a. Hydrographical data

We used temperature and salinity data from the WOCE, the U.S. Climate Variability and Predictability Program (CLIVAR), and other programs, which were downloaded from the CLIVAR and Carbon Hydrographic Data Office website (https://cchdo.ucsd.edu/) on 9 July 2015. These data include 723 sections and 34 705 stations between 2 April 1980 and 17 July 2014. The data quality is controlled by the following criteria:

• There should exist at least two data points between 10 and 40 dbar.
• The maximum measured depth is ≥50 dbar.
• The mean depth resolution is ≤2 dbar; 673 profiles cannot meet this requirement and are ruled out. The remaining profiles have depth resolutions of 1 or 2 dbar, in particular 78.3% of them have a depth resolution of 2 dbar. As discussed below, the relative variance method is less affected by the depth resolution; however, this criterion ensures that the depth resolutions of all profiles are consistent.
• The shallowest measurement depth of a profile is ≤20 dbar.
• The maximum temperature (salinity, density) difference in the depth range between the sea surface and 20 dbar should be less than the maximum respective difference in the depth range from 20 dbar to the deepest depth of the examined profile
• The temperature difference of the whole profile is ≥ 1°C.

These criteria leave a final dataset of 32 686 CTD profiles by removing some profiles that are useless, erroneous, or contain atypical data for the MLD computation. The coverage of these profiles is shown in Fig. 1.

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Spatial distribution of WOCE data used in the study. Data are generally denser over the Atlantic Ocean. Section A05 is marked (red).

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

To assess the impact of depth resolution on the determination of the MLD, we applied different methods to the transatlantic hydrography section at 24.5°N in 2010 (WOCE section A05, ExpoCodes: 74DI20100106), where only one CTD cast was considered at each station and a total of 135 profiles were obtained. The temperature was recorded by Sea-Bird 911plus CTD with a sampling rate of 25 Hz. Some raw data had spurious temperature values occurring slightly beneath the sea surface ( dbar). These unrealistic values were removed when their temperature is 0.02°C larger or smaller than the median temperature in the depth range between 25 and 40 dbar. Then, these corrected profiles are averaged and resampled to generate data of different depth resolutions from 0.04 to 25 dbar for simulating different types of data products.

b. Methodology

As mentioned in section 1, there lies a layer of homogeneous water mixed by wind, convection, and surface flow in the ocean surface. Below the mixed layer, there is pynocline (thermocline), where the density (temperature) gradient is large. One of our motivations is to seek a universal way rather than depending upon subjective thresholds to identify the MLD. The proposed method is designed by using the fact that the range of temperature (salinity or density) is more sensitive than its variation upon the depth change around the low bound of the mixed layer. The detailed algorithm procedure for finding MLD is presented here. Let us consider a profile, , where z is depth (pressure) and ϕ represents temperature, salinity, density, or other variables. An example, shown in Fig. 2a, is the temperature profile at station 65 (24.5°N, 56.7°W) in section A05. For a set of points , , ranging from the first point near the sea surface at depth to the current depth , its standard deviation is estimated as follows:

where is the mean value of over the depth range of interest . While the maximum variation over the same depth range is calculated by

where and are the maximum and minimum values, respectively, within data points . Figures 2b,c show the example δ and σ profiles. Then, the relative variance of ϕ over the depth range is defined as

When k is counted from 1 to N (the total data points of the profile), the χ profile is obtained (Fig. 2d). In the case when is inside the ML, as shown in Fig. 2, both and are less varied, which leads to a less changeable χ_k except for the perturbation of the first few points. While is located below the base of the ML, the variable ϕ changes abruptly, which results in the increased and . However, the increase of is more pronounced than that of , implying that is more sensitive to the variation of ϕ. With increasing , gets ever smaller until it reaches its minimum value, the occurrence depth of which is defined as . This procedure is described as

In some sense, the occurrence of the minimum is similar to the “misalignment problem” in the CTD measurement (Neshyba et al. 1971), where some false inversions occur at the turn points of the salinity and density profiles as a result of the different response times of the temperature and conductivity sensors. As shown in Figs. 2a–d, is slightly deeper than the base of the ML. Next, we trace back from upward to find the MLD accurately. Here, is used because it is less affected by the fluctuation of ϕ, which is indicated in the comparison between the δ and σ profiles in Figs. 2b,c. As has a sharp change at the base of the ML (Fig. 2b), we can locate the corresponding by finding the first depth above z_n1 for satisfying

where is a critical threshold value. A value between 0.1 and 0.5 works well for most of the profiles. For the profile as shown in Fig. 2, is changed by only 2 dbar as increases from 0.1 to 0.5. As z_n2 is not sensitive to the selection of Δ, Δ = 0.3 is selected in the present work. Then, is defined as the MLD, H.

