1. Introduction

Small-scale turbulent mixing of heat in the tropical Pacific is an important component of the sea surface temperature (SST) budget and contributes to changes in SST over a range of time scales. Variations in diapycnal turbulent transport play a role in the seasonal cycle of SST (Moum et al. 2013) and the diurnal cycle in SST (Bernie et al. 2005; Danabasoglu et al. 2006). Modulations in turbulence at time scales of weeks to months associated with intraseasonal Kelvin waves (Lien et al. 1995; McPhaden 2002), the Madden–Julian oscillation (Chi et al. 2014), and tropical instability waves (TIWs) (Menkes et al. 2006; Moum et al. 2009; Inoue et al. 2012) may have an important impact on the mean state of the Pacific Ocean and thus on global climate.

Generated in the eastern tropical Pacific and Atlantic Oceans, TIWs propagate westward with wavelengths of 700–1600 km and periods of 15–40 days (Qiao and Weisberg 1995; Kennan and Flament 2000; Willett et al. 2006; Lyman et al. 2007). TIWs are generated through barotropic and baroclinic instability of the mean equatorial current system (Philander 1976; Cox 1980; Masina et al. 1999; Lyman et al. 2005) and vary seasonally and interannually, obtaining peak amplitudes in boreal fall and winter and in La Niña conditions (Contreras 2002; An 2008). Near the equator, TIW meridional advection of the equatorial fronts results in variations of several degrees Celsius in SST, organized into alternating warm and cold phases. TIWs and their associated vortices [tropical instability vortices (TIVs)] drive lateral heat fluxes that warm the cold tongue by ~1°C month⁻¹ (Menkes et al. 2006; Jochum et al. 2007; Graham 2014), potentially contributing to the asymmetry of the ENSO cycle (An 2008; Imada and Kimoto 2012). Jochum and Murtugudde (2006) suggested that TIWs warm the cold tongue not through typical eddy mixing, but instead through modifications of the air–sea fluxes and vertical entrainment. The modifications of air–sea fluxes by TIWs have received considerable attention (Thum et al. 2002; Seo et al. 2007; Small et al. 2009), while the modifications of vertical entrainment by TIWs have been observed only recently.

These observations have shown that TIWs modulate vertical mixing in the upper Equatorial Undercurrent (EUC) (Lien et al. 2008; Moum et al. 2009; Inoue et al. 2012). Here, the gradient Richardson number (Ri) of the flow is preconditioned to drop below the critical value for shear instability of ¼, where

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is the buoyancy frequency squared, is the buoyancy, g is the acceleration due to gravity, is the potential density, is a reference density, and are the zonal and meridional velocities, respectively. A telltale signature that shear instabilities have been active is for a flow to be in a state of marginal stability, where the Richardson number is close to ¼ and the reduced shear squared,

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is near zero. Indeed, observations show that for three-quarters of the year, the upper EUC is in this state (Smyth and Moum 2013). Using a Lagrangian float, Lien et al. (2008) observed that the vertical entrainment flux at the base of the mixed layer varied with the TIW phase along the equator, with the highest values occurring leading into the TIW warm phase. They attributed this increase in entrainment flux to an increase in the and pointed out that it could not be explained by variations in wind-driven mixing. More recently, Inoue et al. (2012) observed modifications of deep-cycle turbulence (a layer of strong turbulence below the mixed layer that shows a distinct diurnal cycle), mixed layer depth, and the turbulent heat flux with TIW phase using a 2-week time series of microstructure profiles at 0°, −140°E. Their measurements show that the highest values of , dissipation, and turbulent heat flux occur at the leading edge of the TIW warm phase when the meridional velocity switches from northward to southward, consistent with the observations of Lien et al. (2008).

The mechanism through which TIWs modulate small-scale turbulent mixing is not known. Moum et al. (2009) hypothesized that the extra meridional shear added by TIWs can push the marginally stable reduced shear squared above the EUC core over the threshold for shear instability producing stronger turbulence. However, the meridional shear associated with TIWs is significantly weaker than the zonal shear in this region [see Fig. 4 of Inoue et al. (2012)]. TIWs also influence both zonal velocities and the density field near the equator (Lyman et al. 2007; Jochum et al. 2007; Inoue et al. 2012), suggesting that other processes may be at play. It is also unclear if TIW-induced variations in mixing lead to a net cooling or warming of equatorial SSTs. Moum et al. (2009) observed turbulent mixing associated with TIWs that drove surface cooling of 1°–2°C month⁻¹. In contrast, the modeling study of Menkes et al. (2006) found that variations in the vertical turbulent entrainment flux on TIW time scales had a rectified warming influence of 0.37°C month⁻¹. In this article, we identify the mechanism responsible for the modulation of mixing by TIWs in a regional ocean model of the equatorial Pacific with a highly resolved TIW field. These simulations are combined with a one-dimensional (1D) vertical mixing model to quantify the net turbulent heat fluxes associated with TIWs and assess their sensitivity to the vertical mixing parameterization scheme.

The article is organized as follows: Section 2 describes the ocean model and evaluates its performance in the upper EUC in comparison to observations from the Tropical Atmosphere Ocean (TAO) mooring array (McPhaden et al. 1998). Section 3 examines the factors influencing mixing on TIW time scales in the upper EUC, using both the model and TAO data. Section 4 examines the dynamics behind the modulation of mixing by TIWs using a Lagrangian analysis. Section 5 examines the total influence of TIWs on the turbulent heat flux using a simple 1D mixing model of the EUC, and section 6 discusses and summarizes the results.

2. Model and general flow description

This section describes the ocean model setup (section 2a) and its main features, evaluates its performance in comparison to observations (section 2b), and discusses a feature of the observations not well represented in the model, that of marginal stability (section 2c).

b. General flow description and evaluation of model performance

The model produces vigorous TIWs with meridional velocities reaching 0.5 m s⁻¹ on the equator (Fig. 1b) that advect the north equatorial front across the equator, inducing SST variations of the order of 2°C (Figs. 1a, 2), consistent with the findings of Qiao and Weisberg (1995) and Inoue et al. (2012). North of the equator, TIVs centered around 4°N are characterized by an anticyclonic circulation with vorticity close to −f (f is the Coriolis parameter) that induces variations in zonal velocity and stratification (Holmes et al. 2014; Fig. 2). These features and the spatial structure of the TIWs and TIVs are consistent with the observations of Qiao and Weisberg (1995) and Kennan and Flament (2000).

