1. Introduction
Wind-generated near-inertial waves (NIWs) are ubiquitous in the global ocean and play an important role in the oceanic energy budget (D’Asaro 1984, 1985; Alford and Gregg 2001; Alford et al. 2016). In the internal wave spectrum, there is a pronounced near-inertial peak that accounts for half of the energy and a substantial portion of the shear (Ferrari and Wunsch 2009; Alford et al. 2016). Therefore, NIWs have the potential importance in mixing the deep ocean by energy cascade involving wave–wave interactions or by direct wave breaking (Olbers 1976; Gregg et al. 2003; Alford and Gregg 2001).
Because of the slow group velocity, NIWs are more likely to interact with mesoscale eddies than internal tides, resulting in complex propagation trajectories. NIWs are repelled away from cyclonic eddies (CEs) and are trapped within anticyclonic eddies (ACEs) in both horizontal and vertical directions (Lee et al. 1994; Lee and Niiler 1998; Zhai et al. 2005, 2007; Danioux and Klein 2008; Lelong et al. 2020). Kunze (1985) shows that when NIWs are generated inside an ACE with negative relative vorticity, they have frequency below the effective Coriolis frequency, feff, of the surrounding water and cannot propagate outside freely. Here, feff = f0 + ζ/2 is equal to the planetary value of the Coriolis frequency f0 plus one-half of the relative vorticity ζ. Whitt and Thomas (2013) also suggested theoretically that the trapping and the largest amplification of NIWs appear in regions of tilted isopycnals, where the mean flow is strongly baroclinic and the wave frequency is lower than the feff. After being trapped by an ACE, the propagation of NIWs is suppressed due to the vertically weakening of the vorticity field. NIWs finally reach a critical layer in which their group velocity becomes zero with the shrinking of their vertical wavelength, which ultimately promotes turbulent mixing.
However, about how deep the near-inertial energy can penetrate under the influence of ACE, there are still different opinions. Recent observations indicated that although ACEs are favorable for the amplification of the near-inertial energy, they may contribute little to the deep-ocean mixing due to the trapping of NIWs (e.g., Zhang et al. 2018; Xu et al. 2022). In the subtropical northwestern Pacific, Zhang et al. (2018) found that NIWs can seldom penetrate into the deep ocean because most of them dissipate near the critical layer at ∼200-m depth. By using two mooring arrays and cruise transects, Xu et al. (2022) showed that NIWs are trapped at depths around 200 and 315 m due to the critical layer effect. In the Kuroshio extension region with strong baroclinicity and abundant energetic mesoscale eddies, Li et al. (2022) found that the critical layers of the trapped NIWs in ACE are located at 350–650-m depth. In contrast, many studies point to the important role of ACEs in promoting the downward penetration of NIWs and enhancing the diapycnal mixing stirred by the wind-generated NIWs at greater depth. Using numerical simulations, it was found that the leakage of near-inertial energy out of the surface layer is strongly enhanced by the presence of eddies, especially ACE, by which the near-inertial energy can be carried to more than 1500-m depth (Zhai et al. 2005). Based on Argo data, Jing et al. (2011) found that the enhanced downward propagation of near-inertial energy can extend up to 1000-m depth in the northwestern Pacific in the presence of ACEs. Whalen et al. (2018) suggested that the wind-driven mixing is especially strong and can reach 2000-m depth in the presence of energetic ACEs as compared with CEs. Therefore, the propagation of NIWs in the presence of ACEs deserves further study to figure out what causes this difference.
Additionally, the energy exchanges among mesoscale eddies, NIWs, and high-frequency internal waves (HFIWs) is believed to play an essential role in the ocean energy budget (Kunze et al. 1995; Jing et al. 2017; Thomas and Daniel 2020). Polzin (2010) indicated that the mesoscale-internal wave coupling through horizontal interactions is a significant sink of eddy energy, and the estimated energy transfer rate is the same order of magnitude as the dissipation of the internal wave field. By simulations, Barkan et al. (2021) showed that the wave–eddy coupling can effectively transfer up to 25% of the eddy energy forward into the internal wave field. In wave–eddy couplings, Jing et al. (2018) suggested that NIWs make a significant contribution to the energy exchange due to their stronger interaction with mesoscale eddies than HFIWs. With respect to the dissipation of energetic NIWs, evidence for the direct breaking of them is limited (Sun and Pinkel 2012). As a primary pathway, the energy is first transferred to the HFIWs, and then these waves break, leading to turbulence production, dissipation, and mixing (Alford and Pinkel 2000; Gregg et al. 2003; Waterhouse et al. 2014). Sun and Pinkel (2012) suggested that the energy transfer rates from inertial-diurnal internal waves to HFIWs are of similar magnitude to the estimates of turbulent dissipation. During the downward propagation of NIWs trapped by ACEs, existing studies are mainly restricted to the energy transfer between ACEs and NIWs (Li et al. 2022; Xu et al. 2022), while few studies on the energy transfer between NIWs and HFIWs have been conducted.
