1. Introduction
The cold tongue of the eastern equatorial Pacific is a region of intense heat uptake by the ocean. To understand the effect of this heat on sea surface temperature, and therefore on weather and climate, we must understand the turbulent processes that distribute the heat through the water column. Here, we examine seasonal variations of turbulence using a combination of moored observations and large-eddy simulations (LES).
Near the equator, a vertically sheared current structure is maintained by easterly trade winds. These tend to accelerate a westward flow at the surface, which in turn generates a pressure gradient that drives a return flow at depth: the Equatorial Undercurrent (EUC; see Fig. 1a). Stratification is maintained by the combined action of solar heating near the surface and upwelling of cold water at depth, resulting in a pycnocline that coincides roughly with the EUC (Fig. 1b).
Marginal instability and turbulence during 7 days in boreal fall 2008 at 0°, 140°W. (a) Zonal velocity profile showing the core of the EUC. (b) Squared buoyancy frequency. (c) Time-depth dependence of the turbulent kinetic energy dissipation rate ε (Moum and Rippeth 2009) over the same period. Solid (dashed) curve is the base of the DML, defined by a density change of 0.01 kg m−3 (temperature change of 0.04 K) from the surface. (d) Richardson number computed from hourly averaged currents and stratification, with dashed line indicating Ri = 1/4. In (a), (b), and (d), the thick curve is the median at each depth; the band indicates the upper and lower quartiles.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
The diurnal mixed layer (DML; solid curve in Fig. 1c) is the layer adjacent to the surface, which is nearly homogeneous, so that vertical motions are unhindered by gravity (e.g., Moum et al. 1989). As a result, the DML is strongly turbulent, and changes in watermass properties due to surface fluxes are therefore communicated rapidly throughout its vertical extent. Likewise, property changes at the base of the DML are quickly communicated to the surface, with important consequences for sea surface temperature.
A surprising discovery in early microstructure surveys at the equator (Moum and Caldwell 1985; Gregg et al. 1985) was the existence of turbulence in the stably stratified fluid below the DML, often occupying a layer many times thicker than the DML itself (Fig. 1c, where red indicates a large energy dissipation rate, that is, a rapid downscale cascade of turbulent kinetic energy). The diurnal fluctuation of this turbulence suggests that it is triggered by surface fluxes, even though it is found deep below the DML. For this reason it is called the deep cycle (DC). More recently, observations in the equatorial Atlantic cold tongue have shown a similar enhancement of turbulence below the DML (Hummels et al. 2013) as well as a diurnal cycle (M. Dengler 2017, personal communication), indicating that the process is active in both oceans.
Because it coincides with strong thermal stratification, DC turbulence has the potential to drive an intense heat flux (up to 400 W m−2 in the example shown in Fig. 1; see Moum et al. 2009), thereby establishing a thermal link between the DML and the ocean interior. On the seasonal time scale, for example, turbulence exhibits a clear cycle that is causally connected with the seasonal cycle of sea surface temperature (Moum et al. 2013).
The details of DC turbulence in the different seasons are not well established because the shipboard observations that provide the closest look have mostly been conducted in boreal fall (e.g., Fig. 1). A single exception was the Tropic Heat 2 experiment (Moum et al. 1992), conducted in April 1987. DC turbulence was observed but reached only to 50-m depth. Smyth and Moum (2013) examined a decade of χpod microstructure records between the 45- and 70-m depths and found that turbulence, present through most of the year, disappeared almost entirely in boreal spring. In this study, we explore the seasonal cycle of DC turbulence and the factors that control it, using a decade-long time series of moored observations of currents, temperature, and air–sea fluxes, together with LES of the upper ocean. A particular goal is to better understand changes in the deep cycle in boreal spring.
In section 2, we review what is known of the mechanics of DC turbulence, focusing on the role of the marginal instability state (MI; Thorpe and Liu 2009; Smyth et al. 2013; Pham et al. 2013) in a stratified, parallel shear flow. We then examine the diurnal cycles of the surface fluxes and of the mean shear and stratification that affect the DC (section 3). We make use of observational data from the Tropical Atmosphere Ocean (TAO; McPhaden 1995) mooring array and the TropFlux reanalysis (Kumar et al. 2012). Because the mooring data do not show us the turbulence directly, and because χpods do not show details of the vertical structure (particularly in the uppermost 30 m), we perform LES at four points in the seasonal cycle, using the observations to define surface forcing and initial conditions (section 4). The results provide new insights into the temporal and vertical structure of DC turbulence throughout the year. Conclusions are summarized in section 5.
2. Marginal instability and the mechanics of deep cycle turbulence
A central parameter governing turbulence in a stratified shear flow is the gradient Richardson number Ri = N2/S2, where N2 is the squared Brunt–Väisälä frequency and S2 is the squared vertical shear of the mean horizontal current. Linear stability analysis suggests that Ri < 1/4 is a necessary condition for instability growth in the geophysically relevant limit of high Reynolds number Re (Miles 1961; Hazel 1972; Maslowe and Thompson 1971; Smyth et al. 2013). Studies of turbulence in uniform shear and stratification indicate that regimes of turbulence growth and decay are separated by a critical value of Ri (Rohr et al. 1988; Jacobitz et al. 1997), which depends on Re and is consistent with an approach to 1/4 as Re → ∞.
