A new criterion, based on the shallowest extreme curvature of near surface layer density or temperature profiles, is established for demarking the mixed layer depth, hmix. Using historical global hydrographic profile data, including conductivity-temperature-depth and expendable bathythermograph data obtained during World Ocean Circulation Experiment, its seasonal variability and monthly to interannual anomalies are computed. Unlike the more commonly used Δ criterion, the new criterion is able to deal with both different vertical resolutions of the data set and a large variety of observed stratification profiles. For about two thirds of the profiles our algorithm produces an hmix/c that is more reliable than the one of the Δ criterion. The uncertainty for hmix/c is ±5 m for high- (<5 m) and ±8 m for low- (<20 m) resolution profiles. A quality index, QImix, which compares the variance of a profile above hmix to the variance to a depth of 1.5 × hmix, shows that for the 70% of the profile data for which a clearly recognizable well-mixed zone exists near the surface, our criterion identifies the depth of the well-mixed zone in all cases. The standard deviation of anomalous monthly hmix/c is typically 20–70% of the long-term mean hmix/c. In the tropical Pacific the monthly mean anomalies of hmix/c are not well correlated with anomalies of sea surface temperature, which indicate that a variety of turbulent processes, other than surface heat fluxes, are important in the upper ocean there. Comparisons between observed hmix/c and Massachusetts Institute of Techonology/ocean general circulation model/Estimating the Circulation and Climate of the Ocean model simulated mixed layer depth indicate that the KPP algorithm captures in general a 30% smaller mixed layer depth than observed.
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 The mixed layer of the ocean is commonly considered as the region near its surface with vertically quasi-uniform oceanic tracers (temperature, salinity and density) above a layer of more rapid vertical changes. The intense vertical turbulent mixing near the surface penetrates a short distance into the top of the pycnocline and is the cause of the observed vertical uniformity. The vigorous vertical turbulence is generated mainly by the action of the horizontal momentum and the vertical buoyancy fluxes derived from the atmospheric energetic motions. Variations of these fluxes are well documented [e.g., Kalnay et al., 1996; Roads et al., 2003] and these force variations of the mixed layer depth, hmix, on daily to interannual timescales. The heat budget of hmix is of particular interest because it governs the evolution of the sea surface temperature (SST), which is the most important ocean parameter influencing the atmosphere. It is important to know not only the SST evolution but also how deep the homogeneous thermal energy column penetrates into the stratified ocean. The main objectives of this paper are to present a new computation of the global distribution of hmix, to verify the skill of this method based on both global observations and the output of ocean general circulation models (OGCMs) and to analyze the effect of hmix anomalies on seasonal-to-interannual ocean-atmosphere interaction.
 As a point of departure, consider the global coupled atmosphere-ocean model of Alexander et al. . In its locally one-dimensional upper ocean hmix is time dependant and there is a flux of thermal energy out of the bottom of the mixed layer. According to Alexander et al.  the evolution of SST anomalies, δ(SST), is dominated by three terms:
where hmix ≡ 1/η, overbars define a temporal mean, and δ( ) defines the departures from the mean; cp is the specific heat and ρ the density of seawater. The entrainment heat flux is proportional to the product of the entrainment rate we and the temperature jump at the base of the mixed layer ΔT. Therefore changes in δ(SST) are forced by the time integration of highly energetic short-term anomalous variations of surface fluxes, δ(F), the entrainment rate at the base of the mixed layer and, of capital importance for this study, anomalous hmix.
Dommenget and Latif  point out that the variability of hmix on seasonal to longer timescale should also be an important parameter to consider in climate model diagnostics. In our context the second term on the right hand side of equation (1) gives a direct way to estimate the sensitivity of SST evolution in an anomalous hmix field. A monthly mean δ(hmix) anomaly during midlatitudinal spring and summer of 5 m would result in 0.25 K month−1 δ(SST) change (with δ(η = −mix−2δ(hmix) a monthly mean mix = 50 m, = 200 W m−2, cp = 4000 J kg−1 K−1 and ρ = 1026 kg m−3). This monthly change of SST is well within its observational accuracy on a global basis [Reynolds and Smith, 1994]. The closure of Alexander et al.  is the importance of anomalous hmix, beside heat flux variations, during spring and summer north of 20°N.
 The impact of anomalous entrainment out of the mixed layer base to the SST tendency is indirectly through hmix. The stored anomaly of thermal energy beneath the seasonal thermocline can last for many years, which in our context is represented by the third term on the right hand side of equation (1). Thermal anomalies of the previous winter become entrained into the current winter mixed layer as the mixed layer deepens in the fall [Alexander et al., 2000]. The implementation of this “reemergence mechanism” could also directly extend the persistence of winter SST anomalies to several years [Alexander and Deser, 1995; Deser et al., 2003]. Therefore, to quantify, understand and ultimately predict ocean-atmosphere interactions on seasonal-to-climate relevant timescales it is crucial that the sensitivity of anomalous SST to changes of hmix be well modeled in comparison to observations.
 A more complicated set of processes, as wind stress changes that force horizontal currents to advect thermal energy from place to place seem to explain a portion of the observed SST changes in the El Niño–Southern Oscillation (ENSO) regime in the tropical Pacific [Jin, 1997]. Additional changes are caused by the mixed layer depth anomalies, including turbulent fluxes out of the mixed layer bottom [e.g., Wang and McPhaden, 2001], internal waves, and ocean eddies. To be useful in climate analyses and climate modeling, the uncertainty of the hmix must match the uncertainty of SST observations, or in our context, that all terms in equation (1) have comparable errors. The two major reasons for the lack of observed hmix are the insufficient number of observations, both in spatial and temporal senses, to average the effects of internal waves and ocean mesoscale [e.g., Moisan and Niiler, 1998] and that hmix is not a directly measurable quantity. The ocean mixed layer depth hmix is most commonly defined as that depth where the temperature (potential density) has decreased (increased) from the surface value by a constant amount Δ. For climatological profiles on a global scale it is found that the Δ criterion introduces an error of 20 m in hmix [Kara et al., 2000b], which is several times larger than the error of the terms in equation (1) introduced by SST or flux errors. The error in hmix seems to be reduced when applying the threshold method to individual profiles [de Boyer Montégut et al., 2004].
 The main focus in this paper is to establish a new criterion for hmix that can be used globally to study the large-scale processes of thermal energy storage in the upper ocean in both observations and in OGCM simulations. In this respect hmix should be considered in a similar way that the high-quality SST is currently used for a diagnostic of the upper ocean processes. We require from the introduced criterion that it minimizes systematic errors between estimates of hmix based on measured data and the output of OGCMs and that applications of the criterion to both data sets provide comparable statistics for hmix, respectively. This analysis should make model-data comparison more stringent because not only must SST be correct, but also its first vertical moment, or hmix. We use the global historical hydrographic measurements with both high and low vertical resolutions, including the World Ocean Circulation Experiment (WOCE) data set, and a full OGCM output to determine whether calculations of hmix and its variability can be improved.
