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  • View in gallery

    Summary of the instrumentation discussed in this paper. (a),(b) Sensor arrangements on the surface-following, towed platform SurfOtter. The deployments are discussed in section 2. (c) SurfOtter is designed to steer outboard when towed by the ship. Additional deeper measurements are provided by repeat vertical profiling with the Chameleon microstructure profiler.

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    Summary of the 12 days with DWLs that are analyzed in this paper. One example from each cluster is labeled and studied in detail in the text. The strength of the DWL is characterized here by the maximum difference during the day between temperatures at the shallowest sensor and at 8 m. Solar radiation values are included for completeness, but within this dataset there is no relevant correlation between ΔT and solar radiation.

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    Temperature structure and microstructure during the three days used as case studies in this paper. Differing color maps are used to emphasize the differing color scales. Turbulent quantities are inferred from Tt as measured at discrete depths and are shown in (g)–(i). For clarity, only two of the six fast thermistor records on the calm day are shown; zTzmax is the depth of maximum vertical temperature gradient.

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    Example records from the four GusTs at 1530 LT on the moderate day. (a)–(d) Measured temperature [high-passed with a cutoff of 1/(30 s)] and the component induced by surface waves [Eq. (B10)]. Note the differing y-axis limits. (e)–(h) Spectra of Tx from 10-min records (1530–1540 LT). For clarity, the spectra are averaged in logarithmically spaced frequency bins. Fits to the analytical form use the corrected spectra [Eq. (B11)].

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    Enhanced and suppressed turbulence dissipation above and below zTzmax, the depth of maximum temperature gradient. The enhancement or suppression is presented as a logarithmically scaled ratio of εχ to the scaling from Eq. (1). Ratios near that surface that are ≪1, especially on the windy day, are underestimates as discussed in appendix C.

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    DWLs trace counterclockwise paths in log(N)–log(ϵχ) space. Wind speeds increase from left to right panels, and depths increase from top to bottom panels. The steeper dashed lines, constant Ozmidov scale, are a proxy for the vertical size of turbulent eddies. The flatter long-dashed lines indicate constant buoyancy Reynolds numbers. Moving toward the lower-right corner of each panel, the upper two Reb lines (Reb = 1500 and 300) indicate increasing levels of anisotropy discussed by Gargett et al. (1984), and the lowest is a threshold at which turbulent buoyancy fluxes vanish (Reb = 25; Stillinger et al. 1983).

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    Thermodynamic heating evaluated between adjacent GusTs. The negative of ∂H/∂t is plotted so that the left- and right-hand sides of Eq. (2) agree if the thick and thin red lines coincide. Different pairs of GusTs are used for the different days, with the depths of the GusTs indicated in (a), (d), and (g): (a),(b),(d),(e),(g),(h) Ten-minute averages; (c),(f),(i) 10-min averages are binned to 30-min averages. Error bars encompass the minima and maxima within the 30-min bins. Notable features are labeled with a letter and discussed in section 4a. Heat fluxes (and hence divergences) are not calculated when Tz < 0.002°C m−1 [dashed line in (g)].

  • View in gallery

    Vertical structure of turbulent heat transport at representative times of day. Wind speed increases from top to bottom panels. The Jqs profiles (thin, smooth lines) are estimated assuming a two-band exponential decay, whereas Jqt (thick, piecewise lines with symbols) are evaluated at discrete depths. Some profiles of Jqt for the 1600–1700 LT profiles are excluded because Tz is less than 0.002°C m−1.

  • View in gallery

    Interpretation of an averaged, normalized Jqt profile. Data come from the four days on which we can adequately track zTzmax throughout the day (25 and 27 Aug, 29 Sep, and 4 Oct). Gray lines are individual, 10-min profiles normalized and aligned vertically about zTzmax. From the resulting distribution of points, the binned profile is created such that each value is derived from the 30 vertically closest points. Note that the denominator is not a true depth mean but rather is a mean of values from thermistors at discrete depths.

  • View in gallery

    (a) Redistribution of absorbed solar energy by turbulence in a diurnal warm layer. Most solar energy is absorbed in the top few meters leading to strong solar heat flux convergence. Turbulence induced by wind mixes this heat deeper throughout the day. The integrated heat flux convergence in (b) is between the two horizontal lines shown in (a). Vertical slices at 1500 LT are in (c). (b) The observed heat flux and (c) its divergence as described in sections 4a and 4b, respectively. The temperature gradient and turbulent transport in the top O(10) cm are inferred by assuming the turbulent heat flux asymptotically approaches the value of surface heat loss. These details were not measured in this paper.

  • View in gallery

    The temperature record of a thermistor that perfectly follows the sea surface in a nonturbulent wave field. Wave-orbital velocities decay with depth, with higher-frequency components decaying faster. Because the temperature field is advected vertically by wave orbitals while the thermistor parallels the surface, the thermistor traverses isotherms and consequently records a variable temperature.

  • View in gallery

    Examples of periods during which turbulence is small or negligible and the wave-induced temperature signal dominates. (a),(b) From the late morning on the calm day, the record from the shallowest GusT. (c),(d) From early afternoon on the moderate day, the record from the deepest GusT.

  • View in gallery

    Estimating the near-surface range over which ϵχ is questionable because it does not account for TKE input at the surface. Black lines show an analytical result for ϵ(z) under a diffusion–dissipation balance with a surface TKE flux (section 4 of Craig and Banner 1994). Gray lines are all 10-min ϵχ values. We consider near-surface ϵχ values as questionable if they are near or below the analytical curves.

  • View in gallery

    Statistical comparisons of ϵχ derived using two different frequency ranges: (a) two-dimensional histogram of all 10-min ϵχ values from all GusTs for all 12 days in Table 1, and (b)–(d) ratios of the data from (a) separated by the wind speed at the time of measurement.

  • View in gallery

    Slopes of best fit to horizontal temperature gradient spectra from the GusT at 1.7 m deep. (a)–(d) Slopes are from linear fits of log10(ΦTx) against log10(f) over the frequency range corresponding to the periods shown. (e) The maximum coherence max[γ(f)] between the measured temperature and the wave component over the 15–0.5-s band.

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Heat Transport through Diurnal Warm Layers

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  • 1 College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon
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Abstract

Penetration of solar radiation in the upper few meters of the ocean creates a near-surface, stratified diurnal warm layer. Wind stress accelerates a diurnal jet in this layer. Turbulence generated at the diurnal thermocline, where the shear of the diurnal jet is concentrated, redistributes heat downward via mixing. New measurements of temperature and turbulence from fast thermistors on a surface-following platform depict the details of this sequence in both time and depth. Temporally, the sequence at a fixed depth follows a counterclockwise path in logϵ–logN parameter space. This path also captures the evolution of buoyancy Reynolds number (a proxy for the anisotropy of the turbulence) and Ozmidov scale (a proxy for the outer vertical length scale of turbulence in the absence of the free surface). Vertically, the solar heat flux always leads to heating of fluid parcels in the upper few meters, whereas the turbulent heat flux divergence changes sign across the depth of maximum vertical temperature gradient, cooling fluid parcels above and heating fluid parcels below. In general, our measurements of fluid parcel heating or cooling rates of order 0.1°C h−1 are consistent with our estimates of heat flux divergence. In weak winds (<2 m s−1), sea surface temperature (SST) is controlled by the depth-dependent absorption of solar radiation. In stronger winds, turbulent mixing controls SST.

Corresponding author: Kenneth G. Hughes, kenneth.hughes@oregonstate.edu

Abstract

Penetration of solar radiation in the upper few meters of the ocean creates a near-surface, stratified diurnal warm layer. Wind stress accelerates a diurnal jet in this layer. Turbulence generated at the diurnal thermocline, where the shear of the diurnal jet is concentrated, redistributes heat downward via mixing. New measurements of temperature and turbulence from fast thermistors on a surface-following platform depict the details of this sequence in both time and depth. Temporally, the sequence at a fixed depth follows a counterclockwise path in logϵ–logN parameter space. This path also captures the evolution of buoyancy Reynolds number (a proxy for the anisotropy of the turbulence) and Ozmidov scale (a proxy for the outer vertical length scale of turbulence in the absence of the free surface). Vertically, the solar heat flux always leads to heating of fluid parcels in the upper few meters, whereas the turbulent heat flux divergence changes sign across the depth of maximum vertical temperature gradient, cooling fluid parcels above and heating fluid parcels below. In general, our measurements of fluid parcel heating or cooling rates of order 0.1°C h−1 are consistent with our estimates of heat flux divergence. In weak winds (<2 m s−1), sea surface temperature (SST) is controlled by the depth-dependent absorption of solar radiation. In stronger winds, turbulent mixing controls SST.