A striking feature of the mean flow seen in Fig. 1d is that Ri remains near the value 1/4 in a layer extending from the base of the DML nearly to the core of the EUC (roughly −90 < z < −20 m), the layer where DC turbulence is found. This suggests that the flow subject to DC turbulence is in MI (Thorpe and Liu 2009; Smyth and Moum 2013; Pham et al. 2013). The net effect of the trade winds is to reduce Ri, leading to instability and turbulence. Countering this is turbulent mixing, which tends to mix the shear and stratification profiles so as to increase Ri, causing turbulence to decay. We therefore expect Ri to fluctuate about 1/4, as observed. A recent study (Smyth et al. 2017) explores in greater detail the interactions between mean shear, stratification, and turbulence during those fluctuations.
Early attempts to relate equatorial turbulence to Ri (e.g., Peters et al. 1988; Moum et al. 1989) revealed two distinct regimes: a weak turbulence regime in which Ri varied over many orders of magnitude and a strong turbulence regime in which Ri clusters around a value near 1/4 [see Fig. 14b of Peters et al. (1988); Fig. 15 of Moum et al. (1989)]. What was not clear at that time was that this marginally unstable regime is readily identifiable even when Ri is calculated from heavily averaged data, thus sidestepping the biasing problem that normally complicates Ri measurements. Although individual measurements of Ri are biased high by the averaging inherent in the measurements, values in the strong turbulence regime cluster near Ri = 1/4 even when averaging is severe. Smyth and Moum (2013) showed that the unimodal distribution of Ri with peak near 1/4 remains easy to distinguish even when the vertical bin size is increased from 2 to 16 m. In appendix A, we show that the same is true when the time averaging interval is increased from 1 h to 1 day. While our observations to date have focused on the equatorial Pacific, the MI regime has also been identified in the equatorial Atlantic (Wenegrat and McPhaden 2015).
MI is not unique to the equator. Similar Ri statistics have been observed in the most turbulent parts of the Mediterranean Outflow (Nash et al. 2012), the bottom-trapped current in the Romanche Fracture Zone (Van Haren et al. 2014), the Connecticut estuary (Geyer et al. 2010), and the Columbia River plume (Jurisa et al. 2016). These are all examples of stratified turbulence forced by a mean shear that varies slowly on the time scale of the instabilities, leading us to suggest that MI is a property of all such flows.
The deep cycle can be understood as a stratified shear layer in a state of marginal instability that is perturbed by some mechanism that varies diurnally. The identity of this diurnal “trigger” has long been a subject of debate. Recent evidence identifies the daytime surface current (Schudlich and Price 1992; Sutherland et al. 2016) as the trigger. During the day, solar radiation stratifies the uppermost few meters of the ocean, trapping the wind-driven turbulent momentum flux and allowing the acceleration of a strongly sheared surface current (Price et al. 1986; Kudryavtsev and Soloviev 1990). After local noon, the stabilizing influence of solar heating relaxes, and in late afternoon, the surface current becomes unstable. Momentum from the surface current spreads downward, enhancing the shear in the MI layer, reducing Ri and thereby triggering strong turbulence (Smyth et al. 2013; Smyth and Moum 2013; Pham et al. 2013).
The essential mechanism of turbulence resembles Kelvin–Helmholtz (KH) instability, as shown by stability analysis (e.g., Smyth et al. 2013) and by observations of ramps in the temperature signal (Moulin et al. 2016, manuscript submitted to J. Geophys. Res.). In addition, model results (Pham and Sarkar 2010; Pham et al. 2012) have shown KH billows near the base of the mixed layer triggering turbulence in the underlying fluid by the downward ejection of intensely nonlinear hairpin vortices.
Other mechanisms have been proposed. For example, at night in a convective mixed layer, internal waves can be generated by the impact of convective plumes on the mixed layer base, by corrugations of the mixed layer base interacting with the sheared current (Wijesekera and Dillon 1991; Wang and Muller 2002), or by an unstable shear layer overlying a stratified interior (Pham et al. 2009). These internal waves may break to generate turbulence. Energy can also be radiated by KH instability focused at the base of the mixed layer (Sun et al. 1998). Our simulations will show that none of these alternate mechanisms can account for the initiation of DC turbulence. It is possible, however, that these mechanisms are active later in the night, sustaining the deep cycle after the initial burst of turbulence from the descending daytime shear layer has dissipated.
In summary, DC turbulence appears to be controlled by a combination of local shear and stratification (combining to produce the MI state) and surface fluxes of momentum and buoyancy (tipping the MI flow into the unstable regime each evening). In the section to follow, we examine how these contributing factors vary diurnally and seasonally.
3. Processes affecting the deep cycle: Observed diurnal and seasonal variations
a. Data
Surface heat and momentum fluxes were obtained from the TropFlux reanalysis (Kumar et al. 2012). We use daily averages taken between 1 January 1979 and 31 March 2013. To obtain values for the equator at 140°W, data were averaged over latitudes ±0.5° and western longitudes of 139.5° and 140.5°.
Temperatures and currents were obtained from acoustic Doppler current profiles (ADCP), current meters and thermistors mounted on the TAO moorings at the equator at 140°W (McPhaden 1995). Daily, hourly, and 10-min-averaged data were used. To create an hourly averaged dataset, hourly and 10-min-averaged temperatures were combined into the same hourly bins used for the current data. The data used span the period from 11 May 1998 to 30 May 2007.
Daily temperatures and currents are available over the longer period 1 May 1990–26 November 2010. Although the DC is a diurnal phenomenon, we are able to infer much about it using this daily data (appendix A).