 The paper is organized as follows: In section 2 we present the data and discuss the difficulties in estimating hmix from these data; we then introduce a new criterion for estimating an hmix (which we refer to as hmix/c) and test the sensitivity of hmix/c to the assumptions made in our new criterion and also compare our results with the more traditional Δ criterion. In section 3 we introduce a “quality index” and discuss the quality of the observed hmix/c. In section 4 we describe some characteristics of hmix/c and its space and time variability over ocean basins. In section 5 we apply our algorithm for hmix to the output of the Massachusetts Institute of Technology (MIT)/OGCM/Estimating the Circulation and Climate of the Ocean (ECCO) model in an assimilation mode and compare hmix/c to the diagnosed planetary boundary layer depth in the model KPP parameterization. A summary of our results is given in section 6.
2. Problems of Estimating Mixed Layer Depths
2.1. Data Set
 We use the quality controlled hydrographic global measurements from mechanical bathythermographs (MBTs), expendable bathythermographs (XBTs) and conductivity-temperature-depths (CTDs) with high and low vertical resolution (the depth interval is equal to 1–5 m and 20 m, respectively). These are filed by World Ocean Circulation Experiment International Project Office, WOCE Data Products Committee (WOCE IPO)  (and available at http://www.nodc.noaa.gov/woce_v3). The MBT temperature measurements provide the longest (from the early 1950s until the mid-1990s) and the most extensive time series (2.4 million profiles). The vertical resolution of these profiles is irregular and most of the profiles do not extend below 200 m (maximum depth is around 700 m). Almost the same number (about 2.1 million) of temperature profiles from XBTs is available, some of which reach depths of 700–1500 m with an interval of 2 m. However, the XBT data show quite often a gap between the surface value (∼0 m) and the second measuring point (about 10–30 m); this includes a temperature jump of ΔT ∼ 0.2 K which effects the estimation of hmix using a small (ΔT ≤ 0.2°C) threshold criterion. Therefore we start the estimation method of hmix at the second measuring point to avoid artificial mixed layer depth between the first two measuring points. The CTD temperature and salinity profiles (0.4 million) typically extend over the whole water column with a spacing of 1–2 m. XBT and CTD profiles become readily available after the late 1960s (Figure 1).
 We also use the daily temperature profiles from the Tropical-Atmosphere-Ocean(TAO)/Trition (T/T) buoy array (http://www.pmel.noaa.gov/tao). The first buoys were deployed along 110°W in late 1984. The array currently consists of approximately 70 deep ocean moorings (from the surface down to 500 m) with 2°–3° meridional and 10°–15° zonal resolutions, spanning an area between 8°S and 8°N, 137°E and 95°W. To be consistent with our ECCO model-data comparison, we chose the time frame from 1992 to 2001 that covers both the 1992–1995 and 1997–1998 El Niño cycles [Wang and McPhaden, 2001]. Since these data are not continuously distributed in space and time, we normalized monthly means by the number of days of data per month. For our study, data are available from 7 depth levels, at 20 m separation to 140 m, and SST is the temperature at 1 m depth.
 The model data comes from a 1° × 1° 1992–2001, global adjoint run of the ECCO model where the daily average temperature is used to diagnose hmix, using both the Δ criterion as well the criterion introduced below for computing hmix (namely, A. Köhl et al., The ECCO 1 degree global WOCE Synthesis, ECCO Report 20, Nov. 2002, available at http://www.ecco-group.org/reports.html). Daily average ocean planetary boundary layer depths hpbl that are the depths to which surface-forced turbulent mixing penetrates in the “K profile parameterization” (KPP) are also retrieved from the model solution.
2.2. Definitions for the Ocean Mixed Layer Depth hmix
2.2.1. The Δ Criterion and Its Limitations
 Direct measures of turbulence in the near sea surface layer or turbulent dissipation rates [Brainerd and Gregg, 1995] are rare, so the actively turbulent layer cannot be defined on basin scales by direct turbulence observations. Overturning scales of 10 cm could potentially be used, but the very high resolution profile data are not filed in the public data sets. The “mixed layer” in our coarse (>2 m) vertical resolution data is the zone of relatively homogeneous water formed by the history of turbulent mixing, including zones where water masses created by deep convection, “mode waters” [Talley and Raymer, 1982], abound. We endeavor to find the homogeneous layer closest to the surface, believing that we have captured the effect of the most recent, above the daily cycle, mixing events. The simplest concept is to define the ocean mixed layer depth hmix as the depth where the quasi-homogeneous profile of temperature has decreased (potential density increased) from a “reference” value, most commonly the surface value, by a constant amount, Δ (various published Δ criteria are listed by Kara et al. [2000b] and de Boyer Montégut et al. ). According to Lukas and Lindstrom , a potential density based Δ criterion is more reliable than the temperature based, because in many regions the temperature profile does not capture the vertical stratification correctly. However, density profiles show significant areas where hmix is determined by a halocline. “Barrier layer” regions [Sprintall and Tomczak, 1992] are identified by a deeper temperature-based hmix than the density-based hmix; the latter hmix is deeper than hmix based on temperature profiles in “vertically density-compensated areas” [de Boyer Montégut et al., 2004, paragraph 1]. Presently, the available density profiles are about an order of magnitude less than the temperature profiles (e.g., Figure 1). For most parts of this paper we use temperature profiles, unless explicitly indicated. In section 3.1 we address some of the differences that result in hmix computations when using the density profiles.
 We can see from the variety of temperature profiles, which features the complexity of the upper ocean vertical structure (Figure 2) that the stratification depends strongly on the dynamical regions and the seasons. Thus a value of Δ chosen subjectively for one region or season might not be applicable to another region or season. Even if a profile shows a well-defined mixed layer and the ΔT is within the observed temperature range, the depth of the mixed layer is not defined accurately by the Δ criterion (Figures 2b, 2f, and 2h). Kara et al. [2000b] computed hmix from both monthly climatological data of the ocean weather station P in the northeast Pacific and the Levitus climatology [Levitus et al., 1994; Levitus and Boyer, 1994]. Kara et al. [2000b, p. 16,803] concluded “that the inherent variability of hmix only allows for an accuracy of 20 m in 85% of the cases” (using a 0.8°C criterion). Deriving at hmix from in situ profiles, recently, de Boyer Montégut et al.  showed that a threshold of 0.2°C is most appropriate to result in optimal estimates of hmix. Systematic biases of hmix develop when the Δ criterion is applied to idealized temperature profiles that are demonstrated in detail on Figure 3.
 The Δ criterion depth hmix is sensitive to the following.
 1. For the chosen Δ (Figure 3a), the larger Δ, the deeper is hmix (blue lines). Note in particular that this bias is not negligible when small vertical gradients at the base of the mixed layer occur.
 2. For the “reference value” (Figure 3b), the smaller the difference between the temperature of the mixed layer (the reference value) and the one below and the smaller the vertical gradient at the base of the mixed layer, respectively, the larger is hmix. This implies an artificial dependence of hmix on the reference value, or SST.
 3. For the vertical resolution (Figure 3c), linear interpolation in low vertical resolution profiles leads to a smaller value, or shallow bias of hmix. In other words, the lower the vertical resolution, the larger the tendency of hmix to cluster at the levels where the data is sampled and sampling levels become “sticking points” for hmix (see also red lines in Figure 3a).
 4. For the vertical gradient at the base of the mixed layer (especially in the 20-m resolution case) (Figure 3d), the larger the vertical gradient at the base of the mixed layer the smaller (by about 20%) and the “stickier” is hmix.
