Seasonal variations of tidal currents in the deep Timor Passage

Exact knowledge on the seasonal variations of main tidal constituents is beneficial for improving tidal prediction. The semi-annual cycles in K1 and S2 tides are abnormally exaggerated by astronomical P1 and K2 tides, which interferes with our understanding on tidal seasonality. The widely-used tidal inference method in previous studies cannot fully separate astronomical P1 and K2 tides from seasonal P1 and K2 tides due to inaccurate inference relationship. In this study, on the basis of the ‘credo of smoothness’ which indicates that tidal admittances are smooth functions of tidal frequencies, we develop a novel but simple method to address this intractable issue and applied this method to explore the seasonality of tidal currents observed in the deep Timor Passage at the depth of 1800m. We find that the timing and range of seasonal modulations of M2, S2, K1, and O1 tides are distinct. Annual variations in tidal currents are much stronger than semi-annual variations in tidal currents. The annual and semi-annual ranges of M2 tide can reach 2.69 cm/s and 1.51 cm/s, which are largest among main constituents. Although the annual range of K1 tide is only 1.85 cm/s, considering the relatively small amplitude of time-averaged K1 tide (2.87cm/s), K1 the most affected tide by the annual cycle. The seasonal cycles of semi-diurnal tides (M2 and S2) are basically synchronous while those of diurnal tides (K1 and O1) are generally out-of-phase. As a general method, the proposed method can be widely applied to other sea areas to explore local tidal seasonality.


Introduction
Originated from astronomical forcing, tides and tidal currents are omnipresent in the global ocean and fundamental for ocean activities such as maritime logistics and ocean engineering (Amin, 1985;Pan et al., 2022a;Pan et al., 2022b;Wei et al., 2022). The interplay of barotropic tides with rough topography in the stratified ocean can generate baroclinic tides (Wunsch, 1975;Zhao et al., 2019). As an indispensable intermediate process in tide-to-turbulence cascade, baroclinic tides play a vital role in ocean mixing processes (Munk and Wunsch, 1998;Egbert and Ray, 2000;Li et al., 2021). Egbert and Ray (2000) indicated that deep sea mixing needs~2TW energy to maintain deep-water circulation and at least 1TW energy is provided by baroclinic tides. As a result of seasonal changes in ocean environment (such as river flow, ocean stratification and sea ice), tides and tidal currents display significant seasonal variations which have been explored in the global ocean (Corkan, 1934;Foreman et al., 1995;Kang et al., 2002;St-Laurent et al., 2008;Kagan and Sofina, 2010;Georgas, 2012;Müller, 2012;Devlin et al., 2018;Wang et al., 2020;Du and Yu, 2021;Ray, 2022).
The seasonality of tidal currents in the deep sea are mainly derived from ocean stratification and astronomical factors (Xu et al., 2014;Cao et al., 2017;Li et al., 2021). The frequency of K 2 (P 1 ) tide is equal to that of the S 2 (K 1 ) tide add (minus) 2 cycle per year. Therefore, the semi-annual cycles of K 1 and S 2 tides are significantly enhanced due to the existence of astronomical P 1 and K 2 tides. To keep pace with the seasonal variations of M 2 and O 1 tides which are not influenced by nearby astronomical tides, nearly all researches applied tidal inference method to infer and eliminate the contribution of astronomical P 1 and K 2 tides when discussing the seasonal variations of K 1 and S 2 tides. The inference relationship between K 2 (P 1 ) and S 2 (K 1 ) tides can be determined based on the actual tidal constants from observed time series longer than half a year. It should be noted that the observed K 1 tide is nearly astronomical while the observed P 1 tide has two major energy sources: One is the astronomical P 1 tide, the another is the real seasonal variations of K 1 tide originated from semi-annual variations in ocean environment (labeled as the seasonal P 1 tide). Although the astronomical P 1 tide and the seasonal P 1 tide have same tidal period, their amplitudes and phases are totally different because they are forced by distinct physical processes (see section 3 for details). It is well known that astronomical P 1 and K 1 tidal waves have similar physical properties, thus, the astronomical P 1 tide can be simply inferred from the astronomical K 1 tide while the seasonal P 1 tide cannot. The observed P 1 tide is the vectorial synthesis of the seasonal P 1 tide and the astronomical P 1 tide. Similarly, the observed K 2 tide is the vectorial synthesis of the seasonal K 2 tide and the astronomical K 2 tide. Hence, the inference relationship derived from the observed P 1 (K 2 ) and observed K 1 (S 2 ) tide may be problematic due to the interference of the seasonal P 1 (K 2 ) tide.
To the best of our knowledge, there are no valid methods to take the place of the potentially problematic inference method to fully remove astronomical P 1 and K 2 tides from observed P 1 and K 2 tides. The aim of this research is to revisit this noteworthy issue and propose a new method according to the 'credo of smoothness' (Munk and Cartwright, 1966) to solve the problem. The new method is applied to the deep Timor Passage to explore the seasonality of local tidal currents. Our paper is organized as follows. Study area and tidal current observations are introduced in section 2. Section 3 displays the methods and results, followed by the discussions and conclusions in section 4 and section 5, respectively.