The subtropical northwestern Pacific (STNWP) is a region with ubiquitous westward-moving mesoscale eddies (Qiu and Chen 2004) and a very high concentration of tropical cyclones (D’Asaro et al. 2011). Previous observations mainly focused on the propagation of NIWs generated inside an ACE (e.g., Lelong et al. 2020). In this study, we investigate the propagation and energy transfer of NIWs that are generated in regions with positive relative vorticity but subsequently traversed by a westward-propagating ACE. The remaining part of the paper is organized as follows. Section 2 introduces the data and methods. Section 3 presents the downward propagation and trapping of NIWs by a westward-moving ACE. Section 4 discusses the effects of a westward-moving ACE on the propagation of NIWs using a ray-tracing model. In addition, the energy transfers among ACE, NIWs, and HFIWs as well as the implications for mixing are also investigated. The conclusions are given in section 5.
2. Data and methods
a. Data
To investigate the influence of the westward-moving ACE on the propagation of NIWs generated by wind in the STNWP, a subsurface mooring system was deployed at 25°N, 146°E from April 2017 to June 2018 (Fig. 1). The bottom depth at the mooring site was 5692 m. The subsurface mooring system was equipped with one upward-looking and one downward-looking Teledyne RD instrument Workhorse Long Range 75-kHz acoustic Doppler current profiler (ADCP) to measure current velocities in the upper 900-m depth. The sampling time interval of the ADCP was set as 1 h, and the bin size was 8 m. Eight SBE 37-SM instruments (CTD37), designed by Sea-Bird Electronics, Inc., were mounted on the upper 1000 m of the mooring to observe temperature T, salinity, and depth every 10 min. Eleven SBE 56 temperature loggers were also mounted in the upper 700-m depth to intensively measure the temperature (Fig. 1b). On 29 August 2017, Typhoon Sanvu was generated in the STNWP and passed by the mooring position. The typhoon track was obtained from the Japan Meteorological Agency. The wind speed data were obtained from the fifth major global reanalysis produced by the European Centre for Medium-Range Weather Forecasts (ERA5), which has a temporal resolution of 1 h. Sea level anomaly (SLA) and surface geostrophic velocity field data were obtained from altimeter products produced by Ssalto/Duacs, which are distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS). From 20 September 2017 to 30 November 2017, an ACE moved westward and passed by the subsurface mooring system (Figs. 1c–f).
(a) Location of subsurface mooring (red pentagram), with color shading representing the topography. The blue line indicates the track of Typhoon Sanvu. The radius of maximum winds, translation velocity, and minimum distance of subsurface mooring from the typhoon center are indicated. (b) Instruments equipped in the upper 800-m depth of the subsurface mooring. (c)–(f) Spatial distributions of sea level anomaly (SLA) during the westward movement of an ACE. The gray arrows denote geostrophic velocities. The gray dashed lines denote the periphery of the ACE.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
To obtain near-inertial velocities from the ADCP data, a fourth-order Butterworth filter was used for bandpass filtering in the range [0.9 1.15]f0, where f0 is equal to the planetary value of the Coriolis frequency (Chen et al. 2013; Pallàs-Sanz et al. 2016). The background velocities were obtained by applying low-pass filtering with a cutoff frequency of 0.1f0 (Zhang et al. 2018). The HFIW velocities were obtained by high-pass filtering with a cutoff frequency of 6 cpd (Sun and Pinkel 2012), where cpd represents cycles per day. Semidiurnal tides were obtained by bandpass filtering with periods of 10–14 h (Zhang et al. 2018).
In addition to observational data, we also use a reanalysis product (Clementi et al. 2019) modeled by the Met Office Coupled Atmosphere–Land–Ocean–Ice data assimilation (CPLDA) system to analyze the effects of background velocity and stratification on the NIW propagation. The CPLDA reanalysis product is distributed by the CMEMS, which provides data including two-dimensional daily mixed layer depth and three-dimensional daily mean fields of temperature, salinity, and current with horizontal resolutions of 0.25° × 0.25°. Although the subsurface mooring system is equipped with temperature and salinity sensors, the limitation of the vertical resolution in the upper layer makes it difficult to obtain a required stratification profile in the upper 200-m depth. The CPLDA product has 16 layers of data in the upper 200 m, so that a higher vertical resolution profile of stratification can be obtained relative to observations. Before using the CPLDA product, we compared them with observations and found that the reanalysis data can simulate well the potential density and flow fields (see Fig. S1 in the online supplemental material). Therefore, the reanalysis data are utilized to obtain the stratification and relative vorticity below the sea surface.
b. Calculation of vertical wavelength
Wave parameters, such as the horizontal wavenumber kH, vertical wavenumber kz, horizontal wavelength lH, vertical wavelength lz, and vertical group velocity Cgz, can be employed to depict the evolution of NIWs. The kz is calculated by the depth-varying near-inertial velocities at each moment of the ADCP observation. Figure 2 provides a case of calculation of kz. The phase angle, θ = arctan(uni/υni), is derived by the arctangent of the ratio of uni to υni. Figure 2b shows that the phase angle increases with depth, indicating a clockwise (CW) rotation of velocity. Since kz = δθ/δz is the first-order derivative of the phase angle with depth, it can be obtained by a least squares linear fit of the phase–depth profile. In this case, kz is 0.01 rad m−1, corresponding to a lz of 628 m. Thus, lz can be obtained at each moment during the downward propagation of NIWs.
