1. Introduction
Internal tides are internal waves with tidal frequency that are generated in the stratified ocean by barotropic tidal currents flowing over varying topographies, such as midocean ridges, seamounts, continental shelves, and slopes. In the tides-to-mixing cascade, internal tides play an important role in dissipating surface tides (Munk 1997; Egbert and Ray 2000) and enhancing oceanic mixing (Niwa and Hibiya 2001; Rudnick et al. 2003) and, consequently, contribute to large-scale ocean circulations.
One direct and important approach to investigate internal tides is to analyze mooring observations containing currents measured by acoustic Doppler current profilers (ADCPs) and temperature by thermometers. Near the Hawaiian Ridge and Luzon Strait, which are two typical sources of internal tides, internal tides have been detected and studied by many scholars. Rudnick et al. (2003) mentioned that the peak-to-peak amplitude of semidiurnal internal tides near the Hawaiian Ridge could reach 300 m by analyzing mooring temperature data. Chavanne et al. (2010) investigated the semidiurnal internal tides near Hawaii by comparing the observed currents with numerical model predictions and analytical model results. Zilberman et al. (2011) studied the incoherent nature of M2 internal tides at the Hawaiian Ridge by using the current, temperature, and conductivity data, and found that the energy conversion near the ridge crest varied considerably by a factor of 2 (0.5–1.1 W m−2). By analyzing 6-month records of current profiles, the nonlinear energy transfer from M2 internal tides to diurnal waves in the Kauai Channel (Hawaii) was detected by Chou et al. (2014). For the internal tides near the Luzon Strait, Alford et al. (2011) investigated their generation, propagation, and dissipation using two profiling moorings containing ADCPs and CTDs. Results indicated that the peak-to-peak baroclinic velocity and vertical displacement exceeded 2 m s−1 and 300 m, respectively, and that the energy flux exceeded 60 kW m−1 at the spring tide. Klymak et al. (2011) studied the breaking and scattering of internal tides on the continental shelf of the South China Sea (SCS) using mooring data and calculated the energy flux and turbulence dissipation. Using current data obtained from mooring ADCPs, Xu et al. (2011, 2013, 2014) investigated the seasonal variation and multimodal structure of internal tides at three stations in the SCS. In addition, Katsumata et al. (2010), Ross et al. (2014), Tanaka et al. (2014), and Terker et al. (2014) researched the internal tides on the Timor shelf, in the Patagonian fjord, in the Bussol Strait, and on the California continent margin using mooring observations, respectively.
There still exists a problem in the acquisition of full-depth observations because of the water depth at the mooring station, the number and measuring range limitation of ADCPs, etc. Some scholars used linear interpolation (extrapolation) to obtain the missing data beyond the measuring range (Katsumata et al. 2010). However, linear interpolation (extrapolation) seems not to be a good method for the lack of physical explanations. Webb and Pond (1986) put forward a method to extract internal tides in the Knight Inlet using modal decomposition. As a method to separate motions of each mode based on the eigenfunctions of an eigenvalue problem, modal decomposition is widely used in data analysis (Kelly et al. 2010; Buijsman et al. 2010; Klymak et al. 2011; Ma et al. 2013), theoretical (Griffiths and Grimshaw 2007) and numerical models (Heaps 1983), and it performs as well in the estimation of modal motions as the empirical orthogonal function, ridge regression, and optimal estimate (Smith et al. 1985). According to the method in Webb and Pond (1986), up-inlet and down-inlet internal tides in the Knight Inlet were extracted from observations. In the method, observations containing density (temperature and salinity), currents, and barotropic tides are essential because equations of horizontal currents and density perturbations need to be solved simultaneously. However, it is difficult to have all these observations for general moorings, especially in the deep ocean. In addition, criteria about the normal mode fitting are not very clear in Webb and Pond (1986). So, in this study, a general method is put forward to extract tidal currents of each mode and to reconstruct full-depth tidal currents from limited mooring observations by using harmonic analysis and modal decomposition. In the proposed method, only the current data are essential [stratification can be calculated using the World Ocean Atlas 2005 (WOA05) data] and the determination of normal mode fitting is discussed in detail in this study. The paper is organized as follows. Section 2 introduces the mooring data and the method to reconstruct full-depth tidal currents. In section 3, twin experiments (TEs) are carried out to test the feasibility of the proposed method. Based on the proposed method and a 3-month-long time series of acoustic Doppler current data, tidal characteristics of M2 and K1 at a station in the northern SCS are investigated in section 4. Finally, the paper is summarized in section 5.
