Abstract

The effect of equatorial undercurrent (EUC) shear on equatorial upper-ocean mixing is studied using a large eddy simulation (LES) model. This study consists of five numerical experiments of convection with various initial shear profiles: 1) full background shear (EUC shear), 2) same as 1 but with a surface cooling rate reduced by a factor of 10, 3) no shear, 4) stable part the background shear only (velocity constant above 30 m where Ri < 1/4 in experiment 1), and 5) unstable part of the background shear only (velocity constant below 30 m). It is found that flow evolution crucially depends on the background shear. Removal of all or part of the shear profile dramatically degrades the realism of the results. Convection in the mixed layer triggers shear instability, which in turn radiates gravity waves downward into the upper thermocline. Local shear instability can be triggered by downward-propagating internal waves in a marginally stable environment. This local shear instability is the cause of mixing well below the mixed layer. When complete EUC shear is present, internal waves with wavelengths of 200–300 m are generated below the boundary layer, in agreement with observations and linear instability analysis. The total shear profile determines the characteristics of the waves. When the stable shear, or the portion of the shear with Ri > 1/4, is eliminated, the internal waves have smaller wavelengths (about 80 m). When the unstable shear, or the portion of the shear with Ri ≤ 1/4, is eliminated, the intensity of internal waves below the boundary layer is much reduced, but the wavelengths are much larger than the case of convection without shear. In the absence of large-scale forcing to maintain the surface shear, the bulk of the kinetic energy from the mean shear is released in just a few hours after the onset of convection and shear instability. Turbulent kinetic energy budgets with and without shear show some similarities during the early stage of convection but show dramatic differences when the turbulence is fully developed. Namely, the turbulent transport and pressure transport terms are important in the case of convection without shear but are negligible in the case of convection with EUC shear, even though the surface forcing is the same. Local shear instability in a marginally stable mean flow environment is shown to play an important role in transporting heat and momentum into the stratified region below the mixed layer. Turbulence and waves generated by the mean shear instability are shown to be more effective than convective plumes in triggering local instability in the marginally stable region below the mixed layer.

1. Introduction

Great progress has been made in understanding ocean mixing processes in various oceanic environments during the past two decades, mainly from microstructure profiling measurements, tracer release experiments, Lagrangian measurements (floats), and direct measurements of turbulence (e.g., Gregg et al. 1985; Moum and Caldwell 1985; Moum et al. 1992; McPhaden and Peters 1992; Peters et al. 1994; Toole et al. 1994; McPhee and Martinson 1994; Lien et al. 1996; Polzin et al. 1997; Gregg et al. 1999). Three-dimensional numerical simulation, or large eddy simulation (LES), of oceanic turbulence is a new area in physical oceanography. It has a great potential in advancing our knowledge of ocean mixing processes and in providing insights for improving parameterizations of mixing in ocean general circulation models, as demonstrated by recent studies (e.g., Siegel and Domaradzki 1994; Skyllingstad and Denbo 1995; Wang et al. 1996; McWilliams et al. 1997; Wang et al. 1998; Large and Gent 1999; Skyllingstad et al. 1999, 2000; Skyllingstad and Denbo 2001). Numerical simulation is now an indispensable tool for oceanic turbulence research, complementary to field and laboratory experiments.

Equatorial undercurrent (EUC) is ubiquitous in the Pacific and Atlantic Oceans (with occasional disappearance during El Niños; Firing et al. 1983) and is a seasonal phenomenon in the Indian Ocean. The presence of EUC shear is unique to the equatorial ocean boundary layer. There is no atmospheric counterpart. Recognizing the importance of EUC shear on ocean mixing, Pacanowski and Philander (1981) employed a Richardson number–dependent vertical mixing scheme in a numerical model of the equatorial Pacific Ocean to take into account the effect of EUC shear. They found that the thermocline structure was greatly improved with this mixing scheme. Microstructure observations revealed that on the equator, turbulence can penetrate well below the mixed layer during nighttime convection (Gregg et al. 1985; Moum and Caldwell 1985). Gregg et al. (1985) suggested that shear instability of the EUC might be the cause of this so-called deep-cycle turbulence. Using a 1D mixed layer model, Schudlich and Price (1992) were able to reproduce the deep-cycle turbulence when EUC shear was included. It is now well known that the presence of EUC influences equatorial upper-ocean mixing greatly.

Observations often show elevated internal wave activities during nighttime convection (e.g., McPhaden and Peters 1992; Moum et al. 1992; Lien et al. 1996), accompanied by an increase in shear and Froude number (Peters et al. 1994). Observational and theoretical analysis suggest that momentum flux due to internal waves can be substantial (e.g., Dillon et al. 1989; Wijesekera and Dillon 1991; Lien et al. 1996). There have been several numerical studies, aimed at understanding the role of internal waves in the equatorial current system (e.g., Skyllingstad and Denbo 1994; Mack and Hebert). Using linear instability analysis, Sutherland (1996) and Sun et al. (1998) suggested that the observed internal waves can be explained in terms of the most unstable modes of the EUC shear. However, these numerical and linear analyses do not simultaneously resolve 3D turbulence and internal waves. Therefore, it is unclear whether the results are directly applicable to the real ocean.

In a 3D large eddy simulation study, Wang et al. (1998) suggested that local shear instability plays a crucial role in the diurnal cycle of turbulence at the equator. However, their model domain was too small to resolve the observed “long” internal waves with wavelengths of a few hundred meters (e.g., Moum et al. 1992; McPhaden and Peters 1992; Lien et al. 1996). (Long waves here mean the wavelengths are much larger than the mixed layer depth, or the maximum size of turbulent eddies.) Computational resources were the limiting factor in large domain simulations because hundreds of Cray-YMP hours were needed to carry out even the small domain simulations of the diurnal cycle. In this study, we build upon previous studies to gain further insights on the effects of EUC on upper-ocean mixing and wave generation, with a focus on the nighttime convection phase of the diurnal cycle. A much larger model domain is used such that turbulence and long internal waves (or the dominant modes of shear instability) can be simultaneously resolved.

