Detailed observations of the diurnal jet, a surface intensification of the wind-driven current associated with the diurnal cycle of sea surface temperature (SST), were obtained during August and September 2012 in the subtropical Atlantic. A diurnal increase in SST of 0.2° to 0.5°C was observed, which corresponded to a diurnal jet of 0.15 m s−1. The increase in near-surface stratification limits the vertical diffusion of the wind stress, which in turn increases the near-surface shear. While the stratification decreased the turbulent dissipation rate ε below the depth of the diurnal jet, there was an observed increase in ε within the diurnal jet. The diurnal jet was observed to increase the near-surface shear by a factor of 5, which coincided with enhanced values of ε. The diurnal evolution of the Richardson number, which is an indicator of shear instability, is less than 1, suggesting that shear instability may contribute to near-surface turbulence. While the increased stratification due to the diurnal heating limits the depth of the momentum flux due to the wind, shear instability provides an additional source of turbulence that interacts with the enhanced shear of the diurnal jet to increase ε within this shallow layer.
Many processes in the ocean surface boundary layer (OSBL) vary with the diurnal cycle of solar heating and nighttime cooling. The most frequently studied is that of sea surface temperature (SST), which is most pronounced under conditions of low winds and high solar insolation (e.g., Ward 2006). The diurnal variability of SST can be several degrees Celsius and has been shown to affect air–sea fluxes on climatological time scales in subtropical regions (Bernie et al. 2005; Kawai and Wada 2007; Clayson and Bogdanoff 2013). In addition to SST, diurnal variability has been observed in sea surface salinity (Drushka et al. 2014; Asher et al. 2014), momentum and shear (Weller and Plueddemann 1996; Plueddemann and Weller 1999; Cronin and Kessler 2009; Weller et al. 2014), and biogeochemical tracers such as dissolved oxygen and chlorophyll (Nicholson et al. 2015). In comparison, the diurnal variability of turbulence has received far less attention, although it is a key physical component the OSBL.
Diurnal restratification has been observed to reduce the dissipation rate of turbulent kinetic energy (ε; see section 3c for definition) below the restratification layer, leaving behind a remnant layer that is isolated from the surface wind stress (Brainerd and Gregg 1993; Caldwell et al. 1997; Callaghan et al. 2014). Restratification limits the vertical diffusion of the surface wind stress and thus causes the momentum flux to be focused to a shallower layer, causing it to slide with minimal friction (Kudryavtsev and Soloviev 1990). The magnitude of the diurnal jet is typically O(10) cm s−1 and extends to a depth of a few meters (Price et al. 1986; Woods and Strass 1986). Kelvin–Helmholtz (KH) instabilities have been observed to form from the enhanced shear due to the diurnal jet and may be a source of active turbulence (Woods 1968; Soloviev and Lukas 2014).
While there exist many studies that have investigated the response of ε to a destabilizing buoyancy flux (Shay and Gregg 1986; Lombardo and Gregg 1989) and below the diurnal jet (Lombardo and Gregg 1989; Brainerd and Gregg 1993; Caldwell et al. 1997), there have been no studies that the authors are aware of that have investigated the variability of ε within the diurnal jet. If shear instabilities are common, as suggested by Woods (1968), this would provide a source of turbulent kinetic energy (TKE), which would enhance air–sea transfer of moderately soluble gases, such as CO2 and O2 (Lamont and Scott 1970; Jähne and Haußecker 1998; McGillis et al. 2004; Zappa et al. 2007), relative to wind-based parameterizations. Experiments by Khoo and Sonin (1992) demonstrated an appreciable increase in the mass transfer of CO2 due to an increase in turbulence on the water side with zero interfacial shear or wave motion. As diurnal cycles of SST occur predominantly in low and midlatitudes, regions with a positive δpCO2 (the partial pressure of the CO2 in the air minus the partial pressure of CO2 in the water) and hence a source of CO2 to the atmosphere (Takahashi et al. 2009), enhanced turbulence associated with the diurnal jet would act to increase outgassing in these regions.
The diurnal jet in the ocean is analogous to that of the nocturnal jet in the atmosphere, also known as the low-level jet, as both consist of inertial rotations due to a diurnally varying buoyancy flux. However, there are a few key differences that could have large consequences for the dynamics. The nocturnal jet in the atmosphere is generated above the frictional stress layer during nocturnal cooling of the surface layer. This leads to an inertial oscillation generated due to frictional decoupling of the geostrophic forcing (Gill 1982). This mechanism suggests that the low-level jet only requires a nocturnal inversion to occur and will be independent of the wind and the nocturnal boundary layer depth. The diurnal jet in the ocean, on the other hand, is different in that it exists within the stress layer. Therefore, it is expected that the diurnal jet should exist as a wind-driven current that will rotate with the local inertial frequency (Price et al. 1986) and not as a free inertial oscillation (Gill 1982), although inertial oscillations could theoretically be generated from the diurnal variability in the OSBL thickness (D’Asaro 1985).
