1. Introduction
Upper ocean heat content changes daily due to radiative forcing from the sun that creates a thin diurnal warm layer (DWL) extending several meters beneath the sea surface. While sea surface temperature (SST) controls the instantaneous air–sea turbulent heat flux, upper ocean heat content provides a diurnally varying heat reservoir that regulates the phase between solar heat flux and the diurnal SST cycle. The thickness and average temperature of this reservoir, which constitutes the DWL, depend on net surface heating, wind speed, and subsurface ocean mixing (Price et al. 1986; Fairall et al. 1996a; Kawai and Wada 2007).
Daytime increases in near-surface temperature cause near-surface vertical stratification (Fig. 1), which is diffused by vertical mixing. Diffusion of heat happens slowly during the day but then rapidly upon the onset of nighttime cooling and convectively driven mixing (e.g., Anis and Moum 1994). The daily cycle of heating and mixing serves to progressively increase upper ocean heat content by 20 MJ m−2 over the top 10 m during the 5 days shown in Fig. 1, equivalent to several tenths of a degree Celsius. This heat may later be transferred back to the atmosphere during periods of strong surface forcing from westerly wind bursts associated with the Madden–Julian oscillation (MJO) in the equatorial Indian Ocean (Moum et al. 2014), for example, or may even contribute to driving synoptic atmospheric storms, including tropical cyclones (Emanuel 1999). Accurate representation of diurnal variability in the ocean is therefore important in simulation of lower-frequency atmospheric signals (e.g., Lee and Liu 2005; Solomon and Jin 2005; Danabasoglu et al. 2006; Seo et al. 2009; Bellenger et al. 2010).
Detailed temperature structure between 8 and 12 Oct 2011. (a) Net surface heat flux (
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
The developing stratification within the DWL can have competing influences on turbulent mixing. On the one hand, stratification increases the gravitational stability of the system, thereby dampening turbulence. For example, turbulence has been shown to decay by an order of magnitude when the near-surface stratifies, either through surface heating (Brainerd and Gregg 1993; Caldwell et al. 1997) or precipitation (Smyth et al. 1997). At the same time, increased stratification in the DWL is known to trap wind-driven momentum at the base of the surface mixed layer forming a diurnal jet (Thorpe 1978; Price et al. 1986; Cronin and Kessler 2009), with velocity anomalies up to 0.19 m s−1 (Kudryavtsev and Soloviev 1990). Additionally, observations (Callaghan et al. 2014; Sutherland et al. 2016) have shown that within this diurnal jet turbulence is enhanced relative to depths below the DWL.
The enhancement of shear associated with the diurnal jet provides an energy source for turbulence production, an example of which is provided by Pham et al. (2017), who propose that DWL shear acts as the trigger to the deep cycle of turbulence below the equatorial mixed layer. The mechanisms leading to reduced turbulent kinetic energy (TKE) dissipation may vary. For example, Smyth et al. (1997) and Callaghan et al. (2014) attribute turbulence decay to the interruption of the downward advection of convectively generated TKE at the surface by stratification capping associated with rainfall and surface heating, respectively. In contrast, in the cases presented by Sutherland et al. (2016), the reduction of TKE dissipation ε was associated with a decrease in the local shear production of turbulence due to enhanced stratification. In all of these examples, the time-evolving spatial patterns of turbulence in the upper ocean are directly linked to surface forcing with direct implications for air–sea interactions through control on sea surface temperature by subsurface mixing.
Here, we use time series data from the equatorial Indian Ocean to examine the detailed evolution of temperature T and ε within the DWL under light to moderate wind forcing and clear sky conditions (Fig. 1). Our observations show a repeated cycle in ε over the course of 5 days that were representative of these atmospheric conditions. An initial decay of ε at sunrise, which is likely caused by suppression of convection associated with the nascent DWL, is followed by a period of growth in ε through development of the diurnal jet as stratification, and presumably shear, builds in the DWL. Later in the afternoon, shear production and buoyancy suppression roughly balance one another, leading to a near-steady state in ε. Details of the measurements and analysis methods (section 2) are followed by a summary of the observations of temperature (section 3) and turbulence (section 4). We then provide a quantitative evaluation of decay and growth rates of turbulence via an evolution equation for ε (section 5). We close with a short discussion of these results (section 6).