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Determination of MLD using the relative variance method on (a) the potential temperature profile at station 65 (24.5°N, 56.7°W) on 23 Jan 2010 along section A05. The depth of minimum , (blue circle) and the MLD (red circle) are indicated. (b)–(d) Corresponding profiles of , , and , respectively.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

As the ML has a limited thickness, it is much more realistic to seek the ML base at the depth of an intermediate data point rather than at the data point of the deepest depth. In practice, the variable is detrended and the maximum or minimum value of the detrended profile is acquired depending on whether is larger than . Generally, the maximum (minimum) value is sought for most temperature (density) profiles. In most cases, the depth of this maximum (minimum) value is close to the base of the ML, and is selected as twice of this depth. When it is impossible to define with this process, for example, to some irregular profiles, it is set as 600 dbar. This additional process reduces the chance to mislocate the MLD. Note that the detrending process is merely applied to define ; the proposed method is working on the original profile.

a. Influence of noise

Typically, it is unavoidable that the temperature or density profiles contain spurious spikes or inversions formed by various types of instrument noise. We will test whether the new method is sensitive to the random noise in the determination of the MLD. Here, we first construct an artificial temperature profile ranging from 0 to 300 dbar as

As shown in Fig. 3a, this artificial temperature profile is uniform for the upper 72 dbar and is exponentially decaying with increasing depth, corresponding to an MLD of 72 dbar. Next, the temperature profile is added with the normally distributed random noise of amplitude , where ε is the noise level ranging from 0% to 5% and is the maximum temperature difference in the depth range between 0 and 300 dbar. Finally, the relative variance method is applied to identify the MLD. At each noise level, 100 trials are performed and the mean and the corresponding 95% confidence interval are shown in Fig. 3b. The random noise does not have any effect on the new determined MLD at a low noise level (). As a noise gets stronger, the determined MLD seems to gradually deviate from the expected value of 72 dbar and the deviation reaches about 5 dbar at . As shown in the inset of Fig. 3a, it is clearly indicated that the temperature is indeed homogeneous in the upper 77 dbar at ; that is, the base of MLD is moved downward as a result of the influence of random noise.

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(a) Artificial temperature profiles generated by Eq. (6) (gray line) and (black line), where is the normally distributed random noise at . To avoid overlap, is shifted by 0.5°C. (b) The MLD H of the artificial profile in (a) as a function of ϵ. At each ϵ, the error bar of MLD is the confidence interval over 100 trails. (c) Temperature profile as in Fig. 2a (gray line) and the constructed one (black line) with the addition of the normally distributed random temperature noise at . To avoid overlap, the constructed temperature profile is shifted by 0.5°C. (d) The MLD H of the profile in (c), but as function of ϵ. At each ϵ, the fluctuation of MLD is the confidence interval over 100 trails.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

In addition, we add the normally distributed random noise to the temperature profile shown in Fig. 2a. Similar to the abovementioned artificial profile, as shown in Fig. 3d, the MLD determined by the proposed method keeps constant (89 dbar) at a low noise level (). The MLD slightly deviates by less than 2 dbar for the stronger noise level (). At , the noise amplitude reaches 0.32°C, and it is larger than the threshold value (0.2°C) used in a previous MLD work (de Boyer Montégut et al. 2004). In the influences of random noise, comparatively, it seems that the relative variance method exhibits a better performance to the real temperature profile shown in shown Fig. 3c than to the artificial profile shown in Fig. 3a, which should be attributed to the fact that the water mass underneath the ML has a sharper temperature gradient in the real temperature profile. The results shown in Figs. 3b,d suggest that the relative variance method is less affected by noise and it is robust in identifying the MLD.

b. Comparison with other methods

To further evaluate the performance of the relative variance method, a comparison of this method with the six other currently used methods is being done. These methods are described as follows.

