Simulation and Scaling of the Turbulent Vertical Heat Transport and Deep-Cycle Turbulence across the Equatorial Pacific Cold Tongue

D. B. Whitt aNASA Ames Research Center, Moffett Field, California
bNational Center for Atmospheric Research, Boulder, Colorado

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D. A. Cherian bNational Center for Atmospheric Research, Boulder, Colorado

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R. M. Holmes cSchool of Geosciences, University of Sydney, Sydney, New South Wales, Australia
dClimate Change Research Centre, University of New South Wales, Sydney, New South Wales, Australia
eSchool of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia
fARC Centre of Excellence for Climate Extremes, University of New South Wales, Sydney, New South Wales, Australia

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S. D. Bachman bNational Center for Atmospheric Research, Boulder, Colorado

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R.-C. Lien gApplied Physics Laboratory, University of Washington, Seattle, Washington

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W. G. Large bNational Center for Atmospheric Research, Boulder, Colorado

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J. N. Moum hCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Abstract

Microstructure observations in the Pacific cold tongue reveal that turbulence often penetrates into the thermocline, producing hundreds of watts per square meter of downward heat transport during nighttime and early morning. However, virtually all observations of this deep-cycle turbulence (DCT) are from 0°, 140°W. Here, a hierarchy of ocean process simulations, including submesoscale-permitting regional models and turbulence-permitting large-eddy simulations (LES) embedded in a regional model, provide insight into mixing and DCT at and beyond 0°, 140°W. A regional hindcast quantifies the spatiotemporal variability of subsurface turbulent heat fluxes throughout the cold tongue from 1999 to 2016. Mean subsurface turbulent fluxes are strongest (∼100 W m−2) within 2° of the equator, slightly (∼10 W m−2) stronger in the northern than Southern Hemisphere throughout the cold tongue, and correlated with surface heat fluxes (r2 = 0.7). The seasonal cycle of the subsurface heat flux, which does not covary with the surface heat flux, ranges from 150 W m−2 near the equator to 30 and 10 W m−2 at 4°N and 4°S, respectively. Aseasonal variability of the subsurface heat flux is logarithmically distributed, covaries spatially with the time-mean flux, and is highlighted in 34-day LES of boreal autumn at 0° and 3°N, 140°W. Intense DCT occurs frequently above the undercurrent at 0° and intermittently at 3°N. Daily mean heat fluxes scale with the bulk vertical shear and the wind stress, which together explain ∼90% of the daily variance across both LES. Observational validation of the scaling at 0°, 140°W is encouraging, but observations beyond 0°, 140°W are needed to facilitate refinement of mixing parameterization in ocean models.

Significance Statement

This work is a fundamental contribution to a broad community effort to improve global long-range weather and climate forecast models used for seasonal to longer-term prediction. Much of the predictability on seasonal time scales is derived from the slow evolution of the upper eastern equatorial Pacific Ocean as it varies between El Niño and La Niña conditions. This study presents state-of-the-art high-resolution regional numerical simulations of ocean turbulence and mixing in the eastern equatorial Pacific. The results inform future planning for field work as well as future efforts to refine the representation of ocean mixing in global forecast models.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: D. B. Whitt, daniel.b.whitt@nasa.gov

Abstract

Microstructure observations in the Pacific cold tongue reveal that turbulence often penetrates into the thermocline, producing hundreds of watts per square meter of downward heat transport during nighttime and early morning. However, virtually all observations of this deep-cycle turbulence (DCT) are from 0°, 140°W. Here, a hierarchy of ocean process simulations, including submesoscale-permitting regional models and turbulence-permitting large-eddy simulations (LES) embedded in a regional model, provide insight into mixing and DCT at and beyond 0°, 140°W. A regional hindcast quantifies the spatiotemporal variability of subsurface turbulent heat fluxes throughout the cold tongue from 1999 to 2016. Mean subsurface turbulent fluxes are strongest (∼100 W m−2) within 2° of the equator, slightly (∼10 W m−2) stronger in the northern than Southern Hemisphere throughout the cold tongue, and correlated with surface heat fluxes (r2 = 0.7). The seasonal cycle of the subsurface heat flux, which does not covary with the surface heat flux, ranges from 150 W m−2 near the equator to 30 and 10 W m−2 at 4°N and 4°S, respectively. Aseasonal variability of the subsurface heat flux is logarithmically distributed, covaries spatially with the time-mean flux, and is highlighted in 34-day LES of boreal autumn at 0° and 3°N, 140°W. Intense DCT occurs frequently above the undercurrent at 0° and intermittently at 3°N. Daily mean heat fluxes scale with the bulk vertical shear and the wind stress, which together explain ∼90% of the daily variance across both LES. Observational validation of the scaling at 0°, 140°W is encouraging, but observations beyond 0°, 140°W are needed to facilitate refinement of mixing parameterization in ocean models.

Significance Statement

This work is a fundamental contribution to a broad community effort to improve global long-range weather and climate forecast models used for seasonal to longer-term prediction. Much of the predictability on seasonal time scales is derived from the slow evolution of the upper eastern equatorial Pacific Ocean as it varies between El Niño and La Niña conditions. This study presents state-of-the-art high-resolution regional numerical simulations of ocean turbulence and mixing in the eastern equatorial Pacific. The results inform future planning for field work as well as future efforts to refine the representation of ocean mixing in global forecast models.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: D. B. Whitt, daniel.b.whitt@nasa.gov

1. Introduction

Over the last several decades, multiple field campaigns have observed strong turbulence above the equatorial undercurrent in the eastern Pacific Ocean (Gregg et al. 1985; Moum and Caldwell 1985; Peters et al. 1988; Lien et al. 1995; Moum et al. 2009, 2013; Warner and Moum 2019; Smyth et al. 2021). Like upper-ocean turbulence elsewhere in the tropics and subtropics, the diurnal cycle is a dominant mode of variability, but turbulence in the eastern equatorial Pacific is unusual in that it penetrates tens of meters below the base of the surface mixed layer and into the thermocline. This turbulence produces exceptionally strong heat fluxes of O(100) W m2 on average and up to 1000 W m−2 during occasional bursts of intense turbulence in the nighttime and early morning in a stratified layer tens of meters thick (Moum et al. 2013, 2009; Smyth et al. 2021). Hence, this “deep-cycle turbulence” (DCT) drives stronger cooling of the near surface and warming of the thermocline compared to diurnal surface boundary layer turbulence in other areas of the global oceans. DCT thus contributes to sustaining the relatively cool sea surface and net ocean heat uptake in the eastern equatorial Pacific Ocean cold tongue on average (Wang and McPhaden 1999; Moum et al. 2013). DCT also varies with and influences the regional sea surface temperature (SST) dynamics on multiple time scales beyond diurnal, including interannual (Warner and Moum 2019), seasonal (Wang and McPhaden 1999; Moum et al. 2013), and subseasonal (Lien et al. 2008; Moum et al. 2009), although these variations are not as well understood as the diurnal cycle.

If the available data from 0°, 140°W are representative, then turbulent mixing is an important participant in the SST budget and air–sea interaction in the Pacific Ocean cold tongue. However, neither the spatiotemporal variability of ocean mixing nor the physical drivers of variability on time scales beyond diurnal are well observed or understood. In particular, our knowledge of the area and vertical extent of strong turbulent heat fluxes is based almost entirely on extrapolation using parameterizations beyond 0°, 140°W (e.g., Pacanowski and Philander 1981; Holmes and Thomas 2015; Holmes et al. 2019a; Pei et al. 2020; Deppenmeier et al. 2021; Cherian et al. 2021). In addition, none of these parameterized modeling studies present results over a sufficient duration to provide a climatological perspective from a model with sufficiently fine horizontal grid spacing [<10 km horizontal (Marchesiello et al. 2011), and <5 m vertical (Jia et al. 2021)] to fully resolve the mesoscale variations in vertical shear, which significantly modulate mixing (Moum et al. 2009; Inoue et al. 2012; Holmes and Thomas 2015; Cherian et al. 2021). Hence, the broader implications of downward turbulent heat transport and specifically DCT in the cold tongue for global ocean, climate, and Earth system dynamics are not well understood (but see Meehl et al. 2001; Richards et al. 2009; Danabasoglu et al. 2006; Newsom and Thompson 2018; Holmes et al. 2019a,b; Zhu and Zhang 2019; Huguenin et al. 2020; Deppenmeier et al. 2021). In addition, climate models suffer from long-standing and significant biases in their simulation of the SST, thermocline, and circulation in the eastern equatorial Pacific (Li and Xie 2014; Li et al. 2015). Since some biases persist with refinements in model horizontal grid resolution and the mean ocean circulation (Small et al. 2014) and are sensitive to the formulation of the mixing scheme (Meehl et al. 2001; Richards et al. 2009; Zhu and Zhang 2019), it seems plausible if not likely that poor performance of parameterizations of ocean mixing physics (Zaron and Moum 2009) is at least partially responsible for equatorial Pacific biases in climate and Earth system models. Hence, we conducted a regional process modeling study of turbulent heat transport and DCT in the equatorial Pacific Ocean cold tongue as a contribution to a broader effort to conduct a prefield process modeling study of Pacific equatorial upwelling and mixing physics.

In this manuscript, we present new state-of-the-art simulations and new metrics to characterize turbulent vertical heat transport in the Pacific Ocean cold tongue. First, we examine the climatological (1999–2016) spatiotemporal variability of the turbulent vertical heat flux, including the time-mean, seasonal cycle, and aseasonal variability (i.e., all deviations from the mean seasonal cycle) of the daily mean flux, in a relatively fine (1/20° horizontal, 2.5 m vertical) resolution regional hindcast simulation of the eastern equatorial Pacific Ocean with parameterized vertical mixing. The results provide a climatological perspective on the recent finding that global ocean models can simulate DCT (Pei et al. 2020), as well as the finding of and explanation for DCT off the equator in a regional ocean model (Cherian et al. 2021), and complement other climatological studies of mixing in the equatorial Pacific cold tongue focused on different questions, different metrics, and different models with coarser resolution (e.g., Ray et al. 2018; Holmes et al. 2019a; Huguenin et al. 2020; Deppenmeier et al. 2021). The analysis of the regional model also shows that the daily mean turbulent heat transport is logarithmically distributed, thus relatively rare events associated with aseasonal variability on time scales of days to weeks have a strong influence on and spatially covary with the time-mean transport.

