.

The gradient criterion $BG95_D$ estimated similar $h_D$ to $VIS_D$ for the Knorr cruise, but overestimated their values for STRASSE where 25% were rejected due to a low-quality index. Among the difference criterion, K00 presented a high performance in terms of $QI_D$ having less than 10 rejected estimates in total for both cruises, and it was the closest to $VIS_D$. ML97 has an acceptable performance for Knorr, but discrepancies with $VIS_D$ during STRASSE, which is also noticeable in the $QI_D$, reflected a lower performance. This could be attributed to the fact that a larger threshold in temperature is better for estimating the permanent mixed layer depth that is not product of diurnal heating.
The hybrid methods, such as BM04 and HT09, presented a better performance in terms of *QI*<sub>D</sub> for STRASSE than for Knorr. However, 14% of the HT09 *h*<sub>D</sub> estimates were rejected for STRASSE, and HT09 returned the shallowest *h*<sub>D</sub> compared to the other methods. By looking at individual profiles, it is possible to observe that there are several cases where despite the fact that *QI*<sub>D</sub> is close to 1, BM04 and HT09 captured short-range gradients within the mixed layer, underestimating its real value.

There are fewer published estimates of the MLD based on $h_e$. The visual estimate $VIS_e$ has one of the larger number of profiles, which qualified as acceptable for both cruises, with a better performance found for STRASSE than for Knorr, contrary to $h_D$. Among the methods that use a threshold to estimate $h_e$, those with the biggest thresholds (L05 and FS07) are the ones with the biggest differences to $VIS_e$ and also a higher number of rejected $h_e$ estimates. DW90 and BG95<sub>e</sub> have a better correlation with $VIS_e$, and although BG95<sub>e</sub> had a better performance than DW90 in terms of $QI_e$, it tended to overestimate $h_e$.

The method based on the shape of $\epsilon$ profiles (G20) had the highest correlation with $VIS_e$ and had a slightly higher statistical significance in terms of the $QI_e$. This indicates that for spiky and highly fluctuating profiles (typical of $\epsilon$ data), a method based on the shape of the profiles helps to estimate $h_e$ avoiding the spikes and misleading results.

To analyze the accuracy of $h_D$ and $h_e$ in representing the depth of the OSBL, the sources of turbulence that deepen the OSBL were considered. The wind energy ($E_{10}$), the surface buoyancy flux ($B_0$), and the significant wave height ($H_s$) during both cruises were analyzed in order to study the variability of $h_D$ and $h_e$. Overall, $h_D$ and $h_e$ differ substantially for most of the deployments when periods of restratification or ocean warming occurred. This reflects the fact that stratification counteracts the turbulent mixing at the OSBL during these periods. Only under periods of convective turbulent, both variables were comparable. Moreover, two exceptional occasions were found during the Knorr cruise. During deployment 2, $B_0$ dominated the turbulent mixing over wind and wave forcing, occasioning convective turbulence at the beginning of the deployment and stratification at the end. While convection was actively mixing the surface, both $h_e$ and $h_D$ were equal. Subsequently, the buoyancy flux changed its sign and started to produce stratification counteracting the mixing at the ocean surface. Consequently, $h_e$ got shallower but $h_D$ was not affected by this change. On deployment 3, a similar pattern, this time between $h_e$ and the wind energy, was also observed. During this deployment, $E_{10}$ outweighed the effects of the buoyancy flux and waves on mixing the OSBL. As a consequence, $h_e$ followed the wind pattern with deeper values during high wind energy and shallower values during weaker winds. In both situations, it was not possible to determine how those conditions affected $h_D$ since it did not respond directly to the changes in these forcing. Nonetheless, $h_e$ was sensitive to the changes in the sources of turbulence and described the variability of the OSBL with accuracy.

The power of the article from Belcher et al. (2012) largely derives from being able to compare models to observations on a near-global basis. The epsilon-based metric proposed here is nowhere near global due to data sparsity, but during this decade it is anticipated that the number of dissipation measurements in the ocean will increase substantially, especially given current efforts to deploy shear probes on Argo floats. In addition, Sutherland, Christensen, and Ward (2014) showed that the model from Belcher et al. (2012) produced better agreement with the data when $h_e$ was used instead of $h_D$, which is further evidence that a dissipation-based turbulent length scale is more appropriate for describing the dynamics of the OSBL.

A recent study by Sallée et al. (2021) showed that $h_D$ has increased in the summertime by almost 3% per decade. A future study to see if a similar increase in $h_e$ globally would be very relevant in this context.

The current configuration of ASIP cannot be used to study areas where MLD > 100 m due to the limitations of its pressure housing, but a new version currently under development will allow depths to 200 m to be explored. However, ASIP is not the only instrument that can simultaneously measure dissipation and density, and an approach for expanding this study would be to deploy both ASIP and deep ocean microstructure profilers, which could capture both surface turbulent processes as well as abyssal convective processes. These latter measurements are important to achieve a global picture of dissipation-driven MLDs, given that deeper MLDs are often responsible for changes in climate (e.g., Sallée et al., 2021).

The stabilizing influence of restratification when $h_e \ll h_D$ (see Figure 9) is the time when mixing models disagree the most and where Large Eddy Simulation (LES) have explored the least. Much of the submesoscale literature is centered around understanding how dynamical restratification occurs in parameterizations (e.g., Fox-Kemper
& Ferrari, 2008) and biology (e.g., Mahadevan et al., 2012). Having observations of both dissipation and density under conditions of restratification are an advantage for improving LES models and parameterizations.

Since the MLD is the region of the ocean that experiences atmospheric surface forcing, it is more appropriate to use $\epsilon$ for accurately representing the influence of the different turbulence sources. Estimates of MLD based on temperature and density do not follow the variability of these forcing parameters. This suggests that scaling of $h_e$ using surface forcing is a means to provide future improvements in MLD estimates.

## Data Availability Statement

Data are available here: https://doi.org/10.6084/m9 figshare.19376123.v1.

## Acknowledgments

The lead author received support from the Irish Research Council through the Government of Ireland Postgraduate Scholarship Programme (GOIPG/2017/1270). Further support has been provided by the Marine Institute (grant PBA/ME/16/01). This project received funding from the European Union's Horizon 2020 Research and Innovation Programme under grant agreement 821001. Open access funding provided by IReL.

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