1. Difference method (): Since this method is simple and stable in determining MLD, it has been extensively applied. As recommended by de Boyer Montégut et al. (2004), the temperature difference of 0.2°C and the density difference of 0.03 kg m⁻³ (to the reference depth of 10 dbar) are the optimal thresholds when taking the global MLD distribution into consideration. Here, the reference depth is alternatively selected as the sea surface for consistency with the other methods.
2. Difference-interpolation method (Dif Int): A linear interpolation between observed levels is additionally employed to the result of the difference method to determine the MLD more accurately (de Boyer Montégut et al. 2004).
3. Gradient method (): Following the suggestion by Dong et al. (2008), a temperature gradient criterion of 0.025°C dbar⁻¹ and a potential density gradient criterion of 0.0005 kg m⁻³ dbar has been used to determine the MLD.
4. Hybrid method (): From a suite of possible MLD values assembled by the difference and gradient methods, a complicated fitting algorithm is employed to approximate the MLD (Holte and Talley 2009). The detailed algorithm of the method are publicly available.
5. Curvature method (): In this method one examines the second derivative of temperature (density) with respect to depth, referred to in the literature as “curvature” (Lorbacher et al. 2006). For each data point, a curvature value is being estimated and the first (as one scans the profile downward from the sea surface) local maximum of curvature (within some tolerance of one standard deviation of temperature/density) is being defined as the lower MLD base. The algorithm of the method is publicly available.
6. Maximum angle method (Max Ang): According to this method, two linear regressions are applied to the temperature/density profile, one on the data points above and one beneath each data depth. The depth, at which the angle formed by the two aforementioned linear fits is maximum, is identified as the MLD (Chu and Fan 2011).

1) Example profiles

Here, we first show the efficiency of these methods in determining the MLD of four temperature profiles from the section A05. As shown in Fig. 4a (station 65), when the ML and the underneath water column are well separated, all the methods can approximately determine the MLD. The MLD is visually inspected to be 89 dbar. It is estimated to be within the range between 87.9 dbar by the curvature method and 94.0 dbar by the difference method. When the dynamics are complex in the upper ocean, leading to highly stratified water columns occurring in several depth ranges (Fig. 4b, station 69), the determined MLDs by different methods are much scattered. The base of a homogeneous layer is identified by the hybrid, curvature, and relative variance methods at 61, 65, and 59 dbar, respectively. The difference and difference-interpolation methods also approximately capture the MLD, which are 72.0 and 70.5 dbar, respectively. This is consistent with the findings in previous studies (Lorbacher et al. 2006; Holte and Talley 2009); that is, the estimations from the difference and difference-interpolation methods are generally deeper than the actual MLD. The determinations by the maximum angle and gradient methods are 152 and 156 dbar, respectively, which correspond to the top depth of deeper stratified water column. In the case where the temperature changes gradually with increasing depth—for example, in station 13 (Fig. 4c)—the hybrid method identifies the MLD as 13 dbar, which seems too shallow with visual inspection. The curvature method finds the MLD to be 26 dbar. The relative variance method finds the almost homogeneous temperature depth range with the determined MLD of 71 dbar. The MLD determined by the difference and difference-interpolation methods is decided by the threshold values, which are 76.0 and 75.2 dbar, respectively. The maximum-angle and gradient methods capture the top of the deeper stratified water column; their MLDs are estimated to be 126 and 124 dbar, respectively. When the temperature signal gets more complicated in the ML—for example, the temperature inversions occurred at station 60 (Fig. 4d)—the curvature and maximum angle methods are affected by the temperature inversion around dbar and they determine the MLDs to be 7 and 5 dbar, respectively. The hybrid and gradient methods find the same MLD, 11 dbar. The MLDs by the difference and difference-interpolation methods are still affected by the threshold value, being 19 and 18.4 dbar, respectively. The relative variance method looks insensitive to the temperature inversion and determines the MLD to be 77 dbar. These examples show that the relative variance and curvature methods tend to find similar MLDs for the most homogeneous layer, but that the relative variance method is less affected by the small-scale processes, for example, inversion. The difference and difference-interpolation methods give a slightly thicker MLD, and the maximum angle and gradient methods easily overestimate the MLD in most cases. The hybrid method looks to find the more reliable MLD than the difference and gradient methods, probably attributing to its assembled algorithm (Holte and Talley 2009).