To simplify the problem, we neglect large-scale forcing terms such as zonal pressure gradient, upwelling, and mesoscale eddy forcing used in Wang et al. (1998). To simplify the problem even further and to isolate the processes, we use surface cooling as the only forcing. Our modeling strategy is to simulate the early hours of nighttime convection under various vertical shears and surface cooling rates, to make good use of limited computing resources available. The goal is to understand how EUC influences turbulence and wave generation and how shear instability interacts with convection (or surface cooling), with emphasis on the effects of EUC on upper-ocean mixing. Detailed examinations of how internal waves influence/generate mixing in the EUC system will be the subject of a future study.

This paper is organized as follows. The numerical model and experiments are described in section 2. In section 3, we use zonal sections of temperature and vertical velocity to present the basic phenomenology of the involved processes. We demonstrate that the full EUC shear is crucial for both longwave generation and deep penetration of turbulence. Without either the surface unstable shear or the stable shear below the mixed layer, the results are dramatically different. We also address the questions of how shear affects wave energy flux into the deep ocean and how LES results compare with linear theory, observations, and previous modeling studies. In section 4 we discuss how turbulence is affected by the presence of the EUC shear in detail. We look at turbulent entrainment, fluxes, and balances in the turbulent kinetic energy (TKE) budget. In section 5, we examine the Richardson number and its implications for mixing. Conclusions and discussions are presented in section 6.

2. Description of numerical experiments

The LES model used in this study was developed by Moeng (1984). In the absence of large-scale forcing terms and solar radiation, the governing equations are

 
formula

where u = (u, υ, w) is the velocity, p the pressure (normalized by a reference density), α = 0.00026 K−1 the thermal expansion coefficient, g the gravitational acceleration vector, τ the subgrid stress tensor, T the potential temperature, q = (q1, q2, q3) the subgrid heat flux, e the subgrid-scale turbulent kinetic energy, ε the turbulence dissipation, and K a diffusion tensor. The last term of (4) represents the parameterization of subgrid TKE change due to subgrid pressure transport and diffusion. Detailed descriptions of discretization can be found in Moeng (1984) and Sullivan et al. (1996). Description of the subgrid-scale parameterization can be found in Wang et al. (1996, 1998). Both vertical and horizontal components of the earth's rotation are ignored. Periodic boundary conditions are used in the horizontal. The ocean surface is a rigid lid. The open boundary condition of Klemp and Durran (1983) is used at the model bottom to allow vertical propagating internal waves to leave the system.

The model domain is 960 m × 320 m × 270 m in the zonal, meridional, and vertical directions, with a grid geometry of 192 × 64 × 270 for all experiments. A meridional domain size larger than 320 m is preferred, but the computational effort will be significantly greater. So the choice of 320 m in the meridional direction is a compromise of retaining 3D simulations and saving computer time at the same time. Nevertheless, the model domain is 12 times that of the largest domain size used in Wang et al. (1998).

There are no wind stress forcing or other large-scale forcing terms as were used in Wang et al. (1998) and Wang and Müller (1999). The initial temperature profile (Fig. 1a) is integrated from a prescribed buoyancy frequency profile N2 (Fig. 1c), which is zero above 25 m and constant below 180 m, with a maximum value of 0.00025 s−2 at 130 m. The initial velocity profile (Fig. 1b) is described by

 
formula

where z is in meters. Below 240 m, U is set to zero. At the surface, U = −0.22 m s−1. The EUC has a core speed of 0.8 m s−1, at 112 m. The initial temperature and velocity profiles used here are idealized representations of the eastern Pacific Ocean. In reality, the westward surface current and the core speed of the EUC vary considerably and can be stronger or weaker than what we used here. Therefore, the stratification and shear used here are only one possible realization of a large number of possible states. Experiments with different background shear and stratification are desirable but are not conducted here. We should point out that stratification N2 and Richardson number Ri (Figs. 1c,d) are comparable to the observations of Peters et al. (1991). Therefore, qualitatively our model design mimics the reality.

Fig. 1.

Initial horizontal mean profiles of (a) temperature (for all experiments), (b) zonal velocity for experiments 1 and 2, (c) buoyancy frequency squared (all experiments), and (d) Richardson number Ri (for experiments 1 and 2)

Fig. 1.

Initial horizontal mean profiles of (a) temperature (for all experiments), (b) zonal velocity for experiments 1 and 2, (c) buoyancy frequency squared (all experiments), and (d) Richardson number Ri (for experiments 1 and 2)

Five numerical experiments, listed in Table 1, of convection are conducted. A constant surface cooling rate of 200 W m−2 is applied for all experiments except experiment 2, in which a surface cooling rate of 20 W m−2 (or one-tenth that of experiment 1) is applied. In experiments 1 and 2, the full background shear (labeled full shear) is used (Fig. 1b). In experiment 3, no background shear (labeled no shear) is applied. In other words, it is a pure convection experiment. In experiment 4, only the stable part of the background shear is used (labeled stable shear only) and the initial zonal velocity is constant above 30 m. In experiment 5, only the unstable part of the background shear is used (labeled unstable shear only) and the initial zonal velocity is constant below 30 m. Note that instability is a global property of a shear flow. The structure of unstable modes is a function of the whole shear, not just the unstable part of the shear. What we call “unstable shear” or “unstable part of the shear” is only meant to identify the portion of the shear profile with Ri ≤ 1/4. Random noise of small amplitude in horizontal velocity is introduced initially at the top 20 m to instigate the turbulence. The random noise is the same for all five experiments. Therefore, beside the initial background flow, the initial conditions of all experiments are identical.

Table 1.