The motivation for this paper is to investigate the turbulent dynamics within the diurnal jet. What is the effect of the diurnal on the near-surface turbulence? Does the increase in shear due to the diurnal jet increase production of turbulent kinetic energy or does gravity limit the vertical diffusion of the Reynolds stress? Is the diurnal jet susceptible to shear instability? This paper will attempt to address some of these questions with regards to the diurnal increase in stratification and the diurnal jet.
The outline for the paper is as follows: Section 2 provides an overview of the diurnal processes in the OSBL. Data and methods are presented in section 3. Section 4 shows the diurnal variability of temperature and velocity over the duration of the experiment. The diurnal structure of the observations was investigated using a composite day, which is calculated from phase averaging the forcing and ocean response as a function of the local time of day in section 5. An investigation of the relevant depth scale along with comparisons of previous results is shown in section 6. Section 7 presents statistics of the Richardson number, and section 8 investigates the large values of ε near the surface. A summary and discussion of the results are presented in section 9.
2. Overview of diurnal evolution of ε
The diurnal evolution of ε in the OSBL is primarily controlled by the varying surface buoyancy flux of solar heating during the day and surface cooling during the night (Price et al. 1986). The convective cycle during the night has been well described by similarity theory (Shay and Gregg 1986; Lombardo and Gregg 1989), while the details of the stable daytime stratification regime are still relatively unknown due to the lack of observations in these shallow stable layers (Soloviev and Lukas 2014). Presented in this section is a short summary of the diurnal cycle outlining some of the larger gaps in the current understanding.
The OSBL is highly turbulent with the energetics being quantified by the time evolution of TKE. Assuming the flow is horizontally homogeneous, the vertical structure of TKE in the OSBL can be written as (Belcher et al. 2012)
where q is the TKE, is the friction velocity in water, U is the mean horizontal current, is the Stokes drift, is the density of seawater, and , , and are turbulent fluctuations in buoyancy, pressure, and velocity, respectively, where is the vertical turbulent velocity. The buoyancy fluctuation is defined as . The time evolution of TKE that can be seen in (1) is the sum of the (a) = mean shear production, (b) = production from the Stokes shear, (c) = buoyancy flux, (d) = transport of TKE and (e) = the dissipation rate of TKE due to viscosity. In (1), terms (a) and (b) are sources of TKE, (c) can be a source or sink depending on the surface buoyancy flux and the amount of mixing, (d) redistributes TKE and is neither a source or a sink, and (e) is a sink for TKE. In the constant stress layer near the ocean surface, , where τ is the surface wind stress, and , where is the surface buoyancy flux.
In the OSBL, TKE is generally considered to be in a quasi-steady state (Osborn 1980) with ε being balanced by the sum of shear production and buoyancy fluxes. The length scale at which shear production, assuming a logarithmic velocity profile, and buoyancy are balanced is known as the Monin–Obukhov length (Gill 1982), defined as
where κ = 0.4 is the von Kármán constant, and B(z) is the depth-dependent buoyancy flux to an OSBL thickness of z and is defined as
where BR(z) is the depth-dependent shortwave component of the buoyancy flux, and Bt is the turbulent component that includes latent and sensible heat and infrared radiation, as this does not penetrate greater than 1 cm in the ocean (Soloviev and Lukas 2014).
A schematic of the diurnal cycle is depicted in Fig. 1. For depths less than LMO, shear turbulence dominates buoyancy forces, and this region is denoted by the turbulent near-surface region in Fig. 1. Cooling at night causes the surface layer to be gravitationally unstable, generating convection cells that extend to the seasonal pycnocline, creating a well-mixed OSBL (Shay and Gregg 1986). Ocean convection has been well studied, and it has been shown that ε ∝ B0 below the near-surface turbulent region (Shay and Gregg 1986).
In the morning, the rising sun provides a stabilizing buoyancy flux into the OSBL. This buoyancy flux varies with depth (Paulson and Simpson 1977) and, under favorable wind conditions, the near surface of the ocean will begin to restratify. The depth to which significant heating occurs is expected to be related to LMO (Kudryavtsev and Soloviev 1990; Large et al. 1994). Below this restratification depth, a remnant layer exists that is isolated from surface wind forcing (Brainerd and Gregg 1993). A decrease in ε is observed in this remnant layer with an exponential decay in time (Brainerd and Gregg 1993; Caldwell et al. 1997; Callaghan et al. 2014).
