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Improve the Simulations of Near-Inertial Internal Waves in the Ocean General Circulation Models

Zhao Jing Department of Oceanography, Texas A&M University, College Station, Texas, and Key Laboratory of Physical Oceanography, and Qingdao Collaborative Innovation Center of Marine Science and Technology, Ocean University of China, Qingdao, China

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Lixin Wu Key Laboratory of Physical Oceanography, and Qingdao Collaborative Innovation Center of Marine Science and Technology, Ocean University of China, Qingdao, China

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Xiaohui Ma Department of Oceanography, Texas A&M University, College Station, Texas

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Print Publication:
01 Oct 2015
DOI:
https://doi.org/10.1175/JTECH-D-15-0046.1
Page(s):
1960–1970
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Abstract

The near-inertial wind work and near-inertial internal waves (NIWs) in the ocean have been extensively studied using ocean general circulation models (OGCMs) forced by 6-hourly winds or wind stress obtained from atmospheric reanalysis data. However, the OGCMs interpolate the reanalysis winds or wind stress linearly onto each time step, which partially filters out the wind stress variance in the near-inertial band. In this study, the influence of the linear interpolation on the near-inertial wind work and NIWs is quantified using an eddy-resolving (°) primitive equation ocean model. In addition, a new interpolation method is proposed—the sinc-function interpolation—that overcomes the shortages of the linear interpolation.

It is found that the linear interpolation of 6-hourly winds significantly underestimates the near-inertial wind work and NIWs at the midlatitudes. The underestimation of the near-inertial wind work and near-inertial kinetic energy is proportional to the loss of near-inertial wind stress variance due to the linear interpolation. This further weakens the diapycnal mixing in the ocean due to the reduced near-inertial shear variance. Compared to the linear interpolation, the sinc-function interpolation retains all the wind stress variance in the near-inertial band and yields correct magnitudes for the near-inertial wind work and NIWs at the midlatitudes.

Corresponding author address: Zhao Jing, Department of Oceanography, Texas A&M University, O & M Building, Room 612, MS 3146, College Station, TX 77843. E-mail: jingzhao198763@tamu.edu

Abstract

The near-inertial wind work and near-inertial internal waves (NIWs) in the ocean have been extensively studied using ocean general circulation models (OGCMs) forced by 6-hourly winds or wind stress obtained from atmospheric reanalysis data. However, the OGCMs interpolate the reanalysis winds or wind stress linearly onto each time step, which partially filters out the wind stress variance in the near-inertial band. In this study, the influence of the linear interpolation on the near-inertial wind work and NIWs is quantified using an eddy-resolving (°) primitive equation ocean model. In addition, a new interpolation method is proposed—the sinc-function interpolation—that overcomes the shortages of the linear interpolation.

It is found that the linear interpolation of 6-hourly winds significantly underestimates the near-inertial wind work and NIWs at the midlatitudes. The underestimation of the near-inertial wind work and near-inertial kinetic energy is proportional to the loss of near-inertial wind stress variance due to the linear interpolation. This further weakens the diapycnal mixing in the ocean due to the reduced near-inertial shear variance. Compared to the linear interpolation, the sinc-function interpolation retains all the wind stress variance in the near-inertial band and yields correct magnitudes for the near-inertial wind work and NIWs at the midlatitudes.

Corresponding author address: Zhao Jing, Department of Oceanography, Texas A&M University, O & M Building, Room 612, MS 3146, College Station, TX 77843. E-mail: jingzhao198763@tamu.edu

1. Introduction

Near-inertial internal waves (NIWs; Garrett 2001) form a pronounced peak in the ocean current frequency spectrum. They are furnished primarily by the wind work on near-inertial motions in the mixed layer during the passage of traveling storms (Wunsch and Ferrari 2004). Part of their energy is dissipated in the mixed layer due to shear instability, contributing to the mixed layer deepening and sea surface cooling (Greatbatch 1984; Price et al. 1986; Jochum et al. 2013). The remaining energy radiates downward into the thermocline and deep ocean and may play an important role in maintaining the diapycnal mixing there (Munk and Wunsch 1998; Wunsch and Ferrari 2004; Jing and Wu 2014).

The near-inertial wind work and NIWs in the ocean have been extensively analyzed using ocean general circulation models (OGCMs) with the reanalysis wind forcing (e.g., Nagasawa et al. 2000; Furuichi et al. 2008; Rath et al. 2013, 2014; Zhai et al. 2005, 2007, 2009). Currently, many reanalysis products (e.g., the Climate Forecast System Reanalysis) have a temporal resolution of 6 h and should be sufficiently fine to resolve the near-inertial motions at the midlatitudes. However, most of the OGCMs, such as the Regional Ocean Modeling System (Moore et al. 2004; Shchepetkin and McWilliams 2005), interpolate the reanalysis winds or wind stress linearly onto each time step, which partially filters out the wind stress variance in the near-inertial band (Niwa and Hibiya 1999; Nagasawa et al. 2000). In this study, we quantify the influence of the linear interpolation on the near-inertial wind work and oceanic NIWs. It is found that the linear interpolation underestimates the near-inertial wind work and near-inertial kinetic energy in the upper ocean by an amount proportional to the loss of near-inertial wind stress variance. This further weakens the diapycnal mixing in the ocean due to the reduction of near-inertial shear variance. We propose an alternative interpolation method, that is, the sinc-function interpolation. Compared to the linear interpolation, the sinc-function interpolation retains all the wind stress variance in the near-inertial band and yields correct magnitudes of near-inertial wind work and NIWs at the midlatitudes. The paper is organized as follows. The numerical model configurations are introduced in section 2. In section 3, we compare the linear and sinc-function interpolations, especially their influences on the frequency spectrum of wind stress. Model results and analyses are presented in section 4. Conclusions are summarized in section 5 followed by the discussions.