Daily averaged (a) temperature, (b) meridional velocity, (c) stratification, (d) zonal shear, (e) meridional shear, (f) meridional diffluence, and (g) reduced shear squared along the equator on yearday 262.5 of the basinwide ROMS simulation. The dark green lines indicate the depth of the KPP mixed layer, the gray lines indicate the depth of the EUC core, and the black contours are isopycnals at 0.2 kg m⁻³ spacing. High generally occurs at the leading edge of a TIW warm phase and is accompanied by high zonal shear and preceded by meridional diffluence.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (a) temperature, (b) meridional velocity, (c) stratification, (d) zonal shear, (e) meridional shear, (f) meridional diffluence, and (g) reduced shear squared along the equator on yearday 262.5 of the basinwide ROMS simulation. The dark green lines indicate the depth of the KPP mixed layer, the gray lines indicate the depth of the EUC core, and the black contours are isopycnals at 0.2 kg m⁻³ spacing. High generally occurs at the leading edge of a TIW warm phase and is accompanied by high zonal shear and preceded by meridional diffluence.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (a) temperature, (b) meridional velocity, (c) stratification, (d) zonal shear, (e) meridional shear, (f) meridional diffluence, and (g) reduced shear squared along the equator on yearday 262.5 of the basinwide ROMS simulation. The dark green lines indicate the depth of the KPP mixed layer, the gray lines indicate the depth of the EUC core, and the black contours are isopycnals at 0.2 kg m⁻³ spacing. High generally occurs at the leading edge of a TIW warm phase and is accompanied by high zonal shear and preceded by meridional diffluence.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (a) zonal shear, (b) stratification, (c) reduced shear squared, and (d) vertical diffusivity at 72-m depth on yearday 262.5 of the high-resolution ROMS simulation. Also shown are contours of SST at 0.75°C intervals, with the 25° isotherm highlighted in thickness. The dashed line indicates the equator. In (c), three patches of high are visible at the leading edge of TIW warm phases.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (a) zonal shear, (b) stratification, (c) reduced shear squared, and (d) vertical diffusivity at 72-m depth on yearday 262.5 of the high-resolution ROMS simulation. Also shown are contours of SST at 0.75°C intervals, with the 25° isotherm highlighted in thickness. The dashed line indicates the equator. In (c), three patches of high are visible at the leading edge of TIW warm phases.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (a) zonal shear, (b) stratification, (c) reduced shear squared, and (d) vertical diffusivity at 72-m depth on yearday 262.5 of the high-resolution ROMS simulation. Also shown are contours of SST at 0.75°C intervals, with the 25° isotherm highlighted in thickness. The dashed line indicates the equator. In (c), three patches of high are visible at the leading edge of TIW warm phases.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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In this article, we focus on the upper EUC shear layer between the EUC core and the mixed layer, where the shear, stratification, and are largest (Figs. 1c–e,g, 2a–c). This layer contains the turbulent deep cycle and upper-core layers observed by Inoue et al. (2012). These are manifest in the model as an enhancement of the daily averaged vertical diffusivity, which reaches values >10⁻³ m² s⁻¹ at these depths (Fig. 2d). The model contains variations in a number of properties, including the , with TIW phase. These modulations are discussed in section 3.

Profiles of model vertical shear and stratification averaged over the TIW season (September–December) of the last year of the basinwide simulation compare favorably with data taken from the TAO array mooring at 0°, −140°E (Fig. 3). The TAO data are averaged over the neutral ENSO years 1996, 2001, 2003, and 2005. The zonal shear in the model compares well with the TAO data in the upper EUC shear layer (cf. red dashed line with black line in Fig. 3a), but the shear below the EUC core is too weak. The average EUC core depth is 10 m too deep in the model (134 m in ROMS vs 122.5 m in the TAO observations). The stratification is stronger in the surface layer, and the pycnocline is slightly weaker in the model (cf. red dashed line with black line in Fig. 3b). However, the stratification due to salinity is not included in the observations, and the profile of stratification due to salinity from the model (green dashed line in Fig. 3b) suggests that the comparison may be improved if salinity data from the TAO mooring were available. The model compares well with the observations throughout the upper 150 m where data are available. Above 80 m, the mean is close to zero in both the observations and the model, in agreement with the observations of Smyth and Moum (2013). However, the daily averaged does reach positive values on some occasions, as discussed in the next section.

Comparison of mean profiles at 0°, −140°E from the TAO array mooring (black lines) with the basinwide ROMS simulation without and with adjustment to marginal stability (red dashed and blue dotted lines, respectively). The profiles are averaged over the September–December period. The TAO data are averaged over the neutral ENSO years 1996, 2001, 2003, and 2005. (a) Mean zonal velocity shear and (b) mean stratification, where the green dotted line indicates the buoyancy gradient due only to salinity in ROMS. The TAO data are calculated only with temperature. (d) Mean reduced shear squared. RMS temporal variations in the (c) zonal and meridional shear squared, (e) reduced shear squared using only temperature stratification, and (f) temperature stratification. The RMS values are obtained by taking the square root of the total power between the frequencies 1/50 and 1/5 day⁻¹ calculated from the power spectrum of the respective variables in the months September–December. The error bars represent the ±2σ standard error spread over the four neutral ENSO years included in the TAO calculation.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Comparison of mean profiles at 0°, −140°E from the TAO array mooring (black lines) with the basinwide ROMS simulation without and with adjustment to marginal stability (red dashed and blue dotted lines, respectively). The profiles are averaged over the September–December period. The TAO data are averaged over the neutral ENSO years 1996, 2001, 2003, and 2005. (a) Mean zonal velocity shear and (b) mean stratification, where the green dotted line indicates the buoyancy gradient due only to salinity in ROMS. The TAO data are calculated only with temperature. (d) Mean reduced shear squared. RMS temporal variations in the (c) zonal and meridional shear squared, (e) reduced shear squared using only temperature stratification, and (f) temperature stratification. The RMS values are obtained by taking the square root of the total power between the frequencies 1/50 and 1/5 day⁻¹ calculated from the power spectrum of the respective variables in the months September–December. The error bars represent the ±2σ standard error spread over the four neutral ENSO years included in the TAO calculation.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Comparison of mean profiles at 0°, −140°E from the TAO array mooring (black lines) with the basinwide ROMS simulation without and with adjustment to marginal stability (red dashed and blue dotted lines, respectively). The profiles are averaged over the September–December period. The TAO data are averaged over the neutral ENSO years 1996, 2001, 2003, and 2005. (a) Mean zonal velocity shear and (b) mean stratification, where the green dotted line indicates the buoyancy gradient due only to salinity in ROMS. The TAO data are calculated only with temperature. (d) Mean reduced shear squared. RMS temporal variations in the (c) zonal and meridional shear squared, (e) reduced shear squared using only temperature stratification, and (f) temperature stratification. The RMS values are obtained by taking the square root of the total power between the frequencies 1/50 and 1/5 day⁻¹ calculated from the power spectrum of the respective variables in the months September–December. The error bars represent the ±2σ standard error spread over the four neutral ENSO years included in the TAO calculation.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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c. Marginal stability

In the model, the daily averaged occasionally reaches positive values well above the criteria for K–H instability (Figs. 1g, 2c). Smyth and Moum (2013) showed that 6-hourly averaged profiles of the Richardson number in the upper EUC remain in a state of marginal stability, a characteristic not reproduced by the model. In these regions the KPP vertical diffusivity (Fig. 2d) approaches its maximum interior value of K₀ = 2 × 10⁻³ m² s⁻¹ (see section 5). This suggests that the mixing scheme is not able to provide enough mixing at high in order to restrict the to negative or near-zero daily averaged values.

To evaluate the influence of this issue on our results, we constructed an algorithm to enforce marginal stability offline for all regions in the simulations where . The algorithm is similar to the Price et al. (1986) mixing scheme and iteratively applies vertical fluxes of temperature, salinity, and momentum in a Prandtl number one ratio until regions of are removed, subject to the constraints of conservation of heat, salt, and momentum (see the appendix for a description of the algorithm). The resulting profiles have regions of marginal stability (Fig. 4; between 100 m and the surface) bordered by thin regions with strong momentum and buoyancy gradients (Fig. 4; near 100 m), in agreement with theories of eddy-driven mixing (Holzer and Siggia 1994). This algorithm will be used in sections 3 and 4 to evaluate the impact of not enforcing marginal stability on our results. The KPP interior mixing parameterization for shear instability and its relation to marginal stability are examined in more detail in section 5.

