## 1. Introduction

Midlatitude atmospheric storms can efficiently generate inertial oscillations in the surface mixed layer (ML) of the ocean. After oscillations are generated, part of the energy in the ML radiates to the ocean interior as low-mode near-inertial internal waves (Gill 1984; D’Asaro et al. 1995). It has been hypothesized that radiation, propagation, and breaking of near-inertial internal waves are important in maintaining stratification in the abyssal ocean (Munk 1966; Munk and Wunsch 1998).

By applying the slab ML model (Pollard and Millard 1970; D’Asaro 1985) to the global ocean, the estimated global inertial energy flux into the surface ML (hereinafter called the wind power input) is around 0.5 TW (Alford 2001; Watanabe and Hibiya 2002; Alford 2003; Watanabe et al. 2005). Wind power input exhibits seasonal variability related to midlatitude storm activities. Midlatitude storms, whose paths are called storm tracks, display variabilities on time scales longer than the seasonal variation. For example, in the North Pacific, storm tracks develop in autumn and winter. They are affected by the position and strength of the Aleutian low, which shows a temporal variability (e.g., Sugimoto and Hanawa 2009; Di Lorenzo et al. 2015). Sugimoto and Hanawa (2009) suggested that the zonal shift of the Aleutian low on a decadal or longer time scale is associated with the Pacific–North American teleconnection pattern and the meridional shift with the west Pacific teleconnection pattern. Inoue et al. (2017) applied empirical orthogonal function (EOF) analysis to a time series of the wind energy inputs estimated by the slab model. They showed that the long-time-scale variability of the near-inertial motions is also related to the Aleutian low.

Deep ocean current meter moorings also found a seasonal cycle of near-inertial internal wave activity (Alford and Whitmont 2007). Similarly, vertical diffusivities inferred from the global Argo float array, which use vertical density profiles, indicate a seasonal cycle, where the internal wave field is enhanced in the wintertime (Whalen et al. 2012). However, only a small fraction (less than 25%) of the near-inertial energy in the ML propagates into the main thermocline (Furuichi et al. 2008; Alford et al. 2012; Rimac et al. 2016) because most of the energy dissipates in the ML. It is suggested that only low mode inertial waves can propagate into the ocean interior, and internal waves with a higher vertical mode are dissipated in the upper ocean due to the higher vertical shear and slower propagation speed (Alford et al. 2016).

In the past few decades, the role of wind energies on abyssal mixing has been investigated. Wind energy is generally not considered an important factor for driving abyssal mixing because internal-wave energy, which reaches the main thermocline, is much smaller than that caused by tides. On the other hand, the role of the near-inertial energy, which dissipates locally and immediately in the upper ocean (e.g., inside and just below the surface ML) on the climate system is less understood. It is important for mixed layer and subsurface heat budgets due to enhanced entrainments and circulation changes. We also speculate that modulation of high-mode wave energies affects the biogeochemical properties because the energies can influence the bottom of the euphotic layer. Jochum et al. (2013) introduced a parameterization which added near-field mixing effects related to the near-inertial oscillation in the ML generated by storms, and incorporated it into a coupled ocean–atmosphere model. They found that near-inertial waves influence the climate system by deepening the tropical mixed layer, and suggested that it is important to understand tropical near-inertial wave energies qualitatively to reduce uncertainties in the parameterization.

In this study, we adopt the parameterization developed by Jochum et al. (2013) to a coarse-forward ocean general circulation model (OGCM) equipped with a turbulence closure model (Noh 2004) and optimized through data assimilation (Osafune et al. 2015) (section 2). In section 3, we integrate the OGCM with the parameterization for 58 years, describe near-inertial motions and waves reproduced in the OGCM, and compare results with and without the parameterization. We also discuss the possible effects of near-field wind mixing due to storm tracks in the North Pacific, which are modulated by the Aleutian low on a decadal scale. In section 4, we evaluate the parameterization and discuss potential improvements. Specifically, we compare vertical diffusivities inferred from the parameterization and Electro Magnetic Autonomous Profiling Explorer (EMAPEX) float (Teledyne Webb Research; Sanford et al. 2005) measurements. Here, the float measured abrupt propagation of near-inertial internal waves within the North Pacific Subtropical Mode Water (NPSTMW) and shear instability in the layer between NPSTMW and North Pacific Intermediate Water (NPIW) in the Kuroshio Extension (KEx). Finally, we discuss differences between these diffusivities.

## 2. Methods

### a. Parameterization of near-field wind mixing

*ρ*and

*N*

^{2}are the density (kg m

^{−3}) and buoyancy frequency squared (s

^{−2}) at each grid point, respectively.

^{−1}), where

^{−2}) in St. Laurent et al. (2002) and is defined as

*b*

_{fr}and

*l*

_{fr}are the fraction of the energy dissipated in the ML and propagating below, respectively. Here, these are set at 0.7 and 0.5, respectively (Jochum et al. 2013). These values mean that 15% of the near-inertial energy in the ML is used to generate turbulence below the ML (near field) and another 15% radiates away as low-mode internal waves. Low-mode internal waves contribute to the formation of the background internal wave field as well as interior mixing (far field). The

*E*

^{flux}is the near-inertial energy flux into the surface boundary layer,

**⋅**

*τ***U**

_{i}. Here,

**is the wind stress which forces the OGCM and**

*τ***U**

_{i}is the surface velocity of the near-inertial current which is estimated from the time series of outputs by a bandpass filter between 0.8 and 1.2 times of the local Coriolis frequencies. In Jochum et al. (2013), it is estimated as

*k*

_{niw}corresponds to changes in the ML. Here

*α*is a scaling factor to adjust the global annual mean of (3) to that of simulated

**⋅**

*τ***U**

_{i}and was set to 0.05 in Jochum et al. (2013). The bulk kinetic energy of near-inertial motions in the ML, KE

_{ML}, is defined as

*u*

^{i}and

*υ*

^{i}, obtained from the differences between two consecutive time steps (Δ

*t*). The term

*T*is the inertial period. Equations (5) and (6) should work because the OGCM works as a low-pass filter. The computational method is selected to suit the forward time integration because variables in the right-hand side of (5) and (6) are obtained in the same time step. Otherwise, a bandpass filter would be necessary to obtain near-inertial motions.