 The tendency of hmix to cluster at sampling levels seems to be enhanced for small Δ applied to low-resolution profiles and then relatively independent from the stratification below hmix. In the 20-m resolution case, with a vertical gradient at the base of the mixed layer of 0.2 K (10 m)−1, hmix estimated for ΔT = 0.2°C(ΔT = 0.8°C) shows an offset from the ideal hmix by 5 m(4 m); with a vertical gradient of 0.1 K (10 m)−1 the standard deviation yields 4 m(2 m).
 In summary, the above discussion points to three major concerns about the Δ criterion method. First, if we assume that a ΔT = 0.2°C is representative of an adequate criterion for the threshold method [de Boyer Montégut et al., 2004], the estimated hmix is often below the visible point in the profile that clearly marks hmix (Figures 2b, 2f, 2g, and 2h); the resulting deviations can be sometimes in the order of hmix itself (Figure 2f). Second, the complicated dependence of hmix on the vertical resolution is not desirable, especially when comparison studies are made with the low vertical resolution output of OGCMs. Third, because hmix depends on the SST or a reference value and there is no rational choice for this value leads us to be a more than skeptical that the turbulent region of the upper ocean is captured well by the Δ criterion. We are thus motivated to seek a more universal criterion that can be applied to computing mixed layer depths.
2.2.2. Gradient Method and Our Method, a Curvature-Based Criterion
 Consider now the information that can be gleaned from inspection of temperature gradients. Large et al.  suggested using the vertical gradient where hmix is taken to be the shallowest depth where the vertical density gradient is equal to twice the average gradient from the surface to 120-m depth. Since the temperature profiles are often smoothed to 5-m resolution further choice has to be made on how to smooth the salinity spikes or the length scale over which the gradients are computed. Like the Δ criterion, the gradient criterion depends critically on the prescribed gradient threshold; in addition, the threshold is usually kept constant for all space and time (see listing by Kara et al. [2000b]). In comparison with the Δ criterion the vertical gradient-based criterion seems also to be less robust [Brainerd and Gregg, 1995]. Besides the threshold methods (and in absence from direct turbulent measurements), hmix is derived from oceanic profiles, by less conventional methods, like the “integral,” “least squares regression,” and “split-and-merge” methods (taken up and recently introduced, respectively, by Thomson and Fine ).
 The above considerations led us to look for a method that does not depend critically upon the choice of values of a number of parameters which are to take the best compromise to deal with the global ocean. Consider, hence, the information contained in the second derivative, or “curvature,” of the profile. Perhaps the first local extreme value of the curvature of temperature or density profile is a distinguishing feature of the depth to which the most recent mixing events penetrate. An advantage of this approach is that, unlike threshold methods, the estimated hmix is not a linear function of another dynamical quantity, such as the SST for instance and is less dependant on the dynamical region or seasons, respectively. First of all our approach specifies an interval in depth where hmix should be identified which requires the choice of mainly two parameters as boundary conditions. Afterward the approach searches within this interval after the first extreme curvature of the profile and interpolates from its depth to hmix. The detailed description of our approach follows below; Figure 4 accompanies the description considering as example a high- and low-resolution temperature profile. The complete MATLAB code is listed in the auxiliary material.
18.104.22.168. Defining Derived Quantities
 We define the depth z(i) as negative definite and refer to i = 1 as the level closest to the surface. Gradients of the vertical profiles at level i are defined as
where the index i5 m refers to the next deeper level, which is at least 5 m deeper than the level i, for a smoother and more stable estimate of gT. Note, since we are interested in gT(i) only as a boundary condition for the estimation of the level closest to hmix we defined the gradient gT(i) as the gradient below the level i and not as the centered gradient; gT is positive if T decreases with depth (see Figures 4c and 4f). The curvature is
cT(i) is centered, and it is positive if gT above the current level is smaller than gT of the current level (see Figures 4b and 4e).
 As a second boundary condition for the estimation of the level closest to hmix we evaluate the variability of the profile in a 30-m region below the current level, in order to distinguish a near homogeneous region from a region of rapid change. We define σ30(i) as the standard deviation of T over the levels in a 30-m interval below the current level (the current level included) (see Figures 4c and 4f).
22.214.171.124. Truncation Condition
 If max[σ30] < 0.02 K we set hmix = min[z]. Hence, if no region of significant inhomogeneity exist in the profile, then hmix must be below the lowest measured level (these profiles are flagged in the computing algorithm).
126.96.36.199. A First Guess of hmix
 We define a lower limit in the profile, z(imld0), in which the hmix can be found; z(imld0) is the first level with ∣gT(i)∣ > 0.25 max[gT] and σ30(i) > 0.02 K. This level usually refers to the top of the thermocline and is closely below hmix (see Figures 4b and 4c). This range is used to estimate the scale of a significant gradient, which is defined as the standard deviation of gT over the interval [z(1), z(imld0)], similar to the gradient based approach of Large et al. . Additionally, we set the absolute lower limit for to 0.004 km−1 for high (<6 m) and 0.002 km−1 for low vertical resolution profiles, respectively (finite difference measurement errors in the gradient estimation match the half of these empirical values).
188.8.131.52. Level Closest to hmix
 The level closest to hmix, z(imld), is the first local maximum/minimum of cT that falls together with a positive/negative gradient gT. Additionally we ask for two boundary conditions: First, ∣gT∣ > which defines a threshold for a significant local inhomogeneity in the profile. As a second boundary condition we demand that σ30 > 0.02 K. This second condition takes a more global point of view. It assures that z(imld) refers to a point above a region of rapid changes and not just a small-scale intrusion, which is important for high-resolution profiles (z(imld) is marked throughout Figure 4).
 If no extreme value is found in this upper subsection of the profile, we set z(imld) to the first level with gT ≥ 0.7 max [gT]. This condition happens for typically shallow mixed layers. Less than 1% of the profiles are affected if hmix exceeds 40 m.
 For low-resolution data it is important to interpolate hmix between the levels, in the interval [z(imld − 1), z(imld + 1)]. Our interpolation scheme essentially assumes that the profile follows the idealized structure illustrated in Figure 3. Therefore we fit an exponential function fit(z) = C + AeBz to the interval [z(imld), z(imld + 2)] if possible; that means A, B, C ∈ R and B > 0. hmix is defined as the depth z for which fit(z) = Tmix, where Tmix is the mean temperature in the interval [z(1), z(imld)]. Hence hmix is interpolated in the interval [z(imld − 1), z(imld)]. However, in cases where gT(imld − 1) is both less than 10% of gT(imld) and an exponential fit is possible from z(imld + 1), we shift the interpolation into the interval [z(imld), z(imld + 1)] (see example in Figures 4d–4f). Note that such an exponential interpolation my also improve the hmix estimate of a Δ criterion based method.
 If an exponential fit is not possible we use a simple linear function for fit(z) with the gradient ((gT(imld − 1) + gT(imld))/2) and use Tmix = T(imld − 1) as reference. If both an exponential fit is not possible and ∣gT(imld − 1)∣ < ∣gT(imld)∣, the profile does not fit to the idealized structure illustrated in Figure 3. We then use a simple linear interpolation to the reference value Tmix = T(imld − 1) ± δT, where δT is the maximum T difference between two consecutive levels found in the interval [z(1), z(imld−1)].