2. Data and methodology
a. Data
In this study, a 3-month-long (from 1 March 2014 to 31 May 2014) time series of current data obtained from two 75-kHz ADCPs located at a station (21.1°N, 117.9°E) in the northern SCS is used. Water depth at this station is 980 m. Two ADCPs were both positioned at 400-m depth, of which one was uplooking to capture currents from 50 to 370 m and the other was downlooking to capture currents from 430 to 880 m. Current data were recorded with precision of 5 × 10−3 m s−1 and a time interval of 1 h. In addition, temperature between 100- and 400-m depth was also measured by thermistor chains. The observed currents and temperature are shown in Fig. 1. As seen, the observed currents are not full depth due to the measuring range limitation and vertical movement of each ADCP.
Observed current components (a) u and (b) υ (m s−1), positive when pointing east and north, respectively; and (c) temperature (°C).
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Observed current components (a) u and (b) υ (m s−1), positive when pointing east and north, respectively; and (c) temperature (°C).
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Observed current components (a) u and (b) υ (m s−1), positive when pointing east and north, respectively; and (c) temperature (°C).
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
b. Methodology
To obtain full-depth tidal currents and internal tidal currents of each mode for the principal constituents from limited observations, a method is put forward combining harmonic analysis and modal decomposition. Based on the least squares method, harmonic analysis (Fang et al. 1999) is carried out to calculate harmonic constants of tidal currents of the principal constituents. Using these harmonic constants, time series of currents of each constituent are calculated and then used in modal decomposition to calculate full-depth tidal currents of each mode (containing both the barotropic and baroclinic modes). Adding the barotropic tidal currents to all the baroclinic ones, the full-depth tidal currents of each constituent are reconstructed.
3. Twin experiments
In this section, TEs are carried out to test the feasibility of the proposed method. For simplicity, only M2 is considered in the TEs. The process of TEs is designed as follows. 1) Based on the normal modes at the mooring station, full-depth barotropic and baroclinic tidal currents are prescribed, which are then used to compose a 3-month-long time series of M2 tidal currents. The composed M2 tidal currents are regarded as the prescribed “observations.” 2) According to the proposed method, full-depth M2 tidal currents are reconstructed from these “observations.” It should be noted that in order to be consistent with reality, only the observations in the ADCP measuring range (shown in Fig. 4a) are used. 3) By calculating the difference between the prescribed and reconstructed tidal currents, the method can be assessed.
Before TEs are carried out, it is necessary to analyze the stratification and to calculate the normal modes at the mooring station. Figure 2 shows the spring temperature, salinity, and density from WOA05 data (time averaged from March to May). The observed temperature is also plotted in Fig. 2a for comparison (to remove the influence of the mesoscale eddy, which occurred in April, shown in Fig. 1c, the time-averaged temperature in May is regarded as the spring temperature for observations). The observed temperature shows good agreement with that from WOA05 data, so the density obtained from WOA05 data can be regarded as the real value and used to calculate the buoyancy frequency (Fig. 3a). According to Eqs. (6) and (8), φm and ∂φm/∂z are calculated, which are shown in Figs. 3b and 3c, respectively.