We build understanding of the subseasonal part of aseasonal variability in mixing via large-eddy simulations (LES) that are embedded in a regional ocean model so that the simulated turbulence varies in the context of realistic variations in horizontal currents and temperature and atmospheric forcing over time scales from hours to more than a month. These LES address a key source of uncertainty in our regional model and all prior studies of ocean mixing on time scales from weeks to months using models: our regional models and all prior models are based on uncertain mixing parameterizations. Here, the LES are used to study the variability of explicit (rather than parameterized) turbulent mixing and DCT on time scales from days to a month for the first time. Our LES build on prior shorter simulations of diurnal cycles and shorter variability with idealized boundary conditions and forcing (Skyllingstad and Denbo 1994; Wang et al. 1996, 1998; Large and Gent 1999; Wang and Müller 2002; Pham et al. 2013) as well as how the diurnal cycles vary between the four seasons at 0°, 140°W (Pham et al. 2017; Sarkar and Pham 2019). Through both the analysis of the regional model and the LES, we confront the simulations of turbulence with observations and critically evaluate the model representations, albeit only at 0°, 140°W. Future observations are needed to evaluate and constrain modeled turbulence beyond 0°, 140°W in the Pacific cold tongue.

2. Methods

a. Ocean hindcast of the eastern equatorial Pacific, 1999–2016

Climatological statistics of vertical mixing throughout the equatorial Pacific cold tongue are derived from an ocean hindcast of the period 1999 through 2016 in the region from 170° to 95°W and from 12°S to 12°N in a submesoscale-permitting 1/20° configuration (Cherian et al. 2021) of the MITgcm (Adcroft et al. 2004; Marshall et al. 1997). As described previously (Cherian et al. 2021), the model is forced at the surface by fluxes derived from bulk flux algorithms and the JRA-55 based surface dataset for driving ocean–sea ice models (JRA55-do) atmospheric reanalysis (Tsujino et al. 2018) and at side boundaries by daily mean horizontal velocity, temperature and salinity from the Mercator Global Ocean reanalysis and Simulation (GLORYS) 1/12° ocean reanalysis. Solar radiation penetrates and warms the water below the surface, and there are no tides. Vertical mixing is represented by the K-profile parameterization (KPP) (Large et al. 1994), which was compared against and tuned to match LES of partially resolved DCT at 0°, 140°W (Large and Gent 1999). This hindcast is very similar to that of Cherian et al. (2021), where some observational validations are presented. The main technical difference between the two hindcasts, in addition to the different and longer simulated time interval, is that the model grid has a slightly coarser vertical resolution (2.5 m versus 1 m over the top 250 m), because the reduced vertical resolution had a negligible impact on the solutions in short tests and reduced the computational cost. The analysis is conducted on the saved daily mean temperature, salinity, and heat budget diagnostics. See Table 1 for a list of several of the most commonly used metrics to quantify and describe vertical mixing as well as the sections in which they are defined and discussed.

Table 1

A glossary table with definitions and sections where key metrics are defined.

Table 1

b. Large-eddy simulation hindcasts of turbulence over 34 days

To better understand and validate the subseasonal spatiotemporal variability in turbulent mixing on and off the equator, we report results from two 34-day LES that are hindcasts of upper-ocean turbulence in a small 306 m × 306 m × 108 m deep domain during the period from 2 October to 5 November 1985 at 0° and 3°N along 140°W in the equatorial Pacific cold tongue. Unlike the regional ocean hindcast and most other ocean models, the LES explicitly simulates rather than parameterizes the outer scales O(1) m of the turbulence and thus can provide insight into the physics of ocean mixing and DCT. However, the LES has a computational cost that is many orders of magnitude greater than the regional ocean model per unit simulated time and volume, hence the LES must be run for much shorter time intervals and in much smaller domains (Skyllingstad and Denbo 1994; Wang et al. 1996, 1998; Wang and Müller 2002; Pham et al. 2013, 2017; Sarkar and Pham 2019). A detailed description of the LES model is given in the appendix. In short, the LES is forced by variable 6-hourly air–sea fluxes (including a diurnal cycle of penetrating shortwave radiation) and larger-scale (15km) oceanic tendencies, such as advection and the pressure gradient force, derived from a regional ocean hindcast simulation of the entire Pacific cold tongue. The LES forcing is from the parent ocean model ROMS, not MITgcm, because ROMS solutions (based on earlier work of Holmes and Thomas 2015) were available earlier with all the necessary outputs. However, the domain, the horizontal resolution 1/20°, the vertical mixing scheme KPP, the 3-hourly surface forcing (including diurnal cycle of penetrating solar radiation) from JRA55-do are all the same in ROMS and MITgcm, and the mesoscale fields and parameterized mixing dynamics of interest are qualitatively similar [see the appendix for details and compare the results reported in Holmes and Thomas (2015) and Cherian et al. (2021)].

The inclusion of larger-scale oceanic tendencies of temperature and momentum from ROMS are an important novelty in these LES and crucial for sustaining realistic temperature and horizontal velocity profiles over time scales longer than a few days (Qiao and Weisberg 1997). These tendencies also provide a source of subseasonal variability on time scales from days to a month (Holmes and Thomas 2015; Cherian et al. 2021). Hence, an important point of reference is the one previous LES study of the eastern equatorial Pacific that incorporated large-scale tendencies (Wang et al. 1998). In addition to finer grid resolution, comparisons with an off-equatorial domain, and longer (34 versus 6 days) simulations than in Wang et al. (1998), the ocean tendencies used here also differ from those in Wang et al. (1998) in that they are derived from a realistic regional ocean model rather than idealized mathematical formulas. Thus, the large-scale oceanic conditions and related large-scale tendencies (as well as the air–sea fluxes) evolve on time scales from 6 h to 1 month during the simulations, in conjunction with the passage of a tropical instability wave and other mesoscale ocean variability. In addition, there is approximate dynamical consistency between the initial conditions, surface fluxes and interior tendencies, as well as between the LES at 0° and 3°N across this range of time scales. Hence, despite some broken feedbacks between the limited LES domain and the larger-scale ocean and atmosphere, the differences between the LES and the ocean model mean profiles of temperature and zonal momentum are always less than 0.5°C and 0.25 m s−1. That is, the turbulence simulated by LES, the surface fluxes, and the interior tendencies remain approximately consistent as if the LES was part of a two-way coupled regional system rather than an isolated domain throughout the 34-day simulations.

LES outputs include instantaneous statistics, such as the horizontally averaged turbulent vertical fluxes of heat and momentum among others, which are saved irregularly about every 2–5 simulated minutes and additionally binned into daily mean statistics for some analyses (to obtain the data and source code, see data availability statement). Note that all times are in UTC, and the local solar time is about 9 h behind UTC, so solar noon occurs at about 2100 UTC. All daily mean LES statistics, such as daily mean flux profiles, are calculated from 2100 UTC so that the 34 daily means begin and end at about solar noon, beginning on 2100 UTC 2 October 1985 and ending at 2100 UTC 5 November 1985.

c. Evaluation of the LES zonal velocity and temperature by comparison with observations

Comparisons with observations suggest that the LES yield plausibly realistic zonal velocity and temperature simulations with a few exceptions. Mean vertical profiles of temperature and zonal velocity are generally within observed ranges at 0°, 140°W where mooring observations from the Tropical Atmosphere Ocean (TAO) array (McPhaden et al. 2010) are available (Figs. 1 and 2). At 0°, 140°W, there is a clear depth range between about 10 and 75 m where the gradient Richardson number of the horizontally averaged profile, that is the vertical gradient of buoyancy over the squared vertical gradient of horizontal velocity
Rig=N2S2=b/z|uh/z|21/4,
is in a state of marginal instability as observed by Smyth and Moum (2013) (see Fig. 3). The LES results are presented at 3°N for comparison in Figs. 13, although mooring observations are not available at 3°N for validation. The observed annual mean climatology of zonal currents and temperature (Johnson et al. 2002) is plotted for comparison with the LES at 3°N, 140°W, but the observed annual climatology is insufficient to validate October mean profiles in the LES at 3°N because there is significant seasonal, interannual, and subseasonal variability. Perhaps the most notable difference between the two latitudes is that the shear is weaker on average at 3°N than at 0°, and Rig > 1/4 most of the time at 3°N. Hence, marginal instability Rig ≈ 0.25 is intermittent (about 25% of the time) from 20- to 70-m depth at 3°N rather than persistent as at 0°.
Fig. 1.
Fig. 1.

A comparison between the simulated (LES; solid lines) and (a) observed mean temperature and (b) zonal velocity profiles at 0° (blue) and 3°N (red) along 140°W. At 0°, 140°W, the observations (horizontal bars) span the interquartile ranges of all monthly means (September–November only) from the TAO mooring (1988–2018). At 3°N, 140°W, a ship-based annual climatology is plotted (Johnson et al. 2002), but these are more for reference than for validation since there is significant seasonal and interannual variability.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

Fig. 2.
Fig. 2.

Simulated (LES) and observed frequency spectra of (a) temperature and (b) zonal velocity at 25-m depth at 0° (blue) and 3°N (red) along 140°W. Observed spectra are calculated from the moored temperature sensor (10 min instantaneous sampling) and current meter (1-h average sampling) from the months September–November on the TAO mooring at 0°, 140°W for comparison (1988–2018). The observed spectra are calculated in overlapping time windows that are the same length as the LES simulations (with 17% of points overlapped in each window). The 10% and 90% quantile at each frequency (across all of the spectra windows) is plotted in light blue. The black dotted and blue lines are derived from LES: the sampling is instantaneous (averaged over a single time step) every 10 min in (a) or every 1 h in (b) and averaged spatially over a single grid cell/virtual mooring (black dotted) or the entire horizontal extent of the domain (blue). The spectrum from the virtual mooring (black) flattens similarly to the observations from the TAO mooring at frequencies higher than 3 cycles per day due to aliasing in (a).

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

Fig. 3.
Fig. 3.