 For high-resolution (<6 m) profiles we always use the linear interpolation. In order to optimize our interpolation scheme we compared the estimated hmix based on high-resolution profiles to the estimated hmix based on the same profiles but with reduced 20-m resolution, assuming that the high-resolution estimate is the “real” hmix. The examination of many observed low-resolution CTD profiles revealed that both the exponential and the linear fit have a small systematic offset to the real hmix, which we empirically corrected with an additional small linear shift.
 Note that our method is an empirical method based on the visual examination of high-resolution (∼2 m) temperature profiles of the entire global WOCE-CTD data set, a copy of it which was artificially reduced to 20-m resolution profiles and data from a numerical MIT-OGCM integration. The crucial parameters for this method are the boundary conditions for the first local maximum/minimum of cT (section 184.108.40.206). These parameters essentially define the level of inhomogeneity in the profile, which we consider as indications for hmix. Note that unlike a Δ criterion the hmix/c is not a direct function of these parameters and, in turn, these parameters are not a direct function of another physical quantity (such as SST). Therefore small variations of the parameters will not lead to proportional changes in hmix/c, which is one of the advantages of our method. Additional parameters (section 220.127.116.11) refer to the measurement uncertainties. For high-resolution data, the smoothing of the gradients assures that small-scale intrusions do not effect the estimation of hmix/c. We especial took a careful examination of critical regions/seasons such as the winter time deep convection regions in the northern North Atlantic as well as the tropical regions with a rather smooth transition into the thermocline. The parameters we use are empirically chosen, and were optimized to some degree by variations of a single parameter and a visual examination of all profiles for which the hmix estimate was changed.
 To apply the above criterion to salinity or potential density profiles, we multiply the parameters by 10 and 4, respectively, since the ratio of the standard deviation of salinity/potential density in the upper 500 m to the one of temperature tend to correspond to 0.1/0.25.
2.3. Comparison of hmix/c With the Δ Criterion
 A number of significant differences were found between hmix/c and hmix/d, at which the latter uses the 0.2°C criterion following de Boyer Montégut et al. . The histogram of differences between hmix/c and hmix/d was computed from the entire ensemble of CTD-temperature profiles (Figure 5a). The difference is skewed toward larger hmix/d and is not a Gaussian distribution. The mainly positive outliers indicate that hmix/d is systematically larger than hmix/c by 6 m. The 70% interval around the median of the differences is 2–18 m. The differences between hmix/c and hmix/d are much larger than one may have expected assuming that observed profiles are just noisy versions of the profiles in Figure 3 and therefore hmix/d is just systematically shifted deeper due to its finite Δ value. However, the visual inspection of the examples in Figures 2b, 2i, 2f, and 2h, already indicated that larger differences between hmix/c and hmix/d exist, which are due to the complex nature of the profiles. We randomly chose 500 profiles where the relative differences were more than 20% and a visual inspection for these profiles, like the one used in Figure 2, showed that in 63%, hmix/c appeared to render a more realistic value for upper ocean homogeneous depth than did hmix/d, whereby in 10% hmix/d looked to be more reliable. For the remaining 27% it is not definite which one of the two criteria reproduces the adequate hmix (see Figures 2a, 2d, and 2e).
 Since smaller differences at small hmix/c appear to be more significant for the upper ocean heat budget than at larger values we relate the difference between hmix/c and hmix/d to the magnitude of hmix/c (Figure 5b). The general conclusion is the same as for the nonnormalized differences that about one third of all hmix/d are more than 100% larger than hmix/c. The relative difference peaks at 20% (Figure 5b). (For the occurrence of a relative difference of hmix above 100% we refer to Figures 2c and 2f.) The curvature criterion finds small values of mixed layers what the Δ criterion does not. For instance focus on hmix/c in the Northern Hemisphere: When hmix/c is smaller or equal 10 m and/or is in the first half of the year, most of the relative differences exceed the 100% value. When hmix/c is deeper and/or is in the second half of the year the differences that exceed 100% are reduced to one fourth and the remaining distribution of the relative difference peaks at around 20% (not shown).
 The distribution of hmix stays flat until hmix reaches 50 m; for deeper hmix it decays rapidly with increasing hmix (Figure 6); these shape functions are independent of the method used. In addition the curve progression of hmix shows clearly the overestimation of hmix by the Δ criterion if hmix exceeds 50 m.
 To have a better global and temporal coverage, we combine CTD data with XBT data (see Figure 1). In the Northern Hemisphere, spatial pattern of the relative difference between the long-term seasonal mean hmix/c and hmix/d shows some zonal structures (Figures 7a–7d) that reflect the effects of the progression of the seasonal thermocline on the hmix/d computation. In the thermodynamically active cooling regions in winter and spring, poleward of 40°N, the relative differences exceed 20% (Figures 7a–7d) as predicted by comments in the discussion of Figure 5b. A global uniform ΔT of 0.2°C is often too large to capture the small vertical gradients and is often too close to the observed temperature range to be useful. Therefore hmix/d is generally deeper than hmix/c (Figures 7a and 7b). In the midlatitude North Pacific the top of the main thermocline stratification is strong enough that the overestimation of hmix/d is below 20% (Figures 7a and 7b).
 The relative difference of hmix is reduced in both the midlatitude North Pacific and North Atlantic during summer and fall due to the existence of a distinctive seasonal thermocline (Figures 7c and 7d). The seasonal thermocline is much better developed in the western portion of each gyre than in the eastern portion [e.g., Tomczak and Godfrey, 1994].
 The relative difference of hmix using a larger threshold (ΔT = 0.8°C) is much more pronounced even if the main characteristics are similar to the difference using the 0.2°C criterion. One discrepancy is that the relative difference shows a zonal gradient in the midlatitude North Pacific (not shown). Therefore the 0.8°C criterion seems not to be able to capture the observed temperature range between the seasonal and permanent thermocline in the eastern midlatitudinal North Pacific as it does in the western; whereby the 0.2°C criterion does take into account the effects of the seasonal thermocline and we find that the relative difference is less than 10% (Figures 7c and 7d). This supports the finding from de Boyer Montégut et al.  that the 0.2°C criterion is more successful following the seasonal variability of hmix than a more commonly used larger Δ criterion.
 In the tropics around ±20° latitude of the equator the near surface stratification is characterized by a weak seasonal cycle. In this band a definition of a seasonal cycle of stratification in the upper 50–100 m is not well established since the daily cycle of turbulent mixing is typically stronger than seasonal-to-interannual changes. However, hmix/d appears to be in generally larger by about 20% than hmix/c. The water column is weakly stratified just below the surface and the profiles of temperature within the top of the main thermocline are concave. The vertical gradients differ weak until the top of the thermocline is reached (see section 2.2.2), so a smaller hmix is found in these regions of positively curved profiles by searching for the location of the extreme curvature than the Δ criterion. In the southwestern equatorial Pacific the relative difference between hmix/d and hmix/c is larger than 40% throughout the year. Under “normal” (light) wind conditions this region is weakly stratified in temperature until the top of the thermocline; a threshold criterion captures the depth of the top of the thermocline including a barrier layer and overestimates the depth of the well-mixed region, respectively, except during west wind bursts when barrier layers disappear. The sensitivity of the curvature criterion to small-scale inhomogeneities in the temperature profile at the top of a barrier layer explains shallower hmix/c than hmix/d. Studies of Lukas and Lindstrom  seem to confirm the result of a mixed layer that is shallower than previous estimates; their threshold method found nearly isothermal layers that are deeper by about 33% than the real mixed layers as estimated with density profiles. Further comparisons between hmix/c and hmix/d are taken up in sections 3.2 and 4.