Within the OSBL, ε can be modeled as a function of the friction velocity and surface buoyancy flux B0 (Lombardo and Gregg 1989), which we will denote , where
and a and b are constants. Typical ranges for the constants a and b are 0.5 < a < 2 and 0.3 < b < 2 (Shay and Gregg 1986; Lombardo and Gregg 1989; Brainerd and Gregg 1993; Caldwell et al. 1997; Callaghan et al. 2014). Equation (4) states that ε0 will be greater by an offset proportional to B0 relative to shear production during convection and that ε will not be affected by the buoyancy flux during restratification. Equation (4) only applies to the constant stress layer, which can be very shallow during restratification, and may not apply to the remnant layer (Callaghan et al. 2014). Measuring the depth dependence of the Reynolds stress in the OSBL is difficult due to the spectral overlap of turbulent motions and surface wave orbital motion (D’Asaro 2014). In general, the constant stress layer depth is assumed to be the same as the mixed layer depth, but there is little evidence for this, especially for variability on short time scales (Sutherland et al. 2014b). There have been some efforts to define the OSBL using ε, rather than based on density homogeneity, as this should scale more closely with , with some promising results for high-resolution studies (Brainerd and Gregg 1995; Sutherland et al. 2014a,b).
Equation (4) only accounts for surface forcing due to the wind stress and surface buoyancy flux and does not account for other processes that could affect ε such as breaking waves (Craig and Banner 1994), Langmuir circulations (McWilliams et al. 1997; Belcher et al. 2012), shear instabilities (Woods 1968), horizontal processes associated with ocean fronts (D’Asaro et al. 2011), or breaking internal waves (Wain et al. 2015), to name a few. These processes all generate TKE, which would lead to an underestimation of ε relative to (4).
3. Data and methods
Observations of the OSBL were obtained during the Subtropical Atlantic Surface Salinity Experiment (STRASSE) aboard the Navire Océanographique (N/O) Thalassa (Reverdin et al. 2015). This experiment took place during August and September 2012 as part of the larger Salinity Processes in the Upper Ocean Regional Study (SPURS) project. The location of the experiment is shown in Fig. 2. The latitude of the experiment site was about 25.6°N, corresponding to an inertial period of 27.7 h.
a. Meteorological observations
Radiative fluxes and wind speed measurements were recorded aboard the N/O Thalassa. The wind stress and buoyancy flux were calculated using the TOGA COARE 3.0 algorithm (Fairall et al. 2003). Figure 3a shows the 10-m wind speed , direction , and . The wind speed varied between 2 and 10 m s−1 with directions ranging between predominantly from the north to easterly. The surface buoyancy flux followed a typical pattern of surface cooling at night and heating during the day.
b. Wave and current observations
Observations of surface gravity waves and velocity profiles were made with a cloverleaf buoy, the Trèfle, equipped with a downward-facing Teledyne RD Instruments 300-kHz acoustic Doppler current profiler (ADCP), an MTi-G global positioning system (GPS)–motion sensor package manufactured by Xsens, and a Nortek Vector velocimeter mounted at 0.5-m depth to obtain the near-surface velocity. The buoy was tethered to a surface velocity program (SVP) drifter with a 50-m drogue in order to reduce its windage-induced drift. A custom datalogger performed collection and consistent time stamping of the three data streams. The ADCP profiled from 3.5 to 78.5 m with a 1-m depth resolution and an effective range of roughly 75 m.
Wave motions were quantified using the xSens MTI-G GPS motion sensor package, which consists of a GPS and a 6 degrees of freedom inertial motion unit (IMU) sampled at 10 Hz. The GPS and IMU compose an attitude heading reference system (AHRS) that utilizes a Kalman filter to combine the inertial motions and GPS position. The vertical motion was subsequently bandpass filtered using a fourth-order Chebyshev type-II filter with a 40-dB stop band ripple and cutoff frequencies 0.025 and 3 Hz, applied forwards and then backward to eliminate phase delay, resulting in an eighth-order, 80-dB ripple filter. One-dimensional wave spectra were calculated every 30 min from the AHRS heave. The calculated significant wave height , peak frequency , and mean frequency are shown in Fig. 3b.
The Stokes drift as a function of depth can be calculated from the 1D wave spectrum (Kenyon 1969) as
where σ is the frequency, S(σ) is the 1D wave spectrum, and and are the cutoff frequencies chosen as 0.05 and 0.50 Hz, respectively. There are two opposing uncertainties with calculating the Stokes velocity from a 1D wave spectrum using a finite frequency range: the lack of a measured high-frequency spectral tail and the lack of the directional spreading of the wave energy. The former will lead to a systematic underestimate of Us(0) up to 30% (Rascle et al. 2006), with the exact amount dependent on the slope of the spectral tail, while the latter leads to a systematic overestimate of Us(0) up to 30% (Webb and Fox-Kemper 2011), dependent on the directional spread of wave energy. Lacking measurements of both the high-frequency component of the wave spectrum and directional information, we assume that the effects on the Stokes drift should approximately cancel each other. This assumption has been used in previous studies (Gargett and Grosch 2014; Sutherland et al. 2014a) and has the advantage of estimating Us(0) only from observations and not artificially adding a high-frequency component to the wave spectrum.