Study area and data
As a long, deep and narrow trench between the Australian continental shelf and the Timor Island with average depth of 2000m (Figure 1), the Timor Passage is one of the major corridors for the Indonesian Throughflow (ITF). Fresh and warm sea waters from the western Pacific Ocean are transported to the tropical Indian Ocean via the Timor Passage and the Lombok and Ombai Straits, which are important and essential for maintaining the thermohaline balance in the global ocean (Sprintall et al., 2009). The deep current transport through the Timor Passage shows significant semi-annual and annual variations, which are related to remote Kelvin waves from the Indican Ocean and local monsoonal forcing, respectively .
Due to complex coastlines and topography, tides and tidal currents near the Indonesian archipelago are among the most complicated in the global ocean (Ray et al., 2005;Robertson, 2010). The mixing induced by tides has significant influences on ocean ecology and climate system (Sprintall and Revelard, 2014;Katavouta et al., 2022). Based on EOT20 tidal model (Hart-Davis et al., 2021) derived from multi-satellite altimeters, at the observation point, M 2 tide has the largest amplitude (88.39cm), followed by S 2 (48.78cm), K 1 (27.33cm), and O 1 (17.00cm). Local tidal form factor, which is defined by the ratio of the sum of O 1 and K 1 tidal amplitudes to the sum of S 2 and M 2 tidal amplitudes (Pan et al., 2023a), is only 0.32, indicating that local tides are dominated by semi-diurnal tides.
Hourly current observations at depth of 1800m from the mooring (black dot in Figure 1 As shown in Figure 2, eastward tidal currents are significantly stronger than northward tidal currents due to the direction of the Timor Passage. Thus, we decompose eastward and northward currents into currents along and perpendicular to the trench. Only currents along the trench are focused and harmonically analyzed using S_TIDE toolbox (Pan et al., 2018a). It should be noted that to avoid the interference of strong non-tidal background currents on tidal estimation, we use Iteratively Reweighted Least Squares (IRLS) regression (Huber, 1996;Leffler and Jay, 2009) to take place of widely-used ordinary least squares (OLS) regression in the course of harmonic analysis. IRLS regression is much complicated than OLS regression and readers can refer Leffler and Jay (2009) for details. The effectiveness and accuracy of IRLS regression in tidal estimation have been verified by numerous studies (Leffler and Jay, 2009;Matte et al., 2013;Matte et al., 2014;Pan and Lv, 2021;Pan et al., 2022a;Pan et al., 2023a). Local tidal currents are highly non-stationary, with strong intraseasonal vairiability (Figure 2), which deserves further investigation. Table 1 displays tidal constants of major tidal constituents in the deep Timor Passage. Ssa tide with a period of half a year has the largest amplitude (21.65cm/s). which is consistent with Wang et al. (2022). Sa tide with a period of a year has an amplitude of 4.27cm/s. Among semi-diurnal and diurnal tides, M 2 has the largest amplitude (9.05cm/s), followed by S 2 (4.38cm/s), K 1 (2.87cm/s), and O 1 (2.31cm/s).
Although long-period tides like Sa, Msm and Mf have large amplitudes, they are not significant due to low signal-to-noise ratios (SNRs). Generally, the SNR of a significant constituent should be no less than two (Pawlowicz et al., 2002). M 4 tide is the strongest shallow water constituent, with an amplitude of only 0.33cm/s. Figure 3A shows the combination of observed K 1 , O 1 , and P 1 tides. The sum of K 1 and O 1 tides can induce semi-monthly variations (13.66 days) of high tide. Note that P 1 tide can semi-annually modulate K 1 tide, thus, fortnightly variations of high tides ( Figure 3A) are not stationary but modulated by semi-annual cycles. The combination of observed M 2 , S 2 , and K 2 tides also has semi-annually modulated fortnightly cycles ( Figure 3B).
O 1 and Q 1 tidal frequencies are close, which means that O 1 and Q 1 tides have similar physical properties. As a result, tidal phase lags of O 1 and Q 1 tides are very close, and the difference of O 1 and Q 1 tidal phase lags is only 3.76° (Table 1). The difference of K 1 and P 1 frequencies is much smaller than that of O 1 and Q 1 frequencies, which means that the difference of K 1 and P 1 phase lags should be  The location of the mooring (black dot). Water depths are from ETOPO1 dataset (Amante and Eakins, 2009). Pan et al. 10.3389/fmars.2023.1135911 Frontiers in Marine Science frontiersin.org smaller than 3.76°. However, the observed difference of K 1 and P 1 phase lags is as high as 11.32°, which clearly indicates that the observed P 1 tide is not purely astronomical, but contains a nonnegligible contribution of K 1 seasonality. In the next section, we will introduce a novel method which can fully separate the seasonal P 1 (K 2 ) tide from the astronomical P 1 (K 2 ) tide.