ADCP currents have vertical bin size 5 m. Current meters were located at 25, 10, and (over a 9.5-month period) 5-m depth. Near-surface currents not resolved by the ADCP were estimated using this current meter data, together with wind and stratification measurements as described in appendix B. Vertical gridding of the temperature was accomplished in two steps. First, temperatures were averaged into 5-m bins beginning at z = −2.5 m to eliminate spurious large gradients between closely spaced sensors. Temperatures were then interpolated onto the same 5-m vertical grid as the ADCP currents using Akima splines (Akima 1970).
b. Air–sea fluxes
The zonal wind stress is typically negative because of the easterly trades; hence, the vertical flux of zonal momentum is positive (Fig. 2a). Reversals are associated with the El Niño events of 1982 and 1997. The seasonal cycle (Fig. 2b) features a reduction by ~50% in boreal spring.
Air–sea interactions at 0°, 140°W for 1 Jan 1979–31 Mar 2013. All fluxes are positive upward. Dotted curves indicate the quartile range. Filled circles indicate values used in LES (section 4b). (a),(b) Zonal momentum flux (minus the wind stress), (c),(d) nonsolar fluxes (total, latent, longwave radiation, and sensible as indicated), (e),(f) downgoing solar radiation [in (f), curves show the solar flux as observed and as computed using the Lumb (1964) clear-sky parameterization], and (g),(h) cloudiness as calculated from (2).
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
The nonsolar part of the heat flux Qns, consisting of the longwave, latent, and sensible components, generally has a cooling effect on the equatorial oceans (Fig. 2d). Longwave radiation from the ocean (Fig. 2d, dashed) is nearly constant and is approximately 50 W m−2 at 0°, 140°W. The latent flux due to evaporative cooling of the sea surface (solid) is somewhat stronger than the longwave flux, while the sensible heat flux (dotted) is considerably weaker. Both latent and sensible fluxes are roughly proportional to wind speed and therefore decrease in boreal spring with the slackening of the trades.
The solar heat flux Qsw (Fig. 2e) is negative by definition and features prominent reductions in 1982, 1987, 1993, and 1997 associated with El Niño events. The solar flux is treated separately for two reasons. First, for the purpose of forcing the LES model, solar radiation penetrates the surface and therefore appears as a forcing term in the temperature equation, as opposed to the latent, longwave, and sensible fluxes that appear as boundary conditions. Second, the diurnal variation of the solar flux is critical to DC turbulence, and the long-term record includes only daily averages. To remedy this, we must make some more specific assumptions about the solar flux.
The clear-sky irradiance is calculated using the sun’s elevation, the known transmissivity of a dry atmosphere, and some trigonometry (Lumb 1964). Because of Earth’s axial tilt, clear-sky irradiance has a semiannual cycle with maxima at the spring and fall equinoxes (Fig. 2f, dashed curve). The observed solar irradiance at the surface shows a similar semiannual cycle, but the minimum in boreal summer is longer and more pronounced than its wintertime counterpart because cloud cover is greater (as discussed below).
The albedo of the sea surface is small (<5%) when the sun is high in the sky, but larger (up to ~40%) when the sun’s rays strike the ocean at a low angle. Here, we estimate a(t) using the empirical parameterization of Briegleb et al. (1986).
c. Mixed and marginally unstable layers
As discussed above, DC turbulence is identified as turbulence occurring below the base of the DML and varying on a diurnal time scale. We define the DML base z = −hDML(t) as the depth at which temperature first drops to 0.04 K below its surface value. We do not consider the effect of salinity because it is known to be small in this region (e.g., Smyth and Moum 2013), and salinity measurements are sparse. This definition of the DML in terms of temperature is approximately equivalent to the more common definition in terms of density, that is, a density difference of 0.01 kg m−3 from the surface (e.g., dashed and solid curves on Fig. 1). On a typical day (e.g., Fig. 1c), the DML is shallowest in midafternoon and deepest just before sunrise.
On the time scale of days or longer, it is reasonable to regard the DC as turbulence that exists below the deepest extent of the nocturnal DML, identified here as 
We expect to find DC turbulence in a layer below z = −hDML, where Ri is consistent with marginal instability. We therefore define the MI layer as extending from z = −hDML to z = −hMI, the latter being the deepest depth at which Ri < 1/4 (cf. Lien et al. 1995).
Both 
Medians and quartile ranges of (a) the daily maximum DML depth (solid curves, computed as in appendix A) and the MI layer base (dashed), and (b) the MI layer thickness (black). All are based on daily averaged data.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
d. Descent of the daytime surface current
To describe near-surface currents, we examine hourly averaged current meter data obtained at the TAO mooring at 0°, 140°W. A current meter was deployed at 5-m depth from 23 May 2004 to 1 March 2005. Current meters at 10- and 25-m depth operated for a longer period; here, we will use data from the 1-yr period 22 May 2004–21 May 2005. (While records used in the rest of this study are long enough that the diurnal and annual cycles they yield may be taken as generic, the results described in this subsection pertain only to a single year. There is no reason to think that this year is atypical, but other years may differ quantitatively.)
We begin by looking at the diurnal cycle using the shorter period, 3 May 2004 to 1 March 2005, during which all three instruments were operational. The zonal current was generally westward, consistent with the easterly trades, and decreased with depth; average values at 5, 10, and 25 m were −0.20, −0.17, and −0.08 m s−1, respectively.