*α** is an asymptote from 0 to 0.5 if the wind forcing frequency is close to the inertial frequency. Variable

*α*in Eq. (3) should be 1/

*α** according to Jochum et al. (2013). Hence,

*α** is simply selected from the model output [section 3a(1)] and 1/

*α** is used as

*α*.

*F*(

*z*) (m

^{−1}) is defined with a vertical integration equal to one as

*η*is a length scale and

*H*is the bottom depth, so that

*ε*

_{niw}(W kg

^{−1}). Parameter

*η*is set to 2000 m by Jochum et al. (2013). Here, vertical integration of Eq. (8) between −

*H*and −

*h*becomes one (e.g.,

*η*(1 −

*e*

^{−H/η}), by

*η*for simplicity. Depth is positive upward and thus has negative values. Thus,

*ε*

_{niw}decreases toward the deeper layer. Variables

*H*and

*h*are positive. Vertically integrating Eq. (8) between each depth and −

*h*shows that 99% of internal waves energy dissipates at −4790, −920, and −90 m below the ML for

*η*= 2000, 200, and 20 m, respectively (Figs. 1a–c). In addition to the

*η*= 2000 m run,

*η*= 40 m is arbitrarily chosen so that near-inertial energy in the ML is confined within 200 m below the ML, which mimics breaking of higher vertical mode internal waves in the upper ocean (Figs. 1d,e).

### b. Ocean general circulation model

This study uses version 3 of the OGCM from the Geophysical Fluid Dynamics Laboratory (NOAA, United States) Modular Ocean Model (MOM3) (Pacanowski and Griffies 2000), which has been modified for a long-term data synthesis system that derives the Estimated State of Global Ocean for Climate Research (ESTOC; Osafune et al. 2015). The system applies a four-dimensional variational data assimilation method based on a strong-constraint formalism. It searches for the best time-trajectory fit of the OGCM to the observations by optimizing the initial conditions of the model variables and the 10-day-mean air–sea fluxes (heat fluxes, freshwater fluxes, and wind stress) compiled from 6-hourly air–sea fluxes (heat, freshwater, and momentum) of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996).

The quasi-global OGCM domain covers 75°S and 80°N. Horizontal resolution is 1° in both longitude and latitude, while vertical resolution of the 45 levels varies from 10 to 400 m. An additional layer is applied for the bottom boundary layer. Using Green’s function approach (e.g., Menemenlis et al. 2005), parameters related to isopycnal (Gent and McWilliams 1990) and diapycnal (Gargett 1984; Hasumi and Suginohara 1999; Tsujino et al. 2000) diffusivities and other physical parameters have been optimized to better reproduce deep-water masses and abyssal circulation (Toyoda et al. 2015). For the surface boundary layer, the OGCM has used the turbulence closure developed by Noh (2004). Noh’s model applies a Mellor–Yamada-type second-order closure (e.g., Mellor and Yamada 1982) but with different boundary conditions to diagnose the turbulent energy. The ML depth *h* in Eq. (7) is defined as the shallowest depth where buoyancy frequency is largest. Since OGCM uses the closure model, we do not apply the ML velocity modification used in Jochum et al. (2013) and the subsurface parameterization, Eq. (1), is our focus.

We referred to the original ESTOC, which contains 58 years of data from 1957 to 2014 (Osafune et al. 2014) as the control run (CTL). Using the optimized initial condition and air–sea fluxes of ESTOC, we conducted two experiments. The first involved a high-frequency wind run (HFW-nJ) with 6-hourly wind data added. These data were obtained by subtracting the linearly interpolated 10-day mean wind data from the 6-hourly NCEP–NCAR reanalysis. The second added the parameterization of Jochum et al. (2013) to the HFW-nJ, and is hereafter called HFW-J (*η* = 2000 m) and HFW-Jh (*η* = 40 m). In HFW cases, we decreased the high-frequency wind stress in 65°–70°N by a cosine function and turned it off in 70°–80°N for a stability of computations. Although the 6-hourly wind data were insufficient to resolve storms in higher latitudes where the local inertial period is shorter (the detailed studies on spatial and temporal resolutions of wind data are given in Rimac et al. 2013), a correction (e.g., Nagasawa et al. 2000; Watanabe and Hibiya 2002) was not applied. Finally, since we use the prescribed heat and salinity fluxes optimized by data assimilation, temperature and salinity differences induced by the parameterization represent a pure ocean response without damping at the sea surface (Osafune et al. 2020).

## 3. OGCM

### a. Results

#### 1) Near-inertial motions in the ML

Here and in the next section, near-inertial motions and waves produced in the OGCM are described to introduce the parameterization by Jochum et al. (2013). Figures 2a–d depict excitation of inertial motions after a storm event at 49.5°N, 174.5°E using HFW-nJ. The near-inertial current is generated at the surface after *t* = 285 (elapsed hour from the start, 1 January 1957). Within a few inertial periods (inertial period is about 15.8 h at 49.5°N), the velocity in the ML is homogenized. The near-inertial oscillation propagates below the ML after *t* = 360. A comparison between *E*^{flux} from the last term of Eq. (3), _{ML} from Eq. (4) (Fig. 2e) shows that the _{ML} because it integrates positive values as shown in Eq.(3). The ** τ** ⋅

**U**

_{i}is positive (energy is brought into the ocean), has a value of 0.1. Here, we assume that this condition mimics the

_{ML}and

**⋅**

*τ***U**

_{i}> 0 becomes 0.04, which is smaller than the single storm case. This is partly because successive storms damped the preexisting inertial currents in the ML due to the mismatch between wind and current directions.