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Potential temperature profiles (circles with black lines) and the corresponding QI (gray lines) defined in Eq. (7b) with the vertical space dbar at (a) station 65 (24.5°N, 56.7°W), (b) station 69 (24.5°N, 53.9°W), (c) station 13 (27.3°N, 79.2°W), and (d) station 60 (24.5°N, 60.4°W) along section A05. The MLDs H evaluated using different methods are marked: the difference (pink), difference-interpolation (cyan), gradient (green), hybrid (yellow), curvature (red), maximum angle (blue), and relative variance (black) methods.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

To evaluate whether a method is able to cleanly separate the homogeneous layer from the stratified region below, Lorbacher et al. (2006) proposed a quality index (QI),

where is the standard deviation within the ML and is the standard deviation of ϕ over the depth range of 1.5 times the MLD from the sea surface. Based on the definition of QI, it is expected that QI approaches 1 when the ML is well developed. As reported by Lorbacher et al. (2006), QI is above 0.7 in 70% of the World Ocean. To examine whether can reflect the accuracy of the determined MLD, QI in Eq. (7a) is promoted to any depth ,

In the example profile shown in Fig. 4a, it is found that QI corresponds to the maximum value when approaching the base of a highly homogeneous ML. The maximum QI () is obtained when the MLD is determined with the highest accuracy of 89 dbar. For the profile shown in Fig. 4b, QI is larger than 0.95 for the hybrid, curvature, and relative variance methods; 0.71 for the difference method; 0.80 for the difference-interpolation method; and around 0.81 for the maximum angle and gradient methods. QI is relatively small for the profile shown in Fig. 4c: it varies from 0.52 for the hybrid method to 0.80 for the gradient method. For the profile shown in Fig. 4d, QI is 0.77 for the relative variance method, and it is less than 0.60 for the other methods. These examples show that a larger QI value generally corresponds to an MLD that is determined with higher accuracy. Recently, QI has been further refined to categorize the MLD of different quality levels with the aid of an additional parameter (Abdulla et al. 2016).

2) Influence of depth resolution

The raw temperature data in section A05, as shown in Fig. 1, were averaged and resampled to obtain profiles of different depth resolutions dbar (the total number of resolutions is ). At each station we find the MLDs of the temperature profile at different depth resolutions using different methods. As the performance of each method cannot be readily evaluated based on few individual profiles, we perform the statistical analysis on all profiles (). For each method we examine the mean and standard deviation of H and QI of each resolution. In particular, if X refers to H or QI, its mean value at the resolution is defined as

where is X at station j with the resolution . The standard deviation of X at the resolution is defined as

where is the mean value of each profile over all depth resolutions, and it is expressed as

Figure 5 shows —that is, the averaged MLDs (over all 135 profiles) versus the depth resolution —derived with different methods (corresponding to different colors). Overlaid for comparison is also the average visually inspected MLD value of 92.4 dbar. During the visual inspection process, the temperature fluctuation δ, the temperature difference σ, and QI are also considered to ensure a more reliable MLD for each profile.

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(a) The mean MLD of all 135 stations as a function of the vertical space by the difference (pink), difference-interpolation (cyan), gradient (green), hybrid (yellow), curvature (red), maximum angle (blue), and relative variance (black) methods. Mean MLD by visual inspection is marked (red cross). (b) As in (a), but for the standard deviation of MLD as a function of .