Vertical profiles of (a) velocity, (b) temperature, (c) salinity, (d) vertical shear, (e) stratification, and (f) reduced shear squared at 0°, −140°E on yearday 268.5 from the model. The solid lines show the original profiles, and the dashed lines show the profiles after adjustment to marginal stability for unstable regions using the algorithm described in the appendix. Note the reduction in shear and stratification in the unstable region between 40- and 80-m depth, and the consequent accumulation of shear and stratification on the edges of this region.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Vertical profiles of (a) velocity, (b) temperature, (c) salinity, (d) vertical shear, (e) stratification, and (f) reduced shear squared at 0°, −140°E on yearday 268.5 from the model. The solid lines show the original profiles, and the dashed lines show the profiles after adjustment to marginal stability for unstable regions using the algorithm described in the appendix. Note the reduction in shear and stratification in the unstable region between 40- and 80-m depth, and the consequent accumulation of shear and stratification on the edges of this region.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Vertical profiles of (a) velocity, (b) temperature, (c) salinity, (d) vertical shear, (e) stratification, and (f) reduced shear squared at 0°, −140°E on yearday 268.5 from the model. The solid lines show the original profiles, and the dashed lines show the profiles after adjustment to marginal stability for unstable regions using the algorithm described in the appendix. Note the reduction in shear and stratification in the unstable region between 40- and 80-m depth, and the consequent accumulation of shear and stratification on the edges of this region.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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3. Eulerian analysis of TIW modulation

This section examines the variations in shear, stratification, and mixing on TIW time and length scales. Temperature, velocity, vertical shear, and stratification all vary around their mean state along the equator as a function of TIW phase (Figs. 1, 2). The highest , where the most mixing is expected, occurs on the leading edge of TIW-induced warm perturbations (Fig. 2c), highlighted by near-surface bowl-shaped isopycnals (Figs. 1a,g; near −159°, −149°, −141°, and −125°E). This is in agreement with the observations of Lien et al. (2008). This phase of the TIW generally corresponds to a change in meridional velocity from northward to southward (Fig. 1b), in agreement with the observations of Inoue et al. (2012). Each component of the , the stratification, zonal shear, and meridional shear (Figs. 1c–e) varies at TIW length scales along the equator. While the variations in meridional shear (Fig. 1e) are the clearest, they are weaker in magnitude than the variations in zonal shear and stratification and are not clearly correlated with the (cf. Figs. 1e and 1g near patches of high ). While the stratification shows large variations relative to its mean, it appears to be stronger in patches of high (cf. Figs. 1c and 1g). The zonal shear also shows large variations relative to its mean, with strongly negative zonal shear patches closely correlated with high patches (cf. Fig. 1d with Fig. 1g and Fig. 2a with Fig. 2c). This suggests that increases in zonal shear are responsible for the patches of high .

To quantitatively analyze temporal variations in the shears, stratification, and mixing due to TIWs, we calculated their root-mean-square (RMS) variation over the TIW frequency band by taking the square root of the total power between periods of 5 and 50 days from the power spectrum of the time series at each depth at 0°, −140°E (Figs. 3c,e,f). Average RMS values were also obtained from the TAO observations by calculating individual spectra for September–December of each year. The shape of the profiles and the comparative results were found to be insensitive to the frequency window used. The vertical motion of the EUC core did not influence the results discussed below, as tested by calculating the RMS values in EUC core relative depth coordinates (not shown).

There are strong temporal variations in at TIW frequencies (Fig. 3e), with the RMS value approaching the magnitude of the mean (cf. Figs. 3e and 3d). As for the variations with longitude (Fig. 1), the temporal variations in at 0°, −140°E are because of variations in zonal shear squared [ in Fig. 3c] and stratification (Fig. 3f). The variations in meridional shear squared [ in Fig. 3c] at TIW frequencies are much weaker than the other components in both model and observations. This suggests that TIW meridional shear is not directly responsible for the modulation of vertical mixing, contrary to the hypothesis of Moum et al. (2009). Jing et al. (2014) also noted that temporal variations in the amplitude of 2-h–8-day oscillations in the upper EUC, potentially linked to higher-frequency motions and turbulence, were more correlated with variations in the EUC zonal shear than with the meridional shear of TIWs.

The model and observed RMS values agree well below the surface layer (cf. red dashed lines and black lines in Figs. 3c,e,f). The temporal variation in stratification (Fig. 3f) is weaker in the model than the observations, which may be a result of the weaker thermocline in the model (Fig. 3b). The largest discrepancy between model and observations is in the temporal variations in zonal shear squared in the upper EUC between 120 and 40 m (cf. red dashed and black lines in Fig. 3c). The vertical structure is different, with the model zonal shear varying too much in the upper half of this region and too little in the lower half. These differences were traced to the failure of the KPP interior mixing scheme to enforce marginal stability.

RMS profiles calculated using the marginal stability adjustment algorithm (see the appendix) are more similar to the observations (blue dotted lines in Fig. 3). The mean quantities show only small improvements, with a reduction in the mean zonal shear, stratification, and between 75 and 40 m as a result of the additional mixing (cf. blue dotted to red dashed lines in Figs. 3a,b,d). However, the RMS variation in zonal shear squared shows large changes, with a reduction between 85 and 40 m and an increase around 100 m (cf. blue dotted line and red dashed line in Fig. 3c), consistent with the mixing and expulsion of gradients from regions with . Corresponding to the changes in zonal shear squared variations, there are changes to variations in (Fig. 3e), with only minimal changes to variations in stratification (Fig. 3f).

Temporal variations in zonal shear squared and stratification are similar in magnitude (cf. Figs. 3c and 3f), suggesting that they are equally important in driving modulations in the . However, the patches of high , and thus high mixing, are generally associated with high shear magnitudes and relatively high stratification (e.g., compare Figs. 1c,d,g and Figs. 2a,b,c near patches of high ), as opposed to anomalies of low stratification and low shear. This suggests that the patches of high mixing are driven by increases in shear, as opposed to decreases in stratification. This hypothesis is supported by the Lagrangian diagnostic analyses performed in the next section.

4. Lagrangian analysis of high mixing patches

In this section, we examine the dynamical processes responsible for the increase in on the leading edge of the TIW warm phase. We analyze a set of Lagrangian particles that entered a patch of high located at −139°E on yearday 262.5 (Fig. 5). The particles were chosen by first advecting particles seeded within the patch of high backward in time to identify the source regions of the high and then evenly seeding particles across these regions for a forward in time Lagrangian calculation. A subset of these particles was then chosen that had s⁻² and that were located in the upper EUC shear layer between depths of 120 and 30 m on yearday 262.5. The particles satisfying these conditions were sourced from two different water masses. Of all the particles, 88% entered the region of high from the west through the EUC (black EUC particles in Fig. 5). The other 12% of particles entered the region of high from the east from a nearby TIV (white TIV particles in Fig. 5). The TIV particles entered higher in the water column, reflecting the negative vertical shear in zonal velocity in the upper EUC, the westward flow of the southern portion of the TIV, and the warmth and lighter density of the TIV water mass in comparison to the equatorial water.