2. Measurements
The principal observations used in this study were acquired in October 2011 at 0°, 80.5°E during the Dynamics of the MJO field campaign (DYNAMO; Moum et al. 2014) in the equatorial Indian Ocean from a bow-mounted chain of temperature and conductivity sensors (Table 1, Fig. 2). The ship was also equipped with a full suite of meteorological sensors from which surface heat, momentum, and buoyancy fluxes were derived (De Szoeke et al. 2015). Ancillary measurements include those from a shipboard acoustic Doppler current profiler and the Chameleon turbulence profiler (Moum et al. 1995).
Details of T-chain instruments.
Thermistor chain deployed from the bow of R/V Revelle during DYNAMO. The chain was outfitted with two CTDs and eight thermistors as listed in Table 1.
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
The ship maintained position and was oriented into a strong and relatively steady surface flow that brought undisturbed fluid past the bow-mounted sensor suite. Profilers tethered to the fantail are not well suited to measure the details of upper ocean structure because of contamination by the ship’s wake and bow thrusters. For example, simultaneous profiles of temperature from the bow thermistor chain (T-chain) and Chameleon profiler deployed from the stern of the R/V Revelle show significant differences in the upper 10 m (Fig. 3). Stratification clearly evident in T-chain data is absent in the Chameleon record. The uncontaminated T-chain measurements are essential to resolve development of the DWL, which is often shallower than a few meters.1 Eight fast thermistors (FP07, sampling at 10 Hz) and two CTDs (SBE37, sampling at 3 s) were distributed on the bow chain between 1.5 and 7.5 m (Table 1, Fig. 2). SST was measured by a thermistor mounted into a brass body and sealed in a buoyant Tygon tube towed beside the ship at nominally 0.05-m depth (Fairall et al. 1997). Depths of sensors suspended on the bow chain varied (Fig. 1c) according to changes in relative flow speed of ship and currents (±0.5 m) as well as to motion induced by surface gravity waves (±1.0 m).
Simultaneous temperature profiles in the upper ocean from the Chameleon profiler (black) and from the T-chain (blue). Note that three additional CTDs were installed on the T-chain at 8.6, 9.7, and 10.7 m at the above time, which occurred after the time period presented in Fig. 1.
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
3. Evolution of T(z) in the DWL
Much of the DYNAMO record is influenced by freshwater lenses formed by precipitating convective systems of the MJO (Moum et al. 2014), which have a strong effect on near-surface stratification. No freshwater lenses were detected from 8 to 12 October 2011, as these days preceded the first convective systems of the 2011 MJO season. During this period, wind speeds ranged from 2.3 to 8.7 m s−1, and daily averaged values of wind stress τ varied from 0.01 to 0.04 N m−2, representing low to moderate winds. Clouds were present less than 15% of the day, so that the time period was characterized by nearly clear sky conditions.
The diurnal variations in the vertical structure of T followed a clear pattern. The example shown in Fig. 4 is representative of the 5 days examined here. Nighttime cooling mixed the upper 8 m, leaving a T profile with no vertical structure at 0000 UTC in Fig. 4b. Daytime heating progressively increased the heat content in the upper 8 m by increasing the temperature at all depths, most strongly at the surface. The daytime profiles (0200–1000 UTC, corresponding to 0800–1600 local time) showed roughly an exponential decrease with depth indicating weak mixing and a dominant contribution from penetrating solar radiation (usually approximated as an exponential shape decreasing with depth; Ohlmann and Siegel 2000). As surface heating weakened in the afternoon, the near-surface T decreased, as did the stratification. At 1200 and 1400 UTC, the profile exhibited a characteristic hyperbolic tangent structure with a well-mixed layer near the surface indicative of intensified mixing.
Evolution of (a) 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
We take advantage of this evolution of vertical structure and fit exponential and hyperbolic tangent curves to observed temperature profiles as detailed in appendix A. These fits allow for calculation of a smoothly varying vertical derivative Tz on 10-min intervals. This vertical derivative is then used for estimation of ε (appendixes B and C). Note that the curvature of T is poorly constrained above 1.5 m because of a lack of sensors, leading to a high uncertainty in Tz during the day when stratification develops near the surface. Because of the contribution of Tz to our estimate of ε (appendix B), estimates above 1.5 m are considered less reliable than below and are not used for quantitative scalings presented in section 5.