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

In particular, the function that is determined by the difference method deviates gradually from the visually inspected value and gets deepened with increasing , in accord with the finding of Helber et al. (2012). The large deviation at low resolution (large value) is attributed to the fixed threshold, and this shortcoming is partially remedied by the difference-interpolation method. Namely, as shown in Fig. 5a, although the average MLD values increase with increasing for both methods (difference/difference interpolation), assessed by the difference-interpolation method presents a deviation from the reference value (i.e., the visually inspected one) that is overall smaller than the respective deviation that is derived by the difference method. On the other hand, the gradient method is most seriously affected by even though its (over all values) is close to the visually inspected MLD. Comparably, the MLD determined by the difference method varies less with the depth resolutions (Fig. 5b), which is consistent with the result of Brainerd and Gregg (1995); of the hybrid method is slightly smaller than the visually inspected one, but it remains almost unchangeable for different depth resolutions. Furthermore, derived with the curvature method is much smaller and therefore deviates much more from the visually inspected value at small values, as a result of the temperature profiles containing more spikes, which greatly affect this method (Chu and Fan 2011). For the maximum angle method, gradually decreases with increasing , while it tends to approach the visually inspected value, at large (low depth resolution) values. This is contrary to the expected in Chu and Fan (2011), where the method was suggested to be unapplicable for low depth resolution profiles. Finally, the that is determined by the relative variance method presents the best agreement with the visually inspected value with considerable stability from the aforementioned reference value (Fig. 5b). The mean, median, and standard deviation values of MLDs for all resolutions, as listed in Table 1, show consistent results with those shown in Fig. 5.

Table 1.

Statistics of H and QI for all stations along the section A05, showing the corresponding mean , median , and standard deviation values, with different methods. Percentage (P) of all profiles with different resolutions, where the determined MLD is of with different methods, is presented.

Additionally, the dependences of QI, its mean , and standard deviation on the depth resolution are shown in Fig. 6. In particular one can see in this figure that the values that are derived by the difference and gradient methods gradually decrease from about 0.9 to 0.64 with increasing ; this fact suggests that the threshold methods cannot accurately determine the MLD, especially at low depth resolution. of the hybrid method seems better, varying around 0.85, because this method integrates both the difference and gradient methods. The shortcoming caused by the low depth resolution is greatly remedied by the interpolation in the difference-interpolation method; however, the produced is still less than those of the objective methods (relative variance, curvature, and maximum angle methods). These objective methods produce larger ; moreover, the relative variance method determines the MLD of the highest , which is around 0.94 (Table 1). As indicated in Fig. 6b about , the difference-interpolation method is less varied compared to the difference and gradient methods, its stability is close to that of the curvature method, and it is better than that of the maximum angle method. Comparatively, the relative variance method has the smallest () and the gradient method has the largest (). Furthermore, it can be seen in the sixth column of Table 1 that 43%, 74%, 44%, 77%, 83%, 72%, and 89% of the stations along the section A05 have for the difference, difference-interpolation, gradient, hybrid, curvature, maximum angle, and relative variance methods, respectively. Obviously, the hereby introduced method of relative variance is the most reliable of all methods, as it corresponds to the largest of the aforementioned percentage values (89%). This conclusion is consistent with that drawn from the MLD results shown in Fig.5. The difference-interpolation method also works well in determining the MLD, but its small value implies that this method merely gives a coarse representation of the actual MLD.

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(a) The mean quality index of all 135 stations as a function of the vertical space by the difference (pink), difference-interpolation (cyan), gradient (green), hybrid (yellow), curvature (red), maximum angle (blue), and relative variance (black) methods. (b) As in (a), but for the standard deviation of quality index as a function of .

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

a. Probability density functions of global MLD

The probability density functions (PDFs) of the MLD from and are shown in Fig. 7. One of the remarkable features in this figure is that the PDFs of and derived by different methods tend to coincide when and are approximately larger than 20 dbar, a fact implying that the MLD determined by all the aforementioned methods follows the same PDF distribution globally in cases of relatively thick ML. As temperature/density close to the sea surface is easily affected by the diurnal heating cycle and by mass and momentum exchanges with the atmosphere (Price et al. 1986), some MLDs shallower than 10 dbar are less convincing because only very few data points are included within the ML, especially for these WOCE profiles of depth resolution of 2 dbar. Figure 7a indicates that the difference-interpolation and curvature methods are prone to finding anomalously shallow MLDs. As suggested by Lorbacher et al. (2006) and Chu and Fan (2011), the curvature method suffers from abnormal spikes, which might be responsible for the shallower MLD assessed with this method. The hybrid method produces and with peaks around 11 dbar (Figs. 7a,b), possibly attributed to the artificial limits imposed in the calculating procedure (Holte and Talley 2009). The relative variance method on the other hand derives the least probable (with the smallest population) shallow MLDs in comparison with the other aforementioned methods. This can be partially explained by the fact that the method is less affected by the noise signal, for example, temperature inversions in Fig. 4d. Although the PDFs of and of each method have similar distributions, as shown in Fig. 7, they still are distinct when comparing the respective mean values, and , and median values, and . As listed in Table 2, is 10 and 6 dbar larger than for the difference-interpolation and hybrid methods, respectively; and is 1 or 2 dbar larger than for both the curvature and relative variance methods.