Profiles of the median (thick lines) and interquartile range (iqr; thin lines) of the squared vertical shear of horizontal velocity S2, the vertical buoyancy gradient N2, and the gradient Richardson number Rig = N2/S2 (all of the horizontally averaged profiles). Shown are (a),(b) results from the LES at 0°N and (c),(d) results from the LES at 3°N. The dotted vertical line in (b) and (d) indicates Rig = 0.25 for reference.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

The diurnal cycle in temperature and zonal velocity is plausible but on the weaker side of the observed diurnal cycles at 0°, 140°W, for example, as shown at 25 m in Fig. 2. Consistent with observations, the modeled diurnal cycle is stronger at shallower depths (e.g., shallower than 15 m), weak but with a notable peak in the frequency spectra at intermediate depths (e.g., between 15 and 45 m), and difficult to discern from other nearby frequencies in the spectra at deeper depths (not shown). A detailed investigation of the mechanisms controlling the amplitude of the diurnal cycle of the horizontally averaged current and temperature profiles (and all other variables) is left for future work (for prior studies of the diurnal cycle and DCT at 0°, 140°W in LES, see, e.g., Wang et al. 1998; Pham et al. 2013, 2017). This study instead focuses on variability in daily averaged quantities.

The simulated temperature and velocity variance at time scales from days to weeks is generally realistic at 0°, 140°W. For example, the power spectra of temperature and zonal velocity at 25-m depth (Fig. 2) show that variance at periods from a few days to a month is reasonably realistic, but variability at internal wave time scales ranging from a few days to a few hours is consistently weak in the LES relative to the TAO mooring observations (as shown at 25 m). The weakness of internal wave activity at these frequencies is expected (qualitatively) in the LES since the parent ROMS model does not have tides or grid resolution at horizontal scales from 5.5 to 0.3 km (and only 8-m vertical resolution in the upper ocean), where much internal wave activity occurs and from which it cascades down to smaller scales (Gregg et al. 2003). That is, the embedded LES represents only a limited subset of interactions between internal waves, shear instabilities, and turbulence. First, the LES represents the response of small-scale shear instabilities, internal waves, and turbulence at horizontal wavelengths smaller than 300 m to large-scale internal waves (among other processes) at horizontal wavelengths 15km that are resolved by the parent model. Second, the LES represents some interactions between internal waves, shear instabilities, and turbulence at scales from about 1 to 300 m that are generated locally in the domain. In particular, the periodic horizontal boundary conditions allow internal waves to persist in the model domain and propagate vertically through the stratification. However, going beyond the comparison between the simulated (black dotted) and observed (light blue) temperature spectra in Fig. 2a to a detailed investigation of the internal waves and instabilities in the LES and observations (Lien et al. 1996; Smyth et al. 2011; Moum et al. 2011) is left for future work (for some analysis of these topics in other LES, see Pham et al. 2013, 2017).

Finally, the turbulence simulated by the LES is difficult to validate directly since direct observations of the turbulence are so limited in space and time. That said, the simulated turbulence is qualitatively and quantitatively similar to the turbulence observed by Lien et al. (1995) from 4 November to 12 December 1991 (as discussed in more detail below). And, previous studies in simpler model configurations show that the model simulates idealized test cases and turbulent flows with statistics that are consistent with basic conservation constraints (Watkins and Whitt 2020).

3. Spatial patterns, seasonal cycle, and aseasonal variability in the regional hindcast

Our analysis of the regional ocean model begins with the definition of the metrics to be used throughout the results (section 3a), then provides a description of the climatological time-mean spatial patterns (section 3b), seasonal cycles (section 3c), and aseasonal variability (section 3d) of ocean mixing in the model as well as comparisons to observations at 0°, 140°W.

a. Metrics of ocean mixing

We quantify and compare the downward heat flux due to ocean mixing FQ(z), which tends to cool the upper ocean on average, with the net downward surface heat flux Q0net=FQ(z=0)+PQ(z=0) (including turbulent fluxes F and penetrative fluxes P due to solar radiation), which tends to warm the upper ocean on average (Fig. 4). With regard to ocean mixing, we focus on the maximum over depth z of the daily mean downward turbulent heat flux FQmax=maxzFQ(z), where 〈〉 denotes a daily mean (and a horizontal average is implicit, over a single grid cell in the MITgcm and the entire domain in LES). Since the depth zmax at which 〈FQmax occurs varies in time and space, we also quantify zmax and compare it with the mixed layer depth (MLD, defined by the first depth 0.015 kg m−3 denser than the top 10 m) for reference (Fig. 5).

Fig. 4.
Fig. 4.

(a)–(f) Climatological spatial structure and seasonal cycle of downward heat fluxes in a regional ocean model of the equatorial Pacific Ocean cold tongue forced by atmospheric reanalysis from 1999 to 2016. The net air–sea flux Q0net is in (b) and (e), and the maximum flux due to ocean mixing 〈FQmax is in (c) and (f). Panels (b) and (c) are the zonal means from 95° to 170°W with the time-mean subtracted, and (e) and (f) are the time-means. In addition, we quantify the fraction of the zonal distance (a) and time (d) over which there is net cooling of the surface ocean due to air–sea exchange and ocean mixing, that is, Q0netFQmax<0. The flux due to ocean mixing 〈FQmax in (c) and (f) is defined as the maximum (over depth) of the daily mean downward turbulent heat flux, so the zonal and time means are calculated at a depth that varies in time and space that is plotted in Fig. 5.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

Fig. 5.
Fig. 5.

Climatological comparisons between (b),(e) mixed layer depth (MLD] and (c),(f) the depth zmax where the downward turbulent heat flux is maximum (i.e., the depth where 〈FQmax plotted in Fig. 4 occurs). As in Fig. 4, (b) and (c) are the zonal mean anomalies from the time mean, and (e) and (f) are the time-means. In addition, we quantify the fraction of the (a) zonal distance and (d) time over which the MLD is deeper than zmax. The MLD is defined to be the shallowest depth where water is 0.015 kg m−3 denser than the top 10 m in the daily mean density profile (since higher-frequency output is not available).

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

The maximum daily mean turbulent heat flux 〈FQmax, the daily net surface heat flux Q0net, and their difference Q0netFQmax provide useful measures of the significance of ocean mixing relative to the net surface heat flux in the upper-ocean heat and SST dynamics throughout the cold tongue. This is a simplified view because other terms also contribute to the heat budget above 〈FQmax in addition to 〈FQmax and Q0net, including penetration of radiative heat fluxes 〈PQmax below the depth zmax and advection (e.g., Moum et al. 2013). In addition, the precise role of ocean mixing in the heat budget depends on the depth to which the budget is integrated. Vertical mixing is generally significant if the heat budget is integrated vertically over a layer that is closely correlated with SST (Ray et al. 2018). Having stated the caveats, there are two main reasons we focus on 〈FQmax. First, it is intrinsically interesting because it essentially quantifies and bounds the maximum impact that mixing could have on the upper-ocean heat budget. Second, we aim to use 〈FQmax to model the whole vertical profile 〈FQ〉(z) in the upper ocean (see section 4g). The a priori motivation to focus on 〈FQmax in modeling 〈FQ〉〈z〉 is based on a hypothesis that 〈FQ〉(z) can be approximately reconstructed as an interpolation of three points: the surface flux 〈FQ〉(z = 0), a positive subsurface 〈FQmax if it exists, and a point of nearly zero flux at some depth deeper than zmax. In this manuscript, we quantify and parameterize 〈FQmax and then demonstrate that 〈FQmax can be used to predict 〈FQ〉(z), leaving an exposition of the relationships between 〈FQmax and the upper-ocean heat budget formalism to future work (but see Ray et al. 2018).

Although DCT is characterized by strong 〈FQmax and may contribute significantly to the climatological 〈FQmax, we choose not to distinguish DCT from other causes of 〈FQmax via a formal quantitative metric in this manuscript. This is because we want to characterize 〈FQmax across the cold tongue without assumption about the driving mechanisms, and DCT is not ubiquitous across the cold tongue (Cherian et al. 2021). In addition, even though DCT tends to be associated with strong 〈FQmax, it is not known if strong 〈FQmax is always indicative of DCT or why and to what degree 〈FQmax varies from day to day in DCT or otherwise. However, we refer to the turbulence driving the mixing descriptively as DCT where and when we feel the subjective criteria (based on prior studies) are met. In particular, prior studies have identified DCT as strong diurnally modulated turbulence in a marginally unstable stratified shear layer (Rig ≈ 1/4) just below the deepest nighttime MLD [for a recent review, see Cherian et al. (2021)].

b. Time-mean spatial patterns

We begin by characterizing the time-mean 〈FQmax, which contributes to sustaining relatively cool time-mean SSTs and net ocean heat uptake Q0net in the cold tongue by transporting heat downward from the mixed layer to the thermocline (Ray et al. 2018; Holmes et al. 2019a). Consistent with that interpretation, the comparisons between Q0net and 〈FQmax demonstrate that the time-mean surface flux and ocean mixing have similar spatial patterns (r2 = 0.7; Figs. 4e,f). Both Q0net and 〈FQmax are broadly elevated throughout the cold tongue relative to other areas and take similar area-average values between 6°S and 6°N from 95° to 170°W (77 W m−2 for 〈FQmax and 59 W m−2 for Q0net). In addition, both Q0net and 〈FQmax are enhanced by more than a factor of 2 near the equator (e.g., between ±2°) compared to the area means between 6°S and 6°N [Figs. 4e,f; see also Fig. 2 of Cherian et al. (2021) for snapshot plan views].