The surface Stokes velocity was used in conjunction with to calculate the turbulent Langmuir number , which is used as a proxy for turbulence associated with Langmuir circulations (McWilliams et al. 1997). Figure 3c shows the time series for Us(0), , and . Langmuir turbulence is expected to occur when < 0.3 (McWilliams et al. 1997), which is not fulfilled during our observations. We therefore neglect Langmuir circulations as a dominant mechanism for turbulence in our observations.
Velocities in the OSBL are presented relative to a reference depth of z0 = 40 m located just below the seasonal mixed layer, that is,
where the 0 subscript denotes the velocity at the reference depth. Figure 4 shows the mean currents at 40 m along with the velocity anomalies relative to this current. The orthogonal velocity components are arranged in an along-wind U and across-wind V component, where the V axis is 90° clockwise to U. Enhanced near-surface shear is observed every day, consistent with the diurnal jet phenomenon described by Price et al. (1986).
c. Microstructure measurements
Measurements of the turbulent dissipation rate, temperature, and salinity were obtained with the Air–Sea Interaction Profiler (ASIP; Ward et al. 2014). ASIP is an autonomous, upwardly rising microstructure profiler designed to sample the OSBL. A total of 347 profiles from 40-m depth to the surface were obtained over five ASIP deployments. Each deployment ranged from 24 to 48 h in duration with the exception of the fifth deployment, which was only 10 h. The average time between successive profiles during each deployment was about 20 min.
The time–depth evolution of temperature T and buoyancy frequency N2 are shown in Fig. 5. The density ratio in the OSBL, defined by , where α and β are the thermal expansion coefficient and the saline contraction coefficient, respectively, and ∂T/∂z and ∂S/∂z are the vertical gradients of temperature and salinity, respectively, was typically greater than 2, indicative of temperature controlled stratification. In addition, the conductivity signal, from which salinity is calculated, in the upper 10 m was sometimes contaminated from the impact of near-surface detritus striking the conductivity probe, which acted to increase the noise in the measured salinity. Therefore, only the contribution from ∂T/∂z was used to calculate N2 in the OSBL. Each profile of T was sorted to calculate the stable stratification and ensure N2 > 0.
Dissipation rates of ε were measured using two airfoil shear probes mounted on the front of ASIP (Sutherland et al. 2013). Assuming isotropic turbulence, ε can be calculated from profiles of the microstructure shear (Osborn 1974). The microstructure shear was sampled at 1000 Hz, and vertical profiles were divided into 1-s segments where the power spectral density was calculated using Welch’s method. The mean rise velocity of ASIP was 0.5 m s−1, thus giving a vertical resolution for ε of 0.5 m. Details of the processing algorithm for ε can be found in Ward et al. (2014). Figure 5e shows the measured turbulent dissipation rate for the five deployments. In addition, Fig. 5f shows ε normalized by ε0, calculated by (4), which takes into account turbulence generated from the mean shear and convective buoyancy forcing. The constants a and b in (4) are found to a = 0.5 and b = 0.3, which is consistent with previous results (Shay and Gregg 1986; Lombardo and Gregg 1989; Brainerd and Gregg 1993; Caldwell et al. 1997; Callaghan et al. 2014).
The mixed layer depth (MLD) was calculated using the temperature threshold of 0.2°C relative to the temperature at z = −0.5 m (Kara et al. 2000; de Boyer Montégut et al. 2004). The MLD over the five ASIP deployments is shown by the black–white dashed line in Fig. 5. The active mixing layer depth (XLD) is defined as the depth where ε decreases to an assumed background level and is no longer being influenced by surface forcing (Sutherland et al. 2014a). The XLD is calculated assuming a background dissipation of 10−9 m−2 s−3 (Sutherland et al. 2014b). Differences between the MLD and XLD have been shown to be important in scaling observations of ε with surface forcing (Brainerd and Gregg 1995; Stevens et al. 2011; Sutherland et al. 2014a). The XLD is indicated by the black–white dotted line in Fig. 5.
Mean values for the day and night profiles of temperature, current speed, ε, , and Ri = N2/S2 are shown in Fig. 6. The geometric mean is used for ε, N2, and S2. The mean daytime profile was averaged over 1400–1600 local mean time (LMT), calculated such that noon coincides with the peak solar altitude and the nighttime profile was averaged over 0200–0400 LMT. During the day there is an increase in the near-surface temperature coinciding with an increase in the near-surface velocity to a depth of approximately 8 m. There is an associated enhanced ε in this depth range, while ε decreases by nearly two orders of magnitudes below the shallow restratification depth.