To investigate diurnal variations in the zonal current, a composite day was created by subsampling the zonal current into 24 bins according to the local hour of the day and averaging each bin (Fig. 4). At 5-m depth, a diurnal cycle with amplitude 0.10 m s−1 (Fig. 4a, yellow curve) is evident. The current was weakest around 0700 local time (LT). It then increased steadily (i.e., became more negative) through the day, reaching a maximum at 1800 LT, then decreased throughout the night. This is similar to the diurnal surface current measured 2°N of our location by Cronin and Kessler (2009) during the same time period. The current at 10-m depth shows a similar cycle, but with reduced amplitude and with the fastest westward current occurring later in the evening (Fig. 4a, purple curve). At 25-m depth, the cycle is barely discernible (green).
Composite diurnal cycle of zonal current and shear measured at the TAO mooring at 0°, 140°W during 1 year beginning 22 May 2004. (a) Zonal current at 5-, 10-, and 25-m depth. (b) Average vertical shear of the zonal current computed by differencing adjacent sensors. (c) Average vertical shear between sensors at 5- and 10-m depth for four seasons: DJF = December, January and February (boreal winter), and so on. The mean value for each season has been subtracted. (d) As in (c), but for the shear between 10 and 25 m. Annotations give the range of the diurnal cycle.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
The shear measured between the 5- and 10-m instruments shows a distinct peak near 1600 (Fig. 4b, yellow curve). The shear between 10 and 25 m (purple) is weaker and peaks 2 h later. These observations are consistent with the scenario in which shear builds up during the day then, in late afternoon, becomes unstable and mixes downward, triggering DC turbulence. The 2-h time lag between these shears, which can be considered as averages over layers centered at 7.5 and 17.5 m, suggests a descent rate of 5 m h−1, comparable with the 6 m h−1 rate inferred from shipboard measurements during fall 2008 (Smyth et al. 2013).
To investigate the seasonal variation of this diurnal cycle, we subsample the data further into four 3-month intervals representing the seasons: boreal winter (DJF), spring (MAM), summer (JJA), and fall (SON). For this purpose we use data over the full yearlong period 22 May 2004 to 21 May 2005. As we will see later, the mean shear varies significantly with the seasons. Therefore, to facilitate comparison of the diurnal cycle, we define a shear anomaly by subtracting out the mean in each season.
Considering first the shear anomaly between 5 and 10 m (Fig. 4c), we see that the diurnal cycle followed a similar pattern throughout boreal summer, fall, and winter, albeit with slightly lower amplitude in summer (green curve). We cannot include boreal spring in this comparison because of the lack of measurements at 5 m during that season.
The shear anomaly between 10 and 25 m is well resolved in all four seasons (Fig. 4d). In this case, we find very similar results in boreal summer, fall, and winter (weakest, again, in summer), but in spring the cycle differs strongly, both in amplitude and in phase. As in the other seasons, the shear is weakest (least negative) around 0900 LT. However, the daytime acceleration is markedly slower. The maximum (most negative) shear does not occur until 2100 LT, about 2 h later than in the other seasons. This extra time is needed for the daytime stratification to decrease sufficiently that even this relatively weak shear becomes unstable. The subsequent relaxation of the shear is also delayed by about 2 h.
The net shear released after sunset (the difference between maximum and minimum values) is similar in boreal summer, fall, and winter but is smaller by about 1/3 in spring. We will suggest later that this reduction is partly responsible for the springtime reduction of deep cycle turbulence (Smyth and Moum 2013). A similar annual variation of the diurnal surface current has been identified in the equatorial Atlantic, coinciding with MI beneath the DML (Wenegrat and McPhaden 2015). This suggests that the same mechanism controls the Atlantic deep cycle (Hummels et al. 2013).
e. Currents and stratification from the surface to the EUC: Diurnal and seasonal variability
1) Diurnal variability
Currents above z = −30 m are not accurately resolved by the ADCP and are instead estimated using a combination of current meter data, wind speed, and near-surface stratification as described in appendix B. The analyses include every hour in which that estimate is possible, the main limitation being availability of data from the current meters at 10- and 25-m depth. These times range between 11 May 1998 and 30 May 2007, with occasional gaps. Hourly values are evenly distributed among the months, with a minimum of four full realizations for each month. We collect all values for a given month and a given hour of the day (at a given depth) and compute the median.2
Figure 5 shows the result for the month of October. The daytime surface current (Figs. 5a,b) accelerates between 1200 and 1800–2000 LT. The current is stabilized by strong stratification (Fig. 5c) because of penetrating solar radiation. Between 1500 and 2000 LT, Ri decreases (Fig. 5d) and the stratified surface current begins to spread downward. After 1800 LT, a well-mixed surface layer is formed, ultimately extending to about 20 m (Fig. 5c). This picture is consistent with the 7-day average of shipboard measurements described in Smyth et al. (2013) and with the inferred DC mechanism (section 2).
Composite diurnal cycle at 0°, 140°W for the month of October. (a) Zonal velocity, (b) squared shear, (c) squared buoyancy frequency, and (d) Ri. The squared shear is divided by 4 so that the value equals N2 when Ri = 1/4. Dotted curve indicates the descending shear maximum. The solid curve corresponds to the nightly maximum depth of the DML. Each value plotted is the median of hourly values subsampled by depth, month, and time of day. For visual clarity we apply a binomial smoothing filter in time.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
2) Seasonal variability
Seasonal variability is described using a composite year derived from daily averaged temperature, current, and wind measurements. The nightly maximum DML thickness 
Composite annual cycle at 0°, 140°W: (a) zonal velocity, (b) squared shear, (c) squared buoyancy frequency, and (d) Ri. Solid curves represent the monthly means of the daily minimum DML depth and the daily MI depth (the lowest level at which Ri < 1/4). Each value plotted is the median of daily values subsampled by depth and month. For visual clarity we apply a binomial smoothing filter in time.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
In boreal spring, the EUC strengthens and shoals (Fig. 6a), the result of seasonal Kelvin and Rossby waves (Yu and McPhaden 1999). Shear is strongest on the upper flank of the EUC, reaching a maximum in boreal spring and summer (Fig. 6b). During summer, the shear maximum descends along with the EUC core. The base of the MI layer coincides approximately with the shear maximum. The strongest stratification coincides with the EUC core (cf. Figs. 6c and 6a). Near the surface, N2 peaks in March and April, with values up to an order of magnitude greater than in the other months.