Time series of (a) wind stress ^{−1}) with depth, (c) surface meridional velocity of near-inertial currents, (d) cumulative value of ** τ** ⋅

**U**

_{i},

_{ML}(magenta line),

*x*axis indicates the elapsed time from the start of the numerical simulation, 1 Jan 1957. Contour lines in (b) are

*σ*

_{θ}with a contour interval of 0.1 kg m

^{−3}.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of (a) wind stress ^{−1}) with depth, (c) surface meridional velocity of near-inertial currents, (d) cumulative value of ** τ** ⋅

**U**

_{i},

_{ML}(magenta line),

*x*axis indicates the elapsed time from the start of the numerical simulation, 1 Jan 1957. Contour lines in (b) are

*σ*

_{θ}with a contour interval of 0.1 kg m

^{−3}.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of (a) wind stress ^{−1}) with depth, (c) surface meridional velocity of near-inertial currents, (d) cumulative value of ** τ** ⋅

**U**

_{i},

_{ML}(magenta line),

*x*axis indicates the elapsed time from the start of the numerical simulation, 1 Jan 1957. Contour lines in (b) are

*σ*

_{θ}with a contour interval of 0.1 kg m

^{−3}.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

As in Fig. 2, but for 31.5°N.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

As in Fig. 2, but for 31.5°N.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

As in Fig. 2, but for 31.5°N.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Finally, the monthly average *E*^{flux} estimated from the last term of Eq. (3) with *α** = 0.2 (*α* = 5) in January 1957 (Figs. 4a,c) are 2.38 and 0.26 TW by including and excluding the equatorial region, respectively. The estimated global wind energy input, *α** (=0.2) in this study because in Eq. (3) the damping of KE_{ML} is also treated as energy input and *E*^{flux} matches

Horizontal distributions of (a) *E*^{flux} in the HFW-Jh estimated by Eq. (3) and (b) ** τ** ⋅

**U**

_{i}in the HFW-nJ estimated by the bandpassed surface velocities for every 3 h. Log

_{10}of magnitudes are shown. (c) Zonal average of the

*E*

^{flux}(blue line) and

**⋅**

*τ***U**

_{i}(red line).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of (a) *E*^{flux} in the HFW-Jh estimated by Eq. (3) and (b) ** τ** ⋅

**U**

_{i}in the HFW-nJ estimated by the bandpassed surface velocities for every 3 h. Log

_{10}of magnitudes are shown. (c) Zonal average of the

*E*

^{flux}(blue line) and

**⋅**

*τ***U**

_{i}(red line).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of (a) *E*^{flux} in the HFW-Jh estimated by Eq. (3) and (b) ** τ** ⋅

**U**

_{i}in the HFW-nJ estimated by the bandpassed surface velocities for every 3 h. Log

_{10}of magnitudes are shown. (c) Zonal average of the

*E*

^{flux}(blue line) and

**⋅**

*τ***U**

_{i}(red line).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

#### 2) Near-inertial internal waves

After radiation from the ML, near-inertial motions propagate into the ocean interior as near-inertial waves. Figure 5 shows the bandpass-filtered (between Coriolis frequencies of 30° and 35°N) meridional cross section of meridional velocities along 174.5°E. From Fig. 3, near-inertial motions are generated in the ML before *t* = 259 (Figs. 5a,b). The low-mode near-inertial waves dominate these velocity sections (e.g., Nagasawa et al. 2000). Velocity fields indicate that the low-mode near-inertial waves originated from the latitude band propagates southward (Figs. 5c,d). Another storm event is present around *t* = 600 (Fig. 3), but has weaker near-inertial motions in the deeper ML. This event seems to generate low-mode waves in the latitude band (Fig. 5e). The time series of variance, where two-dimensional horizontal wavenumber spectra of the meridional velocities (without a frequency filter) were integrated in the zonal direction in 9.5°–40.5°N and 143.5°E–179.5°W, also indicate the dominance of southward propagation for linear internal waves under influence of the beta effect (e.g., Gill 1984; D’Asaro et al. 1995; Nagasawa et al. 2000 (Fig. 6). The meridional scale of southward-propagating near-inertial waves decreases with time, and waves dissipate through the viscous effect in the OGCM. Due to the low horizontal resolution (1°) and limited vertical resolution, the OGCM cannot reproduce higher vertical modes. Since the OGCM cannot reproduce the higher vertical modes and the vertical mixing parameterization used in the OGCM is independent on the modeled vertical shear, the enhanced shear due to near-inertial currents in the ML does not increase the vertical diffusivities below the ML. Therefore, it is reasonable to introduce the parameterization by Jochum et al. (2013) in the OGCM. Note that this parameterization is not necessary if a numerical model resolves high vertical modes as well as wave–wave interactions and a vertical mixing parameterization considers the effects of enhanced shears through the Richardson number.

Vertical cross sections of (a)–(e) bandpass-filtered (between Coriolis frequencies of 30° and 35°N) meridional velocities every 6 days along 174.5°E. Hours at the top of each panel denote elapsed time from the start of the numerical simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Vertical cross sections of (a)–(e) bandpass-filtered (between Coriolis frequencies of 30° and 35°N) meridional velocities every 6 days along 174.5°E. Hours at the top of each panel denote elapsed time from the start of the numerical simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Vertical cross sections of (a)–(e) bandpass-filtered (between Coriolis frequencies of 30° and 35°N) meridional velocities every 6 days along 174.5°E. Hours at the top of each panel denote elapsed time from the start of the numerical simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of the log_{10} values of the variances of meridional velocities in 9.5°–40.5°N and 143.5°E–179.5°W. Black line shows the wavenumber of −*df*/*dy* × Time at 30.5°N. Geographical coordinate is used to calculate spectra. The *x* axis indicates the elapsed time from the start of the numerical simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of the log_{10} values of the variances of meridional velocities in 9.5°–40.5°N and 143.5°E–179.5°W. Black line shows the wavenumber of −*df*/*dy* × Time at 30.5°N. Geographical coordinate is used to calculate spectra. The *x* axis indicates the elapsed time from the start of the numerical simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of the log_{10} values of the variances of meridional velocities in 9.5°–40.5°N and 143.5°E–179.5°W. Black line shows the wavenumber of −*df*/*dy* × Time at 30.5°N. Geographical coordinate is used to calculate spectra. The *x* axis indicates the elapsed time from the start of the numerical simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

#### 3) Enhanced mixing below the ML: Comparison between four cases

Figure 7 shows the same ML deepening event in Fig. 2 (49.5°N) for the CTL and HFW-nJ cases. Figure 7a shows that the CTL case does not include high-frequency wind forcing and the high-shear squared layer is confined near the surface. ML depth is similar because convection due to sea surface cooling controls ML deepening (Figs. 7a,b) in the wintertime. The thermocline below the ML is thicker in the HFW cases. The difference between the HFW cases is much smaller than those between the CTL and HFW cases (not shown). The vertical profiles of a monthly average (within January 1957) vertical diffusivity estimated in the OGCM show that the diffusivities increase below the ML in the HFW-J and HFW-Jh cases (Fig. 7c), that mimic shear instability due to near-inertial waves (HFW-J) and breaking of higher vertical mode internal waves below the ML (HFW-Jh). Its increase occurs deeper than the ML base due to higher stratification just below the ML. The maximum value is ~1.8 × 10^{−5} m^{2} s^{−1} in the HFW-Jh case.