2. Model configurations

a. Idealized numerical simulations

Most of the wind work on near-inertial motions occurs during the passage of traveling windstorm systems (Watanabe and Hibiya 2002; Alford 2003). To explore the influences of different interpolation methods on the simulation of near-inertial wind work and NIWs, we first design an idealized numerical experiment in which a windstorm of regular shape passes over the otherwise quiescent and horizontally homogeneous ocean. The ocean model used here is the Regional Ocean Modeling System (ROMS), which is a terrain-following primitive equation model based on the hydrostatic and Boussinesq approximations (Moore et al. 2004; Shchepetkin and McWilliams 2005). The model is configured over a 20° × 20° domain centered at 30°N with a uniform depth of 2000 m. A total of 50 vertical layers are used with 19 layers concentrated in the upper 100 m. The horizontal grid size is approximately 9 km × 9 km. Physics parameterization schemes adopted in the ROMS include a K-profile parameterization (KPP; Large et al. 1994) vertical turbulent mixing closure scheme, a biharmonic horizontal Smagorinsky mixing for the momentum, and a Laplacian horizontal mixing for tracer diffusion. A radiation boundary condition is used to radiate out the waves.

The simulation is integrated for 7 days with a time step of 200 s. A windstorm travels from the west to the east at a constant translation speed of 7 m s−1. The wind field associated with the storm is constructed following Price (1983):
e1
e2
where and are tangential and radial wind components, respectively; r is the radial distance from the storm center; = 52 m s−1; and R = 40 km.

The frequency spectrum of the wind stress peaks around the inertial frequency (Fig. 1a), so that energetic NIWs are generated during the passage of the windstorm. The experiment consists of three runs. In the first two runs, the wind is sampled every 6 h and is then interpolated onto each time step using the linear and sinc-function interpolations. We name these two runs I-Linear and I-Sinc, respectively. The last run is the control run (I-Control), in which the wind is sampled hourly and is then linearly interpolated onto each time step. It is found that further increasing the sampling frequency does not make any difference in the simulated near-inertial wind work and NIWs.

Fig. 1.
Fig. 1.

The frequency spectrum of wind stress for (a) the single fast-traveling windstorm in the idealized experiment and (b) the Kuroshio Extension region simulated in R-Control.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

b. Realistic numerical simulations in the North Pacific

The idealized experiment neglects the interactions of NIWs with the background flow. Previous studies suggest that the relative vorticity of the background flow plays an important role in the radiation of NIWs (Kunze 1985; Balmforth et al. 1998; Lee and Niiler 1998; Zhai et al. 2005, 2007). To test whether the conclusion derived from the idealized experiment is also applicable in the reality, we design a more realistic numerical experiment consisting of three runs.

In the first run (R-Control), NIWs are simulated using a coupled regional climate model (CRCM) configured over the entire North Pacific. The CRCM includes the Weather Research and Forecasting (WRF) Model (Leung et al. 2006) as the atmospheric component and ROMS as the oceanic component. The WRF and ROMS are configured both at 9-km horizontal resolution and share the same grids in the overlapped domain to avoid a mismatch between land and ocean surface heat fluxes. The CRCM simulation is initialized on 1 October 2002 from a 6-yr ROMS’ spinup simulation (1997–2002) and integrated for one year. The atmospheric forcing fields of the ROMS’ spinup simulation are derived from the Co-ordinated Ocean–Ice Reference Experiments, version 2 (CORE II) dataset (Large and Yeager 2009) and the open boundaries of the ocean are forced by a 5-day-average Simple Ocean Data Assimilation (SODA) dataset (Carton and Giese 2008). The 6-hourly National Centers for Environmental Prediction (NCEP) reanalysis data (Kanamitsu et al. 2002) are interpolated onto a 9-km WRF grid to provide initial and boundary conditions for the atmosphere. In the CRCM, WRF and ROMS are coupled every hour, which is sufficiently fine to resolve wind stress variance in the near-inertial band.

The second and third runs are the ROMS-only simulations and are referred to as R-Linear and R-Sinc, respectively. The starting date of these two runs is 1 October 2002 with the same initial and boundary conditions as those of R-Control. The hourly heat and freshwater flux and 6-hourly winds obtained from R-Control are used to force the two ROMS-only simulations. In R-Linear and R-Sinc, the 6-hourly winds are interpolated onto each time step using the linear and sinc-function interpolations, respectively. As the winds are the only difference among R-Control, R-Linear, and R-Sinc, comparisons of the NIWs among these three runs provide an opportunity to evaluate the effect of interpolation methods in the realistic simulations.

Previous studies suggest that pronounced near-inertial energy is inputted into the Kuroshio Extension region by the winter storms (Watanabe and Hibiya 2002; Alford 2003). In addition, it is also the region with energetic mesoscale eddies, suggesting strong interactions of NIWs with the background flow here. In the following analysis, we focus on the NIWs in the Kuroshio Extension region (27°–43°N, 144°–164°E) during December 2002–March 2003, as it serves a good case study. Figure 1b shows the corresponding wind stress frequency spectrum simulated in R-Control. The spectrum is relatively flat between 0.02 and 0.3 cpd. Then it rolls off rapidly as .