Daily averaged (10⁻⁴ s⁻²) in the upper EUC at 72-m depth at four different times separated by 4 days from the high-resolution simulation. The contours are SST at 0.75°C intervals, with the 25°C isotherm thicker. Shown are the positions of Lagrangian particles that enter the patch of high at 0°, −140°E on (d) day 262.5. These Lagrangian particles are identified either as EUC particles (893 particles composing 88% of the total), if they enter the region from the west through the EUC, or TIV particles (123 particles composing 12% of the total), if they enter the region from the east.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (10⁻⁴ s⁻²) in the upper EUC at 72-m depth at four different times separated by 4 days from the high-resolution simulation. The contours are SST at 0.75°C intervals, with the 25°C isotherm thicker. Shown are the positions of Lagrangian particles that enter the patch of high at 0°, −140°E on (d) day 262.5. These Lagrangian particles are identified either as EUC particles (893 particles composing 88% of the total), if they enter the region from the west through the EUC, or TIV particles (123 particles composing 12% of the total), if they enter the region from the east.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Daily averaged (10⁻⁴ s⁻²) in the upper EUC at 72-m depth at four different times separated by 4 days from the high-resolution simulation. The contours are SST at 0.75°C intervals, with the 25°C isotherm thicker. Shown are the positions of Lagrangian particles that enter the patch of high at 0°, −140°E on (d) day 262.5. These Lagrangian particles are identified either as EUC particles (893 particles composing 88% of the total), if they enter the region from the west through the EUC, or TIV particles (123 particles composing 12% of the total), if they enter the region from the east.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The following this set of particles (green line in Fig. 6a) increases rapidly in time to values above zero, and thus the presence of high in the high mixing patch is not a result of advection. This increase in is because of a rapid increase in zonal shear squared (black line in Fig. 6a), as the stratification stays relatively constant (blue line in Fig. 6a) and the meridional shear squared is small (red line in Fig. 6a). This is consistent with the Eulerian analysis in the previous section.

(a) The evolution of the reduced shear squared and its components: the zonal shear squared , meridional shear squared , and −4 times the stratification following the particles shown in Fig. 5. (b) An analysis of diagnostics in the zonal shear budget [Eq. (4)] following the particles shown in Fig. 5. denotes vortex tilting, denotes horizontal vortex stretching, denotes the frictional torque, and denotes the baroclinic torque. The error bars represent the ±1σ spread over the particle ensemble, representing the spatial variability in the underlying fields over the span of the particles. The dashed lines in (a) and (b) are the equivalent curves obtained by first sorting every vertical profile in the simulations using the marginal stability adjustment algorithm described in the appendix and then taking finite-difference derivatives according to Eq. (4). The marginal stability–adjusted [red dashed curve in (b)] is calculated as a residual.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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(a) The evolution of the reduced shear squared and its components: the zonal shear squared , meridional shear squared , and −4 times the stratification following the particles shown in Fig. 5. (b) An analysis of diagnostics in the zonal shear budget [Eq. (4)] following the particles shown in Fig. 5. denotes vortex tilting, denotes horizontal vortex stretching, denotes the frictional torque, and denotes the baroclinic torque. The error bars represent the ±1σ spread over the particle ensemble, representing the spatial variability in the underlying fields over the span of the particles. The dashed lines in (a) and (b) are the equivalent curves obtained by first sorting every vertical profile in the simulations using the marginal stability adjustment algorithm described in the appendix and then taking finite-difference derivatives according to Eq. (4). The marginal stability–adjusted [red dashed curve in (b)] is calculated as a residual.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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(a) The evolution of the reduced shear squared and its components: the zonal shear squared , meridional shear squared , and −4 times the stratification following the particles shown in Fig. 5. (b) An analysis of diagnostics in the zonal shear budget [Eq. (4)] following the particles shown in Fig. 5. denotes vortex tilting, denotes horizontal vortex stretching, denotes the frictional torque, and denotes the baroclinic torque. The error bars represent the ±1σ spread over the particle ensemble, representing the spatial variability in the underlying fields over the span of the particles. The dashed lines in (a) and (b) are the equivalent curves obtained by first sorting every vertical profile in the simulations using the marginal stability adjustment algorithm described in the appendix and then taking finite-difference derivatives according to Eq. (4). The marginal stability–adjusted [red dashed curve in (b)] is calculated as a residual.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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To determine the cause of the increase in zonal shear, we perform a Lagrangian budget of zonal shear along the particle tracks. The zonal shear satisfies

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which is equivalent to the equation for the north–south component of the vorticity for the primitive equations, where F_x is the frictional force in the zonal direction. Integrating along a particle path from a time t₀ to t gives

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Thus, a change in zonal shear following a particle can be attributed to horizontal vortex stretching (STR), vortex tilting (TILT), the baroclinic torque (BTOR), or frictional torques (FRIC). We analyze these terms along the particle tracks shown in Fig. 5, where we choose t₀ as day 253.5. The terms are calculated by taking the vertical derivative of the zonal momentum equation diagnostics output of ROMS. The numerical accuracy of these calculations were checked by comparing each quantity against the same quantities calculated directly using finite-difference derivatives of the velocity and buoyancy, which gave closely consistent results (not shown).

The dominant process driving an increase in the magnitude of the zonal shear as the particles enter the high mixing patch is horizontal vortex stretching (STR in Fig. 6b). Vortex tilting and the baroclinic torque are negligible (TILT and BTOR in Fig. 6b). The increase in drives an increase in vertical mixing that acts to decrease the magnitude of the shear, as shown by the positive frictional torque (FRIC in Fig. 6b). However, the KPP mixing scheme does not provide enough mixing to enforce marginal stability against the horizontal vortex stretching. Equivalent Lagrangian curves calculated by taking finite-difference derivatives of the model variables adjusted for marginal stability (using the algorithm described in the appendix) show that the additional mixing provided by this algorithm drives a decrease in stratification and limits the increase in zonal shear (dashed curves in Figs. 6a,b). The strength of horizontal vortex stretching also decreases because it depends on the shear itself (dashed magenta curve in Fig. 6b). However, horizontal vortex stretching remains the driver behind the increase in mixing, which now manifests itself as a low stratification anomaly. Further Lagrangian analyses of other patches of high mixing confirm the ubiquity of this mechanism (not shown).

Horizontal vortex stretching here relies on meridional diffluence [see in Eq. (4)] away from the equator that acts on the existing shear of the EUC to increase it. Positive meridional diffluence driven by the horizontal flow of TIWs is present at the leading edge of the TIW warm phase [Fig. 1f; also see Fig. 3 of Perez et al. (2010)], in the same phase as the high (cf. Figs. 1f and 1g near the patches of high ). The correlation between and in the upper EUC shear layer at 0°, −140°E reaches a maximum at a lag of 2 days of −0.25 (significant at 95%), showing that maximum meridional diffluence leads minimum (maximum magnitude) zonal shear at 2 days. This is consistent with the pathway of Lagrangian fluid parcels in the EUC and the time-integrated nature of the vortex stretching term in Eq. (4), where meridional diffluence must act over time to increase the zonal shear. This is further illustrated with a Hovmöller diagram of the and meridional diffluence, which highlights the strong correspondence between the two variables, with most patches of high mixing preceded by meridional diffluence in longitude and time (Fig. 7).