The median Ri, based on the daily averaged shear and stratification, is near or less than 1/4 in a layer extending from the DML base to 40–70 m below the surface. This MI layer is thinnest in boreal spring and summer, consistent with Fig. 3. Note, however, that the MI layer is present throughout the year. We must therefore revise Smyth and Moum’s (2013) conclusion, based on measurements in the fixed-depth range 45–75 m, that MI vanishes in boreal spring. What happens instead is that the MI layer shoals so that instruments located below 45 m do not detect it.
4. Seasonal variations of deep cycle turbulence via LES
We have seen that marginal instability, a property of DC turbulence, is present year-round. But does the DC actually occur in all seasons? Direct, year-round observations of DC turbulence are now becoming possible with the mounting of χ pods on the TAO moorings (e.g., Moum et al. 2009; Perlin and Moum 2012; Moum et al. 2013; Smyth and Moum 2013). These observations are highly resolved in time but are too sparse in the vertical to provide a comprehensive view of the DC. For example, Smyth and Moum (2013) described the annual cycle of DC turbulence in a fixed layer −75 ≤ z ≤ −45 m but did not describe annual changes in its vertical structure. To complement these ongoing observations, we have conducted LES of DC turbulence in conditions characteristic of the months of April, July, October, and January. Besides showing the time-depth behavior in detail, close examination of the model fields allows us to better understand the mechanics of the deep cycle.
a. Model formulation
Lateral boundary conditions are periodic, with dimensions determined by the geometry of the largest instability that is likely to arise. Observations in the region of interest show wavelike features with a distinct spectral peak with wavelength 150–300 m (Moum et al. 1992; Lien et al. 1996). These were identified by Sun et al. (1998), as resulting from a Kelvin–Helmholtz-like shear instability. Because the dominant spanwise features in shear instabilities tend to be much smaller in scale than streamwise features (e.g., Klaassen and Peltier 1985a), we are able to make the meridional domain scale considerably smaller. LES of turbulence in strongly stratified fluid is a challenge because of the difficulty of resolving the Ozmidov scale. In a comparison of turbulence statistics between LES and microstructure observations, Skyllingstad et al. (1999) used isotropic resolution with grid spacing of 1.5 m and found that turbulence levels in the stratified fluid below the DML base were seriously underpredicted.
Based on the above considerations, the computational domain is chosen to be 480 m by 80 m by 865 m in the zonal, meridional, and vertical directions, respectively, using 384 by 64 by 512 grid points. The grid spacing is 1.25 m in the horizontal directions. In the vertical direction, the grid spacing is uniformly equal to 0.25 m in the top 80 m. In the region below, the vertical grid spacing is stretched at a rate of 3% with the largest spacing equal to 19 m at 865-m depth. A sponge region is employed in the bottom 300 m.
The modeled flow is statistically homogeneous in the horizontal directions. The statistics that are computed by averaging in the horizontal x–y plane are functions of z and t.
b. Surface forcing and initial profiles
Surface boundary conditions are based on typical fluxes of momentum and heat observed at the sea surface. The momentum flux and nonsolar heat flux are constants equal to climatological monthly averages (Figs. 2b,d; Table 1). The diurnally varying solar heat flux, defined by (1), appears as a forcing term in the temperature equation [(7)]. The clear-sky irradiation and albedo are evaluated instantaneously as described in section 3. The cloudiness factor C is a constant equal to the climatological average for each month (Fig. 2h). Figure 7 shows an example of solar and nonsolar heat fluxes applied for the October case.
Surface heat flux for large-eddy simulations. Parameter choices represent typical October conditions.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
The zonal wind stress τw, cloud factor C, maximum value of solar heat flux Qsw at noon, and the nonsolar flux Qns are given in Table 1 for the four simulated cases. The model is initialized with midafternoon profiles of velocity and temperature representative of April, July, October, and January. The current is a combination of information from ADCP, current meters, wind, and DML thickness, as described in appendix B. Each profile is the average of all observed values between 1500 and 1600 LT for the specified month. Midafternoon initialization is chosen to make the turbulence spinup phase as natural as possible, since turbulence is typically weakest at that time of day. The zonal velocity is extrapolated in the region below 180-m depth so as to smoothly approach zero at 300 m and remains zero below that depth (in the sponge layer). Similarly, the temperature in this region is extrapolated assuming a uniform temperature gradient.