3. The influence of interpolations on the frequency spectrum of wind stress

Let represent the zonal or meridional wind stress component with a time interval of = 6 h. Introduce the zero-padding time series (n/M):
e3
where M is any positive integer and Z denotes the integer set. The interpolation of onto finer grids with a time interval of ΔT/M can be expressed by the convolution formula:
e4
where is called the impulse response function and is uniquely determined by the interpolation method. Introduce the frequency response function:
e5
where and are the frequency spectra for (n/M) and , respectively. Term measures the influence of interpolations in the frequency domain. According to the convolution theorem, , where is the Fourier transform of (n/M).
It can be demonstrated that the impulse response function for the linear interpolation is (Bracewell 1965)
e6
The corresponding frequency response function is
e7
where .
Now introduce the sinc-function interpolation with the impulse response function defined as
e8
The corresponding frequency response function is
e9
where 2π/2ΔT is the Nyquist frequency. For the 6-hourly wind stress, rad s–1 (0.5 cpd).

Figure 2 shows the impulse response function and frequency response function for the linear and sinc-function interpolations. The linear interpolation is equivalent to a low-pass filter, as the value of decreases with the increasing frequency (Fig. 2b). The variance at the inertial frequency is reduced by 18% at 20°N, 35% at 30°N, and 50% at 40°N. In contrast, the sinc-function interpolation retains all the variance within 0-. It should be noted that the magnitude of decreases as asymptotically, suggesting that (n/M) can be approximated as zero in practice for n larger than some critical value . Here the critical value is chosen as , in which case varies within 0.95–1.05 except for the Gibbs phenomenon near (Figs. 2b,c). Sensitivity tests suggest that the simulated near-inertial wind work and NIWs exhibit little variations for further increasing the value of .

Fig. 2.
Fig. 2.

(a) The impulse response function (n/M) for the linear (blue) and sinc-function (red) interpolations. (b) The for the linear (blue solid) and sinc-function (red solid) interpolations of winds in the idealized experiment. The blue and red dashed lines represent associated with the linear and sinc-function interpolations, respectively. (c) As in (b), but for the wind forcing in the Kuroshio Extension region simulated in the R-Control.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

In several widely used atmosphere reanalysis products (e.g., CORE 2), the 10-m wind rather than the wind stress is provided. In this case, the OGCMs compute the wind stress internally based on the bulk formula:
e10
where is the surface wind stress; is the density of air on the sea surface; is the wind at 10 m above the sea surface; and is the drag coefficient, which is also a function of . Here the dependence of wind stress on the ocean surface current (Rath et al. 2013) is neglected for simplicity, as it is unlikely to have any substantial impact on the following conclusions. Because of the nonlinear dependence of the wind stress on the winds, cannot be derived analytically. In particular, is not only related to but also depends on the shape of the frequency spectrum of winds. Here we evaluate numerically.

For both the idealized and realistic numerical simulations, is significantly smaller than , suggesting that the linear interpolation of 6-hourly winds leads to additional loss of near-inertial wind stress variance than predicts due to the nonlinear dependence of wind stress on winds (Zhai et al. 2012). Unlike , agrees with reasonably well (Figs. 2b,c). For <1.5 cpd, the discrepancy between and is less than 5%. Therefore, the sinc-function interpolation retains most of the wind stress variance in the near-inertial band at the midlatitudes.

4. Results

a. The idealized numerical experiment

After the passage of the windstorm, energetic NIWs are excited around the storm track (Fig. 3a). There is evident cross-track asymmetry for the storm-generated NIWs with stronger near-inertial kinetic energy to the right of the track. This is mainly because the wind stress vector rotates anticyclonically to the right of the track and thus is potentially able to resonate with the inertial oscillations there (Price 1983). The energetic NIWs enhance the turbulent mixing in the mixed layer through shear instability, leading to significant sea surface cooling (Fig. 3b).

Fig. 3.
Fig. 3.

The surface (a) near-inertial kinetic energy and (b) temperate just after the passage of the windstorm in I-Control. The white solid line represents the track of the storm, and the white dashed lines denoting its edges.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

To examine the influences of different interpolation methods on the simulated near-inertial wind work and near-inertial kinetic energy (see the appendix for their definitions), and are horizontally averaged within 29°–31°N, 10°–11°E, which is located at the center of the model domain. Sensitivity tests suggest that choosing different longitude bands does not have a substantial impact on the following results. The overall magnitudes of and in I-Sinc are very similar to those in I-Control despite some minor differences in the spatial and temporal structures (Figs. 4, 5). The mean ( in the upper 100 m) in I-Sinc is 0.46 W m−2 (0.28 m2 s–2), comparable to 0.52 W m−2 (0.28 m2 s−2) in I-Control (Table 1). In contrast, the linear interpolation leads to much weaker near-inertial response in the ocean (Figs. 4, 5). The mean in the upper 100 m is underestimated by 50%, while the mean is underestimated by 60% (Table 1). Note that the 60% underestimation of is significantly larger than the 35% reduction of wind variance in the near-inertial band due to the linear interpolation at 30°N (Fig. 2b). This mainly results from the nonlinear dependence of the wind stress on the wind. For the fast-traveling windstorm, the linear interpolation of winds leads to a 60% loss of near-inertial wind stress variance at 30°N (Fig. 2b).

Fig. 4.
Fig. 4.

The time series of averaged within 29°–31°N, 10°–11°E in I-Control, I-Linear, and I-Sinc.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

Fig. 5.
Fig. 5.

