Anomalous 18.61-Year Nodal Cycles in the Gulf of Tonkin Revealed by Tide Gauges and Satellite Altimeter Records

Abstract

Understanding nodal tidal characteristics is essential for accurate long-term tidal prediction. Observational nodal evolution of tides is mainly based on tide gauge records in coastal areas which are limited in time and space, thus impeding coherent determinations of basin-wide patterns of tidal variability. In this paper, we indicate the potential of satellite altimeter data to investigate 18.61-year nodal modulations of main constituents in the Gulf of Tonkin. Three tide gauges and multi-source satellite altimeter observations (TOPEX/Poseidon, Jason1, Jason2, and Jason3) revealed that 18.61-year nodal cycles in tidal amplitudes have noticeable deviations from the equilibrium tidal theory in the Gulf of Tonkin. In general, M₂ and N₂ nodal modulations are anomalously larger than theoretical values while K₂, K₁, and O₁ nodal modulations are noticeably smaller than theoretical values. Compared to point-based tide gauges, satellite altimeter records can provide basin-wide features of nodal modulations of main constituents. Although overlapping geographical blocks are applied to eliminate the effect of tidal alias originated from long-period sampling intervals, the estimation of nodal cycles of minor constituents are still questionable. Nevertheless, the methods described here provide a strong foundation for future research on time-varying tidal dynamics using the combination of tide gauges and satellite altimeter data.

1. Introduction

The 18.61-year lunar nodal cycle is due to the retrograde precession of the lunar ascending and descending nodes along the orbital plane of the Moon and can modulate both lunar and lunisolar tides [1]. Tidal equilibrium theory suggests that the M₂ tidal amplitude should be modified by about 3.7% due to the nodal variations, while the amplitude of the K₁, O₁, and K₂ tides should be modified by about 11.6%, 18.7%, and 28.6%, respectively, over a full nodal cycle [2,3]. However, deviations from theoretical values have been observed in numerous regions around the world such as the Gulf of Tonkin and the Gulf of Maine, typically with real nodal modulations smaller than theoretical values, mainly due to the effects of bottom friction and resonance [1,4,5].

The nodal modulation of the semi-diurnal tides and diurnal tides are notable because they can significantly affect extreme sea levels [6,7,8]. Recently, Peng et al. (2019) [9] calculated the contribution of the lunar nodal modulation to monthly high-water levels (HWLs) globally using long-term hourly tide gauge observations from 574 stations distributed worldwide and found that the nodal modulation had a large influence on the monthly HWLs in the Gulf of Tonkin, the Bristol Channel, and the English Channel; changes in tidal range over the nodal cycle are often observed to be up to 30 cm. An accurate assessment of the influence of these long-period tidal modulations is of critical importance since it may allow the better prediction of coastal flood risk over long periods [10]. It is widely recognized that nodal cycles can also exert strong effects on tidal currents as well as vertical mixing, and therefore influence local sea surface temperature (SST), sea surface salinity (SSS), air temperature, and possibly large-scale climate patterns [11,12,13].

To date, nearly all studies on nodal modulations of major tidal constituents employ hourly long-term point-based tide gauge observations which impede the determination of coherent basin-wide patterns of nodal variability of the major tidal constituents. Although satellite altimetry observations provide a consistent and nearly global coverage of oceanic water levels, few studies have used them. Cherniawsky et al. (2010) [13] carried out harmonic analyses on 16-years of TOPEX/Poseidon (T/P) and Jason-1 data in the Pacific and western Atlantic Oceans. They found that the amplitude ratios between the nodal satellites and their parent constituents are often larger than the theoretical values. Though their work is significant, there are a few major limitations to their approach. First, 16 years of satellite records are barely sufficient to adequately separate the nodal satellite constituents (such as M_2n and K_1n) from their parent constituents (such as M₂ and K₁); since the amplitudes of these nodal satellites are relatively small, they are easily contaminated by low-frequency aliased signals and other potential background noise, especially when the length of records (LORs) is short. Furthermore, compared to coastal areas, satellite altimeter observations are better determined in the open ocean. In shallow water regions (especially coastal zones), altimeter data often cannot provide reliable sea level observations due to various factors such as the interference of land topography on the radar signal [14]. Therefore, the results of Cherniawsky et al. (2010) [13] may be questionable in the marginal seas (e.g., Gulf of Tonkin) and they did not compare the altimeter-based nodal cycles with those of nearby tide gauges.

In the last decade, the spatial coverage and accuracy of satellite altimeter observations in the coastal areas have been significantly improved mainly due to new optimized waveform retracking algorithms, improved radar technology, and geophysical corrections [15]. Numerous coastal altimetry products providing a wide variety of processing level are now available to the scientific community. Among these products is the X-TRACK regional dataset, which is developed by the Center of Topography of the Ocean and Hydrosphere in Toulouse and provides freely available and highly accurate along-track water levels that cover all of the coastal ocean [14]. In this paper, we aim to reveal anomalous 18.61-year nodal cycles of main constituents in the Gulf of Tonkin from a combination of three tide gauges and 27 years of T/P-Jason sea level records processed by X-TRACK. The reminder of this paper is structured as follows. The study area and datasets are described in Section 2. Section 3 details the methodology, followed by the results, discussions, and conclusions in Section 4