Parameters used in the simulated cases; Qsw,max indicates the maximum value of solar radiation at noon, and Qdaily is the daily averaged net heat flux that passes through the ocean surface.
c. Deep cycle turbulence
In the LES, the DC turbulence occurs in all four cases as indicated by the dissipation rate ε = 
Time-depth dependence of the dissipation rate ε, showing the shallowest deep cycle layer during the month of April. The black solid curves mark the (top) DML base and (bottom) the depth below which Ri > 1/4.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
1) Seasonal variations
The difference among the cases is found mainly in the nocturnal vertical extents of the DML and turbulent MI layer. The most robust distinction is between April and the other months. Both 
Another striking difference in the April case is the slow descent of the surface shear layer. In the July, October, and January cases, the base of the DML descends rapidly in the evening, then more slowly after midnight. In the April case, the time dependence of the DML base has a sawtooth character, descending at a slow, steady rate throughout the night. The difference is likely connected with the difference in the diurnal surface current. In April, the shear at the base of that current takes longer to go unstable, presumably because of the combination of weaker acceleration by the wind and stronger near-surface stratification (section 3d; Fig. 4d). The maximum daytime shear is also significantly weaker than in the other seasons (Fig. 12b).
Averaged over a day, dissipation profiles (Fig. 9) show a sharp increase at the surface (possibly affected by the absence of surface wave effects in the model), a slight reduction in the depth range of the DML (2–10 m), a maximum near z = 
Profiles of the temporally averaged dissipation rate for all four cases. The profiles are averaged from 1800 LT of day 1 to 1800 LT of day 2. The dots mark the maximum extent of the DML during the night. Dashed curves denote values in the DML, while solid curves are values in the MI layer.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
In terms of the maximum daily averaged value shown in Fig. 9, the January and July cases are nearly equal, while the April and October cases are smaller by a factor of 2. This model result may be compared with the observations of Smyth and Moum (2013, see their Fig. 3b), which showed that turbulence in the layer −75 < z < −45 m reaches a maximum in July–September and then decreases sharply in October.
The shallow deep cycle seen in April is consistent with Moum et al. (2013) and Smyth and Moum (2013). Moored microstructure (χpod) observations reported in those studies show that the dissipation between 30- and 75-m depth is weak during boreal spring, suggesting that the DC is absent during this period. Here, the April simulation shows that the DC exists during this time, but its vertical extent is so thin that it would be difficult to discern in the moored turbulence observations, which are spaced at about 10 m in the vertical beginning near 30-m depth (Moum et al. 2013). At 30 m (the depth of the shallowest χpod), the time-averaged dissipation rate is reduced by nearly an order of magnitude in April relative to the other months (Fig. 9), consistent with the χpod observations.
2) Mechanisms
The mechanism of deep cycle turbulence has been controversial, with local shear instabilities (Sun et al. 1998), breaking internal waves (Wijesekera and Dillon 1991; Moum et al. 1992), and convection (Wang and Muller 2002) all implicated as contributing processes. In our LES, DC turbulence develops from local shear instabilities as illustrated by cross sections of the temperature fields in the April and July cases (Fig. 10). Kelvin–Helmholtz-like overturns are visible at 25-m depth in the April simulation and at 60 m in the July case (Figs. 10c and 10f, respectively). Conditions for shear instability are present at these depths; shear is maximized (Figs. 10a,d), and Ri is less than 1/4 (Figs. 10b,e). The horizontal distance between the overturns is considerably shorter in the July case, consistent with the fact that the shear maxima are thinner. The overturning process is similar throughout all four simulations.
Development of local shear instabilities as revealed by cross sections of the temperature field. Two example cases are shown: 2200 (LT) during the third night of the (top) April and (bottom) July simulations. (a),(d) Mean squared buoyancy frequency and shear and (b),(e) Ri. (c),(f) Temperature at fixed y.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
Local shear instabilities such as those seen in Fig. 10 develop when Ri decreases to subcritical values (Fig. 11; cf. Fig. 5d). In all four cases, Ri varies diurnally in the MI layer, exceeding 1/4 (yellow shading) during the day and dropping below 1/4 (blue) at night. The nocturnal reduction occurs in multiple events corresponding to the bursts of high ε shown in Fig. 8. Nightly episodes of low Ri originate near the surface and spread downward into the MI layer.
Temporal evolution of the gradient Richardson number Ri in all four cases shows values of less than 0.25 in the deep cycle layer at nighttime because of the descending shear layer. Solid curves mark the (top) DML base and (bottom) zMI; vertical dashed lines indicate the times shown in Fig. 10.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
In all cases, both shear (Fig. 12; cf. Fig. 5b) and stratification (Fig. 13; cf. Fig. 5c) show clear diurnal variability near the surface. During the day, shear increases as penetrating solar radiation creates strong, stable stratification, allowing the development of a wind-driven surface current (e.g., Price et al. 1986). In late afternoon, the stable stratification relaxes and the near-surface flow becomes unstable, generating turbulence; visible in Fig. 8 as strong dissipation near the surface occurring each day near 1900 h. As the turbulence mixes the velocity at each successive depth, it enhances the shear in the region directly below it, causing Ri to decrease to subcritical values. The resulting descending shear layer is accompanied by enhanced stratification. The nighttime reduction of Ri (Fig. 11) occurs because the influence of increased shear outweighs that of increased stratification. The nocturnal shear increase in the MI layer also results in multiple bursts of ε and low Ri (Figs. 8 and 11, respectively).
Temporal evolution of squared shear S2/4 shows the daytime enhancement of shear near the surface in all four cases. In the late afternoon, the enhanced shear becomes unstable and results in turbulence. The black solid curves mark the (top) DML base and (bottom) zMI.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
Temporal evolution of squared buoyancy frequency N2. Negative values of N2 because of convective turbulence, are shown in white. Curves mark the (top) DML base and (bottom) zMI.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
While other studies (e.g., Wijesekera and Dillon 1991; Moum et al. 1992) have suggested that breaking internal waves are important in the DC, the present results indicate that DC turbulence is generated by local shear instability. Internal waves are detectable in the stably stratified fluid below the MI layer, but those waves are too weak to break into turbulence. This is consistent with our previous studies of an idealized model of the EUC (Pham et al. 2012, 2013).