3. Effects of Salinity, Vertical Resolution, and a Quality Index for hmix
3.1. Effects of Salinity
 Potential density profiles are computed from the CTD profiles of temperature and salinity. The buoyancy of the upper ocean water column is in a condition where thermal gradients account for 67% of the density variability and salinity for 33% [Speer et al., 1995]. The differences between hmix/c inferred from profiles of temperature (hmix/cT), salinity (hmix/cS) and potential density (hmix/cPD) show (Figure 8) that in 70% of the cases hmix/cT and hmix/cPD differ less than ±4 m (Figure 8a); hmix/cS and hmix/cPD by less than ±5 m and hmix/cS and hmix/cT by less than ±8 m (Figures 8b and 8c). The distribution of the difference between hmix/cT and hmix/cPD has a median very near zero (Figure 8b). In contrast, the difference between hmix/cS and hmix/cPD is skewed toward larger hmix/cPD (Figure 8a); the difference between hmix/cS and hmix/cT shows a small skew toward larger hmix/cT, with a median at −1 m (Figure 8c). In general, changes in temperature and salinity in the surface layer seem to be density compensating that we find larger isothermal layers than isohaline layers, with isopycnal layers in between. For the compensation of horizontal temperature and salinity gradients in the ocean mixed layer on scales from 10 m to 100 km we refer to, e.g., Rudnick and Ferrari .
 The global map where the differences between the CTD profiles from Figure 8 equal 70% shows that the regions with high values (>15 m) occur in the western and central tropical and North Pacific (Figures 9b and 9c). Large differences between hmix/cS and hmix/cPD extend farther northward along the western boundary of the North Pacific (Figure 9c). Larger horizontal gradients of hmix derived from the different techniques appear within eddies of the Kuroshio compared those within the Gulf Stream [Kara et al., 2000a]. Both the deep saline layer as well as the barrier layer act to define hmix/cS in the western tropical Pacific. During the onset of the 1997–1998 El Niño, zonal advection of fresh water from the west was particularly important in the central equatorial Pacific [Wang and McPhaden, 2001], so shallow wind-driven currents form vertical layers of density that not necessarily reflect the local effects due to forcing by rain and heating. In general, regions in the tropics and northwestern Pacific, where large differences between isohaline and isopycnal methods are found, coincide with fresh water barrier layer regions [Sprintall and Tomczak, 1992; Kara et al., 2000a]. There the generation of turbulence is due to both changes in local surface fluxes as well as strong upper ocean currents (Figure 9a).
3.2. Effects of Low Vertical Resolution
 Low vertical resolution of observed or simulated data has a significant impact on estimating hmix (see section 2.2.1). In order to evaluate the effect of low vertical resolution we artificially reduced CTD profiles to a 20-m vertical resolution and estimated hmix based on both low- and high-resolution profiles. The results of this comparison are presented below. Additionally, we estimated hmix based on the output of an OGCM, with low nonlinear vertical resolution (section 5).
 In order to estimate the error of hmix/c computed from low-resolution profiles we assume that hmix/c computed from high-resolution temperature and potential density profiles represent the real depth of the ocean mixed layer. The median of the differences between hmix/c computed from low-resolution profiles and that computed from high-resolution profiles is 5 m; half of the differences lie within ±8 m around the median (Figure 10). This is a significantly smaller value if compared to the accuracy in hmix/d of ±20 m in 85% of the cases found by Kara et al. [2000b] (applying the 0.8°C criterion to monthly climatological profiles). However, the distribution of the differences show a skew toward an overestimation of using low-resolution data as 12% of the low-resolution computations are larger by more than 30 m, but only 1% are smaller by more than 30 m (Figure 10).
 A most remarkable feature in both the Δ as well as the curvature methods of computing hmix in the 20-m vertical resolution and the 2-m CTD-temperature profiles are the clustering, or stickiness of hmix to the sampling levels (Figure 6). The behavior of the distribution function of hmix/c shows a more sinusoidal behavior than hmix/d even if these have similar amplitudes. Therefore the hmix/d has concentration peaks at the sampling levels indicating a stronger clustering around the levels, in contrast to the structure of hmix/c (see Figure 3d).
 In general all the algorithms endeavor to reconstruct a nonlinear parameter hmix from the discrete profile data. From among those algorithms we investigated, perhaps the best choice is one that produces the smallest bias relative to some simple, “continuously” sampled profiles. A linear interpolation between the levels leads to shallow biases, an underestimation of hmix and it seems to cluster calculations around the shallower level of the interpolation interval, respectively (see Figure 3c). This bias decreases with increasing vertical resolution, but does not altogether avoid stickiness around the layers where data sampled. There is some improvement with an exponential interpolator. The curvature-based algorithm dramatically improves the stickiness in the midlatitudes, especially in summer and fall. Stickiness occurs primarily in the tropics where the daily average profiles appear to have smooth transitions from the weakly stratified, strongly turbulent layer to the top of the less turbulent main thermocline (see Section 2.3).
 Note, in general it is difficult to address hmix/c or hmix/d based on low-resolution profiles as the more reliable estimate for the real hmix. Assuming hmix/c from high-resolution profiles to be representative for the real hmix the histogram of differences between the real hmix and hmix/d based on low-resolution profiles shows the same scattering like the corresponding hmix/c (see Figure 6). However, the explained variance (squared correlation coefficient) between hmix/c based on high- and low-resolution profiles amounts to 72% and to 46% for hmix/c based on high- and hmix/d based on low-resolution; this seems to indicate that hmix/c based on low-vertical resolution profiles is more reliable than the corresponding hmix/d. In addition to determining the correct measured level from where to interpolate to estimate hmix/c, the exponential interpolator appears to be responsible for an hmix/c based on low-resolution profiles (even if it clusters around sampling levels) that shows a higher correlation with the real hmix and that hmix/c is not “just somewhere” between two or at one sampling level like the corresponding hmix/d.
3.3. Biases of Climatologies From MBTs and T/T Array
 In order to analyze the sensitivity of hmix/c with respect to temporal averaging we calculated hmix/c from daily MBT-temperature profiles in the western North Pacific in the 1° × 1° area centered at 31°N, 164° (OWS-V) during July 1954. The monthly mean hmix/c, based on daily average data, is 15.8 m (black solid line in Figure 11); in contrast, the mean hmix/c computed from the monthly mean profile is 10.4 m (gray solid line in Figure 11). The resulting difference of 5.4 m is significant because the standard deviation of the daily hmix/c was 5.1 m. The result from de Boyer Montégut et al.  confirm the shallow bias of hmix/c based on averaged profiles compared to hmix/c derived from in situ data. The authors constructed a global climatology of hmix/d (with ΔT = 0.2°C) based on averaged temperature profiles which is 25% shallower than hmix/d estimated from individual profiles.