Numerical Experiments. For all the numerical experiments, we employ a resolution of 5 m in the horizontal and 1 m in the vertical, with a grid geometry of 192 × 64 × 270. The only surface forcing is surface cooling Qo. Full shear means the complete initial profile U(z) shown in Fig. 1b. Stable shear only means the portion of the U(z) that has Ri > 0.25 (below 30 m) and assuming velocity to be constant above 30 m. Unstable shear only means the portion of the U(z) that has Ri ≤ 0.25 (or above 30 m) and assuming velocity to be constant below 30 m

Numerical Experiments. For all the numerical experiments, we employ a resolution of 5 m in the horizontal and 1 m in the vertical, with a grid geometry of 192 × 64 × 270. The only surface forcing is surface cooling Qo. Full shear means the complete initial profile U(z) shown in Fig. 1b. Stable shear only means the portion of the U(z) that has Ri > 0.25 (below 30 m) and assuming velocity to be constant above 30 m. Unstable shear only means the portion of the U(z) that has Ri ≤ 0.25 (or above 30 m) and assuming velocity to be constant below 30 m
Numerical Experiments. For all the numerical experiments, we employ a resolution of 5 m in the horizontal and 1 m in the vertical, with a grid geometry of 192 × 64 × 270. The only surface forcing is surface cooling Qo. Full shear means the complete initial profile U(z) shown in Fig. 1b. Stable shear only means the portion of the U(z) that has Ri > 0.25 (below 30 m) and assuming velocity to be constant above 30 m. Unstable shear only means the portion of the U(z) that has Ri ≤ 0.25 (or above 30 m) and assuming velocity to be constant below 30 m

It would also be helpful to conduct an experiment with only shear instability and no surface cooling to isolate the effects of shear instability. But such an experiment is very difficult to conduct. Turbulence takes a very long time to develop unless significant noise is introduced initially. This raises the question of how much turbulence is due to the initial noise. Although a 20 W m−2 surface cooling is perhaps not small enough (experiment 2), we believe it does shed some light on the limiting case of pure shear instability in the absence of surface cooling.

We will define a few parameters for the convenience of discussion. We must distinguish between the boundary layer depth and the mixed layer depth. Boundary layer h is defined as the depth at which turbulent heat flux is practically zero (say, 5% of the surface heatflux). Approaching to this depth, turbulent kinetic energy also falls dramatically. The mixed layer depth is often defined as the depth at which potential density differs from that of the surface by a small value. A wide range of values has been used in the oceanographic literature. In terms of potential temperature, difference up to 1.0°C has been used. This definition of mixed layer depth is more arbitrary than the definition of the boundary layer depth and the criterion needs to be adjusted under different situations. To facilitate the discussion, we shall use the following definition of mixed layer: the depth at which heat flux is at its local extreme value (positive maximum or entrainment heat flux). This definition of mixed layer depth serves the purposes of separating the well-mixed region from the stratified region below. Although this definition of mixed layer is not foolproof under circumstances that the local extremum does not exist (e.g., under very strong solar heating), it works in the context of the present study. Entrainment layer is defined as the region below the mixed layer and above the bottom of the boundary layer.

An eddy turnover timescale can be defined as τ = h/(Bh)1/3, where B = −αgQo/Cp is surface buoyance flux, Qo is surface heatflux (positive downward), and h is the boundary layer or mixed layer depth. The eddy turnover timescale is 32 min for all experiments except for experiment 2, where the timescale is 68 min, assuming a mixed layer depth of 30 m. Experiments 1–4 are integrated for 8 h, or about 15 eddy turnover times. In experiment 2, turbulence takes a longer time to develop because of the smaller surface-cooling rate. This case is integrated for 10.5 h. To compare with the other cases, we arbitrarily designate hour 2.5 as the equivalent of “time = 0” of the other cases. With this shift, we will be able to line up the sequences of turbulence evolution (say, maximum rate of change of potential energy) with other cases. Since the solutions evolve slowly for the last 4 h of integration for all cases, this shift does not alter our main conclusions.

Horizontally averaged fields of temperature, velocity, and higher-order statistics are archived every minute. Unless otherwise noted, all vertical profiles and time series of a given quantity at a given depth are horizontal averages.

3. Shear instability and internal waves

The premise of LES is that turbulent fluxes due to resolved motions dominate the subgrid fluxes. For a spinup problem, however, there is always a stage when subgrid fluxes dominate the fluxes due to resolved motions. In fact, the resolved motions (initial noise) decay during the first 30–45 min and then increase rapidly and become dominant as the surface is continuously cooled. The resolved TKE and fluxes become dominant after about 1 h of integration. Our experience shows that the mean turbulent statistics are not very sensitive to the initial random noise as long as it is a small fraction of the turbulence generated by the forcing. Namely, the initial velocity noise should be a small fraction of the friction velocity scale in the case of wind-driven turbulence or convective velocity scale w∗ = (Bh)1/3 in the case of turbulence driven by surface cooling.

Figure 2 shows a zonal-depth section of temperature for experiment 1 (full shear) at 10-min intervals for the period 90–120 min and at 210 min. At 90 min, a plume near x = 400 m was felt at the base of the mixed layer (notice the changes of isotherms at 30 m). After 100 min, the interface between the mixed layer and the stratified region starts to deform. Note that the plumes are slanted, tilting westward. This is due to the westward shear of the surface current. The plumes would be more or less vertical if there were no shear. At 110 min, the plume penetrates about 10 m below the mixed layer, causing a splash, or an overturn, on the eastern side (upstream side) of the plume. This overturn develops into smaller-scale overturns as time goes on (Fig. 2d). Besides this initial plume, other plumes also reach the bottom of the mixed layer (e.g., the ripples between x = 500 m and x = 900 m), causing overturns similar to the first one (not shown). Figure 2e (time = 210 min) represents fully developed turbulence.

Fig. 2.

A zonal section of temperature (°C) for experiment 1 (full shear). The color table is repeated five times to reveal small changes in isotherms

Fig. 2.

A zonal section of temperature (°C) for experiment 1 (full shear). The color table is repeated five times to reveal small changes in isotherms

Figure 2 essentially describes the development of mean shear instability and local shear instability under the perturbation of convective plumes and internal waves. Below the mixed layer, there are smaller-scale overturns. For example, there are several small-scale overturns between x = 700 and x = 800 m near the depth of 45 m at 210 min, which are not directly caused by the penetrating plumes and mean shear instability. At 45 m, the mean flow is dynamically stable (i.e., Ri > 0.25). But local shear instability can occur due to perturbations of the mean shear by internal waves, which are generated by the plumes and mean shear instability from above.