Calculating the vertical wavenumber of NIWs at 0600 UTC 1 Oct 2017: (a) profiles of the near-inertial velocities, and (b) unwrapped angle θ of the near-inertial velocity. The black line in (b) is the result of linear fit.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
c. Ray-tracing model
d. Energy exchange rate among mean flow, NIWs, and HFIWs
3. Results
a. NIWs generation and propagation
To show the generation of NIWs, near-inertial velocities in the mixed layer are obtained by a slab model (e.g., Alford and Gregg 2001) to estimate the inertial wind power input into the mixed layer F = τ ⋅ uni (Fig. 3). Here, τ is wind stress, which is calculated from ERA5 wind data following the parameterization of Oey et al. (2006). The damping parameter r = 0.15fo was used in the slab model following Alford and Gregg (2001). During the passage of Typhoon Sanvu from the end of August to 4 September, there was a significant variation in the wind stress, which induced intense near-inertial motions with maximum velocity of ∼0.5 m s−1 in the mixed layer. The winds are weak for the next 40 days. The total near-inertial wind work done by Typhoon Sanvu is 4 × 10−3 W m−2, which is about an order of magnitude larger than the background level.
(a) Zonal (black line) and meridional (blue line) components of wind stress τ computed from ERA5 wind data. (b) Zonal and meridional near-inertial velocities from the slab model. (c) Near-inertial energy flux F from the slab model. The blue shading denotes the wind work done by Typhoon Sanvu.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
After the generation of near-inertial motions in the mixed layer, prominent near-inertial velocities are quickly observed by the subsurface mooring between 1 September and 30 November 2017 (Figs. 4a,b). The near-inertial kinetic energy (NIKE) is computed as
(a) Depth–time plots of (a) near-inertial zonal velocity, (b) near-inertial meridional velocity, and (c) the logarithm of NIKE from 1 Sep to 30 Nov 2017.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
After 11 September, near-inertial energy starts to propagate to deeper layers, and there is an enhancement of the near-inertial velocities and NIKE near 100- and 500-m depth. On 10 October, near-inertial energy propagates to the maximum observed depth (870 m), and some of the near-inertial energy may continue to propagate downward. However, from 15 October to the end of November, some of the near-inertial energy stays near 750-m depth, indicating that this part of the near-inertial energy may encounter a critical layer and cannot continue to propagate downward. As the critical layer depth of a NIW depends on its frequency (Kunze 1985) and wave packets can be seen as a superposition of different frequencies, the critical layer must extend over a certain depth range. Thus, the observed waves reach the top of the critical layer, but it cannot be ruled out that some energy propagated deeper from about 15 October onward.
It is also noted that from 21 to 26 September, at depths of 150–450 m, the vertical profiles of near-inertial velocities and energy are not coherent in the observations. This may be related to horizontal advection by the mean flow and/or wave propagation (Soares et al. 2016). However, except for this, the near-inertial velocity shows a good coherence through the water column. In addition, we examine the characteristics of the wave packets (Text S2 in the online supplemental material) and demonstrate that the observed near-inertial velocities through the waver column are from the wake following the mixed layer forcing event.
Subsequently, to show the energy peaks and rotary characteristics of internal waves, rotary frequency spectra are computed from velocities measured during the period from 1 September to 30 November 2017 (Fig. 5). Relative to the energy peaks of internal tides (D1 and D2), a more pronounced peak appears at local inertial frequency f25. In the upper layer, the NIWs are a little bit superinertial, with frequency greater than the local inertial frequency, and thus have the ability to propagate downward. In addition, NIWs are strongly CW polarized, which is 2–3 orders of magnitude greater than the counterclockwise (CCW) component. Although energy peaks are also found at diurnal and semidiurnal tidal frequencies, they show noticeably weaker polarization than the near-inertial frequency, with energy of the CW component exceeding that of the CCW component by a factor of 5–10.
Rotary frequency spectra of velocity during the period of NIWs downward propagation for the (a) clockwise (CW) and (b) counterclockwise (CCW) components. The spectral power is taken logarithmically. The local inertial frequency f25, diurnal frequency D1, and semidiurnal frequency D2 are marked by the vertical dashed lines in (a) and (b).
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
Furthermore, the vertical wavenumber spectra of velocity and vertical shear are used to examine the propagation direction and the dominant vertical scales of NIWs. In the Northern Hemisphere, downward-propagating wave energy is associated with CW rotations of velocity and its vertical shear with increasing depth, and vice versa for upward-propagating wave energy (Leaman and Sanford 1975). The wavenumber spectra are plotted for the original measured signals and the filtered signals at near-inertial frequency in Fig. 6. The results show that the spectra of original velocity fall off approximately as
Time-averaged vertical wavenumber spectrum of (a) velocity and (b) vertical shear for original measured signals (dashed lines) and near-inertial signals (solid lines) during period from 1 Sep to 30 Nov 2017. The GM76 spectrum is plotted in each panel with a black dashed line.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
b. Enhanced downward propagation and trapping of NIWs
To better depict the evolution of NIWs as they propagate downward, the vertical propagation depth (VPD), vertical group velocity, and vertical wavelength are examined (Fig. 7). VPD can be obtained by fitting the depth of the mean NIKE (gray dots in Fig. 7a). Because near-inertial energy is initially confined within the upper 100-m depth and starts to propagate downward from 11 September, we compute the vertical group velocity and wavelength of the NIWs in the propagation stage. As can be seen in Fig. 4, the propagation of near-inertial energy can be separated into two parts when arriving at the maximum observed depth on 15 October: one part continuing to propagate downward and the other part staying at ∼750-m depth. Therefore, the fit of the VPD is also divided into two parts. One is to fit the depth of the mean NIKE before 15 October and the other is fitting from 15 October onward. Then, Cgz can be obtained by taking the first-order derivative of the VPD with respect to time (Fig. 7b). It shows that during the downward propagation from 11 September to 15 October (stage I), NIWs are first accelerated to propagate downward, with a maximum Cgz of ∼35 m day−1 at the end of September, and then their Cgz is gradually reduced. After 15 October 2017 (stage II), the Cgz becomes zero, clearly indicating that this part of near-inertial energy no longer continues to propagate downward and stalls at a critical layer of 750 m.