Closer inspection highlights several important differences in the climatological spatial structure of 〈FQmax and Q0net. First, 〈FQmax is significantly stronger than Q0net on average in an equatorial mixing band about 2° wide and centered slightly north of the equator that extends zonally through the entire domain (170°–95°W; see Fig. 4d). In this equatorial mixing band, the annual mean surface heat flux Q0net reaches a peak at just over 120 W m−2 at about 110°W and just south of the equator, whereas the downward heat flux due to ocean mixing 〈FQmax reaches a peak of just over 240 W m−2 at 130°W just north of the equator (cf. Figs. 4e,f). In addition, there is net cooling Q0netFQmax<0 over a greater fraction of the year and over more of the zonal distance in the equatorial mixing band, where Q0netFQmax<0 between 50% and 75% of the time (Fig. 4d). In the equatorial mixing band, the depth of the peak daily mean turbulent heat flux zmax ranges from about 90 m at 170°W to 30 m at 95°W (Fig. 5f). In addition, zmax is virtually always deeper than the MLD and ranges from about 20 to 60 m below the base of the mixed layer in the equatorial mixing band (cf. Figs. 5d–f). The deep zmax in the equatorial mixing band is consistent with prior studies showing that mixing is particularly strong and extends to particularly cold isotherms in this band (Holmes et al. 2019a; Deppenmeier et al. 2021). These results are all consistent with the established results that 1) ocean mixing is uniquely strong in the cold tongue near the equator and plays a leading role in the upper-ocean heat budget, 2) the turbulent heat flux peaks in the stratified ocean below the mixed layer, and 3) the intensity of ocean mixing is sensitive to the strong mean vertical shear in the horizontal velocity (e.g., Figs. 1 and 3) that arises from the eastward equatorial undercurrent at depth and westward South Equatorial Current at the surface.

At latitudes between 2° and 6°, both Q0net and 〈FQmax range from about 80 to 0 W m−2 (Figs. 4e,f). The depth zmax is closer to the base of the MLD than in the equatorial mixing band and just 10–30 m deeper than the MLD on average (cf. Figs. 5e,f). There is also a notable meridional asymmetry in net cooling Q0netFQmax<0; ocean mixing is stronger relative to the surface flux more frequently and over a significantly greater area to the north of the equator (50%–70%) than to the south (30%–40%; see Fig. 4d). This meridional asymmetry arises partly because 〈FQmax is stronger, by O(10) W m2, between about 2° and 5°N than between 2° and 5°S, but also partly because Q0net is stronger by O(10)W m2 between 2° and 5°S than between 2° and 5°N. The weaker downward surface heat fluxes Q0net to the north are consistent with warmer SSTs to the north (through their impact on sensible, latent, and longwave surface heat fluxes). In addition, the asymmetry in time-mean mixing 〈FQmax is qualitatively consistent with (but does not prove) the hypothesis that DCT and stronger ocean mixing events north of the equator arise due to stronger vertical shear in intermittent tropical instability waves and vortices that are also more energetic north of the equator as proposed by Cherian et al. (2021) (see Fig. 6b). The meridional asymmetry in mixing may also be a manifestation of a meridional asymmetry in SST in that warmer SSTs to the north may contribute to stronger upper-ocean temperature stratification that facilitates enhanced 〈FQmax.

Fig. 6.
Fig. 6.

The top row shows the hindcast aseasonal daily mean vertical heat fluxes during 2012 and 2013 along the 140°W meridian [(a) net surface flux Q0net, (b) ocean mixing 〈FQmax, and (c) the depth where strongest mixing occurs zmax]. (d)–(f) Maps that quantify the respective aseasonal interquartile ranges over all latitudes and years 1999–2016. Aseasonal variability is defined by subtracting the mean seasonal cycle (i.e., a daily annual climatology, which is averaged over 18 years and then smoothed with a 15-day moving average), from the total signal at each grid point.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

The model results can be validated using multiyear microstructure observations that are available from χpods on moorings at 0°, 140°W, from which an average annual cycle of the turbulent heat flux between 20 and 60 m has been estimated from deployments between 2008 and 2012 (Moum et al. 2013; see also Smyth et al. 2021). Although the observed and modeled time intervals are not identical, we average the model heat fluxes over the same depth range 〈FQ20–60 and compare them with the observations of Moum et al. (2013) in Fig. 7. We find that the modeled annual mean 〈FQ20–60 is somewhat more than a factor of 2 larger than observed (150 versus 66 W m−2). Restricting the model averaging to the observed years (2008–12) does not change this discrepancy. The maximum flux 〈FQmax is another 80 W m−2 higher than 〈FQ20–60, because zmax ≈ 70 m is below the 20–60-m averaging range and the modeled fluxes depend strongly on depth (Fig. 5f). Although it is not fully understood how the time-mean surface heat flux Q0net is mechanistically coupled to the time-mean subsurface flux 〈FQmax, it is interesting in light of their high degree of spatial correlation and similar magnitudes that Q0net is substantially stronger in the model than reported in Moum et al. (2013): Moum et al. (2013) report 55 W m−2 while the modeled mean is twice as large at 110 W m−2. This may indicate that the modeled heat uptake is biased high; this would be consistent with too-strong mixing assuming incomplete compensation for the too-strong mixing by other terms in the heat budget. However, other observational estimates of Q0net are higher than those reported by Moum et al. (2013). For example, Trenberth and Fasullo (2018) report an estimate of about 90 W m−2 for the 2000–16 period, and the model seems to be within the range of various estimates from 2001 to 2010 reported by Liang and Yu (2016) (roughly 60–120 W m−2 at 0°, 140°W; see their Fig. 2). Hence, we do not conclude that the modeled time-mean surface heat flux Q0net in MITgcm is biased, although it is on the higher end of available estimates.

Fig. 7.
Fig. 7.

Climatological annual cycle of the downward turbulent heat flux at 0°, 140°W in the MITgcm regional ocean model, including monthly means at zmax (〈FQmax, thick red) as well as monthly means from 20- to 60-m depth 〈FQ20–60 (thick gray). Corresponding minima and maxima of monthly 〈FQ20–60 (thin gray) and 〈FQmax (thin red) from 1999 to 2016 are included. For comparison, the observational climatology of 〈FQ20–60 from χpods (Moum et al. 2013) is plotted in black circles. The 95% confidence intervals for the monthly mean 〈FQmax from ROMS and LES (roughly October 1985) as well as the TIWE observations (roughly November 1991) are in magenta, green, and blue, respectively. Note, however, that the LES and TIWE are computed as (ρcp/)Fb = 1.37 × 109FbFQ (W m−2), where ρ, cp, and α are the reference density, specific heat, and thermal expansion coefficient of seawater, respectively, g is the acceleration due to gravity, and Fb is the downward turbulent buoyancy flux. Data from two other shorter field experiments (not shown) resulted in means of roughly 400 W m−2 in October/November 2008 (Moum et al. 2009) and 100 W m−2 in November 1984 (Gregg et al. 1985; Moum and Caldwell 1985) (see Fig. 2d of Moum et al. 2009).

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

c. Seasonal cycle

The climatological seasonal cycle is another metric by which 〈FQmax and Q0net are similar at first glance but exhibit notable differences on closer inspection (Figs. 4b,c). Both the seasonal cycles of 〈FQmax and Q0net exhibit significant diversity. Four different varieties are present between 6°S and 6°N: one-peak-one-trough, two-peaks-one-trough, two-troughs-one-peak, and two-peaks-two-troughs, and there are variations in the timing, duration and amplitude of the peaks and troughs (peaks are red and troughs are blue in Figs. 4b,c). In addition, these spatiotemporal structures of the seasonal cycles in 〈FQmax and Q0net are uncorrelated (pattern correlation r2 < 0.01 for zonal-mean seasonal anomalies, i.e., between the fields in Figs. 4b,c).

The phase and amplitude of the seasonal cycle of mixing in the equatorial mixing band is similar to observations at 0°, 140°W, even though the modeled time-mean 〈FQ20–60 is about a factor of 2 higher than observed [see Fig. 7 and Moum et al. (2013)]. In this equatorial band (see Fig. 4c), the seasonal cycle of mixing 〈FQmax is not in phase with and has a larger peak-to-trough amplitude than the surface fluxes Q0net (Figs. 4a–c). In particular, the peak-to-trough amplitudes are about 70 and 140 W m−2 for Q0net and 〈FQmax, respectively. It is notable that the observations reported by Moum et al. (2013) show a somewhat smaller peak-to-trough seasonal cycle in Q0net50W m2, although the phasing is similar to the model. In particular, Q0net is minimum at about yearday 190 and maximum at about yearday 80, whereas mixing reaches a minimum at about yearday 90 and a maximum at about yearday 215. There is also a secondary peak in mixing at about the new year. Hence, there is a strong seasonal cycle in Q0netFQmax, which is negative (net cooling) at 0° along more than 80% of longitudes between 170° and 95°W in the boreal summer and early autumn (Fig. 4a), when the SST cools in the equatorial mixing band (Moum et al. 2013). Conversely, Q0netFQmax<0 at only about 20% of longitudes in boreal spring (Fig. 4a), when the SST warms (Moum et al. 2013). These results highlight again the importance of seasonal variations in ocean mixing for the seasonal cycle of cold tongue SST. The seasonal cycle of the MLD and the depth zmax are highly correlated throughout the cold tongue. In the equatorial mixing band, minima are achieved at about yearday 90 and local maxima at about yearday 210 (Figs. 5b,c; r2 = 0.76). But, the amplitude of the seasonal cycles are relatively modest with peak-to-trough amplitudes of only about 15 and 25 m for the MLD and zmax, respectively.

A qualitatively similar seasonal cycle is found off the equator in Q0net (Fig. 4b), but the off-equatorial seasonal cycle in 〈FQmax (Fig. 4c) is much weaker and has a different phase relative to the equator. In addition, the amplitude of the seasonal cycle in 〈FQmax is notably asymmetric across the equator. There is a much stronger seasonal cycle to the north than the south; for example, the peak-to-trough seasonal cycle amplitude is about 30 W m−2 at 4°N but only 10 W m−2 at 4°S (Fig. 4c). The stronger seasonal cycle in ocean mixing to the north of the equator is qualitatively consistent with (but does not prove) the hypothesis that the seasonal cycle is due at least partially to tropical instability waves, which have greatest variance from boreal summer to winter (Cherian et al. 2021), although precisely quantifying and even determining the sign of the rectified effect of tropical instability waves on ocean mixing is difficult (Holmes and Thomas 2015).

d. Aseasonal variability

Like the dissipation of turbulent kinetic energy (Crawford 1982; Moum et al. 1989; Smyth et al. 2021), the maximum daily mean turbulent heat flux 〈FQmax is highly variable and logarithmically distributed (Fig. 8). Thus, the arithmetic averages of 〈FQmax are significantly influenced by relatively infrequent strong mixing events (in contrast to Q0net). It follows that the processes underpinning the aseasonal variability in general and infrequent strong mixing events in particular are significant for climatological statistics including the time mean. Hence, we conclude this section on the regional climatological statistics by quantifying the aseasonal variability in 〈FQmax and Q0net, both to provide climatological context for and motivate a more detailed discussion of subseasonal variability in 〈FQmax simulated in LES (for discussion of the physics of subseasonal variability in ocean models, see, e.g., Holmes and Thomas 2015, 2016; Inoue et al. 2019; Liu et al. 2019a,b, 2020; Cherian et al. 2021). When plotting (in Fig. 6) and reporting the statistics from the MITgcm results in this section, the aseasonal variability is separated from the full signal (i.e., defined) by subtracting a daily climatology, which is first averaged over 18 years and then smoothed by applying a 15-day moving average. Hence, aseasonal variability includes both interannual and intra-annual time scales.