The time-depth plot of in m2 s–2 averaged within 29°–31°N, 10°–11°E in (a) I-Control, (b) I-Linear, and (c) I-Sinc. (bottom) The time-depth plot of in 10−4 s−2 averaged within 29°–31°N, 10°–11°E in (d) I-Control, (e) I-Linear, and (f) I-Sinc. The white lines denote the mixed layer depth, defined as the depth at which the temperature is 0.2°C colder than the sea surface temperature.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

Table 1.

The time-mean near-inertial wind work , near-inertial kinetic energy in the upper 100 m, and near-inertial shear variance within 50–150 m. The values are derived from the idealized experiment and are horizontally averaged within the box 29°–31°N, 10°–11°E.

Table 1.

The weakened NIWs in I-Linear have an important influence on the diapycnal mixing. The mean near-inertial shear variance between 50 and 150 m in I-Sinc is s−2, almost the same as that in I-Control (Fig. 5; Table 1). But the mean in I-Linear decreases to s−2, 32% smaller than that in I-Control and I-Sinc. The reduced in I-Linear further weakens the diapycnal mixing and leads to reduced mixed layer cooling (Fig. 6). After the passage of the storm, the mixed layer temperature decreases by ~1.3°C in I-Sinc and I-Control, while it only decreases by ~0.7°C in I-Linear.

Fig. 6.
Fig. 6.

(a) The decrease of mixed layer temperature averaged within 29°–31°N, 10°–11°E, and its contribution from (b) diapycnal mixing and (c) advection.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

b. The numerical experiment in the North Pacific

In this section, we examine the influence of different interpolation methods on the near-inertial wind work and NIWs using a more realistic numerical model configured over the North Pacific. Figure 7 displays the annual-mean surface near-inertial current amplitude in R-Control. It agrees well with the observations derived from surface drifters (Fig. 1b in Chaigneau et al. 2008). The near-inertial current amplitude averaged over the North Pacific is 12.3 cm s−1, comparable to 11.5 cm s−1 obtained from the observations (Chaigneau et al. 2008). The most energetic near-inertial currents are found around 43°N collocated with the midlatitude storm tracks. The zonal-mean near-inertial current variance there reaches up to 55 cm2 s−2 and is close to the observed value ~50 cm2 s−2 (Elipot and Lumpkin 2008). These agreements give confidence that the simulated NIWs in R-Control are qualitatively reliable.

Fig. 7.
Fig. 7.

The mean surface near-inertial current amplitude (m s−1) during October 2002–September 2003 in R-Control. The black box denotes the Kuroshio Extension region analyzed in this study.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

The conclusions derived from the idealized experiment are generally applicable to the more realistic experiment with strong interactions between the NIWs and background flow. While the magnitudes of and between R-Control and R-Sinc are very similar with a discrepancy of less than 10%, the linear interpolation leads to significant damping of NIWs (Fig. 8). In particular, the reduction of and in R-Linear is evident throughout the upper 1000 m, suggesting that the underestimation of NIWs by the linear interpolation is not confined to the mixed layer but extends to the deep ocean. The reduced in R-Linear further weakens the diapycnal mixing at the base of the mixed layer (Fig. 9a). This reduces the downward heat flux into the thermocline, leading to a warm (cold) bias in the mixed layer (thermocline) in R-Linear (Fig. 9b).

Fig. 8.
Fig. 8.

The mean and within the Kuroshio Extension region in R-Control, R-Linear, and R-Sinc.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

Fig. 9.
Fig. 9.

(a) The mean vertical diffusivity within the Kuroshio Extension region in R-Linear (blue) and R-Sinc (red) minus that in R-Control. (b) As in (a), but for temperature.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

To examine the latitude-dependent effect of different interpolations on NIWs, we introduce , , and , which are defined as the ratio of zonal-mean , , and in R-Linear or R-Sinc to that in R-Control, respectively. Here, and are further averaged within 0–300 m, where they show the largest values (Fig. 8). In R-Sinc, , , and vary within 0.95–1.20 throughout 27°–43°N (Fig. 10), suggesting that the sinc-interpolation yields correct magnitudes for the near-inertial wind work and NIWs at the midlatitudes. In contrast, , , and in R-Linear are significantly smaller than unity and decrease with increasing latitude (Fig. 10). In particular, and agree reasonably well with , where is the inertial frequency at each latitude. This agreement can be understood by using the simple slab mixed layer model proposed by Pollard and Millard (1970):
e11
where is the mixed layer current, is the seawater density, H is the mixed layer depth, is the wind stress, and r is the frequency-dependent damping parameter modeling the slow downward radiation of near-inertial energy into the thermocline:
e12
where and (Alford 2003).
Fig. 10.
Fig. 10.

The latitudal distribution of , , and , which are defined as the ratio of zonal-mean , , and in R-Linear (blue) or R-Sinc (red) to that in R-Control, respectively. Here and are vertically averaged within 0–300 m. The gray dashed line denotes the frequency response function associated with the linear interpolation.

Citation: Journal of Atmospheric and Oceanic Technology 32, 10; 10.1175/JTECH-D-15-0046.1

Equation (11) can be solved in the frequency domain. According to the Parseval’s theorem, the time-mean and are
e13
e14

As r is much smaller than f, the function is strongly peaked at the inertial frequency. Equations (13) and (14) thus indicate that the time-mean and are proportional to the wind stress variance at the inertial frequency. Therefore, the underestimation of and by the linear interpolation should be proportional to the loss of near-inertial wind stress variance. Note that the interactions of NIWs with the mesoscale eddies are not included in (11). The agreement between the numerical simulations and predictions from (11) implies that the near-inertial wind stress plays a more dominant role than mesoscale eddies in the mixed layer near-inertial energy budget, consistent with the recent findings of Jing and Wu (2014).