Time series of (a) log_{10} of shear squared (s^{−2}) from the CTL and (b) the HFW-J (section 2b). Contour lines are *σ*_{θ} with a contour interval of 0.1 kg m^{−3}. Magenta line is the ML base. The *x* axis is elapsed time from the start of the numerical simulation. Vertical profiles of (c) monthly-averaged diffusivities used to calculate temperature fields, *k*_{T}, from the CTL (blue), HFW-nJ (red dashed), HFW-J (green dashed), and HFW-Jh (magenta dashed) at 174.5°E and 49.5°N.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of (a) log_{10} of shear squared (s^{−2}) from the CTL and (b) the HFW-J (section 2b). Contour lines are *σ*_{θ} with a contour interval of 0.1 kg m^{−3}. Magenta line is the ML base. The *x* axis is elapsed time from the start of the numerical simulation. Vertical profiles of (c) monthly-averaged diffusivities used to calculate temperature fields, *k*_{T}, from the CTL (blue), HFW-nJ (red dashed), HFW-J (green dashed), and HFW-Jh (magenta dashed) at 174.5°E and 49.5°N.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Time series of (a) log_{10} of shear squared (s^{−2}) from the CTL and (b) the HFW-J (section 2b). Contour lines are *σ*_{θ} with a contour interval of 0.1 kg m^{−3}. Magenta line is the ML base. The *x* axis is elapsed time from the start of the numerical simulation. Vertical profiles of (c) monthly-averaged diffusivities used to calculate temperature fields, *k*_{T}, from the CTL (blue), HFW-nJ (red dashed), HFW-J (green dashed), and HFW-Jh (magenta dashed) at 174.5°E and 49.5°N.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of the differences between the yearly average (final 58th year, 2014) temperatures from the F8 CTL and HFW-nJ cases (HFW-nJ minus CTL, Figs. 8a–d) show the effects of high-frequency wind forcing. Water is warmer, except in the Atlantic and Southern Oceans. Differences are typically less than 28°C at the surface (−5-m depth) and subsurface (−105-m depth) (Figs. 8a,b). Differences are smaller (~1°C) in the deeper layers but there are more cooling places than in the shallower layers (Figs. 8c,d). The upper and lower layer should be cooled and warmed, respectively, if surface mixing is enhanced due to the stronger wind forcing and temperature determines stratification. The ML deepening also cannot explain warming below the ML depth. Therefore, we speculate that these changes are due to changes in circulation.

Horizontal distributions of the differences between yearly average (final 58th year, 2014) temperatures (HFW-nJ minus CTL), at (a) −5, (b) −105, (c) −477, and (d) −719 m. Those of HFW-J minus HFW-nJ at (e) −5, (f) −105, (g) −477, and (h) −719 m. Those of HFW-Jh minus HFW-nJ, at (i) −5, (j) −105, (k) −477, and (l) −719 m.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of the differences between yearly average (final 58th year, 2014) temperatures (HFW-nJ minus CTL), at (a) −5, (b) −105, (c) −477, and (d) −719 m. Those of HFW-J minus HFW-nJ at (e) −5, (f) −105, (g) −477, and (h) −719 m. Those of HFW-Jh minus HFW-nJ, at (i) −5, (j) −105, (k) −477, and (l) −719 m.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of the differences between yearly average (final 58th year, 2014) temperatures (HFW-nJ minus CTL), at (a) −5, (b) −105, (c) −477, and (d) −719 m. Those of HFW-J minus HFW-nJ at (e) −5, (f) −105, (g) −477, and (h) −719 m. Those of HFW-Jh minus HFW-nJ, at (i) −5, (j) −105, (k) −477, and (l) −719 m.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Temperature differences between the HFW cases (HFW-J minus HFW-nJ, Figs. 8e–h) show that surface and subsurface waters are cooled (Figs. 8e–g) and it is warmed in the deeper layer (Fig. 8h) of the equator and subtropical regions, which is expected from vertical mixing in the thermally stratified water. Surface and subsurface waters show smaller-scale temperature modulations around the storm track in the midlatitudes where a deeper ML forms. Differences between the HFW cases (HFW-Jh minus HFW-nJ, Figs. 8i–l) show that surface and subsurface waters are cooled (heated) on the western (eastern) side of the Pacific and Atlantic. Those differences are much larger than those between the CTL and HFW-nJ cases. Water is cooled in the deeper layer of the North Atlantic, and mid- and low latitudes of the Pacific and Indian Ocean (Figs. 8k,l). The horizontal distributions of *k*_{niw} in the HFW-Jh case show enhanced mixing at the equator (not shown) as expected in section 3a(1) (Fig. 4). Since the east–west contrast in the Pacific and Atlantic Oceans has appeared in other years, we speculate that vertical mixing generates the pattern through an adjustment process. Due to uncertainties in the equatorial region, we turn off the parameterization between 20°N and 20°S to isolate the effects of the near-field wind mixing associated with the storm track in the mid latitude of the North Pacific in the next section.

### b. Discussion

To visualize the impact of near-field wind mixing on the climate signal, we investigate SST by focusing on the Pacific decadal oscillation (PDO), which is the dominant year-round pattern in monthly North Pacific SST variability (e.g., Mantua et al. 1997). Using observed SST data, which are assimilated in ESTOC, we applied empirical orthogonal function (EOF) analysis to monthly SST anomalies from 1971 to 2014. Here, this first principal component is used as the PDO index (Figs. 9a and 11a). The corresponding EOF shows the characteristic structure of PDO, a negative anomaly in the western-central region, which is surrounded by a horseshoe-shaped positive anomaly (Fig. 9a).