Corresponding author: Kenneth G. Hughes, kenneth.hughes@oregonstate.edu

1. Introduction

Diurnal warm layers (DWLs) form when strong solar radiation and weak-to-moderate winds allow near-surface stratification to develop. In the tropics DWLs appear around 0800 local time (LT), which is 1–2 h after sunrise, as the surface heat flux changes from net ocean cooling to net warming (Martin 1985; Fairall et al. 1996; Moulin et al. 2018). Heat and momentum trapped in this stratified layer cause sea surface temperature (SST) anomalies of O(0.1–1°C) and near-surface velocity anomalies of 0.1–0.2 m s−1 (e.g., Price et al. 1986; Kudryavtsev and Soloviev 1990). The previous night’s mixed layer evolves into a DWL lying above, and decoupled from, the remnant mixed layer below.

Per the optical properties of seawater, more than half of the sun’s heat impinging on the surface is absorbed in the top two meters of the ocean. The subsequent fate of this heat depends on turbulence. Consider two extremes: strong turbulent mixing and no turbulent mixing. In the former case, the sun’s heat is spread throughout the mixed layer and warms each parcel of water by O(0.1°C) by midafternoon. In the latter case, warming is concentrated in the top 2 m and, consequently, more of this heat is likely to be transferred from the ocean back to the atmosphere over a short time scale. In between these extremes heat transport is more complicated. Warming of the lower half of the mixed layer, for example, lags the surface solar forcing by several hours because it depends on the descent of the DWL. Using a large-eddy simulation under 6 m s−1 winds, Sarkar and Pham (2019) show that the lower half of the mixed layer warmed over their week-long simulation (net mean surface heat input to the ocean was 95 W m−2). Yet the turbulent heat transport causing this warming occurred over only four hours each day during the late afternoon to evening.

The timing and magnitude of subsurface turbulent heat flux governs SST and the consequent latent, sensible, and longwave radiative fluxes at the surface. In the tropics, the atmosphere is sensitive to small SST variations (Webster and Lukas 1992). Increases in SST during the day can lead to air–sea heat flux anomalies of 50 W m−2 relative to what they would be if the sea surface remained at its presunrise temperature (Fairall et al. 1996). Understanding these increases can improve numerical weather forecasts (e.g., Pimentel et al. 2008) and inform operational procedures such as corrections to SST based on bulk temperature measurements below the surface from ships (Fairall et al. 1996). The daily changes also add up over longer time scales. Development of the widely used K-profile parameterization (KPP) mixing scheme was motivated in part by a need to ensure physically reasonable mixing on diurnal time scales given its influence over climate time scales (Large et al. 1994).

A necessary step in predicting or parameterizing the evolution of the near-surface temperature field is an observational depiction of the relevant coevolving parameters (stratification, turbulence dissipation, and turbulent heat fluxes) and their responses to atmospheric forcing. Many studies have described the daytime reduction of turbulence dissipation ϵ in the remnant mixed layer immediately below the DWL (e.g., Dillon and Powell 1979; Moum and Caldwell 1985; Caldwell et al. 1997; Pujiana et al. 2018). The DWL, however, behaves differently. Two recent studies measuring microstructure shear from either a freely rising profiler or a glider demonstrate enhanced near-surface turbulence during daytime (Sutherland et al. 2016; St. Laurent and Merrifield 2017). The enhancement is attributed to shear and internal waves induced by the diurnal thermocline. It is this turbulence that drives the daytime descent of the DWL and hence controls the mixed layer heat budget.

Here we characterize near-surface turbulence dissipation ϵ and turbulent heat fluxes using recent measurements from a surface-following platform. Such platforms negate many of the challenges faced by other types of profilers. Vertical, free-falling profilers are unreliable in the top 5–10 m (e.g., Moulin et al. 2018). Gliders and freely rising vertical profilers can measure closer to the surface, but are limited in profile sampling rate (e.g., Greenan et al. 2001; Sutherland et al. 2013). Moreover, ϵ from shear at microscales depends on body speed to the fourth power making profiles sensitive to the orbital velocities of surface waves (e.g., Lueck 2016).

Recent examples of surface-following platforms include a sailboard adapted to measure salinity profiles in the top meter of the ocean (Asher et al. 2014), a trimaran adapted to measure atmospheric turbulence just above the sea surface (Bourras et al. 2014), and “SWIFT” drifters to measure near-surface turbulence and shear (Thomson 2012; Thomson et al. 2019). Like Asher et al. (2014), our platform is towed so as to sample undisturbed water outside the ship’s wake. Although temperature is measured at only discrete depths, the resulting horizontal profiles capture a broader bandwidth of turbulence compared to vertical profiles (Goodman et al. 2006; Klymak and Moum 2007).

Our platform houses several fast thermistors and thereby allows temperature spectra to be measured at several fixed depths (section 2). Turbulence dissipation rates inferred from the spectra differ by more than two orders of magnitude depending on the position relative to the diurnal thermocline (section 3). Combining the inferred diffusivity with vertical temperature gradients, we directly evaluate subsurface turbulent heat fluxes, which play a first-order role in the near-surface heat budget except in calm conditions (section 4). Overall, we aim to diagnose the feedbacks between turbulence and stratification and the relative roles of solar and turbulent heat fluxes in the development of diurnal warm layers.

2. Instrumentation and deployments

SurfOtter is a platform specially designed for measurements of turbulence, temperature, salinity, and velocity in the near-surface ocean (Fig. 1). It was first introduced in Hughes et al. (2020) to observationally assess vertical current shear induced by DWLs. It is a 3-m-long aluminum tube with a 2-m-deep fin that hosts a suite of oceanographic sensors. A cable attached to the bottom of the fin can extend the depth range to 8 m. SurfOtter is towed outside the ship’s wake at speeds of approximately 1 m s−1 in the manner shown in Fig. 1c. The overall design ensures that the water sampled is undisturbed and that the influence of surface waves is minimized relative to ship-mounted sensors.

Fig. 1.
Fig. 1.

Summary of the instrumentation discussed in this paper. (a),(b) Sensor arrangements on the surface-following, towed platform SurfOtter. The deployments are discussed in section 2. (c) SurfOtter is designed to steer outboard when towed by the ship. Additional deeper measurements are provided by repeat vertical profiling with the Chameleon microstructure profiler.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

While SurfOtter was deployed, we continually profiled with the microstructure profiler Chameleon off the stern of the ship, with profiles every 6–12 min (Fig. 1c). For this paper, this additional dataset provides measurements of temperature and salinity at depths of 8 m and below. Above this depth, Chameleon measurements are influenced by ship wake and prop wash as shown by Moulin et al. (2018).

In this paper, we use data from a 2-month field campaign in 2018 at 12°–18°N, 134°E (time zone: UTC + 9). During this period, SurfOtter cumulatively sampled 24 days. Of these, 12 were full-day records (Table 1) with surface warming of >0.05°C and without a confounding salinity influence (i.e., a transient rain layer). On some days we transited around a point within a radius of 10 km. On others we transited in a straight line and moved >100 km in 24 h. By comparison, for the location and season of our campaign, DWLs with an equivalent radius of O(1000) km are not uncommon (Bellenger and Duvel 2009). To minimize horizontal variability, we discuss temperatures as anomalies relative to their values at a depth near or below the base of the DWL.

Table 1.

Summary of the DWLs observed during the 2018 field campaign analyzed in this paper.

Table 1.

We primarily consider three representative days that differ by wind forcing. We designate these with the relative labels “calm,” “moderate,” and “windy.” The respective wind speeds and wind stresses were 2, 4, and 8 m s−1 and 0.005, 0.02, and 0.08 N m−2. The calm day (4 October) had the lowest wind speed of all days during which SurfOtter was deployed. The cable was not present during this deployment (Fig. 1b). The windy day (27 October) is near the wind limit at which DWLs occur (Kudryavtsev and Soloviev 1990; Thompson et al. 2019). The moderate day (29 October) is a point between the two extremes. Of the 12 days in the dataset, there are two low-, six moderate-, and four high-wind-speed days (Fig. 2). Similar three-category classifications were used by Kondo et al. (1979) and Soloviev and Lukas (1997). The role of surface buoyancy flux in dictating whether near-surface stratification can persist under certain winds is detailed by Thompson et al. (2019).

Fig. 2.
Fig. 2.