 From daily average, 20-m resolution temperature profiles of the T/T array we calculate hmix/c and average these to monthly means. The main effect of temporal averaging of a profile appears to be the smoothing of the vertical gradients at the base of hmix and a “broadening” of the seasonal thermocline. As above, we calculate monthly mean profiles and from these hmix/c (see section 2.1). As at OWS-V, monthly averaging of the T/T profiles results in a shallow hmix/c bias. This averaging also appears to enhance the accumulation of hmix/c around the sampling intervals (Figure 12). However, this accumulation is significantly reduced in the monthly average, even in low-resolution profiles, if daily data on hmix/c are available for computing its monthly mean (Figure 12).
3.4. Quality Index, QImix
 Calculating hmix is based on the notion that there exists a near-surface layer with quasi-homogeneous properties in which the standard deviation of the property about its vertical mean is close to zero. Below the depth of hmix, property variance should increase rapidly about its vertical mean. A quality index for hmix by which we can quantify if this assumption is valid is the ratio of the standard deviation of the observed property in the depth range from the surface to hmix to the standard deviation from the surface to the depth of 1.5 × hmix (namely, Figure 13a). The analysis of the observed profiles showed that the depth 1.5 × hmix is a realistic choice for including most of the upper ocean stratification and not intruding into the main thermocline. We subtract the standard deviation ratio from 1.0 so that a value of 1.0 represents high-“quality” computation of hmix and progressively lower values imply that larger volumes of stratified water are present above the level of hmix (equation (4) and Figure 13a). Consider the quantity, QImix, as
where σ( ) denotes the standard deviation from the vertical mean 〈〉 from h1, the first layer near the surface, to the depth of hmix or 1.5 × hmix, respectively.
 The examples of values from different CTD temperature profiles (Figures 13b–13f) demonstrate that if QImix > 0.8 a well defined hmix results. For QImix in the range 0.5–0.8, increased uncertainty of the profile interpretation becomes evident and with QImix < 0.5 no mixed layer interpretation is possible. We also find that QImix for hmix/d is in average 10% smaller than for hmix/c indicating that the curvature methodology is more robust. The spatial pattern of QImix for hmix/c for different seasons shows that in most regions of the world ocean QImix is above 0.7, for about 70% of the profiles (Figure 14). Predictably, QImix has the largest values in the hemispheric summer and fall when a sharp gradient at the base of the seasonal mixed layer is present. The reason why a lower value of QImix occurs where a barrier layer and near the eastern Pacific equator is that in the former a quasi-homogeneous column exists below the barrier layer and strong equatorial upwelling and daily vertical mixing cycles produce significant layering near the equator. On the basis of the stringent criteria introduced here, it makes little sense to refer the equatorial mixed layer as determined from T/T array, or any limited vertically sampled temperature profiles as the region of enhanced turbulence. In subpolar regions, mainly in the North Atlantic during winter, QImix is also seen to fall below 0.5 (Figure 14a); this reflects that convective mixing is not going on and “wriggly” profiles or small vertical gradients exist throughout the water column that make the turbulent layer determination on the basis of profile data somewhat problematic (Figures 2h, 2i, and 13d–13f).
4. Characteristics of Observed Mixed Layer Depth Variability
4.1. Global Features
 The pattern of seasonal mean hmix/c computed from the global individual profile data (Figure 1) shows the familiar negative zonal gradients in equatorial regions throughout the year with shallow hmix/c (<40 m) in the east and deeper hmix/c (>100 m) in the west; the shallow hmix/c in the eastern equatorial Pacific is reproduced in a broader tongue during the first half of the year (Figure 15a, January–March and April–June). In the winter and springtime North Pacific a band of hmix/c < 75 m is found around 30°N from the west coast to 150°W that is shallower than the surrounding hmix/c (Figure 15a, January–March and April–June). This is due to the occurrence of the subtropical front under which the mixed layer remains shallow even in strong winds that produce deep mixed layers to its north and south [Niiler, 1982]. This band disappears in summer where hmix/c < 40 m north of about 25°N (Figure 15a, July–September). A local maximum of hmix/c (>100 m) appears in winter west of Peru near 25°S, 110°W (Figure 15a, July–September). Tomczak and Godfrey  (with ΔT = 0.5°C) find a similar distribution. A radial pattern is found in hmix/c > 100 m that reflects the subtropical gyre structure (Figure 15a, July–September).
 The data distribution at the time of writing this report significantly limits the computation of significant seasonal means of hmix, especially in high latitudes with weak stratification conditions as for example in the Labrador Sea in winter. There the winter data is composed mainly of XBT profiles that sample to 1000 m. These depict obvious mixed layers near the surface, but without salinity compensating effects, the turbulent layer cannot be determined. Convection to the level of Labrador Seawater formation to 1500 m and below cannot be sampled by these relatively shallow temperature profiles. Therefore our results of a long-term seasonal mean hmix/c and its standard deviation are biased by under sampling and the lack of salinity observations in many regions. In the Labrador Sea the bias is toward shallower hmix/c and smaller standard deviations than we believe should occur from a knowledge of the physical processes observed in concentrated field experiments [e.g., Krahmann et al., 2003]. De Boyer Montégut et al.  identify a maximum winter hmix/d (ΔT = 0.2°C) based on individual profiles in deep water formation regions that is significantly shallower that previous estimates. This indicates that variations of hmix are detected instead of deep thermocline movements. However, It goes without saying that an expert oceanographer must consider the data in each ocean region and global pictures, as presented here, must be used only as points of departure for more thorough investigations.
 The historical data (see Figures 1 and 9a) is relatively comprehensive in many regions and we provide the seasonal means and the standard deviation of anomalous monthly mean as a base from which to embark for future research. The standard deviations of anomalous monthly mean hmix/c in the 2° × 2° boxes are defined only if more than 4 values in a calendar month exist. Recall that the variance (standard deviation squared), in addition of direct forcing from the atmosphere, can be produced both by internal waves and ocean eddies that move the mixed layer vertically and horizontally on short timescales.
 The variance of hmix/c is largest (>50 m) in the mode and deep-water formation regions in both hemispheres in winter and also in the Northern Hemisphere spring (Figure 15b, January–March and July–September). However, generally, the variance is largest in the Southern Ocean due to temporal data gaps. Once the seasonal thermocline is formed in midlatitudes, the variability of hmix/c is less then 10 m in summer, increases slightly to the largest value of 20 m in fall (Figure 15b). In the band around 30°N in the Pacific, where hmix/c < 80 m in winter, the variability exceeds 25 m (Figure 15b, January–March). Farther south (around 10°N) hmix/c is about 100 m and its variability <20 m. This is the region of the North Equatorial Counter Current that flows against the prevailing Trade winds and the turbulence is active there throughout the year. The variability of hmix/c in the equatorial Pacific is about 25 m in the west and is less than 10 m in the east over the entire year (Figure 15b) but recall the caveat of interpreting the quasi-homogeneous layer on the equator as a region of active turbulence.
 In general, the pattern of the anomalous monthly mean hmix/c reflects the one from the long-term seasonal mean: the larger the mixed layer depth the larger its monthly mean anomaly. Consider also the standard deviation of hmix/c relative to its seasonal mean value that takes into account also effects of a variety of short-term vertical and horizontal adiabatic heaving of the seasonal and main thermoclines. The largest relative variance (>30%) appears in the northern hemispheric spring when the seasonal hmix/c is small (Figures 15a, January–March, and 15c April–June); the smallest relative variance (<20%) is in the Northern Hemisphere autumn (Figure 15c, October–December). These patterns reflect either a higher sensitivity of the technique for computing hmix/c during the seasonal thermocline formation that occurs during the demise of the seasonal thermocline, or quite simply upper ocean inhomogeneities are large when the mixed layer is small.