Horizontal distributions of (a) the EOF first mode of the observation and the regression coefficient between the PDO index and SST from (b) CTL and (c) HFW-nJ in the North Pacific.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of (a) the EOF first mode of the observation and the regression coefficient between the PDO index and SST from (b) CTL and (c) HFW-nJ in the North Pacific.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Horizontal distributions of (a) the EOF first mode of the observation and the regression coefficient between the PDO index and SST from (b) CTL and (c) HFW-nJ in the North Pacific.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Figures 9b–c show the regression coefficient of the monthly SST anomaly in each experiment on the PDO index. The regression pattern of the CTL is similar to the PDO pattern, indicating that the PDO is well reproduced in the CTL. Although the regression patterns are altered in the HFW cases (e.g., Fig. 9c for the HFW-nJ), they retain the characteristic structure of the PDO. Differences in the regression coefficients within the HFW-nJ and HFW-J (Fig. 10a), which depict the impact of the near-field wind mixing, are positive around the center of action of the PDO at 170°W in the mid latitudes that is shown as the negative spatial peak (e.g., the black box in Fig. 10a), meaning that the regression coefficient is decreased so that the PDO signal is weakened at the surface.

(a) Horizontal distributions of the regression coefficient from CTL (black line) and the differences of the coefficients between the HFW cases (HFW-J minus HFW-nJ, colored area). Vertical cross section along 170°W for (b) HFW-J and (c) HFW-Jh. Colors and black contours are the same as (a).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) Horizontal distributions of the regression coefficient from CTL (black line) and the differences of the coefficients between the HFW cases (HFW-J minus HFW-nJ, colored area). Vertical cross section along 170°W for (b) HFW-J and (c) HFW-Jh. Colors and black contours are the same as (a).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) Horizontal distributions of the regression coefficient from CTL (black line) and the differences of the coefficients between the HFW cases (HFW-J minus HFW-nJ, colored area). Vertical cross section along 170°W for (b) HFW-J and (c) HFW-Jh. Colors and black contours are the same as (a).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

The reason for the decreased regression coefficient is inferred from the vertical cross-section along 170°W (Figs. 10b,c) where the regression coefficient decreases in the deeper depth due to the smaller influence from the sea surface. Applying parameterization decreases (increases) the regression coefficient in the shallower (deeper) depth for both the HFW-J (Fig. 10b) and HFW-Jh (Fig. 10c), implying that water is vertically exchanged by mixing and transfers the PDO signal into the deeper layer. A comparison between the PDO index (Fig. 11a) and the time series of monthly SST anomaly averaged over the black box (Fig. 11b) confirmed that the SST anomaly in the center of action of the PDO where the negative regression coefficient is large also corresponds to a negative correlation. The difference in SST anomalies between HFW cases (HFW-J minus HFW-nJ in Fig. 11c) is negatively correlated with the SST anomaly in the CTL, which is same for the HFW-Jh (not shown). Thus, mixing reduces the cold and warm anomalies in the positive and negative PDOI periods, respectively, consistent with the idea that it transfers the PDO signal into the deeper layer.

(a) PDO index, and time series of (b) SST anomaly from CTL averaged over the black box in Fig. 10a where the composite monthly mean is subtracted, and (c) differences of average SST anomaly between the HFW cases (HFW-J minus HFW-nJ). Blue lines in (a)–(c) show the monthly means, and thick orange lines are the 13-month running mean.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) PDO index, and time series of (b) SST anomaly from CTL averaged over the black box in Fig. 10a where the composite monthly mean is subtracted, and (c) differences of average SST anomaly between the HFW cases (HFW-J minus HFW-nJ). Blue lines in (a)–(c) show the monthly means, and thick orange lines are the 13-month running mean.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) PDO index, and time series of (b) SST anomaly from CTL averaged over the black box in Fig. 10a where the composite monthly mean is subtracted, and (c) differences of average SST anomaly between the HFW cases (HFW-J minus HFW-nJ). Blue lines in (a)–(c) show the monthly means, and thick orange lines are the 13-month running mean.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

These results indicate that near-field wind mixing is important for the climate system, and suggest that the impact of atmospheric forcing on the subsurface ocean can be underestimated in OGCMs that do not include near-field wind mixing. However, the accuracy of the parameterization remains unknown. In the next section, we clarify the parameterization using EM-APEX float measurements in the North Pacific.

## 4. Observations

### a. Results

*Hakuho Maru*(KH-15-4 cruise). The EM-APEX float is a kind of autonomous ARGO-type float equipped with electrodes to measure voltages from seawater motions in Earth’s magnetic field. The float is also equipped with CTD (conductivity, temperature, and depth) sensors. The vertical resolutions of CTD and EM data are 3–4 and 5–6 dbar, respectively. Data are interpolated every 6 dbar. Vertical shear and density gradient are estimated by first differentiation so that the 6-dbar scale vertical gradients are estimated to calculate Richardson number (Ri) as

*N*

^{2}is the buoyancy frequency squared and

*u*and

*υ*are the zonal and meridional velocities from the float, respectively.

On 24 November 2015 at 34.0°N and 140.1°E, one EM-APEX float was deployed. The float moved eastward along the KEx and captured a few storm events until 15 January 2016 (Fig. 12). The continuous sampling mode was applied, and the samplings were acquired using up and down profiling. The float acquired 594 profiles. Sampling depth was changed from 500 to 1200 dbar to follow the vertical propagation of internal waves after a storm event. On 10 December, a storm passed by the float, and the inertial oscillation was observed in the ML. At the base of the ML, the shear increases and Ri drops below 0.25, conducive to shear instability. Similar to OGCM, near-inertial waves begin to propagate below the ML after a few inertial periods. The waves penetrate NPSTMW (150–350 dbar) because the buoyancy frequency is higher than the local Coriolis frequency, even though the stratification of NPSTMW is low. A high-shear layer (not shown) where Ri is less than 0.25 arises between NPSTMW and NPIW (600–800 dbar) after 20 December that persisted over 2 weeks and 460 km, suggesting that near-inertial waves induce mixing in the main thermocline and dissipate both NPSTMW and NPIW after radiation from the ML along KEx.