The day and night probability density function (PDF) and cumulative distribution function (CDF) of ε and are shown in Fig. 7 for three different depth intervals from the surface to −12 m. For depths from −4 m to the surface (Fig. 7a), ε is greater during the day than during the night with a median value that is 3 times greater. For the next depth interval (Fig. 7), the day and night distributions are very similar, with the night having a slightly larger peak and the day having a greater likelihood for values greater than the median. Below −8 m (Fig. 7), ε is most likely to be less during the day than in the night. The PDF and CDF are also shown for (Figs. 7d–f), demonstrating that there is no day–night bias in the upper part of the ocean.
4. Diurnal variability of temperature and velocity
We define the diurnal variability of temperature and velocity in the same manner as Price et al. (1986), that is,
where the r subscript denotes a reference depth, which is chosen to be deep enough to avoid the diurnal signal and shallow enough to not include the seasonal pycnocline. From visual inspection of Fig. 5, a reference depth of zr = −25 m is chosen. A median point filter with a sliding window of 2 h was applied to remove any spikes associated with the seasonal pycnocline contaminating the reference temperature.
Figure 8 shows the evolution of , , and , where is the surface temperature at a depth zs = −0.5 m. The diurnal cycling of upper-ocean temperature is observed to increase in the near-surface temperature by 0.2°–0.5°C during the day. Figure 8a also shows the difficulty in determining a single reference depth as the seasonal pycnocline appears to contaminate on 08/31 and 09/07. The diurnal cycle of is fairly typical over the ASIP deployments with shallow anomalies extending to deeper depths during the day until they are suddenly exterminated by the change to a destabilizing surface buoyancy flux.
Figure 9 shows the diurnal velocity variability over the three Trèfle deployments. The velocity anomalies begin in the along-wind direction in the morning and turn with the local inertial frequency during the day. The velocity anomaly has a similar pattern as the temperature anomaly in Fig. 8 where the anomaly begins in a shallow layer during the morning, descending in the afternoon, and is erased when the surface buoyancy flux becomes destabilizing.
The diurnal velocity response is more clearly seen in the surface velocity, which is averaged over the upper 5 m, as seen in Fig. 10. The magnitude of the diurnal jet is consistently observed to be approximately 0.15 m s−1. This near-uniform diurnal jet magnitude is consistent with the observations of Price et al. (1986), who found that the magnitude should only be dependent on the shortwave radiation and independent of the wind stress. However, there is day to day variability with regards to the timing of the peak of the diurnal jet in addition to the evolution of the along- and crosswind components (Fig. 10).
5. Composite day
Wind variability predominantly occurred over time scales greater than the diurnal period (Fig. 3), allowing for the creation of a single composite day by phase averaging the forcing and response components as functions of the local time of day. The composite day allows for a detailed analysis of the mean diurnal response of the ocean while filtering out processes that were not phase coherent within the diurnal cycle. This method has proven useful in interpreting the mean diurnal response in regions with strong buoyancy forcing (Caldwell et al. 1997; Smyth et al. 2013; Drushka et al. 2014; Sutherland et al. 2014b).
Figure 11 shows the composite day for the surface forcing and ocean response. The phase averaging was performed with a temporal resolution of 1 h and a depth resolution of 1 m. The phase-averaged surface buoyancy flux B0 and wind stress τ are shown in Fig. 11a. The surface buoyancy flux had a clear diurnal structure with little observed variation. Although the wind speed varied from 2 to 10 m s−1 over all the deployments (Fig. 3a), there was no clear diurnal structure in τ around the mean value of 0.06 N m−1 (Fig. 11a).
The near-surface stratification N2 (Fig. 11b) and shear squared S2 (Fig. 11c) were both observed to increase after B0 became stabilizing, with N2 lagging S2 by a few hours. Both N2 and S2 reached near-surface maxima at close to 1400 LMT after which these maxima descended to greater depths. In the afternoon as the depth of the OSBL increased, the magnitude of N2 and S2 both decreased.
As an indicator for shear stability, the gradient Richardson number was calculated:
where N2 and S2 are composite day values, similar to the composite Ri presented by Smyth et al. (2013). When Ri falls below a critical value of 0.25, the flow may become unstable (Miles 1961). Figure 11d shows the logarithm of Ri centered on log100.25 = −0.60, such that the red values correspond to where shear instability was likely to occur.
6. Restratification length scale
Price et al. (1986) introduced two diagnostic depth scales associated with restratification, which they called the trapping depth DT and penetration depth DP. The trapping depth is defined as the mean depth of the temperature anomaly in the OSBL, where the temperature anomaly is defined in (8).