Figure 3 shows zonal sections of vertical velocity at 150 min for all experiments. For experiment 1 (full shear or EUC shear, Fig. 3a), small-scale turbulent motions dominate above 40 m. Below 50 m, wave motions dominate. We find that the internal waves have wavelengths on the order of 320 m (broad spectral peak). Spectral analysis in the frequency domain reveals that the waves have a typical frequency of 1.2 cycle h−1 (or a period of 50 min). This results in a phase speed of −0.11 m s−1, which is consistent with the phase speed estimated from the examination of Hovmöller diagram of vertical velocity (not shown). The internal waves propagate westward relative to the ground. Based on observational analysis, Wijesekera and Dillon (1991) concluded that internal waves generated at the base of the mixed layer have wavelengths of 100–1000 m, frequencies of 0.72–7.2 cph, and a phase speed of about −0.2 m s−1 during the Tropical Heat I experiment (e.g., Gregg et al. 1985). The phase speed is approximately the average mixed layer velocity. In our experiment, the surface velocity is about −0.22 m s−1 initially and the average mixed layer velocity is about −0.11 m s−1, or roughly the phase speed of the internal waves. During Tropical Heat II experiment (e.g., Moum et al. 1992), the wavelengths of the internal waves have a spectral peak in the range of 150–250 m, comparable to the finding of McPhaden and Peters (1992). Using a different dataset, Lien et al. (1996) found that high-frequency internal waves below the mixed layer have wavelengths of less than 200 m, propagating westward. The frequency, however, is higher, at about 3 cph. In a two-dimensional study, Skyllingstad and Denbo (1994) suggested the long internal waves are generated by shear instability. They also found that the wavelengths are typically a few hundred meters, depending on the EUC profiles used (at different times of observation). The wave frequencies are 0.5, 2.5, and 12 cph, respectively, for velocity and temperature profiles measured at 134°W in 1987 and at 140°W in 1984 and in 1987. The initial conditions in our experiments are closer to those of 140°W in 1984 in Skyllingstad and Denbo (1994) than to other profiles used in their study, although the EUC core speed is much smaller in our study.

Fig. 3.

A zonal section of vertical velocity (m s−1) at 150 min for (a) experiment 1 (full shear), (b) experiment 2 (full shear with surface cooling rate reduced by a factor of 10), (c) experiment 3 (no shear), (d) experiment 4 (stable shear only), and (e) experiment 5 (unstable shear only)

Fig. 3.

A zonal section of vertical velocity (m s−1) at 150 min for (a) experiment 1 (full shear), (b) experiment 2 (full shear with surface cooling rate reduced by a factor of 10), (c) experiment 3 (no shear), (d) experiment 4 (stable shear only), and (e) experiment 5 (unstable shear only)

It should be pointed out that what we call internal waves are the periodic structures in the stratified layer below the mixed layer. One can assign wavelengths, phase propagation, and frequencies to these structures. We have not attempted to check whether or not these structures satisfy the dispersion and polarization relations for free internal waves nor have we attempted to separate wave motions from turbulent motions. Well below the mixed layer, the ratio of perturbation temperature gradient variance to mean temperature gradient squared 〈∂T2/∂z〉/〈∂T/∂z2 is much smaller than unity, suggesting the motions are linear. Therefore, our referring to these structures as internal waves is justified.

From linear stability analysis of a simplified shear profile, Sutherland (1996) argued that the observed internal waves can be explained by the most unstable modes of the EUC current system. Mack and Hebert (1997) repeated the analysis of Sutherland (1996) using a more complicated velocity profile, representing the conditions of Tropical Heat II (Moum et al. 1992). They found that the wavelength was in the range of 100–300 m and the phase speed was in the range of −0.8 to −0.1 m s−1. Using observed velocity profiles, Sun et al. (1998) solved the same linear problem. They found that the first two most unstable modes of the EUC shear (which is unstable in the upper ocean) have wave lengths typically of a few hundred meters. For example, at 9 p.m., the first unstable mode has a wavelength of about 300 m and a frequency of 3.6 cph. The second unstable mode has shorter wavelengths (about 200 m) and lower frequencies (from 0 to 0.4 cph). However, the second mode does not always propagate westward. Linear analysis showed that the vertical extent of the most unstable mode is far greater than the depth range of shear instability, or regions with Ri < 1/4 (Sun et al. 1998). This is also seen in our LES solution, in which internal waves have vertical wavelengths much larger than the depth of the boundary layer (Fig. 3a). Our LES results are consistent with linear analysis, observations, and previous 2D modeling studies, although linear analysis does not contain a turbulent boundary layer and the 2D model of Skyllingstad and Denbo (1994) contains little if at all resolved turbulence. Using a 3D model, we were able to simultaneously simulate turbulence and long internal waves in the EUC. We were able to show that shear instability triggered by convective plumes generate the so-called long internal waves in the equatorial current system.

For experiment 2 (Fig. 3b), although the surface cooling is only one-tenth of experiment 1, the internal waves fields are very similar in both amplitude and wavelengths, suggesting shear instability is principally responsible for the generation of these long internal waves. Convection only serves as a trigger/perturbation of the shear instability. For convection without shear (Fig. 3c), the internal waves below the boundary layer (below about 35 m) have wavelengths of a few tens of meters and there is no preferred horizontal direction of wave propagation, as indicated by the intersections of phase lines. This is because constant surface cooling does not generate any zonal or meridional asymmetry. The wavelengths of the internal waves become larger as the depth increases. At 150 m, the phase lines are also more horizontal. At later times as turbulence is fully developed, the horizontal phase lines disappear and wavelengths become smaller (not shown). With stable shear only (Fig. 3d) internal waves below the boundary layer have larger wavelengths than those of experiment 1 but the amplitude is much smaller compared to experiments 1 and 2, suggesting mean shear instability in the surface layer is the cause of the stronger long internal waves in experiments 1 and 2. With unstable shear only, (Fig. 3e), the internal waves, again, have smaller wavelengths, typically of 80 m with westward and upward phase propagation. From this result we conclude that although the stable shear below 30 m is not responsible for the generation of the long internal waves, its presence is crucial for such waves to exist. Without the stable shear, the waves have much smaller scales. In other words, long internal waves are characteristics of the EUC system during nighttime convection. Shear instability in the upper-ocean alone, which is common for wind-driven oceanic boundary layers at all latitudes, is not enough.