Unlike and , in R-Linear is significantly larger than throughout 27°–43°N (Fig. 10c). This is also the case for the idealized experiment where the linear interpolation underestimates the near-inertial wind stress variance by 60% but only by 32% for . Therefore, the response of to the near-inertial wind stress contains more complicated dynamics not included in (11). The smaller difference of between R-Linear and R-Control may be partially attributed to the turbulent dissipation of NIWs, which is neglected in (11) (Plueddemann and Farrar 2006). While most of the near-inertial energy goes into the low-order vertical modes (Gill 1984), a substantial portion of the near-inertial shear variance comes from high-order modes that are subject to significant turbulent dissipation (Alford and Gregg 2001). Previous studies suggest that the turbulent dissipation rate increases much more rapidly than linearly with the shear variance (Large et al. 1994; Polzin et al. 2014). In this case, may increase less rapidly than linearly with due to the significant turbulent dissipation of the shear-containing modes, making in R-Linear larger than .

5. Conclusions

In this study, we quantify the influences of the linear interpolation of 6-hourly winds on the near-inertial wind work and NIWs using ROMS. Two cases are considered. The first is a single fast-traveling windstorm with its wind stress frequency spectrum peaking around the inertial frequency. The second is the realistic wind forcing over the Kuroshio Extension region. Despite the distinct wind fields between the two cases, the same conclusions are reached and summarized as follows:

  1. The linear interpolation of 6-hourly winds significantly underestimates the near-inertial wind work and NIWs in the ocean. The underestimation of the near-inertial wind work and near-inertial kinetic energy is proportional to the loss of near-inertial wind stress variance due to the linear interpolation. The near-inertial shear variance is also reduced by the linear interpolation. But the reduction is less significant compared to that of near-inertial wind work and near-inertial kinetic energy.

  2. The reduced near-inertial shear variance due to the linear interpolation further weakens the diapycnal mixing at the base of the mixed layer and results in a warm bias in the mixed layer.

  3. The proposed sinc-function interpolation retains all the near-inertial wind stress variance and yields correct magnitudes of near-inertial wind work and NIWs at the midlatitudes. It is a better interpolation method than the linear interpolation and should be used in the OGCMs.

6. Discussions

Niwa and Hibiya (1999) proposed a latitude-dependent amplification factor to correct the underestimation of near-inertial wind work and kinetic energy due to the linear interpolation. Compared to this simple amplification factor, the sinc-function interpolation has the following advantages. First, the latitude-dependent amplification factor is only applicable when the wind stress is used to force the ocean. When the surface winds are used, the underestimation of near-inertial wind work and kinetic energy by the linear interpolation is not equal to due to the nonlinear dependence of wind stress on winds (Figs. 2b,c). Second, the latitude-dependent amplification factor is an “offline” correction for the damping of NIWs due to the linear interpolation. It is not able to correct the underestimation of NIW-induced mixing. In contrast, the sinc-function interpolation can be incorporated into numerical models without any difficulty, leading to correct magnitude for the NIW-induced mixing (Figs. 6, 9).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) Key Project (41130859), the National Major Research Plan of Global Change (2013CB956201), and the NSFC–Shandong Joint Fund for Marine Science Research Centers. Z. J. is partly supported by the China Scholarship Council.

APPENDIX

Computation of Near-Inertial Wind Work, Kinetic Energy, and Shear Variance

In this study, the near-inertial current (,) is retained by high-pass filtering the horizontal velocity with a cutoff frequency of 0.8 f, where f is the local Coriolis frequency (Furuichi et al. 2008). The near-inertial kinetic energy and shear variance are defined as and , respectively. The near-inertial wind work is defined as , where and are the high-pass filtered zonal and meridional wind stress components, respectively.

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  • Greatbatch, R. J., 1984: On the response of the ocean to a travelling storm: Parameters and scales. J. Phys. Oceanogr., 14, 59–78, doi:10.1175/1520-0485(1984)014<0059:OTROTO>2.0.CO;2.

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  • Jing, Z., and Wu L. , 2014: Intensified diapycnal mixing in the midlatitude western boundary current. Sci. Rep., 4, 7412, doi:10.1038/srep07412.

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    • Export Citation

  • Jochum, M., Briegleb B. P. , Danabasoglu G. , Large W. G. , Norton N. J. , Jayne S. R. , Alford M. H. , and Bryan F. O. , 2013: The impact of oceanic near-inertial waves on climate. J. Climate, 26, 2833–2844, doi:10.1175/JCLI-D-12-00181.1.

    • Search Google Scholar
    • Export Citation

  • Kanamitsu, M., Ebisuzaki W. , Woollen J. , Yang S.-K. , Hnilo J. J. , Fiorino M. , and Potter G. L. , 2002: NCEP–DOE AMIP-II Reanalysis (R-2). Bull. Amer. Meteor. Soc., 83, 1631–1643, doi:10.1175/BAMS-83-11-1631.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., 1985: Near-inertial propagation in geostrophic shear. J. Phys. Oceanogr., 15, 544–565, doi:10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Large, W. G., and Yeager S. G. , 2009: The global climatology of an interannually varying air–sea flux data set. Climate Dyn., 33, 341–364, doi:10.1007/s00382-008-0441-3.

    • Search Google Scholar
    • Export Citation

  • Large, W. G., McWilliams J. C. , and Doney S. C. , 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary-layer parameterization. Rev. Geophys., 32, 363–403, doi:10.1029/94RG01872.