Fig. 8.
Fig. 8.

(a),(b) Relative probability distributions of the maximum daily mean turbulent heat flux due to ocean mixing 〈FQmax, (c),(d) the daily mean net surface heat flux Q0net, and (e),(f) the depth zmax at which 〈FQmax occurs. Histograms are included for both 0°, 140°W (blue) and 3°N, 140°W (red) for (left) the 18-yr MITgcm simulation as well as (right) the 34-day LES in October 1985 (red and blue histograms) and the 38-day TIWE experiment at 0°, 140°W in November 1991 (dark-blue edged bars). Note that the data from LES and TIWE are computed based on buoyancy fluxes, e.g., (ρcp/)Fb = 1.37 × 109FbFQ.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

First, it may be noted that the minimum and maximum monthly means 〈FQmax across the 18 simulated years (thin red lines in Fig. 7) span a factor of 3–8 or roughly 50–250 W m−2. So, any given monthly mean is reasonably likely to differ from the corresponding monthly climatology by a factor of 2. In addition, time series of aseasonal 〈FQmax along 140°W in Fig. 6b reveal variability in 〈FQmax of hundreds of watts per square meter on time scales from days to months in 2012–13. A qualitative comparison of the modeled distribution of 〈FQmax at 0°, 140°W (Fig. 8a) to the spread of observed daily mean dissipation from χpods in Fig. B1 of Smyth et al. (2021) suggests that there are fewer instances of weak mixing and a narrower distribution of mixing values in the model compared to observations at 0°, 140°W. But, the different vertical averaging precludes a quantitative comparison (see Fig. 7). Aseasonal variability in mixing exhibits a spatial pattern that is similar to the mean (cf. Figs. 6e and 4f), consistent with a logarithmic distribution. In particular, the interquartile range (IQR) of aseasonal 〈FQmax variability reaches 150 W m−2 in the strong equatorial mixing band but drops from 60 to 20 W m−2 at latitudes from 2° to 6°. There is also a notable seasonal cycle to aseasonal variability, which is stronger in boreal autumn than boreal spring (Fig. 6b; cf. Fig. 4c), as well as meridional asymmetry across the equator with larger aseasonal variability to the north than to the south (Fig. 6e). Both the seasonal cycle and meridional asymmetry of aseasonal variability are consistent with tropical instability wave activity (Halpern et al. 1988; Moum et al. 2009; Cherian et al. 2021). There is also notable aseasonal variability in the depth at which maximum ocean mixing occurs zmax (Figs. 8e and 6c,f). The aseasonal variability in zmax has a similar spatial pattern as the time-mean zmax (cf. Figs. 5f and 6f). The IQR of aseasonal zmax variability is about 40 m at 170°W and 10 m at 95°W. This zonal gradient in the aseasonal IQR of zmax is qualitatively similar at all latitudes from 6°S to 6°N, but the IQR is elevated by 10–20 m in the equatorial mixing band relative to other latitudes (Fig. 6f).

Aseasonal variability in Q0net is qualitatively different from aseasonal variability in 〈FQmax (cf. Figs. 6a,b and cf. Figs. 8a,c). First, Q0net is more nearly normally distributed (Fig. 8c), and the IQR varies relatively little across the cold tongue from about 45 to 70 W m−2 (Fig. 6d). In addition, the maximum Pearson’s r2 between aseasonal anomalies in Q0net and 〈FQmax is only 0.15 (at about 2.5°S, 110°W) and the correlations are mostly much smaller (mean r2 = 0.02 and median r2 = 0.01). Hence, the aseasonal net surface heat flux Q0net anomalies do not covary with the aseasonal turbulent heat flux 〈FQmax anomalies in the model (see Fig. 3i of Smyth et al. (2021) for a qualitatively similar observational result at 0°, 140°W).

4. Subseasonal variability on and off the equator in the LES

To build further understanding of the subseasonal variability in ocean mixing and DCT, both on and off the equator, we turn to the LES (see section 2 and the appendix for details). First, section 4a describes how the metrics of ocean mixing (originally defined in section 3a) are applied to the LES and in observational comparisons to the Tropical Instability Wave Experiment (TIWE; Lien et al. 1995). Section 4b summarizes and contextualizes these LES via comparisons with prior results. Then, sections 4c4g quantify the daily mean turbulent buoyancy flux 〈Fb〉, including the vertical extent of strong mixing (section 4c), the energetics of mixing (section 4d), and the covariability of mixing with nonturbulent variables that may facilitate mixing parameterization (sections 4e4g).

a. Metrics of mixing and observational comparisons

Throughout the analysis of the LES we continue to focus on the maximum of the daily mean flux profile, but we shift our focus from the turbulent heat flux 〈FQmax to the turbulent buoyancy flux 〈Fbmax to leverage links with turbulence energetics, in which Fb appears but not FQ (see the appendix for the relevant equations). However, to facilitate comparisons between the LES and the MITgcm simulations and the χpod observations (Fig. 7), we often report
ρcpgαFbFQ,
where ρ is the reference density of seawater, cp is the specific heat of seawater, g is the acceleration due to gravity, and α is the thermal expansion coefficient of seawater. In the LES, the coefficient fraction is constant 1.37 × 109 (W m−2 s3 m−2) (see the appendix for details) and we apply the same constant scaling to produce FQ from the TIWE data in Figs. 7 and 8. At zmax, the relative error in approximating a constant ratio FQ/Fb is roughly
NT2N21,
assuming the turbulent vertical fluxes of temperature and buoyancy can be approximated using local flux–gradient relationships (i.e., downgradient diffusion) and have the same turbulent diffusivity such that
FQFbρcpT/zb/z=ρcpgαNT2N2,
where NT2=gαT/z. The errors from this approximation are small; the 68 days of LES estimates of 〈FQmax yields an estimate for the mean bias of +6% (−7% and +20% at 0° and 3°N, respectively) and a standard deviation of 26% (10% and 30% at 0° and 3°N, respectively).

We explicitly compare the LES results to 38 days of observations of DCT from the TIWE at 0°, 140°W in November–December 1991 (Lien et al. 1995). The TIWE dataset is a uniquely good point of comparison in that it includes a similarly long 38 days of hourly averaged turbulence profiles based on thousands of microstructure casts (roughly 6–7 per hour) as well as relevant ocean velocity and density profiles and surface flux information derived from continuous occupation of a station at 0°, 140°W by two ships. Although turbulent heat and buoyancy fluxes are not directly measured, they are inferred to within about a factor of 2 using the relationship Fb = Γϵ where ϵ is the observed dissipation rate of turbulent kinetic energy and a mixing efficiency factor is assumed to be a constant Γ = 0.2 at depths below 20 m for simplicity (Osborn 1980; Gregg et al. 2018). The maximum of the daily mean turbulent buoyancy flux 〈Fbmax is calculated after first binning hourly mean Fb profiles into daily means 〈Fb〉 at 1-m vertical resolution and then smoothing 〈Fb〉 with a 10-m moving average. The resulting 38-day mean 〈Fbmax(ρcp)/() ≈ 〈FQmax based on the TIWE data is plotted in Fig. 7 and the distribution of the daily means is shown in Fig. 8 for context. As in the analysis of the LES, we apply the assumption of constant 〈FQmax/〈Fbmax to the TIWE observations (in Fig. 7). We estimate that this assumption yields larger but still modest high bias in the 〈FQmax of up to about +30%, which is smaller than the factor of 2 observational uncertainty. Hourly mean velocity and density from the ADCP and CTD, respectively, are extended to the surface by replicating the top reliable value before calculating vertical gradients in horizontal velocity and buoyancy and related derived quantities.

b. Summary and context

We chose to run LES at 0° and 3°N along 140°W in October 1985, which was characterized by neutral oceanic Niño index, so mixing is expected to be reasonably strong but not maximal both at and north of the equator (Figs. 4 and 7) (see also Warner and Moum 2019; Huguenin et al. 2020; Deppenmeier et al. 2021). Tropical instability waves are a dominant cause of subseasonal variability in currents and density in the LES and are also an important driver of aseasonal variations in mixing (e.g., Moum et al. 2009; Cherian et al. 2021). The 34-day simulations are just long enough to span one full tropical instability wave period, but the tropical instability wave spanned by these LES is not especially strong. The peak-to-trough amplitude of the meridional velocity averaged from 25 to 75 m is only 45 cm s−1 at 0° and 88 cm s−1 3°N (Fig. 9). For comparison, the peak-to-trough amplitude of the meridional velocity variability during the TIWE is about 50 cm s−1 (Plate 3 in Lien et al. 1995) and quite similar to the LES at the same site, even though tropical instability waves were weak during the TIWE due to the onset of El Niño conditions. In contrast, Moum et al. (2009) observed strong turbulent mixing in the presence of a strong tropical instability wave with peak-to-trough meridional velocity amplitude of about 1.5 m s−1 at 0°, 140°W during October–November 2008 in La Niña conditions (see also Inoue et al. 2012, 2019).

Fig. 9.
Fig. 9.