Summary of the 12 days with DWLs that are analyzed in this paper. One example from each cluster is labeled and studied in detail in the text. The strength of the DWL is characterized here by the maximum difference during the day between temperatures at the shallowest sensor and at 8 m. Solar radiation values are included for completeness, but within this dataset there is no relevant correlation between ΔT and solar radiation.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

The calm day is representative of large parts of the tropical Pacific and Indian Ocean where winds are weak and daily SST increases of 1 ± 0.5°C are common (Webster et al. 1996; Stuart-Menteth et al. 2003). For the tropics in general (23.5°S–23.5°N), daily-mean wind speed over the ocean has a mean of 6.5 m s−1 (based on the NCEP–DOE reanalysis for 2018; see Kanamitsu et al. 2002); 11% of values are 0–3 m s−1, 33% are 3–6 m s−1, and 39% are 6–9 m s−1. These values, however, overestimate the prevalence of DWLs, which are weaker if clouds occur. If we recalculate with an additional criterion that cloud cover must be <50%, the respective percentages become 5%, 15%, and 15%.

Across the three representative days, differences in the temperature fields are immediately evident (Figs. 3d–f; note the varying color scales). For example, the surface warms by 1°C relative to 8-m depth on the calm day but only 0.05°C on the windy day. Because temperature is well resolved vertically, we consider its vertical gradient Tz (Figs. 3g–i) a continuous quantity. From Tz, we can discern a descent rate of the DWL using zTzmax, the depth of maximum Tz. This is the nominal depth of the diurnal thermocline. Descent rates are approximately 0.3 and 1 m h−1 on the calm and moderate days, respectively. For much of the windy day, zTzmax is below the 8-m depth range of SurfOtter, but we can observe a descent rate of 4 m h−1 in the Chameleon record (not shown).

Fig. 3.
Fig. 3.

Temperature structure and microstructure during the three days used as case studies in this paper. Differing color maps are used to emphasize the differing color scales. Turbulent quantities are inferred from Tt as measured at discrete depths and are shown in (g)–(i). For clarity, only two of the six fast thermistor records on the calm day are shown; zTzmax is the depth of maximum vertical temperature gradient.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

Temperature microstructure from fast thermistors (100 Hz) was measured at discrete depths. Figures 3g–i include the measured temperature rate of change Tt as a qualitative depiction of turbulence that is partly dependent on Tz. Knowing both Tt and Tz, we can infer ϵ. SurfOtter housed two different instruments with fast thermistors: GusTs (Becherer et al. 2020) and TPods. Both are built in-house at Oregon State University and sample temperature and accelerations at 100 Hz. They are almost the same for our purposes here. On the calm day, there were six fast thermistors: two GusTs and four TPods. The latter lack associated pressure sensors, so we interpolate from GusTs when needed (appendix B). On the moderate and windy days there were only four reliable thermistors, all GusTs. Overall, turbulence measurements during the calm day encompassed a smaller depth range, but were better resolved vertically. Much of our analysis focuses on the GusTs, which were at depths of 0.4, 1.7, 4.2, and 7.7 m.

Fast thermistor measurements are continuous for whole deployments. For turbulence calculations, we segment these into 10-min blocks. With each block, we first convert Tt to an equivalent horizontal gradient Tx = Tt/U where U is the 10-min mean flow speed past the sensor. We then calculate the dissipation of thermal variance χ by fitting Tx spectra to a theoretical form for the inertial–convective (IC) subrange (Batchelor 1959) over the frequency range 0.067–2 Hz, corresponding to periods of 15–0.5 s. We finally calculate turbulence dissipation ϵ whenever there is measurable stratification (Tz > 0.002°C m−1) by assuming a steady-state turbulent kinetic energy balance (Osborn 1980). Hereinafter, we denote ϵ as ϵχ to recognize the intermediate calculation of χ. Although ϵχ is a commonly derived quantity (e.g., Moum and Nash 2009; Becherer and Moum 2017; Moulin et al. 2018), its derivation can include study-specific details. Here, these include aspects specific to SurfOtter and the proximity to the sea surface as detailed in the appendices.

3. Diurnal evolution of turbulence dissipation

The changing stratification throughout the day is the predominant factor explaining the daytime evolution of ϵχ. Moulin et al. (2018) describe the evolution at a given depth in three stages: (i) an initial decay of turbulence 1–2 h after sunrise as the near-surface stratifies, (ii) a rapid increase in turbulence once the DWL descends to the depth in question, and (iii) a near-steady state with a three-term balance for turbulent kinetic energy involving shear production, buoyancy flux, and dissipation. In this section, we extend Moulin et al.’s analysis with new observations that clarify the change in microstructure above and below the diurnal thermocline, the role of wind forcing, and the vertical scales of turbulence.

a. Temperature variance above and below the diurnal thermocline

A clear example of the influence of stratification on temperature microstructure occurs at 1530 LT on the moderate day. The top two GusTs at 0.4 and 1.7 m show turbulent temperature fluctuations of 0.01°–0.02°C (Figs. 4a,b). The third GusT at 4.2 m shows fluctuations that are 5 times as large (Fig. 4c). The deepest GusT at 7.7 m shows 0.01°C fluctuations that are not predominantly turbulent. Instead, they result from the background temperature gradient being advected by wave orbital velocities (see appendix B). Overall, these changes in temperature microstructure across the four GusTs, and the associated values of ϵχ, are explained by the depths relative to zTzmax, which is 5 m at this time (Fig. 3h).

Fig. 4.
Fig. 4.

Example records from the four GusTs at 1530 LT on the moderate day. (a)–(d) Measured temperature [high-passed with a cutoff of 1/(30 s)] and the component induced by surface waves [Eq. (B10)]. Note the differing y-axis limits. (e)–(h) Spectra of Tx from 10-min records (1530–1540 LT). For clarity, the spectra are averaged in logarithmically spaced frequency bins. Fits to the analytical form use the corrected spectra [Eq. (B11)].

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

For the top three GusTs (0.4–4.2 m), temperature gradient spectra display +1/3 slopes characteristic of the IC subrange (Figs. 4e–g; see also Fig. D2). In the top 2 m, ϵχ is nearly the same magnitude (7–8 × 10−7 W kg−1). Note that ϵχ does not account for the possibility of turbulence generated by wave breaking. In appendix C, we infer that ϵχ underestimates the true ϵ in the top 1 m in moderate winds. (On the calm and windy days, ϵχ underestimates ϵ in the top 0.5 and 2 m, respectively.) Perhaps surprisingly, ϵχ is larger at the third GusT at 4.2 m despite the stratification being 3–10 times stronger than at the two GusTs above. Presumably there is stronger vertical current shear here induced by the stratification (Sutherland et al. 2016).

Beneath zTzmax, the turbulent signal is weak, digitization noise is evident, and the signal is predominantly wave induced (Fig. 4d). This nonturbulent wave component is excluded from the spectra by converting the “measured” spectra to “corrected” spectra using the coherence between the measured and wave-induced temperatures as described in appendix B. From the corrected spectra, we find that ϵχ at 7.7 m is smaller by a factor of 500 smaller than that at 4.2 m for the period in Fig. 4.

b. Depth and time variability of turbulence

To see the influence of DWLs on upper-ocean turbulence, we compare ϵχ with the value we expect without diurnal stratification. The latter is often predicted based on a wind-dependent scaling:

ϵ*=u*3κz,

where κ = 0.4 is the von Kármán constant and z is measured relative to the free surface. The expression follows from wall layer scaling based on the friction velocity [u* = (τ/ρw)1/2 with τ being the wind stress and ρw being the density of water]. A second term to account for convectively driven turbulence is often included in Eq. (1) (see, e.g., Lombardo and Gregg 1989). In our case, this term is zero between approximately 0700 and 1700 LT when there is a net surface ocean heat input. Further, tests with the extra term showed that it made little difference to our results between 0500 and 2000 LT, although this would not hold if our analysis extended later into the night or to deeper depths.

The stabilizing influence of a DWL limits the vertical transport of stress. Whereas the ϵ* scaling follows from assuming a turbulent length scale of κz, the true length scale in the presence of stratification is likely smaller. More generally, to illustrate the influence of stratification, we compare ϵχ and ϵ*. On the calm and moderate days, we find that below the DWL ϵχ is one–two orders of magnitude smaller than ϵ* (Figs. 5b,d). We also see ϵχϵ* near the surface on the windy day (Fig. 5f), but this may reflect ϵχ being an underestimate as noted in section 3a.

Fig. 5.
Fig. 5.

Enhanced and suppressed turbulence dissipation above and below zTzmax, the depth of maximum temperature gradient. The enhancement or suppression is presented as a logarithmically scaled ratio of εχ to the scaling from Eq. (1). Ratios near that surface that are ≪1, especially on the windy day, are underestimates as discussed in appendix C.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

Above zTzmax but outside the top 1–2 m, we observe either ϵχϵ* or ϵχ > ϵ*. Prolonged periods during which ϵχ > ϵ* occurred on the moderate day (Fig. 5d) at 4.2 and 7.7 m after 1400 and 1600 LT, respectively. Overall, this is similar to Sutherland et al. (2016) who show ϵ ≈ 2–3ϵ* between 1200 and 1800 LT above the base of the DWL. Another example is found in Anis and Moum (1995) who show agreement between ϵ and ϵ* above the diurnal thermocline during the day, but a decrease in the ratio below the thermocline. Smyth et al. (1997) provide an equivalent example for rain-induced salinity stratification. They show an abrupt change from ϵ > ϵ* to ϵ < ϵ* at the depth of maximum stratification at 20–30 m.