Spring (a) temperature, (b) salinity, and (c) density from WOA05 data (gray lines) and observations (black line) at the mooring station.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Spring (a) temperature, (b) salinity, and (c) density from WOA05 data (gray lines) and observations (black line) at the mooring station.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Spring (a) temperature, (b) salinity, and (c) density from WOA05 data (gray lines) and observations (black line) at the mooring station.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
(a) Buoyancy frequency, and parts of (b) φm and (c) ∂φm/∂z at the mooring station.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
(a) Buoyancy frequency, and parts of (b) φm and (c) ∂φm/∂z at the mooring station.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
(a) Buoyancy frequency, and parts of (b) φm and (c) ∂φm/∂z at the mooring station.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
a. TE1: Feasibility of the method
Based on the normal modes shown in Fig. 3c, one barotropic mode and three baroclinic modes are prescribed in TE1 to compose M2 tidal currents, which are shown in Fig. 4. Using the proposed method and limited observations, barotropic and baroclinic tidal currents of each mode are calculated and full-depth M2 tidal currents are reconstructed. Because the total number of normal modes (M) is important in the proposed method, different values of M are used in TE1. Figure 5 displays the reconstructed results and corresponding modes when M = 1 and 2. It is obvious that both reconstructed M2 tidal currents and corresponding modes have a large difference from the prescribed ones, suggesting that when M is smaller than the number of prescribed modes, the prescribed motion cannot be reconstructed accurately. However, when M ≥ 3, the prescribed motion can be reconstructed accurately, indicating that it is possible to reconstruct the prescribed tidal currents using limited observations. Table 1 lists the mean absolute errors (MAEs) between the prescribed and reconstructed M2 tidal currents in the ADCP measuring range corresponding to different values of M. As can be seen, more modes lead to smaller differences until M is equal to or larger than the number of prescribed modes. Then, it is concluded that the proposed method is feasible when sufficient modes are used.
Prescribed (b) barotropic and (c)–(e) baroclinic modes, and (a) composed M2 tidal currents in TE1. Black line in each subfigure represents the instantaneous currents at 0100 local time (UTC+8) 1 Mar 2014. Shadows in (a) denote that the data at the corresponding depths are beyond the ADCP measuring range and are not used in the proposed method.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Prescribed (b) barotropic and (c)–(e) baroclinic modes, and (a) composed M2 tidal currents in TE1. Black line in each subfigure represents the instantaneous currents at 0100 local time (UTC+8) 1 Mar 2014. Shadows in (a) denote that the data at the corresponding depths are beyond the ADCP measuring range and are not used in the proposed method.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Prescribed (b) barotropic and (c)–(e) baroclinic modes, and (a) composed M2 tidal currents in TE1. Black line in each subfigure represents the instantaneous currents at 0100 local time (UTC+8) 1 Mar 2014. Shadows in (a) denote that the data at the corresponding depths are beyond the ADCP measuring range and are not used in the proposed method.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Reconstructed (a),(d) M2 tidal currents and (b),(c),(e)–(g) corresponding modes when (a)–(c) M = 1 and (d)–(g) M = 2.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Reconstructed (a),(d) M2 tidal currents and (b),(c),(e)–(g) corresponding modes when (a)–(c) M = 1 and (d)–(g) M = 2.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Reconstructed (a),(d) M2 tidal currents and (b),(c),(e)–(g) corresponding modes when (a)–(c) M = 1 and (d)–(g) M = 2.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Differences between the prescribed and reconstructed M2 tidal currents in the ADCP measuring range in TE1.
b. TE2: Effect of measurement errors
Because ADCPs have measurement errors in the real observations, the effect of measurement errors should be considered. Based on the prescribed modes in TE1 (Fig. 4), random errors are added to the composed M2 tidal currents (Fig. 6). The maximum errors in amplitudes of u and υ are 0.5 cm s−1, which is comparable to the precision of the ADCP. In TE2, different values of M are used (from 1 to 10). The reconstructed results and differences between prescribed and reconstructed tidal currents corresponding to different values of M are displayed in Fig. 7 and Table 2, respectively. Similar to results in TE1, the reconstructed M2 tidal currents show large differences from the prescribed ones both inside and outside the ADCP measuring range when M < 3. With larger M (M = 3, 4, 5, and 6), the prescribed M2 tidal currents can be reconstructed, suggesting that the method is feasible when the appropriate mode number is used in the proposed method, although measurement errors exist in the observations. Indeed, when M = 3, 4, 5, and 6, the extracted barotropic and first three baroclinic modes are almost the same as the prescribed ones, and the extra modes (e.g., mode 4 when M = 4, mode 4 and mode 5 when M = 5) are too weak to be detected. However, when more modes are used (M = 7, 8, 9, and 10), unreasonable reconstructed results appear beyond the ADCP measuring range, especially in the upper 100 m (Figs. 7g–j). To study the cause of the unreasonable results, Fig. 8 displays the extracted barotropic and baroclinic modes with M = 10. As can be seen, the extracted results of modes 0–3 show large differences from the prescribed ones and unreasonable results of modes 4–8 appear. Comparing these results with those in TE1, it can be concluded that the measurement errors cause the unreasonable results. To investigate how the measurement errors cause the unreasonable results, attention should be refocused on the proposed method. Actually, the appearance of unreasonable results (e.g., Fig. 8) implies the nonuniqueness of the solution of Eq. (11), which might be attributed to the least squares method and/or the nonorthogonality of normal modes (due to limited observing depths) in the proposed method. However, according to results in Table 2, differences between the prescribed and reconstructed tidal currents within the ADCP measuring range almost keep invariant when M > 3, suggesting that the least squares method is not the reason. Then, it can be deduced that the existence of measurement errors and the nonorthogonality of normal modes lead to the nonuniqueness of the solution of Eq. (11) and thus to unreasonable reconstructed results when more modes are used in the proposed method. However, the nonorthogonality of normal modes and measurement errors would always exist because the aim is to reconstruct full-depth tidal currents from limited observations. Therefore, when dealing with real mooring observations, an alternative way is to try as many modes as possible until the unreasonable results appear.