4. Stratification and turbulence within the diurnal warm layer
The depth and magnitude of the daily cycle in the DWL (Fig. 5) reflected the variability in wind stress with daily ranges in SST (shown here with no cool skin correction) varying from 0.5° to 1.3°C. The largest daily range in SST corresponded to the weakest winds (Figs. 5b,c). Peak values of N2 at 2.0 m varied from 2 to 10 × 10−4 s−2 in the late solar afternoon during this 5-day time span (Fig. 5d). Here, the buoyancy frequency,
Time series of (a) 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
The stratification induced by the divergence of the penetrating solar radiation [the product of the surface-incident shortwave radiation (SW), and the time-dependent, depth-dependent solar transmission profile] appears to play a determining role in the decay of nighttime convective turbulence before the change in sign of the net surface heat flux 
Daily evolution of (a) the composite 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
A daily repeated signature in the temporal evolution of near-surface turbulence is clear in Fig. 5. Upon sunrise, the following trends are apparent in the upper 2.5 m:
Coincident with increased stratification following sunrise but before
This decay was followed by a rapid increase of ε by two orders of magnitude over roughly 2 h.
After the period of rapid growth, the rate of increase in ε slowed or plateaued in the afternoon as the diurnal warm layer deepened.
5. Time-dependent representation of ε at 2.0 m
Variation of time scales γ over realistic ranges of N and S. Red depicts growth and blue depicts decay. The boxes bound regimes represented by the values of N and S listed in Table 2. Gray shaded areas show overlap regions of γ within the boxes and γobs.
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
A direct measure of N is derived from our thermistor chain measurements. The range of values (from Fig. 6) for each of decay, growth, and steady-state periods is noted in Table 2. However, we have no direct measure of velocity above ~10 m, thus limiting our ability to constrain S. Instead, we base estimated shear ranges in the three regimes on the results of Sutherland et al. (2016, see their Fig. 14a that also separates into growth, decay, and steady-state conditions below 4-m depth). Sutherland et al. (2016) show diurnally varying averaged values of S normalized by 
Summary of ranges of values used to estimate γ.
In the next sections we assess Eq. (4) using measured values of N and presumed values of S (Table 2 and using the averaged value of 
a. Decay
In the upper 2.5 m, turbulence decays shortly after the sun rises, but before 
The Businger–Dyer (Businger et al. 1971; Dyer 1974) representation of Monin–Obukhov (MO) similarity theory (Monin and Obukhov 1954) predicts that 
b. Growth
During the period of rapid increase in ε noted in Fig. 6, our estimates of γobs range from 7 to 15 min, with a mean value of 11 min. Using the values of S, N deduced for the growth period in Table 2 to compute γ, the box denoted “Growth” in Fig. 7 includes the full range of our observed estimates γobs.
c. Steady state
6. Discussion
The decay of ε beneath the DWL following the onset of surface heating has been described by Brainerd and Gregg (1993). A similar scenario was described by Smyth et al. (1997) following the capping of the ocean’s surface by freshwater from rain squalls. In both cases the subsurface was decoupled from surface forcing by a rapid increase in near-surface stratification, and they both derived a time scale for the decay of ε from a reduced TKE equation in the absence of shear production as in section 5a. Smyth et al.’s (1997) value for the turbulence decay time scale of 0.7N−1 is comparable to our estimate.
Our measurements differ from those described above in that these data are much closer to the sea surface, where a priori we did not expect S = 0 at any time. The observations of Sutherland et al. (2016) show that daytime values of S are significantly 
Subsequent stages of growth and steady state require a source of turbulence. An additional observation suggests this is due to shear instability. During daytime, temperature time series were marked by the presence of asymmetric structures (Figs. 8a,b) that were more commonly observed from late morning to late afternoon (Fig. 6d). Similar features were termed ramps by Thorpe (1978), who suggested that these are the signatures of Kelvin–Helmholtz instability created by wind-generated shear in the presence of stable heating. In our data, temperature ramps2 were observed near the surface at the end of the ε growth period (Fig. 6d); thereafter, they were observed progressively deeper over the top 7 m and with increasing frequency until N2 decreased at day’s end. These features were not observed at night when vertical stratification was negligible. The measured ε was greater in the presence than in the absence of ramps (Fig. 8c). We suspect, then, that the turbulence observed during the steady-state period was generated through shear instability and breaking Kelvin–Helmholtz billows.