The trapping depth is calculated from the vertical integral of (8), that is,
where zs is the surface depth, which we take to be zs = −0.25 m, and . Equation (12) incorporates the diurnal heat anomaly normalized by the surface temperature.
The penetration depth, on the other hand, is the depth scale in the one-dimensional heat equation, that is,
where cp is the specific heat of seawater, and Q0 is the surface heat flux, which can be related to the surface buoyancy flux , and cp is the specific heat of seawater. Price et al. (1986) calculated the penetration depth to be
The penetration depth can be interpreted as the depth to which mixing from surface forcing is present, assumed by Price et al. (1986) to be entirely derived from the wind.
The trapping and penetration depths were also calculated from the composite day values of T and Q, as shown in Fig. 12. Before 1200 LMT, the minimum DP was very similar to the minimum DT, suggesting that heat and momentum are mixed in the same manner, consistent with Price et al. (1986). At 1200 LMT, DT = DP = 10 m, which is larger than LMO but similar in magnitude to the minimum MLD and XLD (Fig. 12).
Another depth scale associated with restratification is the prognostic depth scale of Price et al. (1986):
where is the time scale associated with the buoyancy flux, taken to be half the length of the duration of stabilizing buoyancy flux; Pτ = f−1[2 − 2 cos(fPQ)]1/2 is the acceleration time scale that accounts for rotation; and J is associated with the depth penetration of shortwave radiation. For the composite day there is good agreement between and the minimum values of , , and MLD (Fig. 12). The minimum XLD thickness is greater than by a factor of 2, which will ultimately be associated with how the XLD is defined. Better agreement between and the XLD is obtained if the background threshold is increased by a factor of 5 (not shown), but this threshold greatly underpredicts the boundary thickness during the night.
For mid to low latitudes where diurnal cycling is prominent, and can be written in terms of , that is,
Assuming that an external time scale is not necessary to determine allows for to be replaced by in (16), leading to the Monin–Obukhov scaling of . It is clear from Fig. 12 that and that the time scale associated with solar heating should be taken into account.
Figure 12 shows that not only underestimates the depth of the boundary layer, but it does not demonstrate any of the temporal variability as shown by the MLD and XLD. It appears that mean values of and alone are insufficient to calculate the scale of the boundary layer depth, and a relevant time scale, such as proposed by Price et al. (1986), is necessary to obtain the correct magnitude.
7. Richardson number statistics
The gradient Richardson number, where Ri is defined by (11), is calculated for each profile and the statistical distribution as a function of local time of day is investigated. The vertical resolution of Ri is limited by the vertical resolution of the ADCP currents, which is 1 m. Figure 13a shows the temporal evolution of the peak of the PDF of Ri at various depth intervals. For depths z < −4 m, the PDF of Ri has a peak at 0.25, which is indicative that this region may be predominantly unstable. For greater depths, −4 m < z < −14 m, there is a diurnal structure to the peak of the Richardson number distribution, where Ri is lower during the day than in the evening. The timing of the decrease in Ri varies for each depth interval, with deeper observations decreasing later than shallower ones. Observations of Ri increase again at ≈1800 LMT, coinciding with becoming negative and destabilizing (Fig. 11).
Another important aspect of Ri is the fraction of occurrences where Ri < 0.25, which we will denote . This fraction is calculated at different depth intervals and is shown in Fig. 13b. For z < −4 m, there are a greater number of occurrences of Ri < 0.25 during the day when the diurnal jet is present compared to the same depths at night. This peak is even more apparent when the number of occurrences of Ri < 0.65 (denoted ) are considered, which may be more a more appropriate threshold, as this corresponds with the mode of the distribution of Ri (Fig. 13a) and our estimates are limited by the 1-m vertical resolution of velocity. Figure 13c shows the diurnal evolution of for various depths, which has the same diurnal shape as .
The evolution of stability can be seen from the diurnal evolution of (Fig. 13c). In the morning, the complex interplay of restratification and the diurnal jet act to increase the occurrence of , indicating a greater occurrence of shear instability. The rate and onset of this increase changes with depth with shallower depths increasing with a greater rate and at earlier times than at larger depths. At 1300 LMT, reaches a peak for the −6 m < z < −4 m depth interval, which suggests that the flow is unstable as the median of Ri drops below 0.65. Through shear instability, the flow entrains the underlying flow that acts to decrease in the shallow layer and increase in the deeper layer. This creates a cascade that appears identical to the mechanism described by Smyth et al. (2013), which provides the initial forcing for deep cycle turbulence in the equatorial Pacific. While our Richardson threshold of 0.65 is somewhat arbitrary and larger than the accepted value of 0.25, the fact that the median values of Ri < 1 for our relatively coarse velocity resolution suggests that shear instability is likely to be present.