Prescribed (b) barotropic and (c)–(e) baroclinic modes, (f) errors, and (a) composed M2 tidal currents in TE2.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Prescribed (b) barotropic and (c)–(e) baroclinic modes, (f) errors, and (a) composed M2 tidal currents in TE2.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Prescribed (b) barotropic and (c)–(e) baroclinic modes, (f) errors, and (a) composed M2 tidal currents in TE2.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Reconstructed M2 tidal currents in TE2 when M = 1–10, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Reconstructed M2 tidal currents in TE2 when M = 1–10, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Reconstructed M2 tidal currents in TE2 when M = 1–10, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Differences between the prescribed and reconstructed M2 tidal currents in the ADCP measuring range in TE2.
Extracted (a) barotropic and (b)–(k) baroclinic modes in TE2 when M = 10.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Extracted (a) barotropic and (b)–(k) baroclinic modes in TE2 when M = 10.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Extracted (a) barotropic and (b)–(k) baroclinic modes in TE2 when M = 10.
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
c. Comparison with linear interpolation (extrapolation)
In this subsection, the proposed method is compared with linear interpolation (extrapolation). Based on the observations in TE2, linear interpolation (extrapolation) is used to reconstruct the missing data of tidal currents at 375–475-m (0–80 m) depth, which correspond to the middle (highest) shadow in Fig. 6a. The M2 tidal currents reconstructed with the proposed method are the results of TE2 with M = 3 (Fig. 7c). Tables 3 and 4 show the differences between the prescribed and reconstructed tidal currents at 375–475- and 0–80-m depths, respectively. Although both the proposed method and linear interpolation can reconstruct the missing data with small MAEs at 375–475-m depth, where the prescribed tidal currents vary slightly, all the MAEs caused by the proposed method are smaller than those by linear interpolation. However, at 0–80-m depth, where the prescribed tidal currents vary considerably, the MAEs corresponding to linear extrapolation are much larger than those corresponding to the proposed method, especially for the MAEs of amplitudes of u and υ, suggesting that the proposed method is better than linear interpolation (extrapolation).
Differences between the prescribed and reconstructed M2 tidal currents at 375–475-m depth corresponding to the proposed method and linear interpolation.
Differences between the prescribed and reconstructed M2 tidal currents at 0–80-m depth corresponding to the proposed method and linear extrapolation.
From all the results in section 3, it can be concluded that the proposed method is feasible in extracting tidal currents of each mode and reconstructing full-depth tidal currents from limited observations, and that it is better than linear interpolation (extrapolation).
4. Reconstructed results and tidal characteristics at the mooring station
Observed (a) M2 and (g) K1 tidal currents, and (b),(h) reconstructed results, as well as corresponding modes [(c)–(f) for M2, (i)–(k) for K1].
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Observed (a) M2 and (g) K1 tidal currents, and (b),(h) reconstructed results, as well as corresponding modes [(c)–(f) for M2, (i)–(k) for K1].