Time series of zonal and meridional velocity (color), temperature (white contours; °C), mixed layer depth (MLD; dashed magenta), the depth where the bulk Richardson number Rib = 0.2 (HRib; thin black), and the base of the low-gradient Richardson number layer Rig < 0.35 (HRig; thick black) in the LES at 0° and 3°N along 140°W. All fields are defined from horizontally averaged profiles. The MLD is defined to be the shallowest depth where water is 0.015 kg m−3 denser than the top 10 m in the instantaneous but horizontally averaged density profile. All time tick marks are at 0000 UTC; local solar time at 0°, 140°W is about 9 h behind UTC, so local solar noon is at about 2100 UTC.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

We find that the mixing in the LES qualitatively reflects the seasonal, interannual and mesoscale context. The 34-day mean 〈FQmax in the LES at 0° (about 110 W m−2) is just above the minimum of the 18 October means simulated from 1999 to 2016 in the MITgcm. In addition, the LES parent ROMS simulation with the same KPP mixing scheme as the MITgcm also has a rather low mean 〈FQmax ≈ 140 W m−2 (compared to an October mean of about 275 W m−2 in the MITgcm), suggesting that the large-scale conditions (e.g., shear, stratification, and air–sea fluxes) in the simulated October 1985 are not exceptional but not as conducive to strong mixing as is typically the case from 1999 to 2016. However, the 34-day mean 〈FQmax is still larger than the 38-day mean 〈FQmax from the TIWE observations (77 W m−2) and about 50% above the climatological 〈FQ20–60 (averaged from 20- to 60-m depth) from χpod observations in October. Noting that 〈FQmax/〈FQ20–60 ≈ 1.5–2 in the MITgcm, these results suggest that the mixing in the LES is fairly typical for October. Consistent with this conclusion, the mixing in our LES is also stronger than that simulated in the LES of Sarkar and Pham (2019) (see also Pham et al. 2017), in which the resolved turbulent heat flux was about 60 W m−2 and ϵ ≈ 10−7 m2 s−3 at the maximum MLD over three days in October at 0°, 140°W (compared to 〈FQmax ≈ 110 W m−2 and 〈ϵmax ≈ 3 × 10−7 m2 s−3 here). Conversely, the mixing in our LES is substantially weaker than the especially strong mixing (with time-mean FQ ≈ 400 W m−2 and ϵ ≈ 10−6 m2 s−3) observed by Moum et al. (2009) at 0°, 140°W in the midst of a strong tropical instability wave during October–November 2008 in La Niña conditions. Finally, the time-averaged 〈FQmax in the LES at 3°N, 140°W is about 30 W m−2, that is 1/4 to 1/3 of the magnitude in the LES at 0°, 140°W. This ratio of 〈FQmax at 3°N over 〈FQmax at 0° is approximately consistent with the climatological ratio from 1999 to 2016 found in the MITgcm even though the mixing in the LES is weaker at both latitudes (Fig. 4f).

Consistent with earlier studies, we find that the diurnal cycle is the dominant mode of temporal variability in the turbulence near the surface, and the simulated diurnal cycles at 0°, 140°W exhibit many of the previously observed and simulated features of DCT at that location (Gregg et al. 1985; Moum et al. 1989; Schudlich and Price 1992; Peters et al. 1994; Lien et al. 1995; Wang et al. 1998; Large and Gent 1999; Danabasoglu et al. 2006; Smyth et al. 2013; Pham et al. 2013, 2017; Smyth et al. 2017; Sarkar and Pham 2019; Pei et al. 2020; Cherian et al. 2021). For example, FQ is shown in Figs. 10a and 11a and can be compared to the time series of the dissipation rate of turbulent kinetic energy ϵ observed during the TIWE in Plate 7 of Lien et al. (1995) (ϵ ≈ 5Fb ≈ 4FQ × 10−9 m2 s−3 below the MLD; see also Fig. 12). During the daytime, shortwave radiation stratifies a shallow near-surface layer where wind-driven turbulence is confined and accelerates a near-surface current with strong vertical shear. During the afternoon and early evening, the stabilizing net surface buoyancy flux weakens and eventually becomes destabilizing. The near-surface shear and stratification descend downward toward the highly sheared and stratified but marginally unstable layer below, where Rig ≈ 1/4 (Fig. 12). At the same time, strong turbulent heat and momentum fluxes FQ and Fm as well as dissipation rates ϵ descend downward as well (Figs. 1113; see the appendix for definitions). During nighttime and early morning, turbulence penetrates deeply below the MLD and into the stratified thermocline (i.e., between about 30- and 90-m depth), where downward turbulent heat fluxes FQ reach a subsurface maximum of hundreds of watts per square meter. Strong turbulent momentum fluxes extract kinetic energy from the shear to drive strong heat fluxes and dissipation rates in the thermocline (Figs. 1113; the energetics is quantified in section 4d). The strong turbulence that is energized locally below the MLD often persists there for hours while the extent and intensity of the near-surface turbulence decline with increasing solar radiation in the morning. In addition, on many nights and mornings there are 2–4 bursts of particularly strong turbulence that cause the heat flux to be elevated by up to hundreds of watts per square meter for hours (Fig. 11a) as observed (Smyth et al. 2017).

Fig. 10.
Fig. 10.

Time series of the net surface heat flux Q0net (left axis; blue), the magnitude of the wind stress |τ| (right axis; red), and the subsurface downward turbulent heat flux FQ profiles from October to November 1985 in the LES at (a) 0° and (b) 3°N along 140°W. Overlaid on FQ are the depth at which the bulk Richardson number Rib = 0.2 (HRib; thin black line), the depth of the maximum daily mean downward heat flux zmax (+ symbols), the daily maximum MLD (defined from the horizontally averaged LES density profiles; magenta circles), and the base of the low gradient Richardson layer Rig < 0.35 (HRig; thick black line). The daily mean meridional velocity averaged from 25- to 75-m depth is in blue; the origin is at a depth of 100 m, a 1-m spacing corresponds to 10 cm s−1, and the peak-to-trough amplitudes are about 40 cm s−1 at 0° and 90 cm s−1 at 3°N. For consistency with other results in section 4, we plot (ρcp/)Fb = 1.37 × 109FbFQ (W m−2), where ρ, cp, and α are the reference density, specific heat, and thermal expansion coefficient of seawater, respectively; g is the acceleration due to gravity; and Fb is the downward turbulent buoyancy flux. All time tick marks are at 0000 UTC, but local solar time at 0°, 140°W is about 9 h behind UTC, so local solar noon is at about 2100 UTC. Daily mean statistics (e.g., zmax indicated by + symbols) are calculated from 2100 UTC so that the averages begin and end near solar noon.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

Fig. 11.
Fig. 11.

As in Fig. 10, but zoomed in on a few days in November and with the addition of the MLD (dashed magenta) and the DCT penetration depth zpen (ϵ ≥ 2 × 10−8 m2 s−3; thin green). The MLD is defined to be the shallowest depth where water is 0.015 kg m−3 denser than the top 10 m in the instantaneous but horizontally averaged density profile.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

Fig. 12.
Fig. 12.

As in Fig. 11, but plots show (a),(b) the vertical buoyancy gradient N2; (c),(d) the squared vertical shear S2; (e),(f) Rig = N2/S2; and (g),(h) the rate of dissipation of kinetic energy ϵ. It may be noted that there are a few instances of elevated dissipation 10−8 < ϵ < 10−7 m2 s−3 below the deepest depths of DCT (zpen; green line) in (h) where Rig > 1. However, these instances of elevated dissipation near the bottom are dominated by dissipation of the mean-flow kinetic energy, and the turbulent fluxes and energetics depend strongly on the subgrid-scale parameterization in the LES (A6) and (A7), may be influenced by the bottom boundary, and should be interpreted with caution.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

Fig. 13.
Fig. 13.

As in Fig. 11, but turbulent vertical momentum fluxes projected onto the shear, i.e., (Fm ⋅ ∂uh/∂z)/|∂uh/∂z|.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

At first glance, the diurnal cycles of turbulent heat fluxes FQ at 3°N in Fig. 10b seem to differ qualitatively from those at 0°, consistent with the hypothesis that equatorial turbulence is enhanced relative to turbulence at higher latitudes due to DCT associated with the strong mean shear between the eastward undercurrent and the westward surface South Equatorial Current (Figs. 1a and 9a). However, DCT and strong heat and momentum fluxes do occur at 3°N in conjunction with strong vertical shear of horizontal velocity (Figs. 9b,d), most prominently on 3–5 November when the subsurface turbulence at 3°N exhibits all of the qualitative features described in the previous paragraph in reference to the DCT at 0° (Figs. 1113). In addition, some days in early and mid-October exhibit downward turbulent heat fluxes FQ below the MLD, although the intensity of these subsurface heat fluxes is weaker than most days at 0° and there are no obvious nighttime turbulent bursts. These results add significant new support to the hypothesis that DCT occurs off the equator. Off-equatorial DCT has previously been hypothesized based on ocean model results with fully parameterized DCT (Pei et al. 2020; Cherian et al. 2021) but has not been previously simulated in LES or observed in microstructure. Although the diurnal cycle of DCT remains a topic of interest for future analysis of our LES, this topic has received substantial attention in prior LES studies (Wang et al. 1998; Large and Gent 1999; Pham et al. 2013, 2017; Sarkar and Pham 2019) and we leave further analysis of the diurnal cycle in these LES to future work.