4.2. Equatorial Pacific Ocean From T/T Array
 To view a more detailed description of the seasonal-to-interannual variability of hmix/c, consider the longitude-time diagram of anomalous monthly mean hmix/c computed from the T/T array (Figure 16a); whereas anomalies of monthly mean hmix/c define deviations from the long-term monthly mean climatology of hmix/c. The phase and amplitudes of anomalous hmix/c in the equatorial Pacific are dominated by the 1997–2000 El Niño/La Niña cycle. During the build up of El Niño a positive anomaly of hmix/c propagates eastward where it reaches its maximum after 12 months. During the La Niña period a maximum negative anomaly also spreads eastward as its amplitude decreases. During the termination of La Niña in 2000 a systematic increase of hmix/c occurs in the west and a decrease in the east, with an inflection point 150°W. The El Niño/La Niña cycle of 1992 to 1996 appears to be stationary with no propagations features that can be readily identified. During the build up of El Niño there is a decrease of hmix/c in the west and slight increase in the east; the La Niña period is marked by the opposite behavior with the inflection point again at 150°W. The corresponding SST anomalies (namely, http://www.cdc.noaa.gov/cdc/data.reynolds_sst.html) depict a strong warming of the “cold tongue” in 1997 that coincides with the largest increase of hmix/c. Unlike the hmix/c, the 1997–2000 SST anomalies show no propagation during both El Niño and La Niña phases. The propagating anomalies of hmix/c toward the east might be more related to the propagating character of the top of the main thermocline, for as has been explained above, the computation of hmix/c in the equatorial zone identifies the location where the weak upper ocean temperature gradient changes to a larger value at the top of the main thermocline.
 In the tropical Pacific area that is spanned by the T/T array, the zero lagged correlation between anomalous monthly mean hmix/c and the SST as average over the eastern region is locally positive around 0.6; the largest lagged negative correlation (<0.3) is in a region northwest of the SST box (Figure 17). This negative maximum, though barely significant at the 95% level, moves toward the equator and becomes less extensive zonally as hmix/c leads. The local positive correlation disappears if hmix/c leads by 6 months (Figure 17, left). In terms of equation (1), it is thus apparent that the SST anomaly evolution is not well predicted by a one-dimensional heat storage processes because there should be a maximum of the correlation with hmix/c leading by 3–9 months.
5. Characteristics of ECCO Simulated Mixed Layer Depth Relative to hmix/c
 It is one aim of this manuscript to develop a criterion for hmix that can be comparably applied to not only observational but also to model data. Therefore the following analysis of an OGCM model output does not only compare the model simulations with observations, but also highlights the skill of the estimated hmix. The model study has, in respect to the skill of the hmix estimation, the additional advantage that the true mixed layer depth can be diagnosed directly, which makes the evaluation of the skill of our method estimating hmix more objective. From the 1° × 1° global ECCO state estimation (see section 2.1) we use daily average potential temperature profiles to compute both hmix/c and hmix/d, after which the monthly means are calculated. ECCO uses the K profile parameterization (KPP) for near-surface vertical diffusion coefficients [Large et al., 1994]. The KPP scheme diagnoses an “oceanic planetary boundary layer depth” (hpbl) that is defined as that depth where the “bulk Richardson number” exceeds a critical value (>0.3) [Menemenlis et al., 2005]. This is the “depth of mixing due to turbulent velocities of unresolved eddies” [Large et al., 1994, p. 1524]. Below this depth, in the ocean interior, the K-mixing profiles of the KPP model depend on a number of processes which are: shear instability, background internal wave activity, and static instability. These processes enhance the mixing coefficients and therefore may lead to larger hmix than predicted by hpbl alone. Thus ECCO model computes the time evolution of vertical profiles of upper ocean temperatures both from assimilation of the instrumental observations, like T/T array data, and from the direct application of the KPP upper ocean mixing scheme.
 In general, the ECCO simulated hmix/c and hmix/d are in fact always larger than hpbl. Largest differences are found throughout the year in the band of the T/T array, ±10° around the equator and west of the boundary upwelling regions. On the global scale, hmix/c is usually less than 40% larger than hpbl, but the relative differences between hmix/d and hpbl are often greater than 50% (not shown). Because there are not very many instrumental observations of temperature in the midlatitudes, as expected, the time series of simulated hmix/c, hmix/d and hpbl at many different locations in the midlatitudes display a remarkable correlation between their phases and amplitudes (Figures 18a and 18c); the fluctuations of hmix/c follow closely the high-frequency, daily changes in hpbl. In the hemispheric winters, where the mixed water column achieves largest values, hmix/c appears to capture simulated mixing events of KPP very well. At the same time, a large Δ criterion fails to find the depth of KPP mixing (Figures 18a and 18c), whereas smaller Δ criterion seems to get the right hpbl but shows less high-frequency variability than hpbl. Over the global domain, which excludes the equators, and independent of the season and chosen ΔT, hmix/d is generally larger than hmix/c or hpbl and the distribution function of hmix/d decays in a smoother fashion, particularly for a larger Δ criterion. The distribution functions of all the three turbulent layer depths are similar in shape to the observed distribution function (namely, Figures 6 and 19). The correlation between anomalous monthly mean simulated hmix and hpbl is always positive and for hmix/c it is >0.8 on a global scale (not shown). The corresponding correlation for hmix/d is somewhat weaker, in particular poleward of 40° and in the equatorial region it is less than 0.4.
 Around the equator the daily variations of the modelled vertical mixing depth are not well reproduced by either method of estimating how deep KPP mixes (Figure 18b). This is an indication that in the Pacific, ECCO often prefers to choose the assimilated T/T data for its profiles, on top of which KPP produces a turbulent “quasi-homogeneous” upper layer that does not appear to be very often the layer assimilated from T/T array into ECCO.
 In comparison with the observed hmix/c, the ECCO simulated hmix/c appear to be generally smaller by around 30% for all seasons (Figure 20). Along the equator the simulated hmix/c is even smaller where relative difference between observed and simulated hmix/c exceeds 50% throughout the year (Figure 20). Exceptions to these general patterns are at the mode and deep-water formation regions, in the Kuroshio, Gulf Stream extensions and the convective regions of the Northern Hemisphere in winter (Figure 20a) and in the Southern Ocean during winter and spring (Figures 20c and 20d), where larger simulated hmix/c occur than are observed. We find that while in general the relative differences between the observed and simulated hmix/c were less than 10% this could well be due to the assimilation by ECCO of the data we used to compute the observed values. The hpbl generated by ECCO is very much smaller than the observed value of hmix/c. ECCO mixes the ocean vertically by assimilation, or by a background vertical diffusivity, more than it does by KPP.