(a) Vertical propagation depth (VPD), (b) negative vertical group velocity −Cgz, and (c) vertical wavelength lz of NIWs. The gray dots in (a) are the depth of mean NIKE at each moment, and the blue line is the VPD derived from polynomial fit. The gray dots in (c) represent the vertical wavelength at each moment, and the blue line is obtained by 72-point smoothing.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
Because Cgz is proportional to lz [Eq. (4)], lz and Cgz have a similar behavior. The evolution of NIW propagation is also reflected by the variations in lz. When the near-inertial energy is accelerated downward, the lz gradually increases from ∼80 m on 11 September to ∼500 m on 30 September; while from 1 October onward, the lz decreases and becomes an almost constant value of 250 m in stage II (Fig. 7c).
Next, we elucidate the relation between the NIW propagation and the ACE. In Figs. 1c–f, we show that an ACE with a radius of ∼150 km moves from the east to the subsurface mooring position about ten days after the passage of Typhoon Sanvu, and leaves the mooring at the end of October. For freely propagating internal waves, their frequencies are above the effective Coriolis frequency of the surrounding water (Kunze 1985). Figure 8a shows that with the arrival of the ACE, the relative vorticity gradually decreases and becomes negative from 21 September to the end of October, causing the effective Coriolis frequency in the surrounding waters to drop to less than the intrinsic frequencies of NIWs. Thus, this leads to a large blue-shift of the near-inertial energy, and rapid downward propagation.
(a) Relative vorticity ζ at the sea surface calculated from geostrophic velocities. Also shown are depth–time plot of (b) zonal and (c) meridional velocities obtained by low-pass filtering with a cutoff frequency of 0.1f. The dashed blue line in (b) and (c) denotes the vertical propagation trajectory of NIWs.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
The distribution of vorticity below the sea surface cannot be directly obtained from our observations. However, we can get a preliminary inspection of the vorticity variations along the propagation trajectory of NIWs by the depth–time plots of background velocities combined with the sea surface vorticity (Fig. 8). It is acknowledged that the relative vorticity increases with depth in the core of an ACE due to the vertical weakening of the eddy flow. With the westward movement of the ACE, NIWs enter the core of ACE after the end of September (Figs. 8b,c), thus encountering an increase in effective Coriolis frequency. Using reanalysis data, we can obtain the distribution of relative vorticity below the sea surface (Fig. S2 in the online supplemental material). It also shows that the relative vorticity along the NIWs trajectory increases after the end of September when the NIWs enter the core of ACE. Under this situation, the propagation of NIWs will undergo a trapping process that is similar to that of the near-inertial energy generated inside an ACE. Therefore, the downward propagation of NIWs will be suppressed vertically, and the vertical group velocity gradually decreases (Fig. 7).
The background velocities also record in detail the passage of the westward-moving ACE (Figs. 8b,c). Because the southern periphery of the ACE crosses the mooring, the zonal velocity is always negative and the meridional velocity is first positive and then negative. Taking the 0.1 m s−1 velocity contour as a reference, the base of the ACE can extend to ∼600–800-m depth, at which the vertical variations in velocities are reduced. It is interesting that the critical layer corresponds exactly to the base of the ACE in stage II of the NIW propagation. The effects of ACE on the propagation of NIWs are discussed in detail in section 4a.
4. Discussions
a. Effects of the westward-moving ACE
A ray-tracing model is applied to interpret the differences of the ACE on NIW propagation under the following two situations of NIW generation: (i) NIWs generated outside an ACE but subsequently past through by the westward-moving ACE and (ii) NIWs generated directly inside an ACE. Figure 9 shows profiles of the background velocities (U, V), stratification, and vorticities. Background velocities are time-averaged results during the downward propagation of NIWs and are obtained directly from the ADCP data. Because the uppermost of temperature/salinity observation extends only to ∼150-m depth, the stratification is derived from reanalysis data, which show a good agreement with observations (see Fig. S3 in the online supplemental material).