The objective of this analysis of the LES is to build understanding of the subseasonal variability of the daily mean 〈FQ〉 on time scales from days to weeks, building on our analysis of the regional MITgcm. The distributions of 〈FQmax, zmax, and Q0net in Fig. 8 show how this variability simulated in the LES compares to the variability in the MITgcm and observed in the TIWE data and generally support the suggestion that the LES are representative of fairly typical conditions in October. As explored in more detail in the subsequent sections, a motivating hypothesis (e.g., Cherian et al. 2021; Smyth et al. 2021) is that the spatiotemporal variability in the vertical shear in the upper ocean (which is defined more precisely later, but see Figs. 9 and 12c,d) is perhaps the most important driver of the day-to-day and spatial variability in DCT and 〈FQmax (e.g., in Figs. 8b and 10). This vertical shear is strong on average above the equatorial undercurrent along the equator, but the shear is also highly variable and intermittently strong throughout the cold tongue (e.g., as shown in Fig. 9) due to a variety of interacting equatorial waves and instabilities (Moum et al. 2009; Inoue et al. 2012; Jing et al. 2014; Tanaka et al. 2015; Holmes and Thomas 2015, 2016; Inoue et al. 2019; Liu et al. 2019b,a; Pei et al. 2020; Liu et al. 2020; Cherian et al. 2021). Hence, strong DCT and 〈FQmax vary in time and space and occur intermittently throughout the cold tongue (and at 3°N specifically) when the shear is strong. Over the next few sections, we explore the hypothesis that shear covaries with 〈FQmax on and off the equator and more generally seek to identify covariates that provide information about 〈FQmax without direct simulations or observations of turbulence.

c. Shear, stratification, Richardson numbers, and the vertical extent of strong turbulence

Previous studies have identified the gradient Richardson number of the horizontally averaged profile Rig [defined in (1)] as an important indicator of the occurrence of DCT and strong ocean mixing in the equatorial Pacific (Pacanowski and Philander 1981; Peters et al. 1988; Large et al. 1994; Smyth and Moum 2013). Consistent with these previous studies, we find that Richardson numbers provide some useful information about the spatiotemporal structure and in particular the vertical extent of strong mixing in the LES and the TIWE observations. Below, we show that two Richardson numbers, both of which are based on the horizontally averaged velocity and density profiles, can be used to model the depth zmax where daily mean turbulent vertical heat fluxes 〈FQ〉 are maximum as well as the daily maximum depth zpen to which strong turbulence penetrates. We define zpen based on a constant threshold in the dissipation rate of turbulent kinetic energy ϵ. It is reasonably straightforward to identify a depth zpen from inspection of time–depth series of ϵ or FQ profiles (as in the midlatitudes, see Brainerd and Gregg 1995). After brief trial and error, we identify the shallowest depth where ϵ < 2 × 10−8 m2 s−3 to be a useful threshold applicable to both of the LES (Figs. 1113) and the TIWE observations. For reference, this ϵ threshold corresponds to a turbulent heat flux of roughly 7 W m−2, which is an order of magnitude smaller than typical 〈FQmax and about two orders of magnitude smaller than peak nighttime heat fluxes FQ during turbulent bursts.

The depth zmax varies from about 10 to 70 m at 0° and from 20 to 60 m at 3°N over time scales ranging from days to weeks (black plus symbols in Fig. 10; see also Fig. 8f). The occurrence of zmax deeper than the nighttime MLD is hypothesized to be an indicator of DCT and strong heat fluxes. Consistent with this suggestion, the nighttime maximum MLD is shallower than zmax at 0° on 29 of 34 days and 9 m shallower on average, but the nighttime MLD is deeper than zmax at 3°N on 32 of 34 days and 9 m deeper on average. Qualitatively, we interpret these results as an indication that DCT occurs about 85% of the time at 0°N and about 5% of the time at 3°N, but there is not a one-to-one correspondence between DCT and zmax deeper than MLD as demonstrated on 3–4 November at 0°, 140°W in Fig. 11. Although the nighttime maximum MLD is somewhat correlated with the depth zmax, the relationship is in fact fairly scattered and the nighttime MLD can only explain about 30% of the variance in zmax across both LES. On the other hand, about half of the simulated variance in the depth zmax can be explained by HRib (r2 = 0.5), the depth at which the mean-profile bulk Richardson number Rib = 0.2. Here,
Rib=ΔbHRibΔu2+υt2,
where Δb and Δu are the bulk buoyancy and velocity differences between the depth HRib and the top 0.1HRib, υt is a turbulent velocity scale that depends on the surface forcing as in Large et al. (1994), and the depth HRib is identified iteratively using the default parameters of Large et al. (1994) in an implementation of KPP by Smyth et al. (2002) (Fig. 14a). The inclusion of υt in Rib systematically deepens HRib by 6 m on average, but has marginal and probably insignificant benefit on the best linear model or correlation with zmax (increasing r2 by 15%). The specific threshold Rib = 0.2 was chosen via trial and error. Larger and smaller thresholds for Rib were not as useful for identifying zmax, but there may be room for future refinement of the model for zmax, because half of the variance in zmax is not explained by HRib.
Fig. 14.
Fig. 14.

(a) The depth zmax of maximum daily mean turbulent heat flux is related to the depth HRib at which the bulk Richardson number is 0.2. (b) The daily maximum depth zpen to which DCT penetrates (ϵ > 2 × 10−8 m2 s−3) is related to the low-gradient Richardson layer depth HRig (above which Rig < 0.35).

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

The deepest depth to which DCT penetrates each day zpen also varies significantly from about 40 to 90 m at 0° and from about 35 to 85 m at 3°N (Fig. 10). And again, the Richardson number—in this case the local gradient Richardson number Rig of the horizontally averaged profiles—provides useful information about zpen each day. In particular, we define HRig as the base of the deep-cycle layer, which is defined by a low-gradient Richardson number Rig < 0.35. In practical applications (e.g., to the TIWE data), Rig is noisy and the definition of HRig requires some additional logic and filtering. In particular, the deep-cycle layer is defined by applying a rectangular filter of about 35 h and 35-m depth to a logical field that equals one where Rig < 0.35 and the depth is below the daily maximum HRib. The second threshold based on HRib is necessary because Rig sometimes rises to high values within the weakly stratified turbulent boundary layer above HRib, particularly at 3°N and even fairly deep within HRib during nighttime (Figs. 12e,f). With regard to Rig, a threshold Rig = 0.25 has a theoretical basis that makes it appealing (Miles 1961; Howard 1961; Holt et al. 1992; Rohr et al. 1988), and Rig = 0.25 has been used previously for identifying the base of the deep-cycle layer in observations at 0°, 140°W (Lien et al. 1995; Smyth et al. 2021). However, we found via trial and error that a somewhat larger threshold Rig = 0.35 is more useful across the LES at 0° and 3°N as well as the TIWE observations. Our approach is also supported by the LES of Pham et al. (2017), in which simulated turbulent bursts penetrate below the layer defined by a threshold Rig = 0.25 in DCT as in our LES. A linear regression on HRig, −6 + 1.1HRig has slope near one, intercept near zero, and explains 80% of the variance in the daily maximum zpen (see Fig. 14b).

Finally, it may be noted that these relationships between zmax, HRib, zpen, and HRig are useful beyond the LES. For example, the TIWE observations reveal similar variability and relationships between zmax, HRib, zpen, and HRig as the LES at 0° (cf. blue stars and black + symbols in Figs. 14a,b). And, HRig is also a useful lower boundary for the deep-cycle layer in the MITgcm regional model with DCT parameterized by KPP (section 3), but the threshold has to be increased to Rig = 0.5 (Cherian et al. 2021).

d. (Non)local energetics of 〈Fbmax

To begin to understand why the intensity of 〈Fbmax (and by extension 〈FQmax) varies in time and space, it is useful to consider these variations in the context of the daily mean turbulent kinetic energy budget under the premise that some of the variability in 〈Fbmax is related to variations in the kinetic energy available to drive turbulent mixing (see the appendix for details). In this kinetic energy budget, the tendency or rate of change of turbulent kinetic energy is driven by vertical transport 〈Tmax, shear production 〈SP〉max = 〈Fm ⋅ ∂uh/∂zmax, dissipation 〈ϵmax, and buoyancy flux 〈Fbmax (Figs. 15a,b). Integrated over a full day, the budget is dominated by a net source due to shear production and net sinks due to buoyancy flux and dissipation at zmax. That is, all other terms (tendency and vertical transport) are subdominant in all but one day and contribute less than 20% of the energy for dissipation and buoyancy flux 〈ϵmax + 〈Fbmax when mixing is strong (roughly 〈Fbmax > 10−7.5 m2 s−3; see Fig. 15a). Hence, the shear production of turbulent kinetic energy at zmax 〈SP〉max is highly correlated with 〈ϵmax + 〈Fbmax (r2 = 0.98; Fig. 15a). In addition, when mixing is strong, 〈Fbmax is in approximately constant proportion to 〈SP〉max (about 0.2) and to 〈ϵmax (about 0.25) (Figs. 15a,b). When the buoyancy flux is weaker 〈Fbmax < 10−7.5 m2 s−3, the ratio Rif1=SPmax/Fbmax declines from 5 to 2 as Rig1=S2max/N2max decreases from 5 to 0.5 and 〈Fbmax weakens to 10−8.5 m2 s−3 (Figs. 15b,c). Here, Rif is the flux Richardson number (e.g., Osborn 1980; Venayagamoorthy and Koseff 2016). In addition, the relationship between Rif1 and Rig1 is associated with a relationship between Rig1 and the turbulent Prandtl number Prt1=Rif/Rig, which quantifies how the turbulent diffusivity of buoyancy declines relative to the turbulent viscosity as Rig1 decreases (Fig. 15d). Finally, it is notable that the turbulent kinetic energy budget contains significant nonlocal (transport) contributions at low 〈Fbmax < 10−7.5 m2 s−3. In particular, transport 〈Tmax ≈ 〈Fbmax + 〈ϵmax − 〈SP〉max becomes a more significant and scattered contributor to the dissipation and buoyancy flux, as 〈Tmax/(〈Fbmax + 〈ϵmax) reaches values of 40% and takes both signs (Fig. 15a).

Fig. 15.
Fig. 15.

Relationships between various terms in the daily mean turbulent kinetic energy budget at the depth zmax where the downward turbulent buoyancy flux is maximum (〈SP〉max +〈Tmax ≈ 〈Fbmax + 〈ϵmax; see the appendix for details). The depths zmax are plotted as + symbols in Fig. 10. Buoyancy flux 〈Fbmax is plotted against (a) shear production over buoyancy flux plus dissipation 〈SP〉max/(〈Fbmax + 〈ϵmax) and (b) shear production over buoyancy flux (i.e., the inverse flux Richardson number Rif1=SPmax/Fbmax). The inverse gradient Richardson number of the horizontally averaged profile Rig1=S2max/N2max is shown in color on all four panels and on the y axes against (c) Rif1 and (d) Prt1=Rif/Rig (the inverse turbulent Prandtl number Prt1 is the ratio of the turbulent diffusivity of buoyancy over the turbulent viscosity of momentum). The thick black line in (c) is the 1–1 line, the thin solid line is a fit to LES of a coastal boundary layer under a hurricane by Watkins and Whitt (2020), and the thin dashed line is a fit to atmospheric boundary layer observations by Anderson (2009), which parameterizes the subgrid-scale Prt1 in the LES. The two days with most anomalously low Rif1 [in (b) and (c): Rif1=0.9and1.6] and high Prt1 [in (d): Prt1=0.4and1.8] also have the largest relative nonlocal sources of turbulent kinetic energy 〈Tmax/(〈Fbmax + 〈ϵmax) ≈ 1 − 〈SP〉max/(〈Fbmax + 〈ϵmax) [i.e., the points with lowest values in (a); 0.3 and 0.6]. Plus (+) symbols are from LES at 0° and circles (○) from 3°N.