Nocturnal convection is also evident in each case (Fig. 13, where white areas denote negative N2). Because of the onset of DC turbulence in late afternoon, the nighttime convection cannot form coherent plumes as seen in the LES of Wang and Muller (2002), where the surface forcing consists of a constant cooling and the plumes are found to extend down to 20-m depth. In our simulations, the convective layer deepens to at most 10-m depth in the July case where the nonsolar heat flux is strongest (Fig. 13). In all cases, the convective layer is significantly thinner than the nocturnal DML, in which turbulence is primarily shear driven. A separate study will show that, while not responsible for the initial triggering of the DC, convection can play a role in sustaining the turbulence later in the night.
5. Conclusions
Our goal has been to describe the seasonal variation of deep cycle turbulence in the eastern Pacific cold tongue. Our results are consistent with the notion that deep cycle turbulence is controlled by the combination of
marginally unstable shear above the EUC, and
a daytime surface current whose collapse near sunset tips the underlying flow into the unstable regime (Schudlich and Price 1992; Smyth et al. 2013; Smyth and Moum 2013; Pham et al. 2013).
Moored observations of temperature and horizontal currents reveal that the MI state persists throughout the year, providing one of the factors that create DC turbulence. While the MI layer is present year-round, it shoals during boreal spring together with the EUC. The other factor, the sheared daytime surface current, is consistent through most of the year but is reduced in boreal spring, consistent with the slackening of the trade winds. Based on these observations, we would expect DC turbulence to be both shallowest and weakest in boreal spring.
To test this hypothesis, LES experiments have been conducted using conditions typical of January, April, July, and October. Model results indicate that the deep cycle continues throughout the year, and the essential physical processes that drive it remain the same. However, both the amplitude and vertical extent of the turbulence change with the seasons.
The DC is strongest and deepest in boreal summer and winter (Fig. 9). Down to about 40-m depth, both shear and stratification are similar in the January and July cases (Fig. 5). Below 40 m, shear is stronger in July, but so is stratification, hence Ri is similar in the two cases. Also, January and July both feature strong winds (Fig. 2b) and weak net surface heating (Figs. 2d,f; Table 1).
In April, besides the shoaling of the EUC and the thermocline (caused by large-scale ocean processes), there are two distinct changes in the surface forcing. First, the wind stress diminishes by a factor of 1/2 as the trades relax (Fig. 2b). Second, the net surface warming increases, both because of the springtime maximum in solar radiation (Fig. 2f) and the reduction in evaporative cooling (Fig. 2d) as the winds diminish. Both of these factors affect the daytime surface current, but it appears that the wind reduction is dominant, as the diurnal shear amplitude is reduced by about 1/3 (Fig. 4d). All of these factors would tend to reduce turbulence, except that the shear of the EUC is strong and close to the surface. Therefore, in a shallow layer between the surface and ~30 m, the deep cycle continues unabated (Figs. 8–13).
The October case is intermediate. The EUC and thermocline are at their deepest so that both shear and stratification are relatively weak near the surface, but the MI layer (in which Ri ~ 1/4) is definitely present (Fig. 5). The slight drop in turbulence in October may be connected with diminished shear or with the autumn increase in solar heating (Fig. 2f).
The April result modifies our picture of the springtime turbulence minimum as described in Moum et al. (2013) and Smyth and Moum (2013). Those observations were made at fixed depths, with the uppermost sensor placed at 30 m, and indicated a virtual cessation of turbulence in spring. In contrast, the present results suggest that the springtime DC is focused too close to the surface to be well sampled by the existing χpod array. While its vertical extent is reduced, the process itself continues unchanged.
A future step will be to use these modeling techniques, again in combination with observations, to learn more about the changes in the DC mechanism over the course of the night, from the initial triggering by the daytime surface current to the pulsating behavior observed later in the night (Smyth et al. 2017). Of particular interest will be the role of nocturnal convection in sustaining the deep cycle. The importance of the daytime surface current suggests the intriguing possibility that factors influencing that current, such as cloud cover, rain, and water transparency, may also influence the deep cycle.
Acknowledgments
We are grateful for the support provided by National Foundation Science Grants OCE-1355856 and OCE-1256620. Subsurface mooring data were acquired from web archives provided by the TAO Project Office of NOAA/PMEL. The TropFlux data are produced under a collaboration between Laboratoire dOcanographie: Exprimentation et Approches Numriques (LOCEAN) from Institut Pierre Simon Laplace (IPSL, Paris, France) and National Institute of Oceanography/CSIR (NIO, Goa, India) and supported by Institut de Recherche pour le Développement (IRD, France). TropFlux relies on data provided by the ECMWF interim reanalysis (ERA-Interim) and ISCCP projects.
APPENDIX A
Inferences from Daily Averaged Temperature and Velocity Measurements
a. Maximum DML thickness from daily averaged temperature
For each day when hourly averaged temperature profiles were available, the maximum DML thickness zMLB was computed. The temperature difference between that depth and the surface was computed using the daily averaged temperature. The median difference was 0.16 ± 0.01 K. Based on this, the maximum DML depth, identified from daily averaged temperature profiles, was estimated as the depth where temperature dropped by 0.16 K from its surface value.
b. Inferring marginal instability from the daily Ri distribution
The distribution of Ri depends on the averaging methods applied to the velocity and density measurements. For a given flow, a typical value of Ri is larger if coarser resolution is employed either in space or in time. For example, in a strongly turbulent deep cycle regime, Smyth and Moum (2013) compared Ri distributions with derivatives taken over a vertical spacing of 2 m and after the resolution was coarsened to 16 m. The tendency for Ri to cluster around 1/4 remained clear (Fig. 2 of Smyth and Moum 2013), but values smaller than 1/4 became less common.