This peak in and can be interpreted as a descent rate. For either critical Richardson number , the weighted mean time associated with the peak of is calculated as
to determine the descent rate for . The descent rate is then calculated as the change in depth of . The dashed black–white lines in Fig. 14 shows , which demonstrates a mean descent rate of 2.0 m h−1 for depths z > −12 m. The descent rate associated with was found to be 1.8 m h−1, which is nearly identical to that obtained with . This weighted mean time appears to be associated with the onset of enhanced , as shown in Fig. 14c. Both descent rates are less than the 6 m h−1 observed by Smyth et al. (2013) in the equatorial Pacific. It is uncertain as to the cause of the discrepancy in descent rates, but it might be attributed to the presence of the equatorial undercurrent or due to the inertial rotation of the diurnal jet at our latitude.
8. Enhanced shear = enhanced ε?
The region with the greater number of occurrences of Ri < 0.25 coincided with enhanced (Fig. 14c). The depth of enhanced ε was bounded temporally by and 1800 LMT and spatially by −10 m < z < 0 m, which is precisely where the diurnal jet occurred. The enhancement of ceased rather abruptly at 1800 LMT at all depths, coinciding with changing from stabilizing to destabilizing.
This region of enhanced also coincides with a region of enhanced mean shear (Fig. 11c), which would increase the production of turbulent kinetic energy through (1). The shear is enhanced by a factor of 5 in the descending diurnal jet relative to that expected of wall-bounded shear (Fig. 14a), which is consistent with the estimates of the near-surface shear made by Kudryavtsev and Soloviev (1990) and with our observed values of . Using the observed shear, we can define a new normalized dissipation rate:
where S is the observed shear [instead of assuming the shear follows the law of the wall scaling as in (4) and the same buoyancy offset is chosen when is destabilizing]. In (18), the friction velocity is scaled by to be consistent with (4). Observed values of ε are more consistent with during the day (Fig. 14d) than (Fig. 14c).
Assuming that the primary source for ε is from shear production, then yields an estimate for the in situ Reynolds stress normalized by the surface stress from the wind. Figure 14d shows for the composite day, and it appears that the enhancement in is primarily due to the enhanced shear associated with the diurnal jet. During restratification (and before the onset of shear instability that is shown by the dashed line in Fig. 14), the Reynolds stress, relative to the wind stress, extends to a uniform shallow depth of a few meters and quickly decreases by an order of magnitude as stratification limits the vertical diffusion. However, the peak corresponds with an increase in before the stratification begins to decrease (Fig. 14b), suggesting that this change is not coming from an increase in the vertical diffusivity but instead is a local source of turbulence. This local source is consistent with shear instability transferring energy from the mean shear into turbulent motions. This increase in corresponds with , suggesting that the descending shear layer is leaving turbulence and instabilities in its wake.
9. Summary and discussion
Presented here are observations of the diurnal structure of the ocean surface boundary layer (OSBL) obtained in the subtropical North Atlantic during August/September 2012. High-resolution observations in the OSBL were made with the Air–Sea Interaction Profiler (ASIP), an autonomous vertical microstructure profiler that profiles upward to the free surface. The fact that ASIP is autonomous and vertically rising allowed for accurate microstructure observations far from the effects of any ship-induced contamination to the near-surface turbulent processes. ASIP was used in conjunction with a cloverleaf buoy (the Trèfle) that resolved velocities in the near-surface region. This pairing of instruments allowed for coincident velocity and microstructure measurements of the diurnal jet in the OSBL.
During the campaign, several diurnal warming events were observed due to the low wind speeds and high solar insolation. These diurnal warming events followed a typical pattern with a daytime increase in SST of 0.2° to 0.5°C occurring at close to 1400 local mean time (LMT). The increase in SST was accompanied by an increase in near-surface velocity, that is, a diurnal jet, which began in the along-wind direction each morning and slowly turned cum sole due to the Coriolis force. The magnitude of the diurnal jet was observed to be on the order of 0.1 m s−1, consistent with previous observations near this latitude (Price et al. 1986).
Several length scales were investigated associated with the diurnal cycle of SST. The classic Monin–Obukhov length underestimated the boundary layer thickness compared to several different diagnostic length scales. This could be due to the TKE shear being underestimated by the law of the wall model or that never reaches a steady-state due to the constantly changing surface buoyancy flux during the day. The prognostic depth scale of (Price et al. 1986) , where is a relevant time scale taken to be half of the total daylight hours, was able to predict the minimum boundary layer thickness during the day.