Methodology
The proposed method is based on the 'credo of smoothness' (Munk and Cartwright, 1966) which implies that tidal admittances TABLE 1 Amplitudes and phase lags of major diurnal, semi-diurnal, and shallow water tides estimated from long-term current observations along the trench. SNR means signal-to-noise ratio.

Constituent
Frequency ( (A) The combination of observed K 1 , O 1 , and P 1 tidal currents. (B) The combination of observed M 2 , S 2 , and K 2 tidal currents. Note that the results are estimated from currents along the trench.
are smooth functions of tidal frequencies (Feng et al., 2015;Pan et al., 2023b). Defined by the ratios of observed amplitudes to equilibrium amplitudes (normalized amplitude) and phase differences of observed phases and equilibrium phases, tidal admittances represent the response of astronomical forcing to local topography and coastlines. In general, tidal waves with close periods always have similar responses which means that their admittances should also be close. Note that such smoothness is built on the premise that tides are purely astronomical. The existence of non-astronomical tides may destroy the nature of smoothness, but also provides an opportunity to eliminate nonastronomical tides. Equilibrium tidal amplitudes are obtained via s_equilibrium_tide function in S_TIDE toolbox. Phase differences of observed phases and equilibrium phases (i.e. phase lags) are directly estimated via classical harmonic analysis. The admittances of minor tidal constituents such as J 1 , 2Q 1 , 2N 2 , L 2 are not used because their SNRs are too small (generally less than 0.5) which means that they may be contaminated by strong non-tidal background noises. As shown in Figure 4A, normalized diurnal amplitudes are parabolic functions of tidal frequencies (dash line). Unknown coefficients (i.e. a, b, c) in Eq.(1) can be estimated by ordinary least squares. f is tidal frequency. Cubic polynomials or higher-order polynomials are not recommended to avoid over-fitting (Feng et al., 2015;Pan et al., 2023b).
Due to the interfere of non-astronomical contributions, normalized observed P 1 amplitude significantly deviates from the fitting curve. Similarly, phase differences are also quadratic functions of frequencies ( Figure 4B). The observed P 1 phase difference noticeably deviates from the quadratic curve. By the quadratic interpolation, the normalized astronomical P 1 amplitude and astronomical P 1 phase difference can be calculated (red dots in Figure 4). Based on known equilibrium amplitudes, the astronomical P 1 amplitude (0.90 cm/s) and phase lag (139.04°) are calculated. The astronomical P 1 phase lag (139.04°) is very close to the astronomical K 1 phase lag (142.01°). The ratio of the astronomical P 1 amplitude (0.90 cm/s) to the astronomical K 1 amplitude (2.87cm/s) is 0.314 which is slightly smaller than the equilibrium theoretical value (0.331). At last, subtracting the astronomical P 1 tide vectorially from the observed P 1 tide generates the seasonal P 1 tide ( Figure 5A). The amplitude and phase of the seasonal P 1 tide is 0.181cm/s and 84.4°, respectively.
As displayed in Figure 6A, normalized semi-diurnal amplitudes range from 0.55 to 0.6 except K 2 . Phase differences for semi-diurnal tides are nearly linear ( Figure 6B). Based on the fitting curve and equilibrium amplitudes, the astronomical K 2 amplitude (1.221cm/s) and phase lag (129.72°) are calculated. Via vector operation, the seasonal K 2 amplitude (0.06cm/s) and phase lag (110.93°) are derived. Because the seasonal K 2 tide is very weak, therefore, the observed K 2 tide is nearly same to the astronomical K 2 tide ( Figure 5B).