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PDFs of MLD (a) and (b) from temperature and density profiles using the difference-interpolation (cyan), hybrid (blue), curvature (red), and relative variance (black) methods.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

Table 2.

Global statistics (mean and median values ) of H and QI determined by using difference-interpolation, hybrid, curvature, and relative variance methods on the WOCE data. Percentages— and for the total temperature and density WOCE data, respectively—are presented in the cases that the determined MLD of and from these methods.

Figure 8 shows the PDFs of the MLD difference, , estimated by different methods. The PDF of peaks at zero (within ±1 dbar) in 20% (difference interpolation), 32% (hybrid), 40% (curvature), and 42% (relative variance) of the profiles. Moreover, if one loosens the criterion of coincidence of and by defining that and are identical when dbar, then one derives that the and values that are determined by the difference-interpolation, hybrid, curvature, and relative variance methods are identical for 69%, 76%, 81%, and 79% of the profiles, respectively. These results suggest that is close to for most of the profiles and that this feature can be captured by all four methods. When we take a closer look at 96% of the total profiles around the PDF peak, as shown in Fig. 8b, the PDF symmetry of is different for different methods. It clearly shows that is skewed toward the positive values for the difference-interpolation and hybrid methods, with a skewness of 2.55 and 2.26, respectively. Traditionally, a barrier layer is identified by a deeper than , while a compensated layer is defined when is larger than (Sprintall and Tomczak 1992; de Boyer Montégut et al. 2004). The results shown in Fig. 8b imply that the compensated layers are much less common than the barrier layers with these WOCE data, which is consistent to the global ocean results from the threshold methods (Helber et al. 2012). However, is found to be relatively symmetric for the curvature and relative variance methods, as confirmed by the relatively much smaller values of skewness (0.39 and 0.40, respectively). As reported in Lorbacher et al. (2006), the curvature method tends to capture the depth of the top of a barrier layer, while the difference-interpolation method finds the depth of the top of a thermocline. As an objective method, the relative variance method tends to detect the top of a barrier layer in a temperature profile too. An example of a barrier layer is shown in Fig. 9. It was measured in the equatorial Atlantic Ocean (4.3°S, 34.8°W) on 21 October 1990. Because of the temperature inversion, a barrier layer exists slightly below the depth of 70 dbar, corresponding to the actual MLD. The relative variance and curvature methods identify this depth from both temperature and density profiles. While the difference-interpolation and hybrid methods can capture only the MLD when using the density profile (Fig. 9b), they give much deeper MLDs when using the temperature profile (Fig. 9a).

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(a) PDFs of MLD difference, , using the difference-interpolation (cyan), hybrid (blue), curvature (red), and relative variance (black) methods. (b) As in (a), except for zooming in on the PDF peaks.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

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Example profiles of (a) potential temperature and (b) potential density from the equatorial Atlantic Ocean (4.3°S, 34.8°W) on 21 Oct 1990. Difference-interpolation (cyan), hybrid (blue), curvature (red), and relative variance (black) methods produce MLDs of (a) 94, 94.15, 68.5, and 68 dbar, and (b) 72, 70.7, 67.8, and 70 dbar.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

Figure 10 shows the PDFs of and by four different methods. The PDF peaks around 0.98, implying that these methods are capable of determining the MLD for most of the profiles. The mean, median, and other values of QI with these methods are listed in Table 2. Generally, it indicates that the QI of the objective methods (curvature and relative variance methods) is larger than those of the threshold methods (difference-interpolation and hybrid methods). Postulating that the MLD is accurately defined when its (Lorbacher et al. 2006; Abdulla et al. 2016), the difference-interpolation, hybrid, curvature, and relative variance methods can obtain reliable MLDs from 60%, 60%, 72%, and 80% of total temperature profiles, respectively. Lorbacher et al. (2006) suggested that ML interpretation is impossible when . The difference-interpolation, hybrid, curvature, and relative variance methods can find the MLD of for 82%, 83%, 86%, and 94% of the total temperature profiles, respectively. Similar results can be found from the comparison among these methods in the QI of (Fig. 10b; Table 2). Hence, the relative variance method gives relatively reliable results (with ) for most (nearly all) profiles, thereby proving its superior effectiveness in determining the MLD.