Although there is variability in ϵχ/ϵ* on 10-min scales, the dominant signal at most depths is a change from values of <1 to values of >1 as the DWL descends. This indicates that the diurnal time scale is dominant, consistent with past studies (e.g., Caldwell et al. 1997; Sutherland et al. 2016). At the low latitudes in this study, the Coriolis period is too long to influence the deepening rate (cf. Pollard et al. 1972; Pham and Sarkar 2017).

c. Inferring anisotropy of diurnal warm layer turbulence

A second depiction of DWL turbulence that complements Fig. 5 is the evolution in log(N)–log(ϵχ) space (e.g., Moum 1997). This space maps the progression of the Ozmidov scale: LO = (ϵ/N3)1/2, a measure of the vertical scale of turbulent overturns (Fig. 6). On the calm and moderate days, LO varies by more than two orders of magnitude with minima less than 0.1 m. During morning and early afternoon, this is the dominant vertical turbulent length scale, whereas later in the day LO increases to values comparable to the distance to the free surface. On the windy day, LO fluctuates around 1–10 m, so both stratification and proximity to the surface limit the vertical extent of turbulent eddies. [See Sutherland et al. (2016) for a comparison of other various length scales relevant to DWLs.]

Fig. 6.
Fig. 6.

DWLs trace counterclockwise paths in log(N)–log(ϵχ) space. Wind speeds increase from left to right panels, and depths increase from top to bottom panels. The steeper dashed lines, constant Ozmidov scale, are a proxy for the vertical size of turbulent eddies. The flatter long-dashed lines indicate constant buoyancy Reynolds numbers. Moving toward the lower-right corner of each panel, the upper two Reb lines (Reb = 1500 and 300) indicate increasing levels of anisotropy discussed by Gargett et al. (1984), and the lowest is a threshold at which turbulent buoyancy fluxes vanish (Reb = 25; Stillinger et al. 1983).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

Another indicator of the state of turbulence is the buoyancy Reynolds number Reb = ϵ/νN2. As Reb increases, turbulence likely tends from anisotropy to isotropy. Although this is a continuum, we consider three representative values: 25, 300, and 1500. When Reb is below 25, turbulent buoyancy fluxes become negligible (Stillinger et al. 1983). The larger two values follow from Gargett et al. (1984). For values of Reb ≲ 300, they describe the turbulence as chaotic and “two dimensional,” with the vertical motions being small due to attenuation by the stratification. At larger, but still moderate Reb (≲1500), the turbulence remains anisotropic. Note that we are being necessarily qualitative about anisotropy. Reb can be interpreted in more than one way depending on physical constraints or assumptions about the generation and evolution of stratified turbulence (Gargett 1988). Here we are assuming that the aforementioned relationship holds in our situation. That is, measurements in geophysical-scale flows demonstrate increasing anisotropy, particularly at lower wavenumbers, with decreasing Reb (Gargett et al. 1984; Yamazaki 1990). The value of 25 comes from laboratory experiments but is subject to similar ambiguity.

With the approximate bounds added to Fig. 6, we infer anisotropy on the moderate and calm days during morning and early afternoon. During a subset of these times, turbulent buoyancy fluxes become negligible. This occurs on the calm day at 0900–1400 LT at 1.7 m and on the moderate day at 1100–1600 LT at 7.7 m. Although values of ϵχ are uncertain at such times because the thermistor record is near digitization noise (e.g., Fig. 4h), the signal being near noise level is consistent with vanishing turbulent buoyancy fluxes.

In log(N)–log(ϵχ) parameter space, the evolution of DWL turbulence reduces to a counterclockwise path, a result that holds across the entire dataset (1). This pathway neatly illustrates the sequence shown by Moulin et al. (2018). Nighttime convection leaves small N and comparatively large ϵχ early in the day. The building diurnal stratification quenches ϵχ as N increases. At depths below zTzmax, there is warming by local absorption of solar radiation, but presumably negligible shear from the diurnal jet. This state may last several hours. On the moderate day at 8 m deep, for example, it lasts until 1600 LT. As the DWL descends to the depth in question, ϵχ increases due to shear instabilities, so the path leads to the upper right. Last, the diurnal thermocline deepens past a fixed depth and the path moves to the upper left (small N; large ϵχ). With increasing depth or decreasing wind speed, the range of ϵχ values over the day increases. Ultimately, this depiction is instructive for the following section given the close relationship between ϵχ and turbulent heat flux.

4. Subsurface heat transport

The evolution of turbulence within the DWL governs if and when heat is transported downward and the consequent evolution of mixed layer temperature. In this section, we use the ϵχ(z, t) fields presented in the previous section to calculate turbulent heat fluxes during our three case study days to show when subsurface turbulent heat fluxes on diurnal time scales are a first-order process.

Here we consider thermodynamic heating between discrete depths in which heat content H changes due to divergences of turbulent and solar heat fluxes:

Ht=Jqsz+Jqtz,

where Jq is a heat flux in watts per meter squared that is separated into its solar (s) and turbulent (t) components. The balance excludes the air–sea interface, so latent, sensible, and longwave radiative fluxes do not contribute. Hereinafter, the lack of a superscript s or t implies the sum of the two terms. Downward fluxes are negative (z increases upward). Hence, ∂Jq/∂z < 0 indicates a convergence of heat leading to warming. Everywhere, Jqs/z<0, whereas Jqt/z can have either sign. Expressions equivalent to Eq. (2) often place the minus sign on the right-hand side, but moving it to the left-hand side here simplifies the discussion to follow. Where applicable, figures will show −∂H/∂t (i.e., with the negative sign) to ease comparison between the two sides.

Parameter Jqt is calculated using the measured stratification and diffusivity derived from χ: Jqt=ρcpKTTz, where KT=χ/(2Tz2) (see appendix A). Parameter Jqs is calculated using a two-band exponential decay (Jerlov type IA; Paulson and Simpson 1977) of the measured net surface solar radiation. Jerlov type IA corresponds to moderately clear water. We could have alternatively invoked the nine-band decay used by Paulson and Simpson (1981) and Gentemann et al. (2009), which concentrates absorption near the surface. For example, 40% of solar radiation is absorbed over the top 0.1 m in the nine-band formulation compared to 10% in the two-band formulation. Choosing the nine-band formulation would improve the estimated balance for moderate winds (Fig. 7f; discussed below), but result in a greater imbalance under calm winds (Fig. 7c). Given the latter case is evaluated closer to the surface (1 versus 3 m), this sensitivity is expected. (See Kantha and Clayson (1994) for a discussion on the importance of absorption profiles in a larger range of contexts.)

Fig. 7.
Fig. 7.

Thermodynamic heating evaluated between adjacent GusTs. The negative of ∂H/∂t is plotted so that the left- and right-hand sides of Eq. (2) agree if the thick and thin red lines coincide. Different pairs of GusTs are used for the different days, with the depths of the GusTs indicated in (a), (d), and (g): (a),(b),(d),(e),(g),(h) Ten-minute averages; (c),(f),(i) 10-min averages are binned to 30-min averages. Error bars encompass the minima and maxima within the 30-min bins. Notable features are labeled with a letter and discussed in section 4a. Heat fluxes (and hence divergences) are not calculated when Tz < 0.002°C m−1 [dashed line in (g)].

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

The heat content per unit volume between vertically adjacent GusTs at z1 and z2 is related to the temperature above the mixed layer temperature TML:

H=1z2z1z1z2ρcp(TTML)dz.

One difficulty in this calculation is an appropriate definition and measurement of TML. We use a variable TML from our microstructure profiling to minimize any apparent horizontal advective heat fluxes that can arise due to transiting at ~1 m s−1. However, there is often no clear single value for TML. Such an issue is well illustrated by Fig. 6b of Matthews et al. (2014). Ultimately, we define TML as the median temperature between 15 and 25 m. We also low-pass filter H with a 2-h cutoff before taking the time derivative. Gerbi et al. (2008) also identified a 2-h smoothing as necessary for a similar analysis.

We evaluate Eq. (2) at both fixed depths and fixed times. Fixed depths are preferable for demonstrating similarity between the left- and right-hand sides, whereas vertical profiles at fixed time better demonstrate the predominantly vertical nature of the heat transport. In Fig. 7, we present the left- and right-hand sides of Eq. (2) and also isolate the turbulent component to indicate its comparative magnitude. In Fig. 8, we show vertical profiles.