(a) Float trajectory with bottom topography. Color bar indicates the date in 2013. Time series of (b) zonal velocity, (c) meridional velocity, (d) high-passed (at 1 cpd) near-inertial speed *x* axis is the date in 2013. Contour lines in (b) and (c) are the potential densities. Magenta lines in (d) and (e) are isotherms of 17° and 19°C, indicating the NPSTMW. White lines in (d) and (e) are potential densities of 1026.7 and 1026.9 kg m^{−3}, indicating the NPIW. The magenta line in (f) indicates that Ri is less than 1/4.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) Float trajectory with bottom topography. Color bar indicates the date in 2013. Time series of (b) zonal velocity, (c) meridional velocity, (d) high-passed (at 1 cpd) near-inertial speed *x* axis is the date in 2013. Contour lines in (b) and (c) are the potential densities. Magenta lines in (d) and (e) are isotherms of 17° and 19°C, indicating the NPSTMW. White lines in (d) and (e) are potential densities of 1026.7 and 1026.9 kg m^{−3}, indicating the NPIW. The magenta line in (f) indicates that Ri is less than 1/4.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) Float trajectory with bottom topography. Color bar indicates the date in 2013. Time series of (b) zonal velocity, (c) meridional velocity, (d) high-passed (at 1 cpd) near-inertial speed *x* axis is the date in 2013. Contour lines in (b) and (c) are the potential densities. Magenta lines in (d) and (e) are isotherms of 17° and 19°C, indicating the NPSTMW. White lines in (d) and (e) are potential densities of 1026.7 and 1026.9 kg m^{−3}, indicating the NPIW. The magenta line in (f) indicates that Ri is less than 1/4.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

### b. Discussion

#### 1) Modification of the near-field wind mixing parameterization

*ε*⟩ over this event using the parameterization of Kunze et al. (1990). This parameterization uses the reduced shear squared, ReSh

^{2}= (∂

*u*/∂

*z*)

^{2}+ (∂

*υ*/∂

*z*)

^{2}− 4

*N*

^{2}, in unstable shear (Ri < 0.25), which gives

*ε*over the period, Δ

*z*is 6 dbar, and

*σ*

_{grw}is the growth rate of the shear instability

*σ*

_{grw}= {[(∂

*u*/∂

*z*)

^{2}+ (∂

*υ*/∂

*z*)

^{2}]

^{1/2}− 2

*N*}/4. The average vertical diffusivity from the EM-APEX over the period (13 December–3 January) is

Next, to calculate diffusivities from Eq. (1), the EM-APEX’s velocity data are interpolated every 6 dbar and 6 h and high-pass-filtered at 1 cpd. The filtered EM-APEX’s velocity data are vertically averaged over the ML to estimate KEML for Eq. (3). In Eq. (8), *η* = 2000 and 40 m are used. Figure 13a compares vertical diffusivities obtained by Eqs. (11) and (1). These two estimations show discrepancies. The average vertical diffusivity *k*_{EM} shows higher values in the deeper layer and lower values in the low stratification layer (NPSTMW) because waves are stretched in the low stratification during vertical propagation. For *η* = 2000 m, *k*_{niw} is inversely proportional to the stratification because the structure function *F* defined by Eq. (8) is fairly uniform compared to *η* = 40 m case (Fig. 13c) and *k*_{niw} is smaller than *k*_{EM}. For *η* = 40 m, *k*_{niw} decreases with depth, consistent with *k*_{EM} above 300 dbar, but its magnitude is overestimated. In addition, enhanced mixing is not reproduced by Eq. (1) in both *η* = 2000 and 40 m cases.

Vertical profiles of (a) vertical diffusivities estimated from EM-APEX float data from Eq. (11) (black line), from Eq. (1) (blue line), and from Eq. (1) with Eq. (12) (red line), (b) fraction of shear instability during a storm event, (c) distribution function from Eq. (8) (blue line) and Eq. (12) (red line), and (d) log_{10} value of the buoyancy frequency from the EM-APEX float (black line) and *N*_{0} = 3 cph (black dashed line). In (a) and (c), WKB depth is estimated on the pressure coordinate, and the dashed line for *η* = 2000 m and solid line for *η* = 20 m. In (a), the 95% confidence limit for the diffusivities (black thin line) is obtained by bootstrapping (Efron and Gong 1983). Here, we assume that velocity measurements include random noises, and bootstrap is applied on ⟨*ε*⟩ in Eq. (11) at each layer.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Vertical profiles of (a) vertical diffusivities estimated from EM-APEX float data from Eq. (11) (black line), from Eq. (1) (blue line), and from Eq. (1) with Eq. (12) (red line), (b) fraction of shear instability during a storm event, (c) distribution function from Eq. (8) (blue line) and Eq. (12) (red line), and (d) log_{10} value of the buoyancy frequency from the EM-APEX float (black line) and *N*_{0} = 3 cph (black dashed line). In (a) and (c), WKB depth is estimated on the pressure coordinate, and the dashed line for *η* = 2000 m and solid line for *η* = 20 m. In (a), the 95% confidence limit for the diffusivities (black thin line) is obtained by bootstrapping (Efron and Gong 1983). Here, we assume that velocity measurements include random noises, and bootstrap is applied on ⟨*ε*⟩ in Eq. (11) at each layer.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Vertical profiles of (a) vertical diffusivities estimated from EM-APEX float data from Eq. (11) (black line), from Eq. (1) (blue line), and from Eq. (1) with Eq. (12) (red line), (b) fraction of shear instability during a storm event, (c) distribution function from Eq. (8) (blue line) and Eq. (12) (red line), and (d) log_{10} value of the buoyancy frequency from the EM-APEX float (black line) and *N*_{0} = 3 cph (black dashed line). In (a) and (c), WKB depth is estimated on the pressure coordinate, and the dashed line for *η* = 2000 m and solid line for *η* = 20 m. In (a), the 95% confidence limit for the diffusivities (black thin line) is obtained by bootstrapping (Efron and Gong 1983). Here, we assume that velocity measurements include random noises, and bootstrap is applied on ⟨*ε*⟩ in Eq. (11) at each layer.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

*F*in Eq. (8) should be defined in the WKB stretched coordinate as

*N*

_{0}is set to 3 cph. Similar to above, we set

*η*

_{WKB}= 2000 and 40 m. In the calculation of

*k*

_{niw},

*F*(

*z*

_{WKB}) is calculated in the WKB stretched coordinate, and OGCM’s vertical coordinate

*z*is converted to

*z*

_{WKB}based on Eq. (13). Both are then used to estimate

*F*at

*z*.

Figures 13a–c compare the newly calculated *k*_{niw} and *F* with the EM-APEX *k*_{EM}. For *η*_{WKB} = 2000 m case, there is no difference except for the depth where *N* < *N*_{0} because the vertical change of *F* is small. For *η*_{WKB} = 40 m case, WKB stretching improves the modeled diffusivities in the upper part of NPSTMW but underestimates diffusivities between NPSTMW and NPIW because the stratification is higher than *N*_{0} even in NPSTMW (Fig. 13d) so that |*z*_{WKB}(*z*)| > |*z*|. Although it is possible to choose a different value of *N*_{0} for the improvement, we hypothesize that the enhanced mixing in the deeper layer is caused by a mechanism not considered in the parameterization. In the next section, we investigate a mechanism that would efficiently bring wave energies to the deeper layer and generate shear instabilities between NPSTMW and NPIW.