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PDFs of QI from (a) temperature and (b) density profiles with the difference-interpolation (cyan), hybrid (blue), curvature (red), and relative variance (black) methods.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

b. Global MLD variability

To gain further insight into the effectiveness of the relative variance method (as compared to the effectiveness of the other methods) in accurately determining MLDs, it is worthwhile to additionally examine both the temporal evolution and the spatial distribution of the various MLD estimates. As indicated in Fig. 1, the WOCE data are relatively sparse in the global ocean; hence, a zonal mean is taken. Since the MLDs are characterized by strong seasonal variation, they are further averaged into monthly bins.

Figure 11 shows the spatial and temporal variation of the MLDs estimated by each of the four aforementioned methods. The blank regions indicate the impossibility of reasonable interpolation owing to data sparsity. In particular, we can see that the month–latitude patterns of MLD are similar for the four methods. In these patterns the seasonality of the MLD is well exhibited, and the seasonal variations of the MLD are almost opposite in the northern and southern hemispheres (as expected). The MLD maxima are found in the wintertime between 45° and 60°S, including the Southern Ocean, and between 30° and 60°N corresponding to the North Atlantic Deep Water formation region. The MLD minima are found in the summertime midlatitude regions. The seasonal cycle of MLD in the region of the Antarctic Circumpolar Current is especially pronounced with several hundreds of decibars in winter and dozens of decibars in summer. Moreover, the pattern of temperature-based MLD, is similar to that of density-based MLD, for each method. The corresponding correlation coefficient is found to be 0.76, 0.93, 0.85, and 0.89 for the difference-interpolation, hybrid, curvature, and relative variance methods, respectively. The difference-interpolation method produces a thicker than (Figs. 11a,b), especially in the equatorial region, where the barrier layer prevails. This may be responsible for the low correlation of the patterns between and determined by this method. The difference-interpolation method finds the anomalous shallow in some regions in the wintertime MLD of the Southern Ocean, where the temperature threshold of 0.2°C might be inappropriate. The hybrid method produces matching patterns of and (Figs. 11c,d) because of its algorithm in the fitting of the ML and its underlying water column (Holte and Talley 2009). The averaged and from the relative variance method have very highly correlated patterns (Figs. 11g,h), which further confirm that this method is capable of accurately determining the MLD from temperature profiles even in the presence of a barrier layer. In the summertime, the MLD determined by the curvature method is shallower than those by the other methods, as is also indicated by the statistics in Table 2.

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Monthly variations of zonally integrated MLD, H, from temperature and density profiles with different methods. (a), (c), (e), (g) and (b), (d), (f), (h) are determined by using the difference-interpolation, hybrid, curvature, and relative variance methods, respectively. The blank regions indicate impossibility of reasonable interpolation owing to the data sparsity.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1

Figure 12 shows the spatial and temporal variation of QI from different methods. Lorbacher et al. (2006) suggested that QI has large values in the hemispheric summer and fall when a sharp gradient at the base of the ML is present. This feature can be observed in the and of these methods. A large QI occurs from January to June in the Southern Hemisphere, and from July to November in the Northern Hemisphere. However, the difference-interpolation and hybrid methods occasionally produce low QI values even in summer and fall, especially for (Figs. 12b,d). This can be partly explained by the fact that the threshold methods find only the coarse MLD. Comparatively, the relative variance and curvature methods lead to larger QI values. Figure 12 also shows the QI values are generally less than 0.8 with these methods in the wintertime of the mid- and high-latitude regions, implying that the MLDs of these associated profiles are determined with considerable uncertainty. Even so, the patterns of and show that the objective methods (the curvature and relative variance methods) have better performance in determining the MLD.

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Monthly variations of zonally integrated QI from temperature and density profiles with different methods. (a), (c), (e), (g) and (b), (d), (f), (h) are determined by using the difference-interpolation, hybrid, curvature, and relative variance methods, respectively. The blank regions indicate impossibility of reasonable interpolation owing to the data sparsity.

Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0104.1