Fig. 8.
Fig. 8.

Vertical structure of turbulent heat transport at representative times of day. Wind speed increases from top to bottom panels. The Jqs profiles (thin, smooth lines) are estimated assuming a two-band exponential decay, whereas Jqt (thick, piecewise lines with symbols) are evaluated at discrete depths. Some profiles of Jqt for the 1600–1700 LT profiles are excluded because Tz is less than 0.002°C m−1.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

a. Time series of subsurface thermodynamic heating

We first consider the heating throughout the day between vertically adjacent GusTs (Fig. 7). For the calm day, we consider the top two GusTs (0.4 and 1.7 m); for the moderate day, the middle two (1.7 and 4.2 m); and for the windy day, the bottom two (4.2 and 7.7 m). Each range best encompasses the diurnal thermocline on the respective day (Fig. 3).

For much of the calm day, Jqt/z is small (−20 W m−3), so ∂H/∂t is dominated by Jqs/z (Fig. 7, feature A). At 1220 LT, we transit for 10 min through a feature with inferred horizontal temperature gradients >1°C km−1. Sharp peaks arise in Tz and Jqt, and both Jqt/z and ∂H/∂t change in concert (Fig. 7, feature B). These peaks emphasize the comparatively small Jqt before and after that occurs due to the suppression of turbulence by stratification. Note that, because feature B is only 10 min long, the balance in Fig. 7c is skewed by H being calculated from temperatures smoothed over 2 h.

At 1500 LT on the calm day, ∂H/∂t switched from warming to cooling (feature C; note that −∂H/∂t is the quantity plotted). At the same time, the magnitudes of Jqt at both depths increased. Between 1500 and 1800 LT, the mean Jqt was −80 W m−2 at 0.4 m (feature D). Its magnitude was larger at 1.7 m (−140 W m−2) due to the larger Tz. We attribute the increase in Jqt at both depths to two factors. First, a wind speed increase. The mean wind speeds in the 3-h periods before and after 1500 LT were 1.5 and 2.3 m s−1, respectively (Fig. 3a). This 50% increase is approximately a tripling in energy input (proportional to cubed wind speed) and potentially sufficient to accelerate the DWL and trigger turbulent mixing: Hughes et al. (2020) show that wind speeds ≳2 m s−1 lead to appreciable wind-induced turbulence. Second, as the sun gets lower in the sky in the afternoon, the net surface heat flux reduces or changes sign. This results in either a comparatively smaller barrier to wind-driven mixing or the possibility of a near-surface, countergradient (here, upward) turbulent heat flux (see, e.g., Deardorff 1966). On this day, the surface heat flux changes to net cooling at 1620 LT. After this time, we may be overestimating the downward Jqt at 0.4 m. Between 1600 and 1800 LT, the water at 0.4 m is stratified according to 10-min means of Tz (0.01°–0.02°C m−1), but on shorter time scales it shows evidence of convection. One-minute means of Tz often fail our <0.002°C m−1 criterion. Together, these reasons can partly explain the discrepancy between ∂Jq/∂z and −∂H/∂t between 1500 and 1800 LT, where the latter is 6 times as large. That is, if Jqt at 0.4 m was closer to 0 or of opposite sign, then ∂Jq/∂z would be larger.

In contrast to the calm day, the dominant balance on the moderate day is between Jqt/z and ∂H/∂t. (Because we are considering a deeper pair of GusTs, the magnitude of Jqs/z is smaller). Until 1430 LT, Jqt/z is negative (Fig. 7, feature E) because zTzmax has yet to pass 4.2 m (see Fig. 3h), the depth of the deeper of the two GusTs being considered. The mean Jqt/z until 1430 LT over this 2.5 m depth range is −25 W m−3. Within an hour of each other, ∂Jq/∂z and −∂H/∂t both turn positive and remain so for the rest of the day (feature F). Overall, the left- and right-hand sides of Eq. (2) agree well over the entire 10-h period.

On the windy day, we approach the limits of our ability to derive Jqt because ϵχ is not calculated when Tz < 0.002°C m−1. Between 1000 and 1300 LT, the top of the diurnal thermocline is situated between the lowest two GusTs (Fig. 3i). It is for this part of the day that we can calculate Jqt (feature G). During this time, both Jqt/z and −∂H/∂t are negative. A mean Jqt of −400 W m−2 moves past the GusT at 4.2 m. Approximately one-third of that continues beyond the GusT at 7.7 m. By comparison, the net surface heat flux at this time was −700 W m−2.

b. Vertical heating profiles

Turbulent heat flux convergence near zTzmax is anticipated because of the decrease in ϵ and hence diffusivity near this depth (Fig. 5). This is indeed observed, and is especially clear during the moderate day (Figs. 8e–h) because the maximum Tz coincides with the middle of our measurement range. At 1000 and 1300 LT, Jqt is strongest at the GusT at 1.7 m, which lies just above the strongest Tz. At both of these times, Tz at 4.2 m is only one-half that at 1.7 m, yet Jqt is smaller by a factor of 5–10 because of the suppressed turbulent mixing. Jqt peaks near zTzmax at 1600 LT with a value of −340 W m−2.

In section 4a, we discussed the convergence of heat on the windy day between 4.2 and 7.7 m. Figures 8i–l further elucidate this convergence. In the top 2 m, Tz has a local maximum due to the continual input and absorption of solar radiation near the surface. Jqt, however, is small in the top 2 m due in part to ϵχ likely underestimating the true ϵ near the surface on this day (appendix C). At 1000 and 1300 LT, Tz at 4.2 m is small (0.002°–0.003°C m−1), yet strong turbulence (Fig. 5f) acting on the gradient results in a large Jqt of 300–800 W m−1. With increasing depth between 4.2 and 7.7 m, Tz increases and, consequently, turbulence is suppressed by the stabilizing stratification. Much of the surface heat input converges within this depth range as noted in section 4a.

c. Evaluation and consequences of subsurface heat fluxes

The primary consequence of subsurface turbulent mixing in DWLs is to move the sun’s heat downward away from the surface and thereby modulate diurnal variation of SST and the associated surface heat fluxes. Taking into account climatological ocean and atmospheric processes over the tropical Pacific, Seager et al. (1988) estimated that a 1 W m−2 change in air–sea heat flux is associated with a 0.04°–0.08°C change in SST. A change of, say, 5 W m−1 equates to ~0.3°C. In our dataset, on the calm day, the temperature at 0.2 m (the shallowest temperature logger) exceeded the mixed layer temperature TML by 0.3°C for 8 h (not shown). On the moderate and windy days, this decreased to 5 and 0 h, respectively. Matthews et al. (2014) demonstrate similar relationships in a 3-month-long dataset from the Indian Ocean (see their Figs. 2 and 8).

Although there is a well-established inverse correlation between wind speed and diurnal SST anomalies (e.g., Price et al. 1986; Lukas 1991; Gentemann et al. 2009), the amount of heat mixed downward is typically an unknown. With SurfOtter, we have rare direct measurements of the near-surface turbulent heat flux into the ocean based on accurate mean temperature profiles.

In moderate conditions (4–5 m s−1), we observe daytime subsurface Jqt from −100 to −300 W m−2 (Fig. 8). Corresponding divergences have a magnitude of ~50 W m−3. Our Jqt values are consistent with heat fluxes simulated with a second-order turbulence scheme by Wade et al. (2011) and a large-eddy simulation (LES) by Sarkar and Pham (2019) from −200 to −300 W m−2 at the base of the DWL in the afternoon under 3–5 and 6 m s−1 winds, respectively. In simulations of a DWL requiring a one-dimensional mixing scheme, we recommend a second-order turbulence scheme over other commonly used schemes such as Price–Weller–Pinkel (PWP; Price et al. 1986) and KPP (Large et al. 1994). By virtue of its simplicity, PWP is limited in ability to simulate the vertical structure of heat flux, especially within the bulk mixed layer. KPP is a more sophisticated scheme but is challenged at high vertical resolution during periods of insolation that can cause the boundary layer to shoal to depths of <1 m (Van Roekel et al. 2018). Note that these suggestions, which mirror our earlier suggestions for adequately simulating shear (Hughes et al. 2020), are specific to simulation of DWLs during periods of low wind and near-surface stratification. They do not necessarily hold during nighttime convection or over longer periods.

From observed temperature rates of change, Price et al. (1986) inferred maximum fluxes of −100 W m−2 at 1400–1500 LT at 5 m deep under 4–6 m s−1 winds. With respect to vertical structure, they highlight the distinct middepth maximum (see their Fig. 3d), which they term a “knee.” Its depth closely follows the deepening surface layer. We also observe such a knee in our Jqt profiles. To show this for many profiles at once, we normalize profiles by an approximate depth mean and align them vertically about zTzmax (Fig. 9). This presentation emphasizes the change about zTzmax from cooling to warming when moving from above to below.