The diurnal jet enhances the near-surface shear relative to that expected for a logarithmic velocity profile. One of the main results of this paper is that the increased shear acts to increase ε, presumably through the production of TKE from the interaction of the Reynolds stress and the mean shear. This process is juxtaposed with the increased stratification acting to reduce the vertical diffusion of the Reynolds stress. Taking into account the observed shear, which is approximately 5 times greater than expected for the observed surface stress, leads to a turbulence scaling (which we denote ), which is more consistent with observations. Assuming that shear production is the predominant source of ε, then the ratio can be interpreted as the Reynolds stress normalized by the surface wind stress. Figure 14d shows that the ratio gives a rapid decrease in the Reynolds stress in the remnant layer during restratification. The depth to which the Reynolds stress falls to 10% of the surface wind stress is nearly constant until the early afternoon, where it increases at depth coinciding with a greater occurrence of Ri < 0.25.
This interpretation, that is, that the reduction in ε is due to a decrease in the Reynolds stress, is slightly different to the conclusions of Smyth et al. (1997), who attributed the decay of ε below a stratified layer as a result of a decrease in the vertical advection of TKE, while shear production persisted in the remnant layer albeit cut off from surface forcing. The interpretation of Smyth et al. (1997) was based on a lack of correlation between ε, N2, S2, and Ri within the remnant layer, and therefore a decrease in shear production was deemed unlikely. However, as we have shown, the stratification above the remnant layer limits the diffusion of the Reynolds stress, and it is this that is separated from the surface. The observed exponential decay of ε cut off from surface forcing (Brainerd and Gregg 1993; Smyth et al. 1997; Callaghan et al. 2014) is also consistent with a decrease in the Reynolds stress, as ε is proportional to the Reynolds stress, rather than a reduction in vertical advection of TKE.
The diurnal jet is also accompanied by a greater occurrence of Ri < 0.25, which is indicative of shear instability. While this increased occurrence is limited to 10% to 20% of the observed values of Ri in the upper 10 m, the 1-m vertical resolution of the ADCP could lead to an underestimate of the shear and hence an overestimate of Ri. It does appear that the conditions are favorable for shear instability, but higher-resolution velocity measurements would be required to confirm this. While the enhanced values of ε appear to be associated with the enhanced shear of the diurnal jet, it may be that shear instability may prove to be an integral part of the enhanced turbulence as it will generate turbulence in addition to the wind stress as the stratification does appear to limit the vertical diffusion of momentum. This statement is at least qualitatively consistent with Fig. 14d, where for z > −3 m during the morning, yet increases with depth in the afternoon, coinciding with the greater observations of Ri < 0.65.
One aspect of the diurnal jet that we were unable to clearly address is whether any memory of previous diurnal events exists, as suspected by Woods and Strass (1986), or whether the jet is formed onto a “clean slate” as hypothesized by Stommel et al. (1969). Figure 10 shows a very different evolution for the diurnal jet during each day. Deployment three in particular shows the diurnal jet on 09/07 to have characteristics of an inertial oscillation, while the next day has a very strange response with little along-wind increase. This interaction between the inertial and diurnal response can also be seen in the currents throughout the OSBL (Fig. 4), but a much longer time series than this one would be required to investigate this interaction.
These observations present new insight into the complicated processes associated with shallow, stable boundary layers in the ocean. During conditions of low wind and high solar insolation an interesting feedback mechanism occurs, which acts to increase the turbulence intensity relative to the wind forcing. The increased stratification creates a delicate balance between the increased near-surface shear and the limited vertical extent of the constant stress layer to create slightly enhanced values of ε relative to turbulence in neutral buoyancy conditions. This process could be an important component to the air–sea transfer of momentum, heat, and trace gases in tropical regions as the diurnal jet is expected to occur wherever a diurnal temperature variability is observed. The enhanced ε relative to wind forcing is expected to have an impact on estimates of the air–sea transfer of water soluble gases, such as CO2 and O2 (Lamont and Scott 1970; Ward et al. 2004; McGillis et al. 2004; Zappa et al. 2007), as a wind stress parameterization would underestimate the gas transfer velocity in regions with large diurnal variability. This enhanced turbulence would lead to an increase in the flux of CO2 from the sea to the atmosphere as diurnal cycling predominantly occurs in tropical and subtropical regions where the partial pressure of CO2 is greater in the sea than in the atmosphere.
The authors are grateful for funding support from the National Science and Engineering Research Council of Canada from the scholarship PGSD3-410251-2011, the Office of Naval Research under Award N62909-14-1-N296, and the Norwegian Research Council under PETROMAKS2 233901 Project. LM thanks Olivier Peden and Olivier Ménage for their help with the Trèfle and A. Marsouin and P. Le Borgne from Météo-France for help in processing the radiative fluxes data. The STRASSE cruise and data analysis was supported by LEFE/IMAGO and CNES/TOSCA grants. The authors thank Bill Smyth and one anonymous reviewer for their helpful comments, which greatly improved the manuscript.