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
Observed (a) M2 and (g) K1 tidal currents, and (b),(h) reconstructed results, as well as corresponding modes [(c)–(f) for M2, (i)–(k) for K1].
Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-14-00221.1
KE (J m−2) of each mode and corresponding proportions for M2 and K1.
5. Summary
In this study, a method is put forward to extract internal tidal currents and to reconstruct full-depth tidal currents from limited observations by using harmonic analysis and modal decomposition. Harmonic analysis is used to separate tidal currents of different constituents, and modal decomposition is used to calculate the corresponding full-depth barotropic and baroclinic tidal currents using the least squares method. Adding the barotropic tidal currents to all the baroclinic ones, the full-depth tidal currents of each constituent are reconstructed.
The feasibility and accuracy of the proposed method is tested by twin experiments. Results indicate that the prescribed motion can be reconstructed by adopting the appropriate number of modes in the proposed method. Because of the existence of measurement errors and the nonorthogonality of normal modes, which is due to limited observing depths, too many modes used in the method may cause unreasonable results. Besides, the proposed method is better than linear interpolation (extrapolation) in reconstructing the missing data of tidal currents.
Finally, the proposed method is used to extract the tidal currents and to reconstruct the full-depth tidal currents for M2 and K1 from the 3-month-long time series of acoustic Doppler current data observed at a station in the northern SCS. Results denote that the total KE of M2 is 25% larger than that of K1. For M2, the first baroclinic mode is dominant and is followed by the barotropic mode, and the sum of these two modes accounts for more than 90% of the total M2 KE. For K1, the proportion of the barotropic mode is larger than 90%, which is almost equal to the proportion of the first two modes for M2. As baroclinic modes of K1 are weak, they contribute more to the vertical variation of the current phase rather than to amplitude.
Acknowledgments
We are grateful to Professor Jiwei Tian for providing the mooring observations and to reviewers for their constructive suggestions, which helped to improve the manuscript.
Partial support for this research was provided by the State Ministry of Science and Technology of China through Grants 2013AA122803 and 2013AA09A502, the National Natural Science Foundation of China through Grants 41206001 and 41371496, the National Science and Technology Support Program through Grant 2013BAK05B04, and the Ministry of Education’s 111 Project through Grant B07036.
REFERENCES
Alford, M. H., and Coauthors, 2011: Energy flux and dissipation in Luzon Strait: Two tales of two ridges. J. Phys. Oceanogr., 41, 2211–2222, doi:10.1175/JPO-D-11-073.1.
Buijsman, M. C., Kanarska Y. , and McWilliams J. C. , 2010: On the generation and evolution of nonlinear internal waves in the South China Sea. J. Geophys. Res., 115, C02012, doi:10.1029/2009JC005275.
Chavanne, C., Flament P. , Carter G. , Merrifield M. , Luther D. , Zaron E. , and Gurgel K.-W. , 2010: The surface expression of semidiurnal internal tides near a strong source at Hawaii. Part I: Observations and numerical predictions. J. Phys. Oceanogr., 40, 1155–1179, doi:10.1175/2010JPO4222.1.
Chou, S. H., Luther D. S. , Guiles M. D. , Carter G. S. , and Decloedt T. , 2014: An empirical investigation of nonlinear energy transfer from M2 internal tide to diurnal wave motions in the Kauai Channel, Hawaii. Geophys. Res. Lett., 41, 505–512, doi:10.1002/2013GL058320.
Egbert, G. D., and Ray R. D. , 2000: Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature, 405, 775–778, doi:10.1038/35015531.
Fang, G., Kwok Y.-K. , Yu K. , and Zhu Y. , 1999: Numerical simulation of principal tidal constituents in the South China Sea, Gulf of Tonkin and Gulf of Thailand. Cont. Shelf Res., 19, 845–869, doi:10.1016/S0278-4343(99)00002-3.
Griffiths, S. D., and Grimshaw R. H. J. , 2007: Internal tide generation at the continental shelf modeled using a modal decomposition: Two-dimensional results. J. Phys. Oceanogr., 37, 428–451, doi:10.1175/JPO3068.1.
Heaps, N. S., 1983: Development of a three-layered spectral model for the motion of a stratified shelf sea. I. Basis equations. Physical Oceanography of Coastal and Shelf Seas, B. Johns, Ed., Elsevier Oceanography Series, Vol. 35, Elsevier, 387–400, doi:10.1016/S0422-9894(08)70507-5.