To see how wave energy flux into the deep ocean is affected by the presence of shear we show, in Fig. 4, the vertical kinetic energy (KE) flux at 150 min. Inside the boundary layer, the KE flux is largely due to turbulence. But below the boundary layer, the KE flux is presumably mostly due to internal waves (i.e., wave energy flux). At 150 min, convection with shear in the surface layer (experiments 1, 2, and 5) shows considerably larger wave energy flux than convection without shear (dashed line) or with stable shear only (dotted line, indistinguishable from the dashed line). The case with unstable shear only has the largest wave energy flux above 120 m. Below 120 m, wave energy fluxes with the full EUC shear (experiments 1, 2) are larger. By hour 8, wave energy fluxes for all cases decreased by a factor of 10 or more. The wave energy flux for experiment 4 is considerably larger than other cases below 40 m. So with unstable shear only, the internal waves with smaller waveslengths (Fig. 3e) carry more energy into the deep ocean.

Fig. 4.

Wave energy flux (mW m−2) at 150 min. Note that below about 50 m, dashed line (experiment 3) and dotted line (experiment 4) overlap each other and are indistinguishable from zero

Fig. 4.

Wave energy flux (mW m−2) at 150 min. Note that below about 50 m, dashed line (experiment 3) and dotted line (experiment 4) overlap each other and are indistinguishable from zero

4. Turbulence

In this section, we discuss the properties of turbulence under different velocity profiles. In the presence of full shear, the transfer of kinetic energy of the mean flow to turbulent kinetic energy as a result of the shear instability occurs rather rapidly. Figure 5 shows vertical profiles of zonal velocity and temperature at 1-h intervals, for experiment 1 (full shear). There is a rapid decrease of westward surface currents and surface shear during the first 3 h, accompanied by a rapid increase of westward velocity below 30 m. After hour 3, the changes are rather gradual (Fig. 5a). Temperature profiles show similar characteristics: rapid change in the first few hours, followed by gradual changes during later hours (Fig. 5b).

Fig. 5.

Vertical profiles of horizontally averaged (a) zonal velocity and (b) temperature at 1-h intervals, for experiment 1. The thick lines are initial conditions. For velocity, from left to right, the thin lines represent vertical profiles at hours 1, 2, … , 8. For temperature, from right to left, the thin lines represent vertical profiles at hours 1, 2, … , 8

Fig. 5.

Vertical profiles of horizontally averaged (a) zonal velocity and (b) temperature at 1-h intervals, for experiment 1. The thick lines are initial conditions. For velocity, from left to right, the thin lines represent vertical profiles at hours 1, 2, … , 8. For temperature, from right to left, the thin lines represent vertical profiles at hours 1, 2, … , 8

To quantify the conversion of mean kinetic energy (KE) to potential energy (PE) and turbulent kinetic energy (TKE), Fig. 6 shows the vertically integrated time rate of change of mean kinetic energy and the time rate of change of potential energy for experiments 1 and 2. The steady PE increase due to uniform cooling is removed (this term is proportional to the surface cooling rate). For experiment 1 (full shear), the peak (negative) of mean KE rate of change is about −0.04 W m−2 (Fig. 6a). The PE rate of change due to shear is about 0.01 W m−2, or 25% of the decrease of mean KE rate of change. The other 75% goes into turbulent kinetic energy (including wave kinetic energy) and dissipation of the system. The PE rate of change increase due to shear is 28% of the PE increase due to uniform cooling (subtracted, therefore not shown in Fig. 6b). When the unstable shear above 30 m is eliminated (unstable shear only, experiment 4), the mean KE rate of change is rather small (not shown). As a result, there is almost no difference between experiments 3 and 4 in PE rate of change. In other words, without the unstable part of the EUC shear, there is very little conversion of kinetic energy into potential energy. If shear below 30 m is eliminated (unstable shear only, experiment 5), there is a substantial conversion of mean KE into potential energy. But the PE rate of change is about half that of experiment 1, consistent with a smaller value of mean KE decrease. This means while the stable part of the EUC shear has little influence on mixing when there was no surface shear, it is nevertheless important when the full EUC shear is included. The above results, once again, demonstrate that the interaction of shear and convection is nonlinear. This can be clearly seen from experiment 2, which is the same as experiment 1 except surface cooling is reduced by a factor of 10. The peak mean KE rate of change is about 50%, not 10% that of experiment 1, as the surface heat flux would suggest if the interaction were linear. The PE increase due to shear is 100% of the PE increase due to uniform cooling (not shown), compared to only 28% in experiment 1. So shear instability is more important for mixing under weaker surface cooling conditions.

Fig. 6.

Time series of time rate of change of vertically integrated (a) mean kinetic energy and (b) potential energy, for experiments 1 and 2

Fig. 6.

Time series of time rate of change of vertically integrated (a) mean kinetic energy and (b) potential energy, for experiments 1 and 2

Figure 7 compares turbulent kinetic energy, dissipation, heat flux, and momentum flux of the five experiments at hour 8. Near the surface (above 5 m), all experiments show comparable TKE (Fig. 7a) except experiment 2 (thin solid line), where surface cooling rate is reduced by a factor of 10. Below 10 m, experiment 1 (full shear, thick solid line) shows a considerably higher level of TKE than the pure convection experiment (no shear, dashed line). Experiment 4 (stable shear only, dotted line) has only slightly higher TKE than experiment 3, meaning the stable shear below 30 m has little effect on convection. Experiment 5 (unstable shear only) has TKE comparable to experiment 1 above 40 m but is closer to experiment 3 below (dash–dotted line). Experiment 2 (thin solid line) is closer to experiment 1 (thick solid line) than to other cases below 40 m. These results depict a nonlinear picture of interactions between convection and background shear. On one hand, without unstable shear above 30 m, there is little deep penetration of turbulence below the depth of convective boundary layer (stable shear only). On the other hand, eliminating stable shear below 30 m (unstable shear only) also prevents turbulence from penetrating deeper. The full shear has to be present to cause significant penetration of turbulence into deeper layers. With full shear, even when the surface cooling rate is reduced by a factor of 10, turbulence still penetrates to similar depths.