(a) Time-averaged background velocities (U, V) during the downward propagation of NIWs. (b) Stratification N2 along the trajectory of NIWs (orange) and inside the ACE (blue). (c) As in (b), but for relative vorticity ζ. The orange and blue lines in (b) and (c) also represent depth variations of stratification and relative vorticity in situation i and situation ii, respectively. Inside the ACE, the vertical profiles of stratification and relative vorticity are time averaged from 30 Sep to 30 Oct. Color shading denotes standard deviation.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
The vertical profiles of stratification and relative vorticity under the above two situations are obtained from reanalysis data (Figs. 9b,c). In situation i, the depth-varying profiles are from the depth–time varying stratification and relative vorticity along the trajectory of NIWs. Initially, the NIWs are located outside an ACE, where the relative vorticity near the surface is positive (Fig. S2 in the online supplemental material). With the movement of the ACE over the mooring, the NIWs propagate into the core of the ACE. The relative vorticity along the trajectory of NIWs turns from positive to negative at the depth of 270 m. At 370-m depth, the relative vorticity along the trajectory of NIWs decreases to the minimum value. Then, the relative vorticity gradually increases with increasing depth. In situation ii, the profiles of stratification and relative vorticity are the time-averaged results inside the ACE. In this situation, the relative vorticity increases with increasing depth due to the weakening of the eddy flow. For (i) the vertical gradient of relative vorticity is first positive above 370-m depth and then negative below that depth; for (ii) the vertical gradient of vorticity is always negative. The stratification has no significant difference in these two situations. Above 150-m depth, however, the stratification rapidly decreases, resulting in a relatively large vertical gradient of stratification.
Since different initial frequencies or vertical wavelengths lead to different critical layer depths (Fer et al. 2018), we set three different initial vertical wavelengths in each ray-tracing model run. According to the results in section 3b, the initial vertical wavelengths are set as 80, 100, and 120 m, respectively. In addition, the horizontal wavelength is set as 200 km, which is estimated from Eq. (4) by substituting the vertical wavelength and the vertical group velocity calculated in section 3b.
Figures 10a and 10b display the three trajectories of NIWs with initial vertical wavelengths of 80, 100, and 120 m for the situation i. Similar to the observations, these rays experience two stages including downward propagation and staying at a critical layer depth. During the downward propagation of these rays, the vertical wavelength and vertical group velocity first increase and then decrease with depth, which can reach 500 m and 40 m day−1, respectively. After around 40 days, these rays encounter a critical depth as the vertical group velocity decreases to zero. In addition, the ray-tracing model shows that the critical depth is dependent on the initial vertical wavelength. For the initial vertical wavelengths of 80, 100, and 120 m, the critical depths are 580, 680, and 810 m, respectively, indicating the larger the vertical wavelength, the greater the critical depth. According to Eq. (3), when the vertical wavelength is larger, the wave frequency is also larger, leading to more rapid downward propagation of NIWs. Therefore, it is reasonable that the critical layer covers a certain depth range due to the superposition of different vertical wavelengths of NIWs.
Wave positions obtained from a ray-tracing model, with l1, l2, and l3 representing the initial vertical wavelengths of 80, 100, and 120 m, respectively, for (left) vertical wavelength lz and (right) vertical group velocity Cgz in situations (a),(b) i and (c),(d) ii.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
In Figs. 10c and 10d, we show the trajectories of NIWs generated in situation ii in which the surface relative vorticity is negative and increases with increasing depth. For the three rays with initial vertical wavelength of 80, 100, and 120 m, the critical depths are 120, 190, and 300 m, respectively. Since these rays are initialized inside the core of ACE with strong red-shifted frequencies, they have a slow propagation at depth and establish a critical layer at shallow depths. As described in the introduction, several observational studies found that the critical depth of an inertial chimney can only reach 200–400-m depth in the western Pacific and South China Sea (Zhang et al. 2018; Li et al. 2022; Xu et al. 2022). Our ray-tracing results also suggest that NIWs generated inside an ACE may contribute little to the deep-ocean mixing, because most of the waves stall at 120–300-m depth and can rarely penetrate to the deeper ocean. However, when the NIWs are generated outside an ACE but are subsequently traversed by a westward-propagating ACE, the critical depths are remarkably deeper than those of NIWs generated directly in an ACE.
To further understand the characteristics of NIWs interacting with the eddy, we design two additional ray-tracing experiments by controlling the vertical variation of the relative vorticity and stratification, respectively. In Figs. 11a and 11b, the input parameters of the ray-tracing model are the same as those in Figs. 10a and 10b, except that the relative vorticity below 370-m depth is set as a constant value of ζ = −1.10 × 10−6 s−1. Thus, there is no vertical increase of relative vorticity in this case. Obviously, the rays continuously propagate downward, and the vertical wavelength and group velocity keep increasing with time, indicating that NIWs can propagate freely and there is no critical depth of propagation.
Similar to Fig. 10, but for two controlled experiments: (a),(b) the vorticity control and (c),(d) the stratification control.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
We note that in Figs. 10c and 10d although the increase of relative vorticity with depth suppresses the downward propagation of the NIWs, the rays still have the characteristic of accelerated downward propagation in the depth of around 100–200 m. Stratification is another significant factor that affects the propagation of NIWs (Gill 1984; Alford and Gregg 2001). Figure 9b shows that the stratification quickly decreases in the upper 200-m depth, so we set the stratification as a background constant (N = 8.3 × 10−4 s−1) to examine its effect (Figs. 11c,d). Interestingly, relative to the results in Figs. 10c and 10d, the downward propagation of the rays is further suppressed and can only reach the depth of around 100 m. This indicates that the vertically decreasing stratification can facilitate the downward propagation of NIWs.