Hovmöller plot of the reduced shear squared (color) and meridional diffluence (, contours) at 72-m depth on the equator from the basinwide ROMS simulation. The x axis is longitude, and the y axis is time in yeardays for the months September–December. The thin (thick) contours denote where s⁻¹ (2 × 10⁻⁶ s⁻¹). Note that positive meridional diffluence is often collocated or leads high patches, consistent with horizontal vortex stretching.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Hovmöller plot of the reduced shear squared (color) and meridional diffluence (, contours) at 72-m depth on the equator from the basinwide ROMS simulation. The x axis is longitude, and the y axis is time in yeardays for the months September–December. The thin (thick) contours denote where s⁻¹ (2 × 10⁻⁶ s⁻¹). Note that positive meridional diffluence is often collocated or leads high patches, consistent with horizontal vortex stretching.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Hovmöller plot of the reduced shear squared (color) and meridional diffluence (, contours) at 72-m depth on the equator from the basinwide ROMS simulation. The x axis is longitude, and the y axis is time in yeardays for the months September–December. The thin (thick) contours denote where s⁻¹ (2 × 10⁻⁶ s⁻¹). Note that positive meridional diffluence is often collocated or leads high patches, consistent with horizontal vortex stretching.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Modulation of vertical mixing through horizontal vortex stretching does not require the horizontal circulation of TIWs to be divergent. While TIWs are associated with large-scale vertical motions [as discussed in Jochum and Murtugudde (2006) and Perez et al. (2010)], in our simulations most of the horizontal vortex stretching is a result of the component of the TIW velocity field that is horizontally nondivergent . The physics of the mechanism are summarized in Fig. 8.

A schematic showing how periodic strain driven by the TIW velocity field (red lines) acts on the horizontal vorticity of the EUC (indicated by the large black spirals) to increase or decrease the shear leading to patches of high (small blue spirals) and low turbulent mixing along the equator.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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A schematic showing how periodic strain driven by the TIW velocity field (red lines) acts on the horizontal vorticity of the EUC (indicated by the large black spirals) to increase or decrease the shear leading to patches of high (small blue spirals) and low turbulent mixing along the equator.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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A schematic showing how periodic strain driven by the TIW velocity field (red lines) acts on the horizontal vorticity of the EUC (indicated by the large black spirals) to increase or decrease the shear leading to patches of high (small blue spirals) and low turbulent mixing along the equator.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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On the trailing edge of the TIW warm phase, TIW horizontal strain drives meridional confluence and thus horizontal vortex squashing. Because of the dependence of the vortex stretching on the zonal shear itself [STR in Eq. (4)], meridional confluence does not drive strong reductions in shear. While not the focus of this study, Lagrangian analyses show that patches of low , and thus low turbulent mixing, can be partially attributed to restratification driven by TIW meridional shear acting on the north equatorial front (not shown). This TIW-driven periodic stratification, combined with the variations in mixing itself, may contribute to the variations in stratification at TIW frequencies seen in the Eulerian analysis (Fig. 3f). Furthermore, this emphasizes that the meridional shear of TIWs, while having a limited direct contribution to the , plays a role in driving variations in stratification that influence mixing.

The nonlinear nature of horizontal vortex stretching, whereby vortex stretching drives large increases in shear while squashing drives lesser decreases in shear, suggests that TIWs may increase the average levels of turbulence in the upper EUC. This is examined in the following section.

5. A simple mixing model of the EUC: TIW influence

This section discusses the total influence of the TIW modulation of the shear on the turbulent heat flux in the context of a simple 1D mixing model of the EUC forced by periodic TIW horizontal strain with alternating periods of meridional diffluence and confluence. We quantify this influence by comparing these 1D mixing model runs with and without TIW strain forcing.

a. 1D model setup

We split the zonal flow at 0°, −140°E into two parts associated with the EUC and TIWs, respectively:

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where we assume that the TIW part is horizontally nondivergent and follows a simple 15-day sinusoidal oscillation

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where represents the initial phase. The vertical shape function

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was chosen by fitting the RMS variations in in the TIW frequency band (calculated from the 3D ROMS model in a similar fashion to the profiles in Fig. 3) to a linear function in z. Although we do not have direct observations of to compare to Eq. (7), we would expect this fitted amplitude to be similar to observations, given that the 3D model TIW meridional velocity amplitude and amplitude (Fig. 3c) compare favorably with observations, and the lateral structure of the 3D model TIWs is comparable to observations (Qiao and Weisberg 1995; Kennan and Flament 2000).

With the idealized TIWs, the zonal momentum equation for the EUC portion of the velocity field becomes

View Expanded

where F_x represents a frictional force associated with vertical mixing and surface forcing, and is the pressure associated with the EUC portion of the flow. We assume that the advection terms in the curly brackets can be treated as external forcing in this 1D framework and can be included as part of a large-scale forcing term PA. This term represents the pressure gradient and advective processes that maintain the EUC against vertical mixing and was chosen to balance the wind stress over the total water column. It has a cubic Gaussian shape

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with decay scale d = 120 m, chosen as it matched the form of the 3D model pressure gradient well.

Dropping the from Eq. (8), the above approximations result in a simple 1D diffusion equation model for the EUC velocity , where we also include equations for the temperature and salinity :

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View Expanded

where the first (second) member of the curly brackets is taken for the temperature (salinity) equation. The Newtonian restoring added to the temperature and salinity equations represents the pressures that maintain the stratification against vertical mixing. The main results are not very sensitive to values of the restoring time scale between 1 and 15 days (not shown), and thus r_T = 5 days was chosen. We solve these equations on the same 50-level vertical grid as the 3D ROMS model.

The surface forcing at 0°N, −140°E averaged over the TIW season (September–December) from the 3D ROMS model was used to force the 1D model, including a τ_x = −0.08 N m⁻² zonal wind stress, a free-slip bottom boundary condition, a downward shortwave radiative heat flux of 275 W m⁻², and a combined sensible plus latent heat flux of −180 W m⁻². We found that including a diurnal cycle of shortwave radiation did not significantly change the results discussed here and thus for simplicity we discuss only model runs without a diurnal cycle.

We compare the results of simulations using two parameterizations for the diffusion terms in Eqs. (10) and (11). The diffusivity and viscosity in both schemes are split into components above and below the boundary layer depth (H_BL):

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where the boundary layer depth and boundary layer diffusivity/viscosity are determined in both schemes by the KPP boundary layer model of Large et al. (1994). However, the schemes differ in the interior parameterization for the diffusivity/viscosity , with the first scheme (the KPP scheme) using the KPP interior form used in the 3D ROMS model (Large et al. 1994):

View Expanded

View Expanded

where is the viscosity, is the diffusivity for scalars, m² s⁻¹, , and the background diffusivities are m² s⁻¹ and m² s⁻¹ (Fig. 9). These parameters are consistent with previous parameter choices for this scheme (Large et al. 1994; Large and Gent 1999; Zaron and Moum 2009). The parameter is smaller than values used elsewhere and was chosen as it gave the most appropriate mean EUC depth in the 3D ROMS model as determined by a sensitivity study (not shown).

The interior diffusivities (κ_T) and viscosities (κ_ν) assigned by the KPP (Large et al. 1994) and P&P (Pacanowski and Philander 1981) interior mixing schemes for shear instability as a function of the gradient Richardson number [as in Eqs. (13)–(16)].

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The interior diffusivities (κ_T) and viscosities (κ_ν) assigned by the KPP (Large et al. 1994) and P&P (Pacanowski and Philander 1981) interior mixing schemes for shear instability as a function of the gradient Richardson number [as in Eqs. (13)–(16)].