Fig. 7.

Mean statistics: (a) TKE, (b) dissipation, (c) turbulent heat flux, and (d) momentum flux. Note that TKE here includes turbulence (resolved + SGS) and wave energy

Fig. 7.

Mean statistics: (a) TKE, (b) dissipation, (c) turbulent heat flux, and (d) momentum flux. Note that TKE here includes turbulence (resolved + SGS) and wave energy

Turbulence dissipation (Fig. 7b) tells more or less the same story as TKE. Again, the full shear is important in generating stronger deep turbulence dissipation (experiments 1 and 2, thick and thin solid lines, respectively). With stable shear only or with unstable shear only (experiments 4 and 5, dotted and dash–dotted lines), dissipation is significantly smaller. The turbulence dissipation for experiment 1 should be qualitatively comparable to the observations because the main ingredients of deep-cycle turbulence (nighttime convection in the presence of shear) are included: surface cooling and EUC shear. Lien et al. (1995) found that turbulence dissipation is typically in the range 5 × 10−9–5 × 10−7 (m2 s−3) above the EUC core. Near the surface, dissipation can be as large as 5 × 10−6 due to turbulence generated by surface gravity waves, a process the LES model does not include. During the early hours of convection, turbulence dissipation of experiment 1 is comparable to the observations of Lien et al. (1996). For example, dissipation at 25 m is about 3 × 10−7 at hour 2 and reaches a maximum value of about 8 × 10−7 at hour 3. However, at hour 8 (Fig. 7b), dissipation inside the mixed layer is significantly smaller than the observations of Lien et al. (1996). This is because the shear in the LES model is not maintained. The LES experiments conducted here did not include some of the important equatorial processes such as zonal pressure gradient, upwelling, wind stress, mesoscale eddy forcing, and diurnal heating. The EUC shear is maintained by the balance between wind stress and eastward pressure gradient. The focus of this study is how EUC shear affects mixing. The lack of wind stress is by design, to isolate the processes such that we can gain insights on how a preexisting shear interacts with convection.

Heat flux is shown in Fig. 7c. With full shear, entrainment heat flux (positive extremum, thick solid line) is about twice that of the case without shear (experiment 3, dashed line). The mixed layer depths with full shear or without shear (experiments 1 and 3) are about the same (depths of heat-flux extrema), 37 m for experiment 1 and 35 m for experiment 3. However, the boundary layer depths (depth of negligible heatflux) are different with or without shear. For experiment 1 the boundary layer depth is about 53 m, versus 38 m for experiment 3. The entrainment layer thickness in the presence of full EUC shear is much larger than that of the case without shear. With full shear but with one-tenth of surface cooling (experiment 2), turbulence reaches to almost the same depth (compare thin and solid lines), although the heat flux profiles are significantly different above 50 m. Since a surface cooling rate of 200 W m−2 is not atypical for the real equatorial oceans at night, we conclude that the presence of shear and shear instability is much more efficient in “propagating turbulence” into the stratified region than convective plumes at the equator, at least during the earlier hours of nighttime convection. We shall come back to this point later when discussing the TKE budget.

Momentum flux at hour 8 is shown in Fig. 7d. The vertical transfer of momentum for the case of full shear (experiment 1, thick solid line) is quite significant, with a maximum value of 0.115 w2, 0.02 N m−2 at 30 m. Because of the smaller surface cooling rate, experiment 2 has a smaller momentum flux for most part of the water column (thin solid line). However, below 45 m, the flux is similar to that of experiment 1. This shows that enhanced mixing in the presence of full EUC shear is due to shear instability. There is no net vertical momentum transfer (or horizontal average is zero) for convection without shear (experiment 3, not shown). Below about 50 m, the momentum flux of experiment 5 (unstable shear only, dash–dotted line) is significantly larger than that of other experiments. This once again shows that wave energy is more efficiently transmitted below the boundary layer in the absence of the stable shear below 30 m. With stable shear only, the momentum flux is smaller than that of experiment with unstable shear only (compare dotted and dash–dotted lines), which has smaller momentum flux than that of the full shear case. However, one still needs to account for the effect of mixing due to the stable shear. It is a challenge to correctly parameterize the effects of a marginally stable shear on boundary layer mixing in 1D models.

We show next how background shear influences turbulent kinetic energy budget. Figure 8 compares TKE budgets of experiment 1 (full shear) and experiment 3 (no shear) averaged for hours 3 and 8. At hour 3, the time rates of change are significant for both experiments (thin solid lines, Figs. 8a,b). For convection without mean shear, the dominant source of turbulence is the buoyancy production term B = αgwT′〉, (dash–dotted line, Fig. 8b), above 20 m. Below 20 m, the divergence of the turbulent transport T = −∂〈½(uu′ + υυ′ + ww′)w′〉/∂z (thick dashed line) is the primary source of turbulence. Since the vertical integral of this term is nearly zero (it would be identically zero if the boundary condition at the model bottom were w = 0, not the radiation boundary condition used), TKE is transported from the mixed layer region to the stably stratified region. Although surface cooling is the only forcing, the presence of EUC shear makes shear production of TKE (S = −〈uw′〉∂U/∂z, thin dashed line) the dominant term for the most part of the boundary layer for experiment 1, except near the surface, where buoyancy flux is dominant, and near the bottom of the boundary layer, where turbulent transport is equally important (Fig. 8a). Note that the shear production below the surface can be as much as three times that of the TKE production due to surface cooling. Turbulence dissipation is stronger in experiment 1 than in experiment 3 throughout the boundary layer, and it coincides with strong shear production in experiment 1. The large time rate of change in TKE for experiment 1 is also due to shear production. It is noteworthy that buoyancy production a few meters below the surface exceeds the surface value at the early stage of convection for both cases. This phenomenon is often observed in simulations where transition to turbulence is occurring, such as transition from daytime heating to nighttime convection after sunset during a diurnal cycle.

Fig. 8.