#### 2) Possible mechanism of the internal wave breaking

In section 4a, we implicitly assumed that shear instabilities between NPSTMW and NPIW are generated by near-inertial waves upon applying the parameterization of Jochum et al. (2013). However, it is possible that the observed low Ri, which is an indicator of the shear instability, is created by trapping near-inertial internal waves caused by spatial changes in the vertical component of the relative vorticity or vertical shear (e.g., Kunze 1985; Whitt and Thomas 2013). In this scenario, vertical propagation is limited within the negative relative vorticity or high vertical shear areas. Consequently, the low-mode internal wave energy is also trapped and used to drive vertical mixing (e.g., Lee and Niiler 1998; Zhai et al. 2009). Therefore, this cannot be represented by the parameterization used in this study.

To examine the possibility of trapping, ray paths of near-inertial waves generated in the ML near the KEx are calculated during the EM-APEX measurements (appendix). To calculate the geostrophic flow fields and relative vorticities, we use monthly mean global ocean eddy-resolving reanalysis (1/12° horizontal resolution and 50 vertical levels, Global_ReAnalysis_phy_001_030) data from December 2015, which was provided by the Copernicus Marine Environment Monitoring Service (CMEMS) in the European Union (http://marine.copernicus.eu/). Absolute sea surface height and a trajectory of the EM-APEX float (Fig. 14a) indicate that the F14 float moved along the south side of the KEx, consistent with the presence of NPSTMW in the EM-APEX float time series (Fig. 12). One cross-section normal to the KEx is used to represent the density and velocity structures during the period when low Ri was observed in late December. For simplicity, it is assumed that neither density nor velocity structures change along the southeastward current. The coordinate system is rotated 45° in the clockwise direction and used to solve the ray equation (appendix). The vertical shear estimated from the model velocities (Fig. 14b) is almost the same as that estimated from the geostrophic balance in the KEx (not shown). We use modeled velocity to estimate the relative vorticity *ζ*_{g} and inverse geostrophic Richardson number

(a) Trajectory of the EM-APEX float on the sea surface height of the CMEMS reanalysis data. Magenta line shows the cross section used in this study. Color bar indicates date in 2013. Vertical cross section of the (b) monthly mean speed, (*U*^{2} + *V*^{2})^{1/2}, (c) vertical component of the relative vorticity normalized by local Coriolis frequency, (d) *ω*_{min}. Green and magenta contours are 1 + *ζ*_{g}/2 and *ω*_{min}/*f*, respectively; *y* = 0 km corresponds to the southwest end of the cross section in (a).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) Trajectory of the EM-APEX float on the sea surface height of the CMEMS reanalysis data. Magenta line shows the cross section used in this study. Color bar indicates date in 2013. Vertical cross section of the (b) monthly mean speed, (*U*^{2} + *V*^{2})^{1/2}, (c) vertical component of the relative vorticity normalized by local Coriolis frequency, (d) *ω*_{min}. Green and magenta contours are 1 + *ζ*_{g}/2 and *ω*_{min}/*f*, respectively; *y* = 0 km corresponds to the southwest end of the cross section in (a).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

(a) Trajectory of the EM-APEX float on the sea surface height of the CMEMS reanalysis data. Magenta line shows the cross section used in this study. Color bar indicates date in 2013. Vertical cross section of the (b) monthly mean speed, (*U*^{2} + *V*^{2})^{1/2}, (c) vertical component of the relative vorticity normalized by local Coriolis frequency, (d) *ω*_{min}. Green and magenta contours are 1 + *ζ*_{g}/2 and *ω*_{min}/*f*, respectively; *y* = 0 km corresponds to the southwest end of the cross section in (a).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

Ray tracing indicates that near-inertial internal waves freely propagate below the NPIW if *ζ*_{g} at the wave’s starting point is close to 0 or positive while waves can satisfy the trapping condition in the negative *ζ*_{g} area (Figs. 14e,f). It is known that near-inertial waves can be trapped if the intrinsic frequency is equal to the minimum internal wave frequency (e.g., Whitt and Thomas 2013). Note that the contribution of the vertical shear on the trapping is important near the KEx. However, the spatial change of *ζ*_{g} is more important in the south of the KEx (Fig. 14f). From ray tracing, we conclude that the shear instabilities between NPSTMW and NPIW are generated by the near-inertial wave trapping where even the low-mode wave’s energy is used for mixing. Therefore, it is suggested that a new parameterization is necessary to represent near-field wind mixing near the western boundary currents and eddies, where the fronts and the trapping of near-inertial waves containing low modes are expected.

In western boundary regions, the Antarctic Circumpolar Current and tropics, anticyclonic areas (*ζ*_{g} < 0) are often found at the sea surface (Fig. 15a). When *ζ*_{g} is negative at the sea surface, frequencies of near-inertial motions in the ML are decreased. Then, near-inertial waves radiated from the ML are trapped in the deeper layer where an intrinsic frequency of waves corresponds to the minimum near-inertial frequency *ω*_{min} (see appendix) as seen in the KEx. Here, it is assumed that the depth of the deepest layer, where *ω*_{min} normalized by the local Coriolis frequency (*ω*_{min}/*f*) is smaller than 0.95, can be regarded as the possible trapping layer (e.g., Fig. 5 in Whitt and Thomas 2013). It is shown that the layer depth becomes deeper in those *ζ*_{g} < 0 areas (Fig. 15b). Therefore, it is suggested that a new parameterization which considers the trapping of near-inertial waves would be required in a part of the world oceans. Note that *ζ*_{g} and *ω*_{min}/*f* were estimated from the monthly mean CMEMS reanalysis data of December 2013 as same as the KEx above. The depth was chosen between a depth that was 20 m deeper than the MLD (to avoid the nongeostrophic high shear zone below the ML, e.g., Fig. 7b) and −1062 m. When |Ro_{g}| > 1, *ω*_{min}/*f* was not used.