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The interior diffusivities (κ_T) and viscosities (κ_ν) assigned by the KPP (Large et al. 1994) and P&P (Pacanowski and Philander 1981) interior mixing schemes for shear instability as a function of the gradient Richardson number [as in Eqs. (13)–(16)].

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The second scheme replaces with the form of Pacanowski and Philander (1981) (we refer to this scheme as the P&P scheme):

View Expanded

View Expanded

where m² s⁻¹ and . These parameters are consistent with previously cited values (e.g., Pacanowski and Philander 1981; Blanke and Delecluse 1993; Li et al. 2001). The P&P parameterization should be more successful at enforcing marginal stability than the KPP parameterization, as it has a steeper increase in diffusivity as the Richardson number passes ¼ and thus can provide relatively more mixing at low Richardson numbers (Fig. 9). At low Richardson numbers the P&P scheme has a Prandtl number greater than 1 unlike KPP and thus preferentially mixes momentum over buoyancy to more efficiently increase the Richardson number. At high Richardson numbers both schemes have Prandtl numbers greater than 1.

The simple 1D mixing model runs were initialized with the 3D ROMS model profiles at 0°, −140°E averaged over the TIW season (September–December) and run for 200 days. We choose the tenth TIW period between days 135 and 150 for our analysis because by this time the results are not sensitive to the initial phase of the strain .

The simple 1D model shows similar qualitative results to the 3D ROMS model. During periods when (Figs. 10a,b), the zonal shear increases, peaking in magnitude 3–4 days after the maximum strain (Figs. 10c,d). High zonal shear is accompanied by low Richardson numbers (Figs. 10g,h), resulting in large diffusivities (Figs. 10i,j) that mix out the stratification (Figs. 10e,f) through the divergence of the turbulent heat flux (Figs. 10k,l):

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where is the specific heat of seawater. As the TIW strain switches sign to become negative ( in Figs. 10a,b), the shear is reduced both through vortex squashing and as a result of the mixing itself. The T-S restoring and the surface heat fluxes then act to restore the stratification profile before the TIW strain switches sign and a new phase of increased mixing begins.

Time–depth plots of simple 1D mixing model runs using (left) KPP and (right) P&P over days 135–165. The (a),(b) imposed TIW periodic strain, (c),(d) zonal shear, (e),(f) stratification, (g),(h) inverse Richardson number, (i),(j) diffusivity, and (k),(l) vertical turbulent heat flux. The dark green lines indicate the depth of the KPP mixed layer, the gray lines indicate the depth of the EUC core, and the black contours are isopycnals at 0.2 kg m⁻³ spacing.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Time–depth plots of simple 1D mixing model runs using (left) KPP and (right) P&P over days 135–165. The (a),(b) imposed TIW periodic strain, (c),(d) zonal shear, (e),(f) stratification, (g),(h) inverse Richardson number, (i),(j) diffusivity, and (k),(l) vertical turbulent heat flux. The dark green lines indicate the depth of the KPP mixed layer, the gray lines indicate the depth of the EUC core, and the black contours are isopycnals at 0.2 kg m⁻³ spacing.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Time–depth plots of simple 1D mixing model runs using (left) KPP and (right) P&P over days 135–165. The (a),(b) imposed TIW periodic strain, (c),(d) zonal shear, (e),(f) stratification, (g),(h) inverse Richardson number, (i),(j) diffusivity, and (k),(l) vertical turbulent heat flux. The dark green lines indicate the depth of the KPP mixed layer, the gray lines indicate the depth of the EUC core, and the black contours are isopycnals at 0.2 kg m⁻³ spacing.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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In these simple 1D mixing model simulations, the P&P scheme does a better job at enforcing marginal stability than KPP, restricting the Richardson number to larger values (cf. Figs. 10g and 10h). The P&P diffusivity is able to reach higher values when the Richardson number is low (cf. Figs. 10i and 10j) and preferentially mixes momentum over buoyancy when the Prandtl number is greater than 1. This results in lower maximum turbulent heat fluxes for P&P compared to KPP (cf. Figs. 10k to 10l). The turbulent heat fluxes averaged over a TIW period (Fig. 11) are also lower for P&P compared to KPP (cf. black and blue lines in Fig. 11a) for model runs both with periodic TIW strain [α(z, t) as in Eq. (6); solid lines in Fig. 11a] and runs without TIW periodic strain [α(z, t) set to zero; dashed lines in Fig. 11a].

The vertical turbulent heat flux [J_q, Eq. (17)] averaged over days 135–150 for simple mixing model runs with [α(z, t) as in Eq. (6); solid lines] and without [α(z, t) = 0; dashed lines] TIWs. Shown are (a) runs with nonlinear TIW stretching, where the TIW body force depends on the velocity itself α(z, t)u(z, t), and (b) linear TIW stretching, where the TIW body force depends on a mean velocity . Areas are shaded according to whether adding TIWs increases (blue) or decreases (orange) the turbulent heat flux averaged over a TIW period. Adding nonlinear TIW stretching increases the turbulent heat flux in all cases with the magnitude of the increase depending on the parameterization for shear instability used.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The vertical turbulent heat flux [J_q, Eq. (17)] averaged over days 135–150 for simple mixing model runs with [α(z, t) as in Eq. (6); solid lines] and without [α(z, t) = 0; dashed lines] TIWs. Shown are (a) runs with nonlinear TIW stretching, where the TIW body force depends on the velocity itself α(z, t)u(z, t), and (b) linear TIW stretching, where the TIW body force depends on a mean velocity . Areas are shaded according to whether adding TIWs increases (blue) or decreases (orange) the turbulent heat flux averaged over a TIW period. Adding nonlinear TIW stretching increases the turbulent heat flux in all cases with the magnitude of the increase depending on the parameterization for shear instability used.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The vertical turbulent heat flux [J_q, Eq. (17)] averaged over days 135–150 for simple mixing model runs with [α(z, t) as in Eq. (6); solid lines] and without [α(z, t) = 0; dashed lines] TIWs. Shown are (a) runs with nonlinear TIW stretching, where the TIW body force depends on the velocity itself α(z, t)u(z, t), and (b) linear TIW stretching, where the TIW body force depends on a mean velocity . Areas are shaded according to whether adding TIWs increases (blue) or decreases (orange) the turbulent heat flux averaged over a TIW period. Adding nonlinear TIW stretching increases the turbulent heat flux in all cases with the magnitude of the increase depending on the parameterization for shear instability used.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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b. The influence of TIWs on the turbulent heat flux

Inclusion of TIW periodic strain results in an increase in the magnitude of the turbulent heat flux averaged over a TIW period (Fig. 11a). The increase is restricted to depths below 75 m for the KPP simulation (cf. black solid and dashed curves in Fig. 11a), while for P&P, the heat flux increases at all depths (cf. blue solid and dashed curves in Fig. 11a). The rectified change in turbulent heat flux R at a given depth is defined as

View Expanded

where the overbar represents an average over a TIW period, and overbarⁿ represents an equivalent average in the run without TIWs. There are a number of nonlinearities in the system that are potentially responsible for rectification :

1. the influence of variations in stratification, both through the dependence of the Richardson number on stratification and the dependence of the heat flux on the product of the diffusivity and the vertical temperature gradient ;

2. the nonlinear dependence of the diffusivity on the Richardson number and through it on the shear, which differs between the two mixing schemes ;

3. the influence of mixing on the underlying shear distribution [even if the diffusivity depends linearly on the shear (i.e., removing nonlinearity 2), the mixing acts preferentially at high shears and thus can result in rectification ]; and

4. the nonlinear influence of vortex stretching on the shear .