    • Search Google Scholar
    • Export Citation

  • Lee, D.-K., and Niiler P. P. , 1998: The inertial chimney: The near-inertial energy drainage from the ocean surface to the deep layer. J. Geophys. Res., 103, 7579–7591, doi:10.1029/97JC03200.

    • Search Google Scholar
    • Export Citation

  • Leung, L. R., Kuo Y.-H. , and Tribbia J. , 2006: Research needs and directions of regional climate modeling using WRF and CCSM. Bull. Amer. Meteor. Soc., 87, 1747–1751, doi:10.1175/BAMS-87-12-1747.

    • Search Google Scholar
    • Export Citation

  • Moore, A. M., Arango H. G. , Di Lorenzo E. , Cornuelle D. B. , Miller A. J. , and Nielson D. J. , 2004: A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of a regional ocean model. Ocean Modell., 7, 227–258, doi:10.1016/j.ocemod.2003.11.001.

    • Search Google Scholar
    • Export Citation

  • Munk, W., and Wunsch C. , 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, doi:10.1016/S0967-0637(98)00070-3.

    • Search Google Scholar
    • Export Citation

  • Nagasawa, M., Niwa Y. , and Hibiya T. , 2000: Spatial and temporal distribution of the wind-induced internal wave energy available for deep water mixing in the North Pacific. J. Geophys. Res., 105, 13 933–13 943, doi:10.1029/2000JC900019.

    • Search Google Scholar
    • Export Citation

  • Niwa, Y., and Hibiya T. , 1999: Response of the deep ocean internal wave field to traveling midlatitude storms as observed in long-term current measurements. J. Geophys. Res., 104, 10 981–10 989, doi:10.1029/1999JC900046.

    • Search Google Scholar
    • Export Citation

  • Plueddemann, A. J., and Farrar J. T. , 2006: Observations and models of the energy flux from the wind to mixed-layer inertial currents. Deep-Sea Res. I, 53, 5–30, doi:10.1016/j.dsr2.2005.10.017.

    • Search Google Scholar
    • Export Citation

  • Pollard, R. T., and Millard R. C. Jr., 1970: Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res. Oceanogr. Abstr., 17, 813–821, doi:10.1016/0011-7471(70)90043-4.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., Garabato A. C. N. , Huussen T. N. , Sloyan B. M. , and Waterman S. , 2014: Finescale parameterizations of turbulent dissipation. J. Geophys. Res. Oceans, 119, 1383–1419, doi:10.1002/2013JC008979.

    • Search Google Scholar
    • Export Citation

  • Price, J. F., 1983: Internal wave wake of a travelling storm. Part I: Scales, energy budget and observations. J. Phys. Oceanogr., 13, 949–965, doi:10.1175/1520-0485(1983)013<0949:IWWOAM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Price, J. F., Weller R. A. , and Pinkel R. , 1986: Diurnal cycling: Observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing. J. Geophys. Res., 91, 8411–8427, doi:10.1029/JC091iC07p08411.

    • Search Google Scholar
    • Export Citation

  • Rath, W., Greatbatch R. J. , and Zhai X. , 2013: Reduction of near-inertial energy through the dependence of wind stress on the ocean-surface velocity. J. Geophys. Res. Oceans, 118, 2761–2773, doi:10.1002/jgrc.20198.

    • Search Google Scholar
    • Export Citation

  • Rath, W., Greatbatch R. J. , and Zhai X. , 2014: On the spatial and temporal distribution of near-inertial energy in the Southern Ocean. J. Geophys. Res. Oceans, 119, 359–376, doi:10.1002/2013JC009246.

    • Search Google Scholar
    • Export Citation

  • Shchepetkin, A. F., and McWilliams J. C. , 2005: The regional oceanic modeling system (ROMS): A split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modell., 9, 347–404, doi:10.1016/j.ocemod.2004.08.002.

    • Search Google Scholar
    • Export Citation

  • Watanabe, M., and Hibiya T. , 2002: Global estimates of the wind-induced energy flux to inertial motions in the surface mixed layer. Geophys. Res. Lett., 29, 1239, doi:10.1029/2001GL014422.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and Ferrari R. , 2004: Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36, 281–314, doi:10.1146/annurev.fluid.36.050802.122121.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Greatbatch R. J. , and Zhao J. , 2005: Enhanced vertical propagation of storm-induced near-inertial energy in an eddying ocean channel model. Geophys. Res. Lett., 32, L18602, doi:10.1029/2005GL023643.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Greatbatch R. J. , and Eden C. , 2007: Spreading of near-inertial energy in a 1/12° model of the North Atlantic Ocean. Geophys. Res. Lett., 34, L10609, doi:10.1029/2007GL029895.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Greatbatch R. J. , Eden C. , and Hibiya T. , 2009: On the loss of wind-induced near-inertial energy to turbulent mixing in the upper ocean. J. Phys. Oceanogr., 39, 3040–3045, doi:10.1175/2009JPO4259.1.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Johnson H. L. , Marshall D. P. , and Wunsch C. , 2012: On the wind power input to the ocean general circulation. J. Phys. Oceanogr., 42, 1357–1365, doi:10.1175/JPO-D-12-09.1.

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  • Greatbatch, R. J., 1984: On the response of the ocean to a travelling storm: Parameters and scales. J. Phys. Oceanogr., 14, 59–78, doi:10.1175/1520-0485(1984)014<0059:OTROTO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Jing, Z., and Wu L. , 2014: Intensified diapycnal mixing in the midlatitude western boundary current. Sci. Rep., 4, 7412, doi:10.1038/srep07412.

