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 K₂ (P₁) tide is equal to that of the S₂ (K₁) tide add (minus) 2 cycle per year. Therefore, the semi-annual cycles of K₁ and S₂ tides are significantly enhanced due to the existence of astronomical P₁ and K₂ tides. To keep pace with the seasonal variations of M₂ and O₁ tides which are not influenced by nearby astronomical tides, nearly all researches applied tidal inference method to infer and eliminate the contribution of astronomical P₁ and K₂ tides when discussing the seasonal variations of K₁ and S₂ tides. The inference relationship between K₂ (P₁) and S₂ (K₁) 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 K₁ tide is nearly astronomical while the observed P₁ tide has two major energy sources: One is the astronomical P₁ tide, the another is the real seasonal variations of K₁ tide originated from semi-annual variations in ocean environment (labeled as the seasonal P₁ tide). Although the astronomical P₁ tide and the seasonal P₁ 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 P₁ and K₁ tidal waves have similar physical properties, thus, the astronomical P₁ tide can be simply inferred from the astronomical K₁ tide while the seasonal P₁ tide cannot. The observed P₁ tide is the vectorial synthesis of the seasonal P₁ tide and the astronomical P₁ tide. Similarly, the observed K₂ tide is the vectorial synthesis of the seasonal K₂ tide and the astronomical K₂ tide. Hence, the inference relationship derived from the observed P₁ (K₂) and observed K₁ (S₂) tide may be problematic due to the interference of the seasonal P₁ (K₂) 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 P₁ and K₂ tides from observed P₁ and K₂ 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.

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 (Wang et al., 2022).

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 Révelard, 2014; Katavouta et al., 2022). Based on EOT20 tidal model (Hart-Davis et al., 2021) derived from multi-satellite altimeters, at the observation point, M₂ tide has the largest amplitude (88.39cm), followed by S₂ (48.78cm), K₁ (27.33cm), and O₁(17.00cm). Local tidal form factor, which is defined by the ratio of the sum of O₁ and K₁ tidal amplitudes to the sum of S₂ and M₂ 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 ) located in the southeast of the Timor Passage (122.9598°E, 11.3683°S) as part of the INSTANT program are analyzed. The Timor Passage mooring observations cover the period from January 1, 2004 to December 20, 2006. However, there are numerous missing values during January 1, 2004 to June 25, 2005. Thus, we only use 18 months observations from June 25, 2005 to December 20, 2006 to ensure the robustness and reliability of the results. The completeness of studied current data can reach 98.55%. More details of mooring observations can be found in Sprintall et al. (2009).

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, M₂ has the largest amplitude (9.05cm/s), followed by S₂ (4.38cm/s), K₁(2.87cm/s), and O₁(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). M₄ tide is the strongest shallow water constituent, with an amplitude of only 0.33cm/s. Figure 3A shows the combination of observed K₁, O₁, and P₁ tides. The sum of K₁ and O₁ tides can induce semi-monthly variations (13.66 days) of high tide. Note that P₁ tide can semi-annually modulate K₁ tide, thus, fortnightly variations of high tides ( Figure 3A ) are not stationary but modulated by semi-annual cycles. The combination of observed M₂, S₂, and K₂ tides also has semi-annually modulated fortnightly cycles ( Figure 3B ).

O₁ and Q₁ tidal frequencies are close, which means that O₁ and Q₁ tides have similar physical properties. As a result, tidal phase lags of O₁ and Q₁ tides are very close, and the difference of O₁ and Q₁ tidal phase lags is only 3.76° ( Table 1 ). The difference of K₁ and P₁ frequencies is much smaller than that of O₁ and Q₁ frequencies, which means that the difference of K₁ and P₁ phase lags should be smaller than 3.76°. However, the observed difference of K₁ and P₁ phase lags is as high as 11.32°, which clearly indicates that the observed P₁ tide is not purely astronomical, but contains a non-negligible contribution of K₁ seasonality. In the next section, we will introduce a novel method which can fully separate the seasonal P₁(K₂) tide from the astronomical P₁ (K₂) tide.

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 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 P₁ and K₂ tides anomalously exaggerate the semi-annual cycles in K₁ and S₂ tides. The method of tidal inference which is widely used in previous studies cannot fully separate astronomical P₁ and K₂ tides from seasonal P₁ and K₂ 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 P₁ and K₂ tides can be obtained via the interpolation. The seasonal P₁ (K₂) tide has totally different amplitude and phase compared to the astronomical P₁ (K₂) 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 semi-annual ranges of M₂ tide are largest among main constituents. O₁ tide has the smallest annual range while K₁ has the smallest semi-annual range. The peak times of seasonal variations of M₂ and S₂ tides are generally consistent while those of K₁ and O₁ 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.

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

HP: Data curation, Conceptualization, Methodology, Visualization, Writing – original draft, Writing – review & editing. JS, TX, FT: Writing – review & editing. ZW: Writing – review & editing, Supervision, Resources, Funding acquisition. All authors contributed to the article and approved the submitted version.