Fig. 9.
Fig. 9.

Interpretation of an averaged, normalized Jqt profile. Data come from the four days on which we can adequately track zTzmax throughout the day (25 and 27 Aug, 29 Sep, and 4 Oct). Gray lines are individual, 10-min profiles normalized and aligned vertically about zTzmax. From the resulting distribution of points, the binned profile is created such that each value is derived from the 30 vertically closest points. Note that the denominator is not a true depth mean but rather is a mean of values from thermistors at discrete depths.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

To put Jqt values into context, the surface heat fluxes for the three days discussed were 40–50 W m−2 for longwave radiation, 50–100 W m−2 for latent heat, and 1–2 W m−2 for sensible heat. The solar radiative flux averaged between 0700 and 1700 LT is from −400 to −600 W m−2. Consequently, subsurface turbulent mixing is a major component in the surface heat budget, second only to absorption of solar radiation. However, when wind speeds are insufficiently strong to promote mixing, the heat tendency is a balance between solar radiation and surface heat loss to the atmosphere. This low-wind state has also been observed by Halpern and Reed (1976), Stramma et al. (1986), and Hodges and Fratantoni (2014). The lack of wind-induced mixing in such cases leaves SST more sensitive to quantities such as air–sea humidity difference or the absorption profile of solar radiation. Hence, a simple regression between diurnal SST anomalies and wind speed does not work at low wind speeds (see Hughes et al. 2020, and references therein).

Using our observed heating time series and profiles, and aided by an idealized simulation detailed by Hughes et al. (2020), we schematize the heating within a DWL in Fig. 10. Primarily representative of moderate wind conditions, the figure summarizes cases when the DWL descends by O(10) m over the day. Before sunrise, convection leads to an upward turbulent heat flux throughout the mixed layer. After the net surface heat flux changes sign, the near surface stratifies and turbulent heat fluxes throughout the mixed layer decay. Much of the incoming solar energy is absorbed close to the surface and then transported downward by turbulent mixing. Turbulent heat fluxes are largest in the top half of the diurnal thermocline in the afternoon and evening owing to the optimal combination of turbulent diffusivity and vertical temperature gradient. The decrease in diffusivity in the lower part of the thermocline causes a convergence of turbulent heat flux and hence warming.

Fig. 10.
Fig. 10.

(a) Redistribution of absorbed solar energy by turbulence in a diurnal warm layer. Most solar energy is absorbed in the top few meters leading to strong solar heat flux convergence. Turbulence induced by wind mixes this heat deeper throughout the day. The integrated heat flux convergence in (b) is between the two horizontal lines shown in (a). Vertical slices at 1500 LT are in (c). (b) The observed heat flux and (c) its divergence as described in sections 4a and 4b, respectively. The temperature gradient and turbulent transport in the top O(10) cm are inferred by assuming the turbulent heat flux asymptotically approaches the value of surface heat loss. These details were not measured in this paper.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

5. Conclusions

Heat flux convergence in the near-surface tropical ocean includes solar and turbulent components. The latter was measured here using fast thermistors and vertical temperature gradients from a temperature array. The total convergence was corroborated by measuring the temporal rate of change of heat from the array. This agreement between two independent measurements was made possible by a new platform SurfOtter. Its ability to follow the surface, together with its suite of sensors, enabled a new method to derive near-surface turbulence dissipation and turbulent heat fluxes. Temperature spectra were fit across and above frequencies corresponding to surface waves. This large bandwidth minimized uncertainty in the calculation of the turbulent quantities.

The diurnal thermocline clearly suppressed the vertical extent of turbulence. Specifically, the depth of the peak temperature gradient, zTzmax, coincided with a large change in turbulence dissipation ϵχ. Below this depth, ϵχ decreased by one to two orders of magnitude relative to a wind- and depth-dependent scaling. Because of the feedbacks between the temperature gradient and turbulence dissipation, which are both time and depth varying, the two quantities coevolve in a complex manner. By invoking the log(N)–log(ϵχ) parameter space, we distilled their evolution to a counterclockwise path and inferred departures from isotropic turbulence.

There is also evidence that turbulent heat fluxes shut off completely in the remnant mixed layer. This was quantitatively demonstrated through calculation of the buoyancy Reynolds number. It was also apparent in the raw temperature records. Under 4 m s−1 winds at 7.7-m depth, for example, the temperature record showed digitization noise superimposed on the signal due to surface waves. By comparison, at the same time but at 4.2 m, turbulence dominated (Fig. 4). This depth variation in turbulence, and hence turbulent diffusivity, has a perhaps curious consequence for the diurnal temperature gradient. Although we normally regard turbulence as something that helps mix away gradients, it is not so simple when diffusivity is spatially variable. In such a case, the consequent convergence can maintain gradients in the presence of the otherwise diffusing influence of turbulence. Such convergence was evident in vertical profiles throughout our dataset.

Turbulence, or lack thereof, governed whether downward heat fluxes during the afternoon were appreciable and thereby whether diurnal SST anomalies were O(0.1)°C or O(1)°C . On the moderate and windy days, turbulent heat fluxes were hundreds of watts per meter squared. Such values would be rare in much of the ocean. In diurnal warm layers, however, large turbulent heat fluxes are commonplace given that, in the top 2 m of the ocean on a clear-sky day in the tropics, a daytime mean of O(300) W m−2 of solar energy is absorbed and that heat has to be redistributed vertically.

Acknowledgments

This research is part of the PISTON project: Propagation of Intra-Seasonal Oscillations, funded by the Office of Naval Research. Shipboard measurements described in this paper were taken aboard RV Thomas G. Thompson. Craig Van Appledorn, Kerry Latham, and Pavan Vutukur designed and constructed SurfOtter and the turbulence instruments it housed. Sally Warner processed the Chameleon dataset. Meteorological data were collected and provided by NOAA’s Earth System Laboratory (Chris Fairall, Byron Blomquist, Ludovic Bariteau, Simon de Szoeke, and Elizabeth Thompson). Comments from two anonymous reviewers helped to clarify several aspects of this paper.

Data availability statement

Datasets for each of the 12 days that are described in Table 1 are available online (kghughes.com/data and http://dx.doi.org/10.5281/zenodo.3894910). These include the temperature fields, the full thermistor records, the derived turbulent quantities, and many other auxiliary data.

APPENDIX A

Deriving Turbulence Dissipation from Fast Thermistor Records

The spectrum of temperature gradient at small, turbulent scales is separated into three distinct subranges. At the largest turbulence scales, an IC subrange occurs in which the downscale turbulent transfer is unaffected by molecular processes. With decreasing length scale, the effects of viscosity and then diffusivity become leading order, resulting in the viscous–convective and viscous–diffusive subranges. In this paper, we infer ϵχ by fitting measured temperature spectra to an analytical form for the IC subrange. Good agreement overall in ϵ values derived from IC fits and from fits to higher-frequency subranges has been demonstrated by Zhang and Moum (2010) and Becherer and Moum (2017). There are times, however, when the two estimates disagree for reasons we do not yet understand.

With a measurement of the mean flow speed past the sensor U, temporal temperature gradients can be converted to horizontal temperature gradients as Tx = Tt/U. We exclude any influence of wave orbital velocities in this conversion for three reasons. First, we tow SurfOtter through the water at speeds greater than orbital velocities. Second, the variance of Tt in the 15–0.5-s band can span three orders of magnitude in a day. This dominates the variance of Tx relative to the effects of orbital velocities. Third, the 200-m-long cable used to tow SurfOtter takes a catenary shape, providing elasticity at surface wave frequencies. Hence, to some extent, SurfOtter can behave in a Lagrangian manner, which minimizes the orbital velocities relative to the thermistors.

With Tx calculated, we use the following expressions to derive ϵχ:

ΦTx=(2π)4/3CT(2ΓTz2N2)1/3χ2/3U4/3f1/3and
ϵχ=χN22ΓTz2,

where ΦTx is the spectra of Tx, CT = 0.4 is an empirical constant, and Γ = 0.2 is the mixing efficiency. Further details are found in earlier papers (Zhang and Moum 2010; Becherer and Moum 2017; Moulin et al. 2018). Here, stratification is only temperature dependent (N2 = gαTz, where α is the thermal expansion coefficient). Without stratification, the method represented by Eqs. (A1) and (A2) does not hold. The equations can be rearranged to show that ϵχTz3/2. We therefore limit calculation of ϵχ to periods of measurable stratification (Tz > 0.002°C m−1).