(a),(b) Time series of temperature at four depths showing examples of temperature ramps, and (c) normalized distribution of ε in the presence (green, N = 697) and absence (gray, N = 5063) of ramps. Median values are indicated by inverted triangles at the top of (c). The Kolmogorov–Smirnov test confirms a statistically significant difference between distributions (p ≪ 10−4).
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
We have argued that our estimates of γ based on parameter values listed in Table 2 are consistent with γobs, as indicated in Fig. 7 for periods of both decay and growth. We have also indicated that our estimates of ε based on thermistor measurements from our bow chain are consistent with nearby estimates based on shear probe measurements from our profiler Chameleon (appendix C). As well, during the steady state period, the range of observed values of N and estimates of S in Table 2 overlap the γ = 0 regime in Fig. 7, and are therefore consistent with our proposed scenario.
However, a significant inconsistency arises in our application of Eq. (5), where we found that our estimate of ε is ≈
The depths of DWL and mixed layer in relation to our thermistors is shown in Fig. 5e. The flaw in our definition of mixed layer depth (MLD) is obvious, as the mixed layer was clearly stratified. However, these definitions of DWL and MLD help to distinguish the behavior of ε(t) within the DWL and above the MLD from that below. The behavior of ε(t) discussed in section 4 roughly holds for all of the sensors above the MLD, although we have only emphasized that at 2.0 m in Fig. 6c. Below the MLD the behavior is less clear, in part because the lack of stratification very early in the day precludes estimating ε then. But once the MLD shallowed above sensor depth, ε decayed and remained small during the day, increasing only when the DWL again deepened, and reaching a maximum when MLD was deeper than the sensors. At this time, the warming associated with the increase in ε was presumably due to mixing down of warm water from above.
While we have identified three principal regimes in the evolution of daytime turbulence within the DWL, there are several aspects that we have not addressed, all of which can be seen in Figs. 5 and 6. First, during the decay period a consistent pattern emerges in which there appear to be two stages of decay; slowly, immediately after sunrise and then more rapidly following the change in sign of 
In sum, we have shown a consistent pattern of mixing in the diurnal warm layer, in this case associated with the strong insolation of the tropics. While the signal may be more subtle away from the tropics, the general pattern ought to hold elsewhere, save for a poleward-increasing Coriolis veering of the diurnal jet. The phasing between air–sea fluxes and the ocean’s response, which in part is determined by this repeatable cycle in subsurface mixing, may need to be considered in improvements to diurnal warm layer parameterizations that include more realistic representations of mixing.
Acknowledgments
This work was funded by the National Science Foundation (1059055) and by the Office of Naval Research (N0001-10-1-2098). Data used in this research are available on the DYNAMO website (https://www.eol.ucar.edu/field_projects/dynamo). M. Neeley-Brown, R. Kreth, and A. Perlin provided technical support. The surface flux dataset obtained on the R/V Revelle for this experiment was kindly provided by Jim Edson, Simon de Szoeke, and Chris Fairall. We acknowledge the constructive comments of Ren-Chieh Lien and an anonymous reviewer.
APPENDIX A
Continuous Temperature Profiles from Point Measurements
All fast thermistors were reconciled with SBE37 measurements during periods of nighttime convection, when it was assumed that all sensors recorded the same temperature, by applying constant offsets of up to 0.05°C to fast thermistors as needed. All data were then averaged onto a common 600-s time grid and regressed at each time step onto two types of vertical curves based on observed conditions (as described in section 3) and mean-square error (MSE) between the measurements and fits. Regressions on the selected forms were used to create a gridded temperature product with 10-min resolution in time and 0.1-m resolution in depth (Fig. 1d). This gridded product was used to calculate layer depths (see below) and vertical derivatives of temperature (Tz), which are used in calculation of χ and εχ.
Illustration of temperature profile regression forms from (a) Texp inspired from the heat equation (red) regressed onto observations (black) and from (b) Ttanh derived from a hyperbolic tangent form (blue) regressed onto observations (black). The blue diamond represents the center of curvature of the tanh profile.