To understand the underlying mechanism responsible for the above observed and ray-tracing results, we perform dynamic analysis on the dispersion relation of NIWs based on the theory of wave–mean flow interaction (Kunze 1985). Considering the relationship between the variation of wavenumber and frequency Δ(kz)/Δt = −Δ(ω)/Δz (Lighthill 1978), temporal variation of the vertical wavenumber Δ(kz)/Δt following a wave packet mainly depends on the vertical gradients of the relative vorticity Δ(ζ)/Δ(z) and stratification Δ(N2)/Δ(z) (Kunze 1985; Kunze et al. 1995). Specifically, when the vertical gradient of the relative vorticity or stratification is less than zero [i.e., Δ(ζ)/Δ(z) < 0 or Δ(N2)/Δ(z) < 0], the vertical wavenumber is increasing with time [i.e., Δ(kz)/Δt > 0], resulting in vertical wavelength and group velocity decreasing with time [i.e., Δ(lz)/Δt < 0 and Δ(Cgz)/Δt < 0]. The reverse is true for positive vertical gradient of relative vorticity or stratification [i.e., Δ(ζ)/Δ(z) > 0 or Δ(N2)/Δ(z) > 0].
Recalling the results in Figs. 9b and 9c, the vertical gradient of stratification is positive above the depth of 200 m. Thus, it will promote downward propagation of NIWs. For situation ii, the vertical gradient of vorticity is negative, resulting in the trapping of downward-propagating NIWs. For situation i, the vertical gradient of vorticity is first positive above the depth of 370 m and then negative below the depth. Therefore, it will first promote the downward propagation of NIWs and then suppress their propagation, ultimately trapping them at a certain depth. Therefore, relative to NIWs directly generated inside an ACE, the critical depth for those generated outside an ACE but are subsequently crossed by this ACE will be larger.
Furthermore, to understand the propagation of NIWs in a more realistic scenario, Fig. 12 shows the ray trajectories using depth–time varying stratification and relative vorticity. Seven different initial vertical wavelengths (80, 90, 100, 110, 120, 130, and 140 m) were set in the ray-tracing model. Interestingly, it is found that the ray-tracing results using depth–time variations of stratification and relative vorticity have similar propagation characteristics to those using variations along the trajectory of NIWs (Figs. 10a,b). These rays also experience two stages including downward propagation and staying at a critical layer depth (Figs. 12c,d). This further demonstrates that NIWs generated outside and passed over by an ACE will propagate to deeper depths than those directly generated inside an ACE.
Depth–time plots of (a) stratification N2 and (b) relative vorticity ζ. The dashed blue line in (a) denotes the vertical propagation trajectory of NIWs from observations. Also shown are the trajectories of rays in depth–time space with colors representing the (c) vertical wavelength lz and (d) vertical group velocity Cgz of near-inertial waves and l1, l2, l3, l4, l5, l6, and l7 representing the initial vertical wavelengths of 80, 90, 100, 110, 120, 130, and 140 m, respectively.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
b. Energy exchange among an ACE, NIWs, and HFIWs
As shown in Fig. 4, the enhancement of near-inertial velocity and NIKE is found after the arrival of an ACE in comparison with that in the initial stage of NIWs generated by Typhoon Sanvu. In addition, the presence of near-inertial energy is found to be quite long after it enters the critical layer, which can hold on for about 45 days. Therefore, we will discuss the energy exchanges among the ACE, NIWs, and HFIWs.
1) Energy exchange between the ACE and NIWs
The energy exchange rate between NIWs and the ACE, P [Eq. (5)], is displayed in Fig. 13. During the downward propagation of NIWs (stage I), the calculated P is mainly positive with a maximum value of 13.1 × 10−9 m2 s−3, indicating energy transfer from the ACE to NIWs (Fig. 13a). After temporal averaging of P from 20 September to 15 October (Fig. 13b), it clearly shows a net energy transfer from the ACE to NIWs in the depth ranges of 50–180 m and 230–500 m, which basically corresponds to the depth ranges of the enhancement of near-inertial velocities and NIKE (Fig. 4). The magnitude of the averaged P in stage I is ∼8 × 10−10 m2 s−3, which is about 1–2 orders of magnitude lager than the results of Jing et al. (2018), and can provide energy of 7.1 × 10−4 W m−2 for the upper 870-m depth (
Energy exchange rate P between the ACE and NIWs: (a) depth–time plot of P, (b) time-mean P during the period of downward propagation of NIWs between 20 Sep and 15 Oct, and (c) depth-mean P from 600- to 750-m depth. The dashed gray line in (a) denotes the vertical propagation trajectory of NIWs.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
Note that in addition to the mechanism of the energy transfer from ACE to NIWs that could explain the enhancement of NIKE near the surface about 16 days after the passage of Sanvu, two other mechanisms could also potentially contribute to the observed enhancement of the NIKE. One is the amplification of near-inertial energy in the ACE. This mechanism has been widely reported (Kunze 1985; Jaimes and Shay 2010; Asselin and Young 2020; Lelong et al. 2020). It is suggested that the negative vorticity of anticyclonic eddies can trap and amplify the near-inertial energy. The other is the spatial difference of wind work. In the Northern Hemisphere, the near-inertial motion is clockwise, so the clockwise rotating wind can resonantly enhance near-inertial motions (Price 1981). Thus, relative to the NIKE at the left side of the typhoon track, the NIKE at the right side is stronger (Figs. S6a–d in the online supplemental material). During the passage of Typhoon Sanvu, the mooring site is located at the left side of the typhoon track and the ACE is at the right. The time mean NIKE at the ACE is ∼4 times larger than that at the mooring site (Fig. S6e). Therefore, with the arrival of the ACE at the mooring site, the near-inertial energy is likely to be enhanced due to the strong near-inertial energy trapped in the ACE. However, how much near-inertial energy is induced by this mechanism has not been quantified due to the limitations of single site observation.