We infer ϵχ over 10-min segments and fit over the frequency range corresponding to periods of 15–0.5 s. Segments of this length are necessary for an accurate estimate of the coherence between measured and wave-induced temperatures (appendix B). Although 10 min is longer than one eddy time scale, we will demonstrate that the resulting spectra display the expected spectral slope (appendix D). Further, by using the IC range here rather than higher-frequency subranges, we avoid having to correct for thermistor response at high frequencies.

In the main text, and in what follows, there are parallels with a recent study by Moulin et al. (2018), who limited their spectral calculations to frequencies below those of surface waves. They used frequencies corresponding to 60–15-s periods, which is similar to the band Gerbi et al. (2008) used to calculate heat and momentum fluxes from cospectra of temperature and velocities. Development of SurfOtter was motivated, in part, to improve on the measurements used by Moulin et al. (2018) with measurements that are less affected by waves. As we will show, these new measurements permit fitting temperature gradient spectra over a much larger frequency range. In addition, they yield a more accurate measurement of Tz. For ϵχ and Jqt, an accurate measurement of Tz is as important as an accurate measurement of χ (Stevens and Smith 2004).

APPENDIX B

The Influence of Surface Waves on Temperature Records

By following the surface, SurfOtter minimizes the influence of surface waves on its temperature records, which is the primary obstacle to fitting spectra over the IC subrange. Although not completely eliminated, the remaining influence of waves on measured spectra can be predicted and subsequently removed. Here, we describe the reason for, and the method to predict, the wave-induced component of the measured temperature.

If SurfOtter perfectly follows the surface, a thermistor at a nominal depth d will always be this distance from the sea surface. However, wave orbital velocities induce a depth-dependent vertical advection of isotherms. Consequently, the thermistor traverses isotherms thus recording a variable temperature (Fig. B1). Here we describe two methods to predict the temperature signal introduced by surface waves for a thermistor at a distance d from the sea surface η(t). Both methods are based on a background temperature field advected by the wave orbital velocities.

Fig. B1.
Fig. B1.

The temperature record of a thermistor that perfectly follows the sea surface in a nonturbulent wave field. Wave-orbital velocities decay with depth, with higher-frequency components decaying faster. Because the temperature field is advected vertically by wave orbitals while the thermistor parallels the surface, the thermistor traverses isotherms and consequently records a variable temperature.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

Assume a monochromatic wave η(t) = A cos(−2πft). In a frame of reference in which the sea surface has a vertical position of z = 0 and is negative below, the position Z(z, t) of fluid parcels beneath the wave is given by

Z(z,t)=η(1ekz)+z,

where k is the wavenumber.

The background temperature profile Tb(z) is assumed to be constant in time over several wave periods. It is described by a first-order Taylor expansion about z = −d:

Tb(z)=T0+Tz(z+d).

Here, T0 and Tz are determined from a fit using two temperature sensors above z = −d and two below (or only one below for the deepest GusT). A second-order Taylor expansion is possible and was tested but makes minimal difference to the following analysis.

Vertical advection of Tb by the waves is equivalent to evaluating Eq. (B2) at Z from Eq. (B1):

T(z,t)=T0+Tz[η(1ekz)+z+d].

In our frame of reference where z = 0 at the surface, the thermistor depth is constant, z = −d. Therefore, evaluating Eq. (B3) at z = −d gives the temperature that would be recorded by a thermistor beneath the monochromatic wave field:

T(d,t)=T0+Tzη(1ekd).

Because we started by assuming z = 0 at the sea surface, we ensured that the depth decay term ekd is always less than 1. Had we started by using 0 to define the mean water level, then the depth decay term would have been ek(η−d), which gives decay terms greater than 1 when η < d. Smit et al. (2017) provides a thorough discussion of these issues as well as a method that predicts wave kinematics near the surface that is accurate to second order. Their complete method is beyond the scope of this study, but we note that our approach is equivalent to the deep water limit of their first-order correction [their O(ϵ) solution].

To move from monochromatic waves to a broadband wave field, we invoke the Fourier transform F of η and the wavenumber to frequency conversion from linear wave theory:

F[η(t)]=η^(f)and
k=4π2f2/g.

Products involving η and the depth-decay term are undertaken in frequency space, and the result is converted back with the inverse Fourier transform F−1. This is effectively a linear filtering process (e.g., Wheeler 1970). Equation (B4) becomes

T(d,t)=T0+TzηTzF1(η^zd),

where zd = exp(−4π2f2d/g) is the (nondimensional) depth-decay term.

The second method to predict the temperature variance introduced by surface waves is to make use of the similarities between pressure and temperature measured by SurfOtter. The pressure expected at a distance d from the surface is given by Eq. (A4) of Hughes et al. (2020):

p(d,t)=ρgdρgη+ρgF1(η^zd).

Note the similarity with Eq. (B7). The second and third terms are identical except for the coefficients: −ρg in Eq. (B8) and Tz in Eq. (B7). Therefore, we can predict the wave-induced temperature variability from the measured pressure as

Ttwave=ptρgTzand
Twave=pρgTz+T0,

where T0 is now a constant of integration, which goes to zero when we high-pass filter signals to remove frequencies lower than ~1/(30 s).

The quantity −pt/ρg is in some sense a vertical velocity. Indeed, on moorings, Ttwave=wTz with w from integration of an accelerometer oriented vertically (Moum and Nash 2009; Zhang and Moum 2010). On SurfOtter, however, −pt/ρgw (see Fig. 3 in Hughes et al. 2020). The difference is that −pt/ρg is a vertical velocity relative to the free surface. At the surface, p is presumed constant (atmospheric pressure), so pt = 0. Otherwise, the thermistor and pressure sensor nominally stay a fixed depth beneath the free surface, but the depth variation of orbital velocities leads to a velocity relative to the sensors.

We test the two methods on two 30-s segments during two periods for which surface waves clearly influence the temperature records (Fig. B2). To encompass extremes, the two examples use the shallowest and deepest GusTs. In both examples, the two predictions (the dark blue and black curves) are similar. For the shallow GusT, the predictions agree only somewhat with the measured temperature because the latter includes a nonnegligible turbulent component. Conversely, for the deeper GusT, the measured temperature is predominantly wave-induced variability. This deeper record is also smoother because of the depth decay of the higher frequencies. Of the two methods, the second method using the measured pressure is preferable and used in this paper. It is both easier to implement and advantageous in that is does not need to assume that sensors stay exactly at a fixed distance from the sea surface.

Fig. B2.
Fig. B2.

Examples of periods during which turbulence is small or negligible and the wave-induced temperature signal dominates. (a),(b) From the late morning on the calm day, the record from the shallowest GusT. (c),(d) From early afternoon on the moderate day, the record from the deepest GusT.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

With the prediction of Twave established, its effect can be removed. A seemingly simple approach would be to subtract it from the measured temperature. Such a method, however, is sensitive to slight inaccuracies in the measurement of Tz or small phase differences between the measured p and T. Here, removing Twave spectrally is a superior option. To do so, we first calculate the spectral coherence between Tt and Ttwave. This is frequency dependent and quantified by γ(f), the magnitude-squared coherence (hereinafter, simply coherence). A value of 1 at a given frequency indicates that the two signals are perfectly in phase at that frequency. A value of 0 indicates the two signals are unrelated. To calculate γ in practice requires averaging over several shorter segments within the overall block. Here, we divide each 10-min block into 20 segments, each of which is 30 s long.

Following Levine and Lueck (1999) and Zhang and Moum (2010), we define a corrected spectrum in terms of γ:

Φcorrected=(1γ)Φmeasured.

Results presented in this paper are derived from corrected spectra unless otherwise stated.

APPENDIX C

Uncertainty in ϵχ Values near the Surface

Implicit in the derivation of ϵχ [Eqs. (A1) and (A2)] is the assumption of a steady-state turbulent kinetic energy (TKE) balance between shear production Ps and removal by buoyancy production Jb and dissipation ϵ (Osborn 1980):

Ps=Jb+ϵ.

This assumes both temporal variability and vertical flux of TKE are negligible. The results in section 3 suggest the latter is the larger issue. Hence, we consider here the effect of a vertical TKE flux.

Divergence of F, the vertical TKE flux, would change the balance to

Fz+Ps=Jb+ϵ.

Parameter F can result from (i) a vertical redistribution or (ii) an input of TKE at the surface.

Based on simulations, we expect vertical redistribution to be small during conditions conducive to DWL formation. A large-eddy simulation by Wang (2001) under 4 m s−1 winds demonstrated a shear–buoyancy–dissipation balance. Turbulent transport was largely canceled by pressure transport (see their Fig. 1b). Klein and Coantic (1981) demonstrated the same balance with a second-order turbulence model (see the daytime panel of their Fig. 19).