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
The above fits were applied for depths greater than 1.75 m (i.e., between the location of the first and second fast thermistor), and thus hold for gradients used in calculation of χ at depths greater than or equal to 2.0 m (i.e., at all locations except the uppermost thermistor). Above 1.75 m, we found that the curvature between the sea snake and the first thermistor was not always well modeled by the above descriptions. At these times, we applied an iterative process where the fits above were merged with an exponential fit constrained by the sea snake as well as subsurface sensors in order to minimize the total MSE. This process only influences the gradient at the uppermost fast thermistor and thus influences the magnitude (although not the overall evolution trend) in χ at this location.
The Ttanh and Texp curves were selected based on time of day and the MSE between the regressions and the measurements. From 0000 to 0600 UTC, Texp was presumed more representative unless the MSE of Ttanh was smaller. From 0600 to 2300 UTC, Ttanh was presumed more representative unless the MSE of Texp was smaller. Each form accounted for roughly 50% of 10-min profiles over the 5-day period. Note that the curvature of Ttanh around its inflection point yielded a better fit than Texp starting around 0600 UTC even in the absence of a mixed near-surface layer.
APPENDIX B
Mixing Estimates from Bow Chain Temperature Measurements
To estimate χT from the observed temperature time series, we first convert temporal derivatives to spatial derivatives using Taylor’s hypothesis (
Temperature time series and their derived turbulence quantities from spectral fits. (a),(b) Ten minutes of fast (10 Hz) temperature data from a thermistor at 6.0 m; (c),(d) spectra of Tx (gray) fit to the Batchelor spectrum (black) between frequencies 0.0183 and 0.066 Hz (dashed vertical lines). The resulting χT and εχ estimates are given in (c) and (d). Panels (a) and (c) represent an example of high temperature variance dissipation, and (b) and (d) represent an example of low temperature variance dissipation. The bounds represent 90% confidence based on the 90% range of fits indicated by the dotted lines bracketing the spectra.
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
The above assumptions prohibit calculation of ε during convection, and εχ was not computed when Tz > −10−3 °C m−1. (Note that Tz > 0 corresponds to an unstable density profile over the observation time period when salinity was not an important contributor to density.) Values of ε exceeding 10−5.8 or smaller than 10−12 m2 s−3 were also removed. Collectively, removed data points account for roughly 60% of the total dataset.
The method outlined above follows directly from algorithms developed for processing of moored turbulence meters (χpods) in the viscous-convective (Moum and Nash 2009) and inertial-convective subranges (Zhang and Moum 2010; Becherer and Moum 2017). The results have been validated against microstructure profiling measurements using shear probes by Perlin and Moum (2012) and Pujiana et al. (2018).
In the main text εχ has been replaced with ε for convenience.
APPENDIX C
Comparison of Bow Chain and Profiler Measurements of Turbulence
In Fig. C1, we compare the turbulent kinetic energy dissipation rate from the T-chain at 6.0–6.6 m, εχ, to Chameleon profiler estimates εSh, based on airfoil probe measurements averaged between 8 and 10 m. This Chameleon depth range was determined to be the shallowest range consistently outside the ship’s wake. The profiler was deployed approximately 78 m downstream of the T-chain, corresponding to 1.5–2.0 min of drift time, which is shorter than the time between Chameleon profiles (6–10 min).
Time series of the dissipation of turbulent kinetic energy evaluated from Chameleon shear probes, (εSh), averaged between 8 and 10 m, and evaluated from T-chain (εχ) averaged between 6.0 and 6.6 m.
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0170.1
The magnitude and variability of εχ and εSh are roughly comparable (Fig. C1). The slight phase lag of εSh relative to εχ may be due to the depth-dependent time response of the turbulence to surface forcing.
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The depth of the DWL was defined as the greatest depth at which the time rate of change in temperature Tt exceeded 1.3 × 10−5 °C s−1. This threshold level was based on the average heating rate of SST 1 h after sunrise. The DWL depth was shallowest at sunrise when it was reset to the surface and deepest near sunset. MLD was defined as the depth where T(z) deviated from its surface value by 0.25°C. MLD was deepest at sunrise and shallowest around local noon.
Temperature ramps were identified by the skewness of the lateral temperature gradient Tx. Three-second temperature time series were first low-pass-filtered at 0.066 Hz (15 s) to remove surface wave signals, and skewness of Tx was then computed from hour-long time records. Ramps were identified by Tz > 0.02°C m−1 and hourly skewness of Tx > 0.35.