Seasonal variations of main tidal constituents
The seasonality of main tidal constituents can induce minor constituents whose frequencies are near main constituents. For example, the annual modulation of K 1 tide can induce S 1 and PSI 1 tides, whose frequencies are w K1 -w Sa and w K1 +w Sa , where w K1 and w Sa mean the frequencies of K 1 and Sa tides, respectively. The semiannual modulation of K 1 tide can induce P 1 and PHI 1 tides, whose Normalized tidal amplitudes (A) and phase differences (B) for main diurnal tides at depth of 1800m from the mooring. Black dots are observed tidal admittances while red dots are interpolated admittances. Dash lines are determined via ordinary least squares.
frequencies are w K1 -2*w Sa and w K1 +2*w Sa . The annual modulation of S 2 tide can induce T 2 and R 2 tides, whose frequencies are w S2 -w Sa and w S2 +w Sa . Like P 1 and K 2 , T 2 tide can also directly obtain considerable energy from astronomical forcing. According to the fitting curve in Figure 6, the astronomical T 2 amplitude (0.256cm/s) and phase lag (122.93°) can be calculated. The observed T 2 amplitude and phase lag are 0.578cm/s and 182.03°, respectively. Through vectorial operation, the seasonal T 2 amplitude (0.498cm/s) and phase lag (208.22°) are obtained.
The combination of S 1 and PSI 1 tides represents the annual cycle of K 1 tide while the combination of P 1 (astronomical contribution removed) and PHI 1 tides represents the semiannual cycle of K 1 tide (Figure 7). The seasonal variations of M 2 , S 2 , and O 1 tides can be obtained in similar ways (Figures 7, 8). As shown in Figures 7, 8, seasonal variations of four main constituents are significant and their features are distinct. M 2 tide has the largest annual range (2.69cm/s), followed by S 2 (1.85cm/s), K 1 (1.85cm/s), and O 1 (0.93cm/s). Considering the (A) The vectorial synthesis of the seasonal P 1 tide (blue arrow) and the astronomical P 1 tide (red arrow) generates the observed P 1 tide (black arrow). (B) Same as A, but for K 2 tide.

FIGURE 6
Same as Figure 4, but for semi-diurnal tides. Pan et al. 10.3389/fmars.2023.1135911 Frontiers in Marine Science frontiersin.org relatively small amplitude of K 1 tide (2.87cm/s), it is the greatest affected tide by the annual cycle. The range of the semi-annual cycle is much smaller than that of the annual cycle. M 2 tide has the largest semi-annual range (1.51cm/s), followed by S 2 (0.72cm/s), O 1 (0.45cm/s), and K 1 (0.27cm/s). Among four major tidal constituents, O 1 tide has the largest ratio of the range of the semi-annual cycle to tidal amplitude (0.195), which means that the semi-annual cycle has the strongest influence on O 1 tide. The annual variations of M 2 and S 2 tides are precisely synchronous (Figure 8). Both of them peak at the end of September while reach the minimum value in early April. Compared to semi-diurnal tides, the annual variations of K 1 and O 1 tides are basically synchronous ( Figure 7). K 1 tide peaks in early December while reach the minimum value at the end of May. The annual variation of O 1 tide has a delay of about one month compared to that of K 1 tide.
The semi-annual variations of K 1 and O 1 tides are generally opposite. The semi-annual variation of K 1 reaches the minimum value at the end of August and peaks in early December while that of O 1 peaks in mid-August and reaches the minimum value in mid-December (Figure 7). The semi-annual variations of M 2 and S 2 tides are generally synchronous while a delay of about 20 days exists (Figure 8). The semi-annual variation of M 2 reaches the minimum value at the end of December and peaks at the end of September while that of S 2 peaks in early September and reaches Annual (A) and semi-annual (B) variations of M 2 tidal currents. Annual (C) and semi-annual (D) variations of S 2 tidal currents. Pan et al. 10.3389/fmars.2023.1135911 Frontiers in Marine Science frontiersin.org the minimum value in early December. Figures 7, 8 indicate that tidal response to seasonal changes in ocean environment is frequency-dependent.

Application to surface tides
The proposed method is not limited to deep currents but can also be applied to surface tides because the principle of smoothness is generally credible for all tidal signals. Surface tides at the mooring also have noticeable seasonal variations. Figures 9, 10 display tidal admittances for main semi-diurnal and diurnal tides which are totally different. Tidal admittances for diurnal tides are parabolic functions of tidal frequencies while those for semi-diurnal tides are nearly linear functions. The structure of functions should be related to the local topography and coastline which can influence tidal propagation, reflection, refraction, and dissipation. It is obvious that diurnal tides and semi-diurnal tides which have vastly different periods and wave lengths must show distinct tidal responses to the astronomical forcing in the same sea areas.
Like tidal currents, the seasonality of surface tides makes P 1 and K 2 tidal admittances deviate from the fitted curves. Based on the method described above, the seasonal P 1 (K 2 ) tide can be separated from the astronomical P 1 (K 2 ) tide. The seasonal (astronomical) P 1 tide has an amplitude of 0.46 (8.44) cm and a phase lag of 34.32°( 175.64°) while the seasonal (astronomical) K 2 tide has an amplitude of 1.49 (13.69) cm and a phase lag of 37.02°(126.56°). The ratio of the astronomical P 1 amplitude (8.44cm) to the astronomical K 1 amplitude (27.33cm) is 0.309 which indicates that tidal inference using the equilibrium theoretical value (0.331) may be not accurate enough even in the deep sea.