The horizontal distributions of (a) Ro_{g} *= ζ*_{g}/*f* at the sea surface (*z* = −0.5 m) and (b) depth of the deepest layer where *ω*_{min}/*f* < 0.95 except for 10°N–10°S.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

The horizontal distributions of (a) Ro_{g} *= ζ*_{g}/*f* at the sea surface (*z* = −0.5 m) and (b) depth of the deepest layer where *ω*_{min}/*f* < 0.95 except for 10°N–10°S.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

The horizontal distributions of (a) Ro_{g} *= ζ*_{g}/*f* at the sea surface (*z* = −0.5 m) and (b) depth of the deepest layer where *ω*_{min}/*f* < 0.95 except for 10°N–10°S.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0281.1

## 5. Conclusions and discussion

The role of near-inertial wind energy, which dissipates locally below the surface ML, on the climate system is studied by adopting the parameterization developed by Jochum et al. (2013) to a coarse-forward OGCM forced by assimilated forcing fields (ESTOC; Osafune et al. 2015) and integrated it for 58 years with the 6-hourly wind data of the NCEP–NCAR reanalysis (Kalnay et al. 1996). We compared the parameterization with an EM-APEX float observation. Conclusions are summarized as follows:

The OGCM reproduces near-inertial oscillations in the ML after storms as well as subsequent radiations. Near-inertial waves radiating from the ML are low-mode internal waves whose meridional wavenumbers are influenced by the beta effect.

The ratio,

${\alpha}^{*}={\displaystyle \int}\left(\partial {\text{KE}}_{\text{ML}}/\partial t\right)dt/{\displaystyle \int}\mathbf{\tau}\cdot {\mathbf{U}}_{i}dt$ , in the ML was examined with the OGCM. The wind energy input to the World Ocean,$\int}\mathbf{\tau}\cdot {\mathbf{U}}_{i}dA$ , during January 1957 is 0.22 TW and monthly average global$\int}{E}^{\text{flux}}dA$ estimated from the last term of Eq. (3) with*α** = 0.2 in January 1957 is 0.26 TW. Both exclude the equatorial region (5°N–5°S).Above estimated

*E*^{flux}was used to diagnose vertical diffusivities due to near-field wind mixing, which enhances the diffusivities below the ML. A comparison between four cases (CTRL, HFW-nJ, HFW-J, and HFW-Jh in section 2) in the final year suggests that high-frequency winds cause warming above the main thermocline. Parameterized diffusivities*k*_{niw}increase around the equator, consistent with Jochum et al. (2013). Since the east low–west high SST contrast in the Pacific and Atlantic Oceans in the HFW-Jh case can be caused by an adjustment to vertical mixing around the equator, better understanding of near-inertial motions in the equatorial region is required.We investigated the possible effects of near-field wind mixing due to storm track activities in the North Pacific, which are modulated by the Aleutian low on the decadal scale. We applied the parameterization, excluding 20°N–20°S, to isolate effects of the Aleutian low. The regression coefficient to the PDO index decreases and increases in the shallower and deeper depths, respectively, at the center of the PDO, implying that vertical mixing transfers the PDO signal into the deeper layer.

Due to the possible importance of near-field wind mixing on the climate system, we evaluated the parameterization with EM-APEX measurements. The EM-APEX float observed abrupt and local propagation of the near-inertial internal waves within NPSTMW and shear instabilities in the layer between NPSTMW and NPIW in the KEx for 460 km along its trajectory. Even after considering the WKB stretching, vertical diffusivities inferred from the parameterization do not reproduce enhanced diffusivities in the deeper layer inferred from EM-APEX float measurements. Ray tracing of near-inertial waves near the KEx indicates that these instabilities may be caused by wave trapping where low-mode energies are also dissipated. Therefore, vertical diffusivities due to trapping near fronts need to be parameterized separately.

Previous studies have concluded that near-inertial internal wave energies are not important for driving abyssal mixing because low-mode wave energies, which can reach the main thermocline, are small. However, this study shows that wave energies can change the subsurface temperature fields and that the effects are modulated according to storm tracks, possibly on decadal time scales. Additionally, trapping of near-inertial waves enhances local mixing in the main thermocline. Hence, near-inertial waves should be treated separately in the parameterization because this trapping process dissipates low-mode wave energies (e.g., Kunze 1985). Consequently, a new parameterization for the trapping near fronts needs to be developed. It is possible to trap and amplify the kinetic energy of waves through an interaction with a mean flow (e.g., Kunze et al. 1995; Lee and Niiler 1998). Finally, since the EM-APEX float did not measure turbulence directly, testing the parameterization by Jochum et al. (2013) using in situ microstructure measurements should be investigated in the future. We speculate that, as stated by Jochum et al. (2013), understanding near-inertial motions near the equator is important to better understand the climate system.

## Acknowledgments

We are indebted to Dr. Tadashi Hemmi for helping the numerical simulation and to Dr. Takeyoshi Nagai for deploying the EM-APEX during the KH-15-4 cruise. The manuscript was improved by comments by two anonymous reviewers and Dr. Eric Kunze. This study was supported by a Grant-in-Aid for Scientific Research on Innovative Areas (MEXT KAKENHI-JP15H05817, JP15H05818, and JP15H05819). A part of this study has been conducted using E.U. Copernicus Marine Service Information.

## APPENDIX

### Ray Path of Near-Inertial Internal Waves

*f*

_{eff}≈

*f*+

*ζ*

_{g}/2,

*ζ*

_{g}= ∂

*υ*

_{g}/∂

*x*− ∂

*u*

_{g}/∂

*y*, and

*k*,

*l*, and

*m*are the zonal, meridional, and vertical wavenumbers, respectively. Trapping of inertial waves occurs when

*ω*is equal to the minimum frequency, which is defined by Whitt and Thomas (2013) as

_{g}=

*ζ*

_{g}/

*f*,

**k**= (

*k*,

*l*,

*m*), ∇ = (∂/∂

*x,*∂/∂

*y,*∂/∂

*z*), and

*ω*

_{E}is the Euler frequency, and wave positions,

To solve the ray equations, we assume that the current is only in the zonal direction, the cross current section is in the meridional direction, and *f* is a constant. Since our goal is to demonstrate the possibility of inertial-wave trapping between NPSTMW and NPIW, we arbitrarily set the meridional wavelength to 50 km to numerically integrate the ray equation because this value corresponds to the width of the negative relative vorticity region and a vertical wavelength of 100 m. We also set a zonal wavelength to 0 to avoid a Doppler shift.

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