To examine energy transfer in the critical layer of NIWs propagation, we plot the depth-averaged P below 600 m in Fig. 13c. As compared with the energy transfer rate in stage I, it is found that the calculated P is negative in stage II (Fig. 13c). The time-averaged P during stage II is −1.6 × 10−10 m2 s−3, indicating that the energy transfer is from NIWs to the ACE, but an order of magnitude smaller than that from the ACE to NIWs during the downward propagation of NIWs.
2) Energy exchange between NIWs and HFIWs
Furthermore, the energy exchange rate between NIWs and HFIWs, G [Eq. (6)], is shown in Fig. 14. Interestingly, during the downward propagation of NIWs (stage I), the calculated G is remarkably weaker than that in the period of NIWs stalling in the critical layer (stage II; Fig. 14a). By calculating the time-averaged G in stage I (Fig. 14b), we find that there is no preferred direction of energy transfer between NIWs and HFIWs. In contrast, during the period of NIWs stalling in the critical layer (stage II), the calculated G is mainly positive, indicating that the energy transfer is mostly from NIWs to HFIWs (Fig. 14a). The depth-averaged G below 600 m further shows the pronounced energy transfer from NIWs to HFIWs in stage II in comparison with that in stage I (Fig. 14c). The time-averaged G during stage II is 3.7 × 10−9 m2 s−3, which is 23 times as large as the energy transfer from NIWs to the ACE (Fig. 13c). Therefore, the near-inertial energy trapped in the critical layer mainly cascades forward to HFIWs, rather than inversely cascades to the background flow.
As in Fig. 13, but for energy exchange between NIWs and high-frequency internal waves G.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
Comparing energy exchanges among the ACE, NIWs, and HFIWs, it is 2.1 × 103 J m−2 from the ACE to NIWs during stage I (
c. Diapycnal mixing in the critical layer
To show the time–depth variation of fine-scale shear and diapycnal mixing, a longer time series is provided in Fig. 15. Relative to the shear square S2 before 25 September, Fig. 15a shows the enhancement of S2 at the depth of ∼750 m from 25 September to 30 November. Figure 15b is the inverse gradient Richardson number, log10(Ri−1/4), Ri−1 = S2/N2. Shear instability develops when Ri is less than 0.25 (Thorpe 2005). However, the 10-m-resolution S2 and coarser N2 data do not resolve the small-scale structures associated with turbulent mixing, resulting in a value of Ri that is no longer 0.25 for the criterion of instability (Munk 1981; Canuto et al. 2001). Fer et al. (2018) suggested that Ri equal to 10 can be suggestive of mixing events in the Lofoten Basin Eddy. Martínez-Marrero et al. (2019) assumed that turbulent mixing could be induced when Ri was less than 1 in an ACE. However, no matter what the critical value of Ri is in the present coarse resolution data, the increase of Ri−1 during the period when the NIWs stall in the critical layer implies higher probability of turbulence production, which ultimately dissipates the energy of the trapped NIWs.
Depth–time plots of (a) shear squared S2 and (b) inverse gradient Richardson number log10(Ri−1/4), from 1 Aug to 15 Dec. The blue lines denote the depth ranges for calculating the shear wavenumber spectrum in the GHP parameterization. The dashed gray line denotes the vertical propagation trajectory of NIWs. (c) TKE dissipation rate ε processed by daily averaging and four-point smoothing. (d) Near-inertial (ni), semidiurnal (D2), and subinertial (si) frequencies components of shear squared.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
In addition, the trapped NIWs can also lose their energy via nonlinear interactions among internal waves, which can be estimated by the GHP fine-scale parameterization (Henyey et al. 1986; Gregg 1989; Polzin et al. 1995; Kunze et al. 2006). To obtain shear wavenumber spectrum, we choose a shear profile of the bottom 256-m-thick layer of the observation, which roughly represents the depth range of the critical layer (marked by the blue lines in Figs. 15a,b). Turbulent kinetic energy dissipation, ε, is displayed in Fig. 15c. It shows an enhancement of ε when NIWs enter the critical layer. From 25 October to 11 November, ε is reduced by the weakening of NIWs. From 15 October to 30 November, the time-averaged ε is 6.7 × 10−9 m2 s−3. Note that, after NIWs transfer energy toward waves with higher frequencies and smaller scales, these waves are no longer trapped but can radiate away freely and finally break to produce turbulent mixing. In addition, since the energy of internal waves propagating outward from the eddy critical layer is not equal to that propagating inward from outside, the principle of statistical stationarity on which the GHP parameterization is based is not fulfilled. By comparing microstructure and fine-scale observations in an ACE in the STNWP, it was found that the dissipation rate obtained from the GHP parameterization might be overestimated or underestimated (Chen et al. 2021). Nevertheless, the results of GHP parameterization can still be used to infer that the energy transfer from the trapped NIWs within the critical layer to the smaller-scale internal waves increases, even though our purpose is not to quantify them.