The influence of temporal variations in stratification and the temperature gradient on the heat flux (nonlinearity 1) is minimal. While the vertical gradient in stratification is large, its temporal variations are weaker than the corresponding variations in shear (cf. Figs. 10e,f with Figs. 10c,d) and thus do not have a strong influence on the temporal variations in the Richardson number (cf. Figs. 10e,f with Figs. 10g,h). In addition, dT/dz averaged over a TIW period is similar to dT/dz from the no TIW runs (not shown), and the distribution over a TIW period of the turbulent heat flux at a fixed depth matches the distribution of diffusivity closely (cf. blue distributions in Figs. 12e,f to Figs. 12c,d), implying minimal influence by variations in dT/dz.

Histograms of the magnitude of the (a),(b) zonal shear, (c),(d) diffusivity, and (e),(f) turbulent heat flux over a TIW period at 75-m depth for simple 1D mixing model runs with the (left) KPP and (right) P&P mixing schemes. The blue (green) histograms show the results of runs with nonlinear (linear) TIW stretching. The distribution means are indicated with arrows at the top of the figures, including the mean for runs without TIWs (red arrows). The skew of each distribution is indicated in the upper right of each panel. The distribution of negative turbulent heat flux is plotted to facilitate comparison with the diffusivity distributions.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Histograms of the magnitude of the (a),(b) zonal shear, (c),(d) diffusivity, and (e),(f) turbulent heat flux over a TIW period at 75-m depth for simple 1D mixing model runs with the (left) KPP and (right) P&P mixing schemes. The blue (green) histograms show the results of runs with nonlinear (linear) TIW stretching. The distribution means are indicated with arrows at the top of the figures, including the mean for runs without TIWs (red arrows). The skew of each distribution is indicated in the upper right of each panel. The distribution of negative turbulent heat flux is plotted to facilitate comparison with the diffusivity distributions.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Histograms of the magnitude of the (a),(b) zonal shear, (c),(d) diffusivity, and (e),(f) turbulent heat flux over a TIW period at 75-m depth for simple 1D mixing model runs with the (left) KPP and (right) P&P mixing schemes. The blue (green) histograms show the results of runs with nonlinear (linear) TIW stretching. The distribution means are indicated with arrows at the top of the figures, including the mean for runs without TIWs (red arrows). The skew of each distribution is indicated in the upper right of each panel. The distribution of negative turbulent heat flux is plotted to facilitate comparison with the diffusivity distributions.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The nonlinear dependence of the KPP and P&P diffusivity on the Richardson number and shear (nonlinearity 2) has a strong influence on the turbulent heat flux averaged over a TIW period and is responsible for the differences between the two mixing schemes. Given that variations in stratification are minimal, this nonlinearity is well captured by considering the dependence of the diffusivity on the shear at an average value of the stratification for each mixing scheme (Fig. 13). Using a Taylor series expansion of the functional dependence of the diffusivity on the shear , about the shear averaged over a TIW period , it can be shown that to first order in the shear oscillation variance var

View Expanded

where is the rectification effect of this nonlinearity, and is the curvature of G. Thus, as a consequence of the positive curvature of the P&P curve (Fig. 13), the distribution of P&P diffusivity over a TIW period at an example depth is positively skewed (0.61 at 75 m; blue distribution in Fig. 12d), resulting in a positive rectification effect (cf. no TIW to nonlinear TIW means in Fig. 12d). On the other hand, the negative curvature of the KPP curve (Fig. 13) results in a negatively skewed KPP diffusivity distribution over a TIW period (−0.32 at 75 m; blue distribution in Fig. 12c) and would tend to result in a negative rectification effect. However, the total rectification R is also influenced by the other two nonlinearities, the influence of the mixing on the shear distribution (nonlinearity 3) and the nonlinearity in vortex stretching (nonlinearity 4), that can cause differences between the shear averaged over a TIW period and the shear for the no TIW run and influence the oscillation amplitude [var].

The KPP and P&P interior diffusivities (κ_T) and viscosities (κ_ν) as a function of the shear for a stratification of N² = 1.42 × 10⁻⁴ s⁻², close to the stratification at 75 m averaged over the period shown in Fig. 5 for both the KPP (N² = 1.37 × 10⁻⁴ s⁻²) and P&P (N² = 1.47 × 10⁻⁴ s⁻²) runs. The corresponding values of the Richardson number are shown on the top axis. A linear diffusivity scale is used to highlight the different curvatures of the two schemes.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The KPP and P&P interior diffusivities (κ_T) and viscosities (κ_ν) as a function of the shear for a stratification of N² = 1.42 × 10⁻⁴ s⁻², close to the stratification at 75 m averaged over the period shown in Fig. 5 for both the KPP (N² = 1.37 × 10⁻⁴ s⁻²) and P&P (N² = 1.47 × 10⁻⁴ s⁻²) runs. The corresponding values of the Richardson number are shown on the top axis. A linear diffusivity scale is used to highlight the different curvatures of the two schemes.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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The KPP and P&P interior diffusivities (κ_T) and viscosities (κ_ν) as a function of the shear for a stratification of N² = 1.42 × 10⁻⁴ s⁻², close to the stratification at 75 m averaged over the period shown in Fig. 5 for both the KPP (N² = 1.37 × 10⁻⁴ s⁻²) and P&P (N² = 1.47 × 10⁻⁴ s⁻²) runs. The corresponding values of the Richardson number are shown on the top axis. A linear diffusivity scale is used to highlight the different curvatures of the two schemes.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0209.1

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Nonlinearities 3 and 4 have counteracting influences on the shear distribution. The mixing preferentially acts at high shear (for both mixing schemes), tending to negatively skew the shear magnitude distribution and result in a rectified decrease in the shear magnitude averaged over a TIW period compared to runs without TIWs. The nonlinearity of vortex stretching induces stronger vortex stretching at high shears, positively skewing the shear magnitude distribution and resulting in a rectified increase in the shear magnitude averaged over a TIW period compared to runs without TIWs. For P&P, the skew of the shear magnitude distribution is small (0.07; blue distribution in Fig. 12b) and so is the rectified change in mean shear magnitude (cf. no TIW to nonlinear TIW means in Fig. 12b), suggesting that mixing and vortex stretching compensate. While for KPP, the skew of the shear magnitude distribution is positive (0.38; blue distribution in Fig. 12a), and the rectified change in mean shear magnitude is positive (cf. no TIW to nonlinear TIW means in Fig. 12a), suggesting that the vortex stretching nonlinearity overpowers the mixing nonlinearity for KPP. However, interpreting the impact of nonlinearities 3 and 4 on the heat fluxes based solely on these qualitative arguments is potentially misleading as they can interact with each other and with nonlinearity 2 associated with the curvature in G.