Katsumata, K., Wijffels S. , Steinberg C. R. , and Brinkman R. , 2010: Variability of the semidiurnal internal tides observed on the Timor Shelf. J. Geophys. Res., 115, C10008, doi:10.1029/2009JC006071.
Kelly, S. M., Nash J. D. , and Kunze E. , 2010: Internal-tide energy over topography. J. Geophys. Res., 115, C06014, doi:10.1029/2009JC005618.
Klymak, J. M., Alford M. H. , Pinkel R. , Lien R.-C. , Yang Y. J. , and Tang T.-Y. , 2011: The breaking and scattering of internal tides on a continental slope. J. Phys. Oceanogr., 41, 926–945, doi:10.1175/2010JPO4500.1.
Ma, B. B., Lien R. , and Ko D. S. , 2013: The variability of internal tides in the northern South China Sea. J. Oceanogr., 69, 619–630, doi:10.1007/s10872-013-0198-0.
Munk, W., 1997: Once again: Once again—Tidal friction. Prog. Oceanogr., 40, 7–35, doi:10.1016/S0079-6611(97)00021-9.
Niwa, Y., and Hibiya T. , 2001: Numerical study of the spatial distribution of the M2 internal tide in the Pacific Ocean. J. Geophys. Res., 106, 22 441–22 449, doi:10.1029/2000JC000770.
Ross, L., Perez-Santos I. , Valle-Levinson A. , and Schneider W. , 2014: Semidiurnal internal tides in a Patagonian fjord. Prog. Oceanogr., 129, 19–34, doi:10.1016/j.pocean.2014.03.006.
Rudnick, D. L., and Coauthors, 2003: From tide to mixing along the Hawaiian ridge. Science, 301, 355–357, doi:10.1126/science.1085837.
Smith, J. A., Mooers C. N. K. , and Robinson A. R. , 1985: Estimation of quasi-geostrophic modal amplitudes from XBT/CTD survey data. J. Atmos. Oceanic Technol., 2, 491–507, doi:10.1175/1520-0426(1985)002<0491:EOQGMA>2.0.CO;2.
Tanaka, Y., Yasuda I. , Osafune S. , Tanakab T. , Nishiokad J. , and Volkove Y. N. , 2014: Internal tides and turbulent mixing observed in Bussol Strait. Prog. Oceanogr., 126, 98–108, doi:10.1016/j.pocean.2014.04.009.
Terker, S. R., Girton J. B. , Kunze E. , Klymak J. M. , and Pinkel R. , 2014: Observations of the internal tide on the California continental margin near Monterey Bay. Cont. Shelf Res., 82, 60–71, doi:10.1016/j.csr.2014.01.017.
Webb, A. J., and Pond S. , 1986: A modal decomposition of the internal tide in a deep, strongly stratified inlet: Knight Inlet, British Columbia. J. Geophys. Res., 91, 9721–9738, doi:10.1029/JC091iC08p09721.
Xu, Z., Yin B. , and Hou Y. , 2011: Multimodal structure of the internal tides on the continental shelf of the northwestern South China Sea. Estuar. Coastal Shelf Sci., 95, 178–185, doi:10.1016/j.ecss.2011.08.026.
Xu, Z., Yin B. , Hou Y. , and Xu Y. , 2013: Variability of internal tides and near-inertial waves on the continental slope of the northwestern South China Sea. J. Geophys. Res., 118, 197–211, doi:10.1029/2012JC008212.
Xu, Z., Yin B. , Hou Y. , and Liu A. K. , 2014: Seasonal variability and north-south asymmetry of internal tides in the deep basin west of the Luzon Strait. J. Mar. Syst., 134, 101–112, doi:10.1016/j.jmarsys.2014.03.002.
Zilberman, N. V., Merrifield M. A. , Carter G. S. , Luther D. S. , Levine M. D. , and Boyd T. J. , 2011: Incoherent nature of M2 internal tides at the Hawaiian Ridge. J. Phys. Oceanogr., 41, 2021–2036, doi:10.1175/JPO-D-10-05009.1.