TKE budget averaged for (a) and (b) hour 3 and (c) and (d) hour 8 for experiments 1 (full shear) and 3 (no shear). The variables shown are time rate of change of total TKE (dE/dt), shear production (S = −〈uw′〉∂U/∂z, for experiment 1 only), turbulence dissipation (D = −ε), buoyancy production (B = αgwT′〉), divergence of pressure transport (P = −∂〈pw′〉/∂z), and divergence of turbulent transport [T = −∂〈½(uu′ + υυ′ + ww′)w′〉/∂z]. Zonal mean flow is denoted by U, velocity fluctuations (deviations from horizontal averages) are denoted by (u′, υ′, w′). Triangle brackets represent horizontal averages

Fig. 8.

TKE budget averaged for (a) and (b) hour 3 and (c) and (d) hour 8 for experiments 1 (full shear) and 3 (no shear). The variables shown are time rate of change of total TKE (dE/dt), shear production (S = −〈uw′〉∂U/∂z, for experiment 1 only), turbulence dissipation (D = −ε), buoyancy production (B = αgwT′〉), divergence of pressure transport (P = −∂〈pw′〉/∂z), and divergence of turbulent transport [T = −∂〈½(uu′ + υυ′ + ww′)w′〉/∂z]. Zonal mean flow is denoted by U, velocity fluctuations (deviations from horizontal averages) are denoted by (u′, υ′, w′). Triangle brackets represent horizontal averages

At hour 8, the TKE budget for experiment 1 is similar to that of experiment 3 (Figs. 8c,d) above 20 m. This is because the surface shear in experiment 1 is dramatically reduced after 8 h of mixing. But below 20 m, the balances are very different. Divergence of turbulent transport is the major production term of turbulence for experiment 3 (thick dashed line). For experiment 1 (full shear), shear production (S) is basically the only production term of turbulence. The divergence of pressure transport and turbulent transport (P = −∂〈pw′〉/∂z and T) are very small. So under the same surface forcing condition, the TKE budget is a function of the background state.

5. Gradient Richardson number

The gradient Richardson number Ri of the horizontally averaged flow is computed for all experiments except experiment 3, where no background shear is applied. Figure 9 shows gradient Richardson number of the mean flow as functions of time and depth. There are rapid changes of Ri for the period of hours 2–4, for experiments 1, 2, and 5. After about hour 4, Ri changes gradually. For experiments 1, 2, and 5, although the shear above 30 m is unstable initially, the development of turbulence from the shear instability does not occur at the beginning of integration because the amplitude of the noise introduced initially is rather small. Only when the surface is continuously cooled and convective plumes are formed does shear instability begin to have significant effects. Shear instability caused changes in Ri as deep as 50 m for experiment 1 (full shear, Fig. 9a). A reduction of surface cooling rate by a factor of 10 did not prevent turbulence reaching more or less the same depth in experiment 2 (Fig. 9b), indicated by the changes of Ri to the depth of 50 m. For experiment 4 (Fig. 9c, stable shear only), Ri below 40 m shows almost no changes at all. For experiment 5 (Fig. 9d, unstable shear only), there are also no significant changes in Ri below 40 m. The fact that the boundary layer depth is somewhat deeper in experiment 4 (stable shear only) than experiment 3 (no shear) suggests that stable shear below the mixed layer is conducive to turbulence (see Fig. 7c).

Fig. 9.

Gradient Richardson number of the mean flow for experiments 1, 2, 4, and 5. The dotted lines depict the depths at which Ri = 0.25. Negative Ri contours are omitted

Fig. 9.

Gradient Richardson number of the mean flow for experiments 1, 2, 4, and 5. The dotted lines depict the depths at which Ri = 0.25. Negative Ri contours are omitted

Although the stable shear below the mixed layer is not the original cause of deep mixing for experiments 1 and 2, its existence is crucial for deep mixing to occur. This suggests that local shear instability might be occurring, triggered by disturbances (turbulence and waves) generated by the mean shear instability above. We computed pointwise Ri in the stably stratified region. Figure 10 shows time series of local Ri distributions at various depths (in the regions of stable mean flow). The ordinate shows the percentage of grid points at a given depth having local Ri ≤ 0.25, or percentage of unstable points. At 35 m (solid line), there is a rapid increase of unstable points near hour two. The number of points with local Ri ≤ 0.25 exceeds 60%. At 40 m and 45 m, there are also rapid increases of the number of unstable points (dashed and dotted lines), although the increase at 45 m is somewhat more gradual than at 35 m and 40 m. At 50 m (near the bottom of the boundary layer), the number of unstable points is below 10% before about hour 6 but increases to 20% at hour 8. These results suggest that turbulence below the mixed layer in the stratified region is closely associated with local dynamic instability of the mean flow. It is noteworthy (not surprising) that local shear instability is most likely to occur when the gradient Richardson of mean flow (or simply mean Richardson number) is near the critical value. There is a sharp fall off in the number of gridpoints that show local shear instability from 45 to 50 m. From Fig. 9a, mean Ri is about 0.3–0.4 at 45 m and is about 0.5–0.6 at 50 m. This suggests that under typical oceanic conditions, local shear instability is more likely to occur when mean Ri < 0.4.

Fig. 10.

Time series of distributions of local gradient Richardson number at the depths of 35, 40, 45, and 50 m. The ordinate shows the percentage of the grid points at a given depth that has Ri ≤ 0.25

Fig. 10.