6. Summary and Discussion
 In this study we established a new criterion to identify the depth to which homogenization occurs in the upper layer of the ocean, hmix/c. The criteria uses, together with some boundary conditions, the first maximum of curvature of the temperature and density profiles to identify hmix. Two advantages of this approach are as follows: (1) The empirical specified parameters are independent from another dynamical quantity, such as the SST. (2) The estimated hmix is not a linear function of any of the empirical parameters, as it is the case for hmix/d based on a Δ criterion. The method has been developed empirically on the basis of a global data set of around 5 million MBT, XBT, and CTD profiles as well as the output of the MIT-OGCM, the 1° × 1° global ECCO state estimation. This curvature criterion appears to be more universally applicable in both data and model computations of the mixed layer depth, hmix, compared to an algorithms that utilize a Δ criterion. We made an effort to prove the curvature criterion as systematic as possible, which includes, amongst others, the application of the criterion to both idealized and individual observed profiles with different vertical resolution, the definition of a quality index, QImix, for hmix and comparisons to hmix based on simulated profiles.
 For simplified, but realistically shaped profiles, the curvature criterion, when combined with the exponential interpolation routine, eliminates the weaknesses of the Δ criterion. In about 70% of the observed high-resolution CTD profiles, the hmix/d that are based on the Δ criterion show the systematically overestimation compared to hmix/c based on the curvature criterion.
 For about two thirds of all individual profiles we found hmix/c to be more reliable in separating regions of low- (well mixed) and high- (stratified) temperature variance. Using a ΔT = 0.2°C (following de Boyer Montégut et al. ) hmix/d appears to be superior of hmix/c in 10% of the profiles. For the remaining percentage it is not definite which one of the two criteria selects the optimal quasi-homogeneous layer. In general, a Δ criterion seems to encroach into the stratified layer below depending upon the value Δ chosen; a single value of Δ produces hmix/d with dependencies on the ocean region and season of the year. For example, if an appropriate value of Δ for the midlatitude summer is used, in higher latitude hemispheric winter and spring, hmix/d values are often too large as this algorithm wrongly identifies a significant part of the top of the weak main thermocline as belonging to the mixed layer. Even a, relatively small, global uniform ΔT = 0.2°C does often not capture small vertical gradients indicating the base of deep convective regions.
 As an indicator of the stability of hmix/c we calculated differences from hmix/c, based not only on temperature profiles but also on salinity and potential density profiles. The differences between the temperature and salinity hmix/c and those inferred from potential density profiles are smaller than 4 m on a global average, with no appreciable bias. This leads to an average uncertainty range of hmix/c about ±5 m for high- (<5 m) resolution profiles.
 A quality index, QImix, was introduced that determined whether a methodology would be able to cleanly separate the homogeneous layer depth from the stratified region below. In effect, it computes the anticorrelation between the stratification above and below the calculated mixed layer, with a value of 1.0 if no stratified water is above the depth of the mixed layer computation. QImix/c was large (>0.7) for the bulk (>70%) of the profiles and supports the relatively small uncertainty range for hmix/c computed from high resolution profiles. The curvature algorithm was able to capture the depth of the quasi-homogeneous zone independent of the stratification below.
 The uncertainty range for the 20-m resolution profiles was estimated to be ±8 m. An additional problem of using these low-resolution profiles was that there appears to be an accumulation of hmix estimates around each of the levels at which data is sampled. The exponential interpolation in our curvature-based algorithm was not able to completely bypass this accumulation. However, the algorithm does not exhibit accumulation at sampling levels during summer and fall in midlatitudes where large gradients between the mixed and the submixed layer occur. In the tropics, the accumulation problem persists. Assuming that hmix/c based on high-resolution profiles is the real hmix the significant higher explained variance (squared correlation coefficient) indicates that hmix/c based on low-resolution profiles is more reliable than the corresponding hmix/d.
 Temporal averaging of the profiles, before computing a homogeneous layer depth, systematically underestimates hmix. Furthermore, it enhances the accumulation of hmix around the sampling levels. This occurs because temporal averaging smoothes the vertical gradients and broadens the depth interval of the seasonal thermocline, which in turn reduces the sensitivity of computing the depth of the extreme curvature. We conclude that the main concerns in the computation of a reliable hmix are not only low vertical resolution of the profiles, but also data sparseness so the average of hmix did not equal hmix from the ensemble average profile.
 With the data set at our disposal it was possible to determine the variability of hmix/c on seasonal-to-interannual scales in several ocean regions from observations alone. The absolute standard deviation of monthly mean hmix/c on the global scale for different seasons reflects the pattern of seasonal mean hmix/c. The largest values (>50 m) of the absolute variability were in the mode and deep-water formation regions in both hemispheres in winter (and partly in the northern hemispheric spring) with large hmix/c and small values for small hmix/c. The relative standard deviation of monthly mean hmix/c is most pronounced (>50%) in the eastern equatorial Pacific where there is a small monthly mean hmix/c and weak absolute variability throughout the year. This also reflects the high sensitivity of small hmix/c to variations caused by horizontal or vertical processes that move the thermocline by adiabatic processes. The relative standard deviation of hmix/c ranges between 20 and 70%.
 To highlight the effect of anomalous monthly mean hmix/c on the climate variability, we focused on the tropical Pacific and calculated the correlation between anomalous monthly mean hmix/c and SST; this correlation indicates that basically the variability of hmix/c on seasonal-to-interannual scales is decoupled from the anomalous SST changes. Therefore the processes that change the SST must be related more to horizontal movements of water masses than to processes of local storage of thermal energy.
 Analysis with ECCO simulated temperature profiles first of all revealed that the estimated hmix/c does follow the models true mixing depth, hpbl, relatively close. This indicates that hmix/c is a good proxy for the models mixing depth and it appears to be a better estimate of hpbl than the traditional hmix/d. We found also a weaker correlation between hpbl and hmix/d than for the corresponding hmix/c. Comparisons of observed hmix/c with ECCO simulated hmix/c as well as with the KPP diagnosed oceanic planetary boundary layer depth hpbl confirm that in the Pacific equatorial zone ECCO produced a weakly stratified upper layer more from assimilated T/T observations of temperature rather than from turbulence created by its imbedded KPP. The hpbl was significantly smaller than the quasi-homogeneous layer depth in the ECCO simulations. Secondly, even though observed hmix/c was significantly larger than hpbl, also in midlatitudes, there was a strong correlation between them on monthly anomaly timescales (>0.8). This implies either that the ECCO vertical diffusivity below the KPP planetary boundary layer plays an important role in diffusing heat from the mixed layer into the seasonal and main thermocline below, or that the correlation is set up by the ECCO assimilation of the same profiles we are using for verification. We found that a weaker correlation occurred between hmix/d and hpbl.
 It goes without saying that we are looking very much forward to the profile data set generated from the ARGO project with which to further test the applicability of our, and other, mixed layer computations. However, if all the ARGO data is assimilated, model testing for upper ocean mixing via computation of mixed layer depth will be moot.
 This study benefited from a poster presentation at the WOCE and beyond conference in San Antonio, Texas, November 2002. We would like to thank the anonymous reviewer and James Richman for their valuable and helpful comments. In this respect we thank as well Johannes Karstensen. The paper was motivated by discussions with Russ Davis. Supported in part through ONR (NOPP) ECCO grants N00014-99-1-1049. This is a contribution of the Consortium for Estimating the Circulation and Climate of the Ocean (ECCO) funded by the National Oceanographic Partnership Program.