    • Search Google Scholar
    • Export Citation

  • Jochum, M., Briegleb B. P. , Danabasoglu G. , Large W. G. , Norton N. J. , Jayne S. R. , Alford M. H. , and Bryan F. O. , 2013: The impact of oceanic near-inertial waves on climate. J. Climate, 26, 2833–2844, doi:10.1175/JCLI-D-12-00181.1.

    • Search Google Scholar
    • Export Citation

  • Kanamitsu, M., Ebisuzaki W. , Woollen J. , Yang S.-K. , Hnilo J. J. , Fiorino M. , and Potter G. L. , 2002: NCEP–DOE AMIP-II Reanalysis (R-2). Bull. Amer. Meteor. Soc., 83, 1631–1643, doi:10.1175/BAMS-83-11-1631.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., 1985: Near-inertial propagation in geostrophic shear. J. Phys. Oceanogr., 15, 544–565, doi:10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Large, W. G., and Yeager S. G. , 2009: The global climatology of an interannually varying air–sea flux data set. Climate Dyn., 33, 341–364, doi:10.1007/s00382-008-0441-3.

    • Search Google Scholar
    • Export Citation

  • Large, W. G., McWilliams J. C. , and Doney S. C. , 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary-layer parameterization. Rev. Geophys., 32, 363–403, doi:10.1029/94RG01872.

    • Search Google Scholar
    • Export Citation

  • Lee, D.-K., and Niiler P. P. , 1998: The inertial chimney: The near-inertial energy drainage from the ocean surface to the deep layer. J. Geophys. Res., 103, 7579–7591, doi:10.1029/97JC03200.

    • Search Google Scholar
    • Export Citation

  • Leung, L. R., Kuo Y.-H. , and Tribbia J. , 2006: Research needs and directions of regional climate modeling using WRF and CCSM. Bull. Amer. Meteor. Soc., 87, 1747–1751, doi:10.1175/BAMS-87-12-1747.

    • Search Google Scholar
    • Export Citation

  • Moore, A. M., Arango H. G. , Di Lorenzo E. , Cornuelle D. B. , Miller A. J. , and Nielson D. J. , 2004: A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of a regional ocean model. Ocean Modell., 7, 227–258, doi:10.1016/j.ocemod.2003.11.001.

    • Search Google Scholar
    • Export Citation

  • Munk, W., and Wunsch C. , 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, doi:10.1016/S0967-0637(98)00070-3.

    • Search Google Scholar
    • Export Citation

  • Nagasawa, M., Niwa Y. , and Hibiya T. , 2000: Spatial and temporal distribution of the wind-induced internal wave energy available for deep water mixing in the North Pacific. J. Geophys. Res., 105, 13 933–13 943, doi:10.1029/2000JC900019.

    • Search Google Scholar
    • Export Citation

  • Niwa, Y., and Hibiya T. , 1999: Response of the deep ocean internal wave field to traveling midlatitude storms as observed in long-term current measurements. J. Geophys. Res., 104, 10 981–10 989, doi:10.1029/1999JC900046.

    • Search Google Scholar
    • Export Citation

  • Plueddemann, A. J., and Farrar J. T. , 2006: Observations and models of the energy flux from the wind to mixed-layer inertial currents. Deep-Sea Res. I, 53, 5–30, doi:10.1016/j.dsr2.2005.10.017.

    • Search Google Scholar
    • Export Citation

  • Pollard, R. T., and Millard R. C. Jr., 1970: Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res. Oceanogr. Abstr., 17, 813–821, doi:10.1016/0011-7471(70)90043-4.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., Garabato A. C. N. , Huussen T. N. , Sloyan B. M. , and Waterman S. , 2014: Finescale parameterizations of turbulent dissipation. J. Geophys. Res. Oceans, 119, 1383–1419, doi:10.1002/2013JC008979.

    • Search Google Scholar
    • Export Citation

  • Price, J. F., 1983: Internal wave wake of a travelling storm. Part I: Scales, energy budget and observations. J. Phys. Oceanogr., 13, 949–965, doi:10.1175/1520-0485(1983)013<0949:IWWOAM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Price, J. F., Weller R. A. , and Pinkel R. , 1986: Diurnal cycling: Observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing. J. Geophys. Res., 91, 8411–8427, doi:10.1029/JC091iC07p08411.

    • Search Google Scholar
    • Export Citation

  • Rath, W., Greatbatch R. J. , and Zhai X. , 2013: Reduction of near-inertial energy through the dependence of wind stress on the ocean-surface velocity. J. Geophys. Res. Oceans, 118, 2761–2773, doi:10.1002/jgrc.20198.

    • Search Google Scholar
    • Export Citation

  • Rath, W., Greatbatch R. J. , and Zhai X. , 2014: On the spatial and temporal distribution of near-inertial energy in the Southern Ocean. J. Geophys. Res. Oceans, 119, 359–376, doi:10.1002/2013JC009246.

    • Search Google Scholar
    • Export Citation

  • Shchepetkin, A. F., and McWilliams J. C. , 2005: The regional oceanic modeling system (ROMS): A split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modell., 9, 347–404, doi:10.1016/j.ocemod.2004.08.002.