Time series of distributions of local gradient Richardson number at the depths of 35, 40, 45, and 50 m. The ordinate shows the percentage of the grid points at a given depth that has Ri ≤ 0.25

Having described the statistics of local shear instability at various depths below the mixed layer (Fig. 10), we now turn to horizontal distributions of local shear instabilities. Shown in Fig. 11 is the evolution of local shear instability at 45 m. The dark areas represent grid points where local Ri ≤ 0.25. At 120 min, there are basically two regions where local Ri ≤ 0.25, near x = 310 and x = 420 m (Fig. 11a). At the depth of 45 m, the mean flow is stable (see Fig. 9a). So the local shear instability at this depth is caused either by overturnings due to mean shear instability above or by the perturbation of internal waves, which are generated at the base of the mixed layer due to mean shear instability and convective plumes. At x = 310 m, there is evidence that the local Ri ≤ 0.25 is due to the mean shear instability above, as suggested by the overturning near x = 300 m and z = 40 m from the temperature field (see Fig. 2d). At x = 420 m, however, it is evident that the local shear instability is caused by the internal wave perturbation of the mean flow, since there is no overturning near x = 420 m and z = 45 m extending from inside the mixed layer down to the stratified layer (Fig. 2d). By 130 min, the areas of two regions grew and at the same time were advected eastward by the EUC (Fig. 11b). The eastward movement is consistent with the speed of EUC at 45 m, which is 0.1 m s−1. Near x = 250 m, another region of local shear instability occurred. By 140 min, about 10% of the areas at 45 m have local Ri < 0.25 (Fig. 11c). The two earliest patches of local shear instability shown in Fig. 11a can be traced to this time as well, near x = 400 m and x = 500 m, respectively. By 150 min, about 20% of the areas have local Ri ≤ 0.25. By 180 min, about 40% of the areas have local Ri < 0.25. Beyond 180 min, the total area where local Ri ≤ 0.25 does not change significantly (see dotted curve in Fig. 10). In summary, local shear instability is widespread in the stratified region below the mixed layer when turbulence is fully developed.

Fig. 11.

Snapshots of horizontal distributions of local shear instability, measured by local Ri ≤ 0.25 (dark areas), for experiment 1 (full shear)

Fig. 11.

Snapshots of horizontal distributions of local shear instability, measured by local Ri ≤ 0.25 (dark areas), for experiment 1 (full shear)

6. Conclusions and discussion

In this process study, we investigated the effects of the equatorial undercurrent shear on upper-ocean mixing and internal wave generation. We purposely neglected the large-scale forcing terms used in previous studies and wind stress in order to simplify the problem and to study the processes at work. The neglect of the wind stress enabled us to study how fast the kinetic energy of the mean shear is released after the onset of convection and shear instability.

This is perhaps the first LES study in which turbulence and long internal waves (with wavelengths of a few hundred meters) are simultaneously resolved in the EUC system. These long internal waves are generated during convection only when the full EUC shear is present. They appear under a wide range of cooling rates from 20 to 200 W m−2. We showed that these waves are caused by shear instability but they extend into the stable region. The total shear profile determines the characteristics of the waves. If the unstable surface part of the shear is eliminated (stable shear only), long waves with much smaller amplitudes are generated. If the stable part of the shear is eliminated (unstable shear only), waves of much shorter wavelengths are generated. These findings are consistent with observations and the linear stability analysis. Namely, the vertical structure of the most unstable mode extends far below the region of instability with the largest amplitude occurring, in fact, below the region of instability (Sun et al. 1998).

The full EUC shear is crucial for upper-ocean mixing. Removal of the full shear or part of the shear results in dramatic degradation of realism of the results. Without the unstable part of the shear, turbulence does not penetrate deep enough and internal waves are very weak. More specifically, without surface unstable shear, the system behaves very much like convection without shear. However, without the stable shear, turbulence also does not penetrate deep enough. For deep-cycle turbulence to occur, the stable shear below the mixed layer is also necessary. This demonstrates the highly nonlinear nature of interaction between convection and background shear.

We also found that local shear instability is more likely to occur when the mean Richardson number is below 0.4. The fact that experiment 5 (unstable shear only) has a much shallower boundary layer than the experiments with a full shear profile shows that local shear instability is a very important part of the mixing process. Turbulence can exist only in the dynamically stably stratified region when local shear instability is ubiquitous. When the mean Richardson number rises above 0.4, the number of grid points having local Richardson numbers below 0.25 is drastically reduced and turbulence ceases to exist. We should point out that Wang et al. (1996) found that turbulence exists near Ri = 1. We attribute this difference to that fact that EUC shear is absent in Wang et al. (1996). Turbulence behaves differently in an environment with a preexisting shear (EUC) and in an environment where shear is a result of the boundary layer solution. For example, Ri jumps from being close to zero at the base of mixed layer to being close to infinity over a short vertical distance at the bottom of the boundary layer in Wang et al. (1996). This sharp gradient in Ri is a result of wind-driven turbulent entrainment (engulfment) into a quiescent stratified fluid. Therefore, intermittent turbulence exists over a wide range of Ri. On the other hand, if a preexisting shear has a Ri profile such that 1/4 < Ri < 1 covers a large vertical distance (Ri = 1 occurs at the depth of about 75 m in this study), the only mechanisms for turbulence to develop in regions of Ri ≈ 1 that are far outside of the boundary layer are wave and local shear instability in the EUC system (assuming the absence of double diffusive processes). Waves generated by convective plumes and shear instability might not be strong enough to perturb the mean flow such that local Ri < 1/4. An important implication of this is that equatorial turbulence cannot be adequately parameterized based on the value of Ri alone. Large-scale forcing, surface fluxes, and background states should all be taken into account. Of course, internal waves generated by large-scale atmospheric forcing can cause intermittent turbulence and mixing in the ocean interior where Ri can be much larger than unity. This is, however, outside the realm of the present study.

There are two main conclusions that we draw from all these findings: (i) Local shear instability in the marginally stable flow environment plays a crucial role in transporting heat and momentum into the stratified region below the mixed layer. (ii) Turbulence generated in the surface layer and internal waves generated at the base of the mixed layer by means of shear instability are more effective than convective plumes in triggering local shear instability in the marginally stable region below the mixed layer.

Acknowledgments

Funding to D. Wang is provided by the Frontier Research Systems for Global Change through its sponsorship of the International Pacific Research Center (IPRC). Peter Müller's research is supported by the Office of Naval Research. Numerical computations were carried out on a 24-cpu Cray-SV1 at the IPRC. We thank the two anonymous reviewers for their comments, which helped to improve the presentation of the manuscript.

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Footnotes

Corresponding author address: Dr. Dailin Wang, IPRC/SOEST, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, HI 96822. Email: wangd@soest.hawaii.edu

*

Soest Contribution Number 05855 and IPRC Contribution Number 00109.

+

International Pacific Research Center is partly sponsored by the Frontier Research System for Global Change.