To further analyze the enhancement of fine-scale shear, the total shear is decomposed into near-inertial, semidiurnal, and subinertial components (Zhang et al. 2018). Figure 15d shows the shear components averaged within the depth from 600 to 870 m. The subinertial shear is the smallest, implying the contribution of the mean flow to the vertical shear is small. Although the near-inertial shear is comparable in magnitude to the semidiurnal shear in August, it increases when NIWs reside in the critical layer. After the end of November, near-inertial shear drops to the background level. Thus, the enhancement of shear within the critical layer is dominated by the near-inertial shear.
5. Conclusions
In this study, the influence of a westward-moving ACE on the propagation of NIWs is investigated by subsurface mooring and reanalysis data. Unlike previous studies, the observed NIWs are generated outside an ACE and then interact with the arriving ACE. Initially, the near-inertial energy generated by Typhoon Sanvu is confined above the upper 100-m depth. With the arrival of the ACE, the NIWs start to propagate downward. The vertical wavelength and group velocity increase rapidly to the maximum magnitude of ∼500 m and ∼35 m day−1, respectively. When entering the core of the ACE, the vertically propagating NIWs undergo a trapping process and ultimately stall at a critical layer. Although the bottom of the critical layer is not resolved due to the limit of the maximum observed depth, the top of the critical layer is observed, which basically corresponds to the base of the ACE at around 750-m depth. Therefore, the promoted downward propagation and trapping of NIWs induced by the westward-moving ACE is observed.
By using a ray-tracing model, it is found that when NIWs are generated outside an ACE but subsequently encounter the arriving ACE, the propagation depths are much greater than those for NIWs directly generated inside an ACE. Dynamic analysis indicates that when the vertical gradient of vorticity is less than zero [i.e., Δ(ζ)/Δ(z) < 0], the downward propagation of NIWs will be inhibited, and vice versa. For NIWs generated outside an ACE but subsequently entering the ACE, the vertical gradient of relative vorticity is first positive and then negative. Therefore, the ACE will first promote and then suppress the downward propagation of NIWs, ultimately trapping them at a greater depth. By using depth–time varying stratification and relative vorticity, ray-tracing experiments further demonstrate that NIWs generated outside and passed over by an ACE can propagate to deeper depths than those directly generated inside an ACE.
During the downward propagation of NIWs, the net energy transfer from the ACE to NIWs plays an important role in the enhancement of the near-inertial energy, which provides energy equivalent to ∼71% of the wind power input. For the energy transfer between NIWs and HFIWs, it is basically balanced in the time-averaging sense. However, after the NIWs entering the critical layer, the near-inertial energy lost to HFIWs is about one order of magnitude larger than that lost to the ACE. Previous studies suggested that the duration of trapped waves in the critical layer is about 10–30 days (Pallàs-Sanz et al. 2016; Kawaguchi et al. 2016, 2020). Here, we find that the trapped NIWs can stall in the critical layer for ∼45 days, partly due to the energy extraction from the ACE. Meanwhile, by the GHP fine-scale parameterization, it is found that the interactions among internal waves are enhanced within the critical layer. Thus, the trapped NIWs transfer their energy toward smaller-scale waves that subsequently break into turbulence. The enhancement of Ri−1 in the critical layer indicates that the loss of the trapped near-inertial energy for furnishing diapycnal mixing also includes shear instability, by which the trapped NIWs directly break to mix the water.
Interactions between NIWs and a westward-moving ACE are illustrated in Fig. 16. Relative to the propagation depth of NIWs directly generated inside an ACE (label i), that of NIWs generated outside and then encountering a westward-moving ACE (label ii) is deeper, thereby promoting turbulent mixing in the deeper ocean. Given the fact that mesoscale eddies are characterized by westward movements, our observations and findings can further contribute to the understanding of the propagation paths and the energy budget of NIWs in the real ocean.
A schematic of NIWs propagation modified by a westward-moving ACE. In label i, NIWs are generated inside the ACE; in label ii, NIWs are generated outside the ACE and then encounter the arriving ACE. Color shading indicates relative vorticity ζ. Profiles of the vertical gradients of ζ and stratification N2 are shown on the right-hand side of the schematic. The positive vertical gradients of ζ or N2 promote the downward propagation of NIWs, and the negative vertical gradients of ζ or N2 suppress it.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0226.1
Acknowledgments.
Insightful and constructive comments from Dr. Sylvia Cole and two anonymous reviewers greatly help to improve the paper and are thus gratefully acknowledged. The authors thank the scientists and crews of the R/V Science for their assistance in acquiring the observation data. This work was jointly supported by the National Key Research and Development Program of China (Grants 2022YFC3104105 and 2022YFB3205305), State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences (Project LTO2307), National Natural Science Foundation of China (Grant 42176025), Natural Science Foundation of Shandong Province (ZR2023QD077), and QDBSH20220201032 from Qingdao Municipal Bureau of Human Resources and Social Security.
Data availability statement.
The data used in this paper are available online (sea level and geostrophic velocity data: https://resources.marine.copernicus.eu/?option=com_csw&view=details&product_id=SEALEVEL_GLO_PHY_L4_NRT_OBSERVATIONS_008_046; CPLDA reanalysis product: https://resources.marine.copernicus.eu/?option=com_csw&view=details&product_id=GLOBAL_ANALYSISFORECAST_PHY_CPL_001_015; ERA5 reanalysis data: https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-single-levels?tab=overview; typhoon tracks: https://www.jma.go.jp/jma/jma-eng/jma-center/rsmc-hp-pub-eg/besttrack.html). Subsurface mooring data will be made available on request.
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