    • Search Google Scholar
    • Export Citation

  • Watanabe, M., and Hibiya T. , 2002: Global estimates of the wind-induced energy flux to inertial motions in the surface mixed layer. Geophys. Res. Lett., 29, 1239, doi:10.1029/2001GL014422.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and Ferrari R. , 2004: Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36, 281–314, doi:10.1146/annurev.fluid.36.050802.122121.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Greatbatch R. J. , and Zhao J. , 2005: Enhanced vertical propagation of storm-induced near-inertial energy in an eddying ocean channel model. Geophys. Res. Lett., 32, L18602, doi:10.1029/2005GL023643.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Greatbatch R. J. , and Eden C. , 2007: Spreading of near-inertial energy in a 1/12° model of the North Atlantic Ocean. Geophys. Res. Lett., 34, L10609, doi:10.1029/2007GL029895.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Greatbatch R. J. , Eden C. , and Hibiya T. , 2009: On the loss of wind-induced near-inertial energy to turbulent mixing in the upper ocean. J. Phys. Oceanogr., 39, 3040–3045, doi:10.1175/2009JPO4259.1.

    • Search Google Scholar
    • Export Citation

  • Zhai, X., Johnson H. L. , Marshall D. P. , and Wunsch C. , 2012: On the wind power input to the ocean general circulation. J. Phys. Oceanogr., 42, 1357–1365, doi:10.1175/JPO-D-12-09.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The frequency spectrum of wind stress for (a) the single fast-traveling windstorm in the idealized experiment and (b) the Kuroshio Extension region simulated in R-Control.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) The impulse response function (n/M) for the linear (blue) and sinc-function (red) interpolations. (b) The for the linear (blue solid) and sinc-function (red solid) interpolations of winds in the idealized experiment. The blue and red dashed lines represent associated with the linear and sinc-function interpolations, respectively. (c) As in (b), but for the wind forcing in the Kuroshio Extension region simulated in the R-Control.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The surface (a) near-inertial kinetic energy and (b) temperate just after the passage of the windstorm in I-Control. The white solid line represents the track of the storm, and the white dashed lines denoting its edges.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    The time series of averaged within 29°–31°N, 10°–11°E in I-Control, I-Linear, and I-Sinc.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    The time-depth plot of in m2 s–2 averaged within 29°–31°N, 10°–11°E in (a) I-Control, (b) I-Linear, and (c) I-Sinc. (bottom) The time-depth plot of in 10−4 s−2 averaged within 29°–31°N, 10°–11°E in (d) I-Control, (e) I-Linear, and (f) I-Sinc. The white lines denote the mixed layer depth, defined as the depth at which the temperature is 0.2°C colder than the sea surface temperature.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) The decrease of mixed layer temperature averaged within 29°–31°N, 10°–11°E, and its contribution from (b) diapycnal mixing and (c) advection.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The mean surface near-inertial current amplitude (m s−1) during October 2002–September 2003 in R-Control. The black box denotes the Kuroshio Extension region analyzed in this study.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The mean and within the Kuroshio Extension region in R-Control, R-Linear, and R-Sinc.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    (a) The mean vertical diffusivity within the Kuroshio Extension region in R-Linear (blue) and R-Sinc (red) minus that in R-Control. (b) As in (a), but for temperature.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    The latitudal distribution of , , and , which are defined as the ratio of zonal-mean , , and in R-Linear (blue) or R-Sinc (red) to that in R-Control, respectively. Here and are vertically averaged within 0–300 m. The gray dashed line denotes the frequency response function associated with the linear interpolation.

Fig. 1.

The frequency spectrum of wind stress for (a) the single fast-traveling windstorm in the idealized experiment and (b) the Kuroshio Extension region simulated in R-Control.
Fig. 2.

(a) The impulse response function (n/M) for the linear (blue) and sinc-function (red) interpolations. (b) The for the linear (blue solid) and sinc-function (red solid) interpolations of winds in the idealized experiment. The blue and red dashed lines represent associated with the linear and sinc-function interpolations, respectively. (c) As in (b), but for the wind forcing in the Kuroshio Extension region simulated in the R-Control.
Fig. 3.

The surface (a) near-inertial kinetic energy and (b) temperate just after the passage of the windstorm in I-Control. The white solid line represents the track of the storm, and the white dashed lines denoting its edges.
Fig. 4.

The time series of averaged within 29°–31°N, 10°–11°E in I-Control, I-Linear, and I-Sinc.
Fig. 5.

The time-depth plot of in m2 s–2 averaged within 29°–31°N, 10°–11°E in (a) I-Control, (b) I-Linear, and (c) I-Sinc. (bottom) The time-depth plot of in 10−4 s−2 averaged within 29°–31°N, 10°–11°E in (d) I-Control, (e) I-Linear, and (f) I-Sinc. The white lines denote the mixed layer depth, defined as the depth at which the temperature is 0.2°C colder than the sea surface temperature.
Fig. 6.

(a) The decrease of mixed layer temperature averaged within 29°–31°N, 10°–11°E, and its contribution from (b) diapycnal mixing and (c) advection.
Fig. 7.

The mean surface near-inertial current amplitude (m s−1) during October 2002–September 2003 in R-Control. The black box denotes the Kuroshio Extension region analyzed in this study.
Fig. 8.

The mean and within the Kuroshio Extension region in R-Control, R-Linear, and R-Sinc.
Fig. 9.

(a) The mean vertical diffusivity within the Kuroshio Extension region in R-Linear (blue) and R-Sinc (red) minus that in R-Control. (b) As in (a), but for temperature.
Fig. 10.

The latitudal distribution of , , and , which are defined as the ratio of zonal-mean , , and in R-Linear (blue) or R-Sinc (red) to that in R-Control, respectively. Here and are vertically averaged within 0–300 m. The gray dashed line denotes the frequency response function associated with the linear interpolation.
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