The duration of the Smyth and Moum (2013) dataset was too short to allow an assessment of the result of coarsening the temporal resolution. Because of the length of the mooring records, we are able to make that assessment here by comparing results using hourly and daily averaged data (Fig. A1).
Probability distribution function for Ri calculated from daily (solid) and hourly (dashed) averaged currents and temperature. Depth range is 20–90 m. Legend shows median and sextile ranges.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
There is reason to anticipate that the difference in Ri due to averaging will not impede our ability to discriminate between regimes in which MI is and is not present. The effect of the loss of shear variance in the hourly to daily frequency band is always to increase estimates of Ri. In the strongest turbulence, where the true value of Ri tends to be small, these high frequencies are relatively energetic (i.e., the spectrum is broadband), so their absence from the shear signal will lead to the most severe overestimates of Ri. Conversely, in high- Ri flow, the spectrum is redder and hence the overestimate of Ri is less severe. The overall effect of lost shear variance then is to diminish the apparent difference in typical Ri values between strongly and weakly turbulent flows. We conclude from this that if such a difference is detected, it is probably not an artifact of the use of daily averaged data.
In both hourly and daily cases, derivatives are calculated at 5-m vertical spacing. We use hourly data collected at 140°W between September 1996 and November 2010. The statistics include all values of Ri in the depth range 20–90 m. Using hourly averaged data, we find that the median value of Ri is 0.31 (Fig. A1). When the daily average is applied, the median increases only slightly to 0.32. The mode near 1/4 remains well defined. Values smaller than about 0.2 tend to be replaced by values near the mode. The fraction of values less than 1/4 decreases from 41% with hourly data to 37% with daily data.
APPENDIX B
Estimating Near-Surface Currents
Hourly ADCP data are available with 5-m vertical spacing over several years. These data are good only to an uppermost depth that can be 40, 35, or (occasionally) 30 m. To characterize the deep cycle, we need information about currents closer to the surface. For this, we make use of current meters at 25-, 10-, and 5-m depth, wind speed, and near-surface stratification. Described in this appendix is a method, arrived at after considerable trial and error, which makes use of all of these information sources to estimate near-surface currents.
The mixed layer thickness H is defined slightly differently depending on whether hourly or daily data are used. For hourly data, H is simply the DML thickness hDML, defined as above by a temperature difference of ΔT = 0.04K from the surface. If daily data are used, the temperature difference is ΔT = 0.046K. The latter value is determined by requiring that the inverse layer thickness 1/H, determined from daily data equal, on average, the value obtained from hourly data. The constant Cw was determined using hourly data from the interval 23 May 2004 to 1 March 2005, during which the 5-m current measurement is available so that the coefficients of (B1) are fully determined. By trial and error, we arrived at the value Cw = 0.02.
The validity of this method can be gauged by estimating the shear at 7.5-m depth from (B3) and also as the difference between the 5- and 10-m current meters during the period when the 5-m instrument was operational (Figs. B1a,c). The comparison shows a clear correlation (r = 0.64). On average, the shear given by (B3) is high (less negative) by 2%. For individual hourly estimates, the root-mean-square error is 51%. For comparison, we also show estimates using the simpler method of extrapolating linearly from the uppermost two ADCP bins (Figs. B1b,d). The resulting shear is too negative, on average, by 45%, with a root-mean-square error exceeding 100%.
Hourly averaged shear ∂U/∂z at 7.5 m estimated (a),(c) from (B3) and (b),(d) from a linear fit to the uppermost two ADCP bins vs that measured using current meters during a 9.5-month period when both systems were operational.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
For the present purpose, a more direct test of (B3) is to estimate the diurnal and annual cycles of the shear at 7.5 m (Fig. B2). For the diurnal cycle, estimates from (B3) lie close to the mean of the observed values at all hours, the exception being that the late afternoon peak is underestimated by ~15% (Fig. B2a). The root-mean-square error for the 24 estimates is 17%. To test the applicability of (B3) to daily averaged data, we construct a composite annual cycle (Fig. B2b). Note that no data are available for March and April. For the remaining 10 months, the estimates track the general trend of the measurements, with root-mean-square error of 24%. For all but 2 of the 10 months in which measurements are available, the estimate lies within one standard deviation of the observed mean.
Shear at 7.5 m estimated from (B3) vs that measured using current meters during a 9.5-month period when both systems were operational. (a) Composite diurnal cycle from hourly values. (b) Composite annual cycle from daily values.
Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0008.1
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The binning has no effect on Qsw(t, 0), as it is already based on daily averaged data.
We show the median rather than the arithmetic mean because the statistics of S2 and Ri are highly non-Gaussian, and the mean is therefore a poor descriptor of central tendency. The variables U and N2 are more nearly Gaussian and the choice of mean or median is less critical. For consistency, Fig. 6 shows the median of each field. The Ri is computed from hourly shear and stratification before the median is taken.
The full 3-day simulation is shown to illustrate the persistence of the diurnal cycle. Later times (day 3 in particular) may not be quantitatively precise because of model drift.