To quantitatively evaluate the importance of the vortex stretching nonlinearity compared to the other nonlinearities 1–3, we replace the regular nonlinear TIW stretching [ in Eq. (10)] with linear TIW stretching [, where was the mean velocity profile between days 135 and 150 from the simulations with nonlinear TIW stretching]. In simulations performed with linear TIW stretching, the TIW body force no longer depends on the magnitude of the instantaneous velocity, and thus without mixing a simple symmetric sinusoidal oscillation in shear at each depth would result. Indeed, in these linear runs, the skew of the shear magnitude distributions becomes negative as a result of the mixing nonlinearity 3 (cf. the skews of the blue and green distributions in Figs. 12a and 12b). These changes result in modifications to the distributions of the diffusivity and the turbulent heat flux (cf. blue and green distributions in Figs. 12c–f). In particular, the turbulent heat flux averaged over a TIW period in simulations with linear TIW stretching is now reduced in comparison to the nonlinear TIW stretching runs (cf. Figs. 11a and 11b). For KPP, linear TIW stretching results in a rectified decrease in the heat flux, as a consequence of both the negative curvature of the KPP G curve (Fig. 13) and the mixing nonlinearities. For P&P, the linear TIW stretching results in no change in the average turbulent heat flux compared to simulations without TIWs, likely as a result of the cancellation between the mixing nonlinearity and the nonlinearity associated with the positive curvature of the P&P G curve (Fig. 13). These results emphasize that the dependence of horizontal vortex stretching on the shear itself is the critical nonlinearity that gives rise to the rectified increase in turbulent heat flux R > 0 when TIWs are added.

In the context of this simple 1D mixing model for the EUC, TIWs increase the turbulent heat flux averaged over a TIW period through horizontal vortex stretching. Averages of between 50 and 100 m (Table 1), indicative of the total change in heat content induced by turbulent mixing between this depth layer and the surface, show that TIWs cool the upper ocean in both parameterizations examined here. However, the magnitude of the cooling varies between parameterizations. Using the P&P parameterization, which more realistically enforces marginal stability, TIWs increase the turbulent heat flux averaged between 50 and 100 m by 30%. Runs conducted using the curves of Peters et al. (1988), which have very rapid increases in diffusivity as the Richardson number decreases below ¼, show a further enhancement of the turbulent heat flux by TIWs to 40% (Table 1). This suggests that the rectified TIW influence is larger for parameterizations that are more successful at enforcing marginal stability and whose diffusivity formulas are more consistent with observations.

Table 1.

Turbulent heat flux (W m⁻²) averaged over days 135–165 between 50 and 100 m from the simple 1D mixing model runs.

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While the details of the simple 1D mixing runs are sensitive to the parameters in the mixing schemes, the influence of TIWs on the heat flux is not sensitive to these parameters, only to the shape and curvature of the curve. Sensitivity studies where the KPP maximum diffusivity was doubled to 4 × 10⁻³ m² s⁻¹ (as in Large and Gent 1999), the KPP critical Richardson number Ri₀ was changed to 0.8 (as in Large and Gent 1999), and the P&P maximum diffusivity was halved to 5 × 10⁻³ m² s⁻¹ (as in Ma et al. 1994) show only small changes to the influence of TIWs on the turbulent heat flux (Table 1).

6. Summary and discussion

Recent observations (Lien et al. 2008; Moum et al. 2009; Inoue et al. 2012) have shown that TIWs modulate vertical mixing in the upper EUC, with potential implications for the role of TIWs in the equatorial SST budget. Using a regional ocean model, we have examined the mechanisms through which TIWs modulate the , a proxy for mixing. Contrary to previous hypotheses, our results suggest that the varying TIW meridional shear plays a minor direct role in modulations of mixing (Fig. 3c). Instead, we find that TIW horizontal strain modifies the zonal shear of the upper EUC through horizontal vortex stretching (Fig. 8), acting to increase vertical mixing leading into the TIW warm phase. By including the effects of TIW horizontal strain in a simple 1D mixing model of the EUC, we further showed that adding TIWs can result in a rectified increase in the turbulent heat flux and thus sea surface cooling in this simple context. The amount of sea surface cooling induced by TIWs is larger for parameterizations that are more successful at enforcing the observed (Smyth and Moum 2013) physical constraint of marginal stability. This result has potential implications for the SST budget in ocean models using Richardson number–based mixing parameterizations, such as many of the phase 5 of CMIP (CMIP5) ocean models (Huang et al. 2014).

Horizontal vortex stretching here relies on TIW horizontal strain interacting with the vertical shear in the upper EUC. This coupling of vertical and horizontal processes is a signature of submesoscale physics, which have been found previously to play a role in the dynamics of TIWs and TIVs (Holmes et al. 2014). MacDonald and Chen (2012) found that a similar mechanism involving lateral spreading at a river outlet was important in enhancing stratified shear turbulence in a coastal scale flow. The meridional diffluence that drives vortex stretching acts frontolytically on the equatorial fronts to decrease the horizontal buoyancy gradient. However, the constraint of thermal wind balance is weakened near the equator where the Coriolis parameter is weak. The tilting of planetary and relative vorticity that would normally act (in the midlatitudes) to decrease the shear and maintain thermal wind balance in response to the decreasing horizontal buoyancy gradient does not act strongly here. Thus, this process is distinctly different from the frontogenetic intensification of shear and turbulence discussed by Skyllingstad and Samelson (2012).

In the context of a simple 1D mixing model of the EUC, TIWs increased the turbulent heat flux by ~30 W m⁻² (Fig. 11a; Table 1), which is of similar magnitude to the TIW lateral heat fluxes estimated in Jochum and Murtugudde (2006). For a 30-m-deep mixed layer, this additional heat flux drives an additional cooling of −0.6°C month⁻¹, a significant fraction of the 1°–2°C month⁻¹ cooling implied by the turbulent heat flux observations of Moum et al. (2009) and comparable to the O(1)°C month⁻¹ TIW lateral heating rate (Menkes et al. 2006; Jochum et al. 2007). Thus, the additional mixing-induced cooling effect of TIWs could potentially compensate for TIW lateral heating. However, this result must be considered with caution given the simplified nature of the 1D mixing model and the parameterized mixing.

We have not considered here any rectified influence of TIWs on the surface heat fluxes. The studies of Thum et al. (2002) and Seo et al. (2007) found variations in the latent heat flux of ~50 W m⁻² (°C)⁻¹ of TIW-induced SST change, with the sensible heat flux changes being significantly smaller. Seo et al. (2007) showed that the rectified influence of these changes averaged over a TIW period was negligible (<1 W m⁻²), and thus we expect the increase in turbulent heat flux due to TIWs to dominate any TIW-induced changes in air–sea fluxes.

Our results, combined with the study of Zaron and Moum (2009), suggest that there are several improvements that could be made to the interior KPP mixing parameterization scheme to better model the equatorial oceans and the influence of TIWs on vertical mixing. The inclusion of the KPP boundary layer scheme has been shown to significantly improve simulations of the tropical Pacific compared to the original Pacanowski and Philander (1981) scheme (Li et al. 2001). Our modeling work and observations of curves (i.e., Fig. 1 of Zaron and Moum 2009) suggest that using the interior schemes of Pacanowski and Philander (1981) or Peters et al. (1988) combined with the KPP boundary layer model (Large et al. 1994) may perform better than the original KPP scheme with respect to enforcing marginal stability and thus modeling the influence of TIWs on vertical mixing. This type of scheme has been used in the HadCM3 climate model (Gordon et al. 2000), although the authors did not discuss the reasons for this choice. However, in any future vertical mixing sensitivity study, it must be noted that the strength of TIWs themselves is influenced by vertical mixing, rendering the interpretation of results difficult (Chen et al. 1994).