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

In summary, when mixing is strong (〈Fbmax > 10−7.5 m2 s−3), the energetics are dominantly local to the depth zmax with shear production balanced by dissipation plus buoyancy flux and nearly constant Rif ≈ 0.2 and Rig ≈ 0.25 both on and off the equator. However, the energetics of 〈Fbmax in general (including weaker values) are more complex: the energetics are approximately local on average, but nonlocal (transport) contributes 10%–40% to the energetics on many days and takes both signs. In addition, Rif systematically varies with Rig, both of which take values substantially higher than the canonical values (Rif ≈ 0.2 and Rig ≈ 0.25) on most days at 3°N. At 0°, the canonical DCT and local dynamics are the norm, but at 3°N the canonical DCT and local dynamics are the exception rather than the norm. The simulated energetic relationships encapsulated in relationships between Rif, Prt, and Rig (Figs. 15c,d) are qualitatively consistent with observations in the atmospheric boundary layer (Anderson 2009), a previous LES of ocean turbulence under a hurricane in the coastal midlatitudes reported by Watkins and Whitt (2020), and direct numerical simulations (Venayagamoorthy and Koseff 2016). However, it still remains somewhat uncertain whether the relationships modeled here in the LES are in any sense universal, especially given the significance of nonlocal (transport) dynamics at weak 〈Fbmax.

e. Scaling 〈Fbmax based on the horizontally averaged velocity and buoyancy profiles

Building on the result that 〈Fbmax varies in concert with other metrics of the turbulence energetics such as the shear production and dissipation rate, this section demonstrates how the intensity of 〈Fbmax (and by extension 〈FQmax) covaries with readily measured or simulated nonturbulent variables such as horizontally averaged velocity and buoyancy profiles as well as the surface momentum and buoyancy fluxes. In a second step, we evaluate scaled predictions of 〈Fbmax derived from the LES results by applying the scaling to the independent TIWE observations.

We begin by quantifying the relationship between the mean profile Rig and the intensity of mixing at zmax motivated by popular existing parameterizations of the local intensity of turbulent diffusion as a function of Rig (Pacanowski and Philander 1981; Peters et al. 1988; Large et al. 1994). We find that the simulated inverse Richardson number Rig1 at zmax can explain most of the simulated variability in 〈Fbmax across the LES at both 0° and 3°N [Fig. 16a; r2 = 0.6 for the regression log10(〈Fbmax) ∼ 〈S2max/〈N2max]. On the other hand, Rig1 on its own does not explain the temporal variability in 〈Fbmax very well at 0° in either the LES (r2 = 0.2) or the TIWE observations (r2 = 0.0). These results are consistent with the hypothesis that Rig is a useful predictor of the intensity of mixing across a range of Rig that includes marginal instability (1Rig0.25, as at 3°N) but a poor predictor of the intensity of mixing when marginal instability is either persistent (Rig ≈ 0.25, as at 0°N) or marginal instability never occurs and Rig > 1 is always very large (for background on marginal instability, see Thorpe and Liu 2009; Smyth and Moum 2013; Smyth 2020). For better comparison with previous studies, we also show that variations in the effective turbulent diffusivity of buoyancy at zmax (Kb = 〈Fbmax/〈N2max) are more weakly correlated with Rig1 [r2 = 0.2 for log10(Kb) ∼ 〈S2max/〈N2max in LES; r2 = 0.0 in TIWE] and thus not well explained by Rig1 (Fig. 16b) or Rig-based parameterizations (Pacanowski and Philander 1981; Peters et al. 1988; Large et al. 1994). However, it may be noted that the underlying variables in the regressions for Kb and 〈Fbmax are actually the same, 〈S2max, 〈N2max, and 〈Fbmax, which suggests that the relatively poor correlation between log10(Kb) and Rig1 may be improved by simply reformulating the predictor function of 〈S2max and 〈N2max. Indeed, a general two-variable linear regression of log10Kb on log10S2max and log10N2max yields an r2 = 0.6 for log10(Kb) ∼ log10[〈S2max(〈N2max)−3/2]. In summary, although the LES yield results that are loosely consistent with previous studies (e.g., Fig. 16b), there is significant room to improve parameterizations of ocean mixing in the cold tongue. That is, Rig is useful but certainly not sufficient to explain all of the spatiotemporal variability in 〈ϵmax or 〈Fbmax in the eastern equatorial Pacific (Moum et al. 1989; Zaron and Moum 2009). Other variables and combinations of variables likely contain valuable information about 〈Fbmax in DCT and in general across the cold tongue.

Fig. 16.
Fig. 16.

Relationship between Rig1=S2max/N2max and (a) 〈Fbmax and (b) Kb = 〈Fbmax/〈N2max at zmax (i.e., at the depths indicated by the + symbols in Fig. 10). Averaging diffusivity directly in (b) yields quantitatively different results but qualitatively the same conclusion that Kb is at best weakly related to Rig. Overlaid in (b) are parameterizations of turbulent diffusivity as a function of Richardson number from Pacanowski and Philander (1981) (PP) Peters et al. (1988) (PGT), and Large and Gent (1999) (KPP).

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1

In an attempt to refine our understanding of the mean-profile properties that drive temporal variations in 〈FQmax ∼ 〈Fbmax, we conduct a more general multivariable linear regression analysis with the aim of identifying an optimal power law product (e.g., a product of the generic form cxaybzd…, with variables x, y, z… and constants a, b, c, d… to be determined) to model the maximum buoyancy flux 〈Fbmax as a function of horizontally averaged and readily measured (and modeled) properties, including surface fluxes and the horizontally averaged profiles of velocity and density but without a priori knowledge of the depth zmax at which 〈Fbmax occurs. Although a formulation as a power law may seem arbitrary, this choice is motivated by two factors. First, many familiar mixing models are expressed as a product of terms (e.g., a diffusivity times a gradient, or a mixing efficiency times a momentum flux times a shear; e.g., Fig. 16b) and are therefore power laws. In addition, 〈Fbmax is thought to be logarithmically distributed (see section 3d and Fig. 8), and power laws are readily amenable to linear regression after applying a log transform.

Numerous variables were considered in the regressions, but we only highlight two low-complexity models that we identified. First, the most useful variable that we identified for modeling the combined LES output from 0° to 3°N is the vertical shear Sb. In particular, if Sb is a bulk shear defined by a least squares linear fit to the daily mean and horizontally averaged velocity profile from HRig to 5-m depth, then we find that Sb alone can explain about 70% of the daily variance in 〈Fbmax from both the LES at 0° and 3°N (Fig. 17a; 〈Fbmax ≈ 3.1 × 10−6|Sb|0.88; r2 = 0.7 in log10 space ignoring the TIWE data). In an encouraging result, independent validation of the Sb scaling of 〈Fbmax on the TIWE data is quite good (r2 = 0.5 with little mean bias) and even better than the LES at 0°N alone (r2 = 0.2). In addition, including the TIWE data in the regression in Fig. 17c has little impact on the optimal linear model, which seems fairly robust with relatively narrow confidence intervals on the parameters (cf. Figs. 17a,c). However, the model fit to the LES 〈Fbmax can be improved substantially by adding the surface friction velocity due to the wind stress u*=|τ|/ρ as a variable (τ is the wind stress vector). The optimal linear model based on these two variables Fbmax0.16|Sb|0.98u*2.05 explains about 90% of the LES variance and 70% at 0° or 3°N alone (Fig. 17b). In independent validation on the TIWE data, the two-variable model explains only 40% of the TIWE variance and also has a slight mean bias (Fig. 17b). Including the TIWE observations in the two-variable regression in Fig. 17d leads to a fairly substantial change in the optimal two-variable model 0.0017|Sb|0.92u*1.2 and somewhat reduces the correlation at 3°N in the LES but reduces the mean bias in the TIWE data and slightly improves the corresponding correlation (cf. Figs. 17b,d). These results suggest that although wind stress certainly provides useful information about 〈Fbmax, the available data (including 108 days spanned by the LES and TIWE) is only marginally sufficient to provide a robust linear model based on both Sb and u*.

Fig. 17.
Fig. 17.

Maximum daily mean turbulent buoyancy flux 〈Fbmax scales with (a),(c) oceanic bulk vertical shear Sb and (b),(d) even more closely with a product of Sb and the magnitude of the surface wind stress |τ|=u*2ρ. The scalings are obtained via linear regression on the LES output in (a) and (b), which includes 34 days at 3°N (black open circles) and 34 days at 0°N (black +), or on the 68 days of LES output plus 38 days of TIWE data (blue asterisks) in (c) and (d). Hence, the TIWE observations serve as an independent validation of the regressions in (a) and (b) and constrain the regressions in (c) and (d). The predictors include Sb, which is derived from a linear fit to the mean velocity from HRig to 5-m depth (thick black lines in Fig. 4), and the friction velocity u*=|τ|/ρ. All variables are log-transformed and Pearson’s r in the panel titles is calculated in log space. The various diagonal black lines indicate where the data are along the 1–1 line, within a factor of 2, and within a factor of 3. With 95% confidence intervals, the scalings are as follows: (2–6) × 10−6|Sb|(0.7–1.0) in (a) , (1200)×102|Sb|(0.91.1)u*(1.62.5) in (b), (2–6) × 10−6|Sb|(0.8–1.0) in (c), and (0.031.3)×102|Sb|(0.81.0)u*(0.91.6) in (d).

Citation: Journal of Physical Oceanography 52, 5; 10.1175/JPO-D-21-0153.1