Limitation of the proposed method
Near-inertial currents are generated by ubiquitous changing wind stress (Munk and Wunsch, 1998;Hu et al., 2023). The frequency of near-inertial currents is near F (i.e. Coriolis frequency), which can be expressed as following: Where L is latitude while w is the angular velocity of the earth rotation. It is obvious that the period of near-inertial currents changes with latitude. At 26.45°N/S, 27.61°N/S, 29.82°N/S, 30.00°N /S, the periods of near-inertial currents are same to the periods of Q 1 , O 1 , P 1 , and K 1 tides, respectively. Also, at 70.98°N/S, 74.48°N/S. 85.78°N/S, the periods of near-inertial currents are same to the periods of N 2 , M 2 and S 2 tides. Therefore, at these latitudes, nearinertial motions can contribute to semi-diurnal and diurnal tides, and the credo of smoothness may be interfered. Note that no nearinertial motions can contribute to K 2 tide.
In addition, in the development of the principle of smoothness, Munk and Cartwright (1966) did not consider the potential influence of tidal resonance which may influence the smoothness of tidal admittances. Hence, care must be taken when applying the proposed method to resonant sea areas, such as the Gulf of Tonkin in the South China Sea, which is well-known for strong diurnal resonance (Pan et al., 2022a;Pan et al., 2023a).

Conclusions and summary
Tides and tidal currents display noticeable seasonal variability in numerous sea areas especially in the river estuaries and polar regions. Knowledge on tidal seasonality is fundamental for accurate tidal prediction which is beneficial for substantial human activities Same as Figure 4, but for surface tides at the mooring. Tidal constants are derived from EOT20 model. Pan et al. 10.3389/fmars.2023.1135911 Frontiers in Marine Science frontiersin.org in the ocean like navigation and ocean engineering (Müller et al., 2014;Pan et al., 2018a;Pan et al., 2018b;Gan et al., 2021;Pan and Lv, 2021;Wei et al., 2022). Due to different tidal periods and wave lengths, the seasonal variations of main tidal constituents are distinct. The existence of astronomical P 1 and K 2 tides anomalously exaggerate the semi-annual cycles in K 1 and S 2 tides. The method of tidal inference which is widely used in previous studies cannot fully separate astronomical P 1 and K 2 tides from seasonal P 1 and K 2 tides. In this research, a novel but simple method based on the 'credo of smoothness' is developed to solve this nettlesome problem. Since tidal admittances are smooth functions of frequencies, astronomical P 1 and K 2 tides can be obtained via the interpolation. The seasonal P 1 (K 2 ) tide has totally different amplitude and phase compared to the astronomical P 1 (K 2 ) tide. We applied the proposed method to explore the seasonality of tidal currents observed in the deep Timor Passage at the depth of 1800m. It is found that the timing and range of seasonal variations of four main constituents are discrepant. The annual and semiannual ranges of M 2 tide are largest among main constituents. O 1 tide has the smallest annual range while K 1 has the smallest semiannual range. The peak times of seasonal variations of M 2 and S 2 tides are generally consistent while those of K 1 and O 1 tides are basically not synchronous. Except tidal currents in the deep sea, our method is also suitable for surface tides. It is expected that the proposed method can be widely used in the exploration of tidal seasonality in the global ocean.

Data availability statement
The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding author.

Funding
This study is jointly supported by the Laoshan Laboratory (No. LSKJ202202700), the National Natural Science Foundation of China (NSFC) Projects (42206022,42076024,42076023), the Global Change and Air-Sea Interaction II (Contact No.GASI-01-ATP-STwin), the China Postdoctoral Science Foundation (2022M713677) and the Qingdao postdoctoral application research project (QDBSH202108).

Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Same as Figure 6, but for surface tides at the mooring. Tidal constants are derived from EOT20 model. Pan et al. 10.3389/fmars.2023.1135911 Frontiers in Marine Science frontiersin.org