Whether there exists an appreciable input of TKE at the surface is a matter of debate, especially in low-to-moderate winds. Brainerd and Gregg (1993) observed that, above the diurnal thermocline, ϵ was much larger than predicted by a wind-speed-dependent scaling (Lombardo and Gregg 1989), implying an input of TKE. Indeed, observations by Gerbi et al. (2009) from a tower in 16-m-deep water under winds averaging 7 m s−1 suggested that a TKE flux is necessary to close the turbulent energy budget. In developing DWL simulations, however, Pimentel et al. (2008) argued to forgo surface TKE fluxes. Similarly, using a free-rising turbulence profiler during 5 m s−1 winds, Callaghan et al. (2014) did not observe any persistently enhanced near-surface ϵ due to wave breaking.

Over the depths on which it acts, a surface TKE flux will cause ϵχ to underestimate the true ϵ. For example, using a second-order turbulence model with a surface TKE flux and a stabilizing buoyancy flux, Noh (1996) found that the shear–buoyancy–dissipation balance holds within the diurnal thermocline, but not above it (see his Fig. 4). In our case, this implies that ϵχ will underestimate ϵ near the surface but will be accurate within the region of strong Tz. Consequently, turbulent heat fluxes will also be accurate in this region, which is where they are strongest (section 4).

Our intent here is not to incorporate ∂F/∂z into the calculation of ϵχ but rather to estimate the depths affected by, and the magnitude of, a surface input of TKE insofar as it guides us on the depths over which to treat ϵχ as questionable. For this, Craig and Banner (1994) provide a convenient framework. Their intent was to account for breaking waves and so their expression may be questionable at low-to-moderate wind speed, but it does scale in a reasonable way (u*3).

For an unstratified fluid, Craig and Banner (1994) considered two separate analytical models: balances between (i) a TKE flux and ϵ with a surface input and (ii) shear production and ϵ without a surface input. The former predicts a depth profile of ϵ [their Eqs. (25)–(27)] of

ϵ(z)=2.4u*3αz0n(z0z)n1,

where n = 2.4 and z0 is set to 0.1 m. The parameter α defines the TKE flux at the sea surface:

F=αu*3atz=0.

A wide range of estimates for α (5–250) appear in the literature (e.g., Klein and Coantic 1981; Noh et al. 2004; Feddersen et al. 2007). Craig and Banner (1994) used α = 100.

The z−3.4 dependence in Eq. (C3) implies that the effect of a surface TKE flux decays rapidly. Craig and Banner (1994) found that for unstratified water under 9 m s−1 winds, the transition from the near-surface region with a predominantly diffusion–dissipation balance to a predominantly shear production–dissipation balance occurred at 0.6 m. We find similar depths when comparing ϵχ with Eq. (C3) (Fig. C1). The black curves denoting the analytical diffusion–dissipation balance exceed ϵχ in the top 0.5 m during the calm day and the top 1–2 m on the windy day.

Fig. C1.
Fig. C1.

Estimating the near-surface range over which ϵχ is questionable because it does not account for TKE input at the surface. Black lines show an analytical result for ϵ(z) under a diffusion–dissipation balance with a surface TKE flux (section 4 of Craig and Banner 1994). Gray lines are all 10-min ϵχ values. We consider near-surface ϵχ values as questionable if they are near or below the analytical curves.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

Using an LES to simulate diurnal thermocline formation, Noh et al. (2009) showed that wave breaking has a much larger effect on the TKE budget than Langmuir circulation. Based on this study, uncertainty induced by Langmuir circulation, a process not considered here, is presumed to be second order compared to uncertainty associated with F.

Ultimately, our analysis suggests that (i) care is needed in interpreting ϵχ in approximately the top meter and (ii) turbulent diffusion of TKE from the surface can explain why ϵχ profiles do not increase approaching the surface as is observed when ϵ is measured by shear probes.

APPENDIX D

Comparing Spectra over Differing Frequency Ranges

Because SurfOtter reliably follows the surface (Hughes et al. 2020), we can obtain robust results by fitting spectra over a larger band than the 60–15-s band used by Moulin et al. (2018). In theory, a fit to the inertial–convective range could extend to frequencies up to those at which turbulence transitions to a viscous–convective regime. This occurs near a wavenumber k* (cpm) (Dillon and Caldwell 1980):

k*=C*ϵ1/4ν3/4/(2π)

with values for the constants C* ≈ 0.05 and ν = 1 × 10−6 m2 s−1. To estimate a lower bound for k*, we use typical low values of ϵ = 10−8–10−6W kg−1 and oncoming flow speed of 0.4 m s−1. Together, these give k* = 3–8 cpm, or an equivalent frequency of f* = 1–3 Hz.

In choosing our upper frequency bound, in addition to our estimates of f*, we consider a previous study focused on temperature spectra beneath surface waves. Specifically, Stevens and Smith (2004) demonstrate −5/3 temperature spectra indicative of inertial–convective turbulence from a profiler rising vertically at 0.1 m s−1 beneath waves. Their frequency range of 1–10 Hz corresponds to a wavenumber range of 10–100 cpm. From this, it appears that the expression for k* above underestimates where the transition occurs. Ultimately, we do not expect our measured spectra to include any of the viscous–convective subrange if we apply an upper frequency bound of 2 Hz.

We choose 1/(15 s) as a low-frequency cutoff, rather than using 1/(60 s) as Moulin et al. (2018) did. There are two benefits to excluding the 60–15-s band here. First, we exclude length scales of several tens of meters, which are often outside the turbulent cascade. [Moulin et al. (2018) faced a trade-off between including these larger, potentially nonturbulent scales while still retaining a reasonable bandwidth over which to fit spectra.] Second, with separate bands we independently test the validity of the 60–15-s band. We will show this lower-frequency band to be adequate for capturing patterns of variability, albeit with ϵχ generally underestimated by a factor of 3–5, assuming our values from the higher-frequency band are the benchmark.

There is clear correlation between log10(ϵχ)calculated over the two different frequency bands. For all values from all thermistors for all 12 days in Table 1, R2 = 0.72 (Fig. D1a). At higher wind speeds (>6 m s−1; Fig. D1b), the magnitudes of the two ϵχ values are similar, with a median ratio of 3.0 (the 60–15-s band underestimates relative to the 15–0.5-s band). With moderate winds (3–6 m s−1; Fig. D1c), the median ratio increases to 5.0. This increase occurs because turbulent length scales decrease with decreasing wind, so the 60–15-s band now includes a larger portion of nonturbulent scales, which have an associated lower spectral level. For example, Figs. 4e–h show that the measured spectrum starts to fall off relative to the +1/3 slope with decreasing frequency below ~1/(10 s). The median does not continue to increase moving to lower winds (<3 m s−1; Fig. D1d). Rather, the distribution of the ratios widens indicative of a weaker correlation between ϵχ from the two bands.

Fig. D1.
Fig. D1.

Statistical comparisons of ϵχ derived using two different frequency ranges: (a) two-dimensional histogram of all 10-min ϵχ values from all GusTs for all 12 days in Table 1, and (b)–(d) ratios of the data from (a) separated by the wind speed at the time of measurement.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

For 10-min segments, the 15–0.5-s band encompasses 1160 spectral coefficients. Because coefficients are linearly spaced in frequency, 600 of the 1160 are within 1–2 Hz. Conversely, the 60–15-s band encompasses only 30. For the higher-frequency band, the distribution of slopes from linear fits to log10(ΦTx) against log10(f) for the GusT at 1.7 m is centered near the expected value of +1/3 (Fig. D2b). The medians of the three days shown are within 0.1 of +1/3. For the lower-frequency band, the medians are well above +1/3 at 0.6–0.8 and the distribution is much wider (Fig. D2d).

Fig. D2.
Fig. D2.

Slopes of best fit to horizontal temperature gradient spectra from the GusT at 1.7 m deep. (a)–(d) Slopes are from linear fits of log10(ΦTx) against log10(f) over the frequency range corresponding to the periods shown. (e) The maximum coherence max[γ(f)] between the measured temperature and the wave component over the 15–0.5-s band.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0079.1

The tight distribution of slopes in the higher-frequency band makes it possible, if desired, to use noticeable deviations from this slope as a flag for periods when the fit is questionable. For example, between 1000 and 1600 LT on the calm day, the best-fit slope oscillates noticeably (Fig. D2a). During this time, the GusT at 1.7 m deep is below zTzmax. Consequently, the record is dominated by waves, akin to the example in Fig. 4d. Alternatively, we can use the maximum coherence, max[γ(f)], as a flag. For the period in question, this is 0.5–0.9 on the calm day but 0.2–0.3 on the two other days (Fig. D2e). Ultimately, we do not flag any data here because high coherence or poor fits occur when ϵχ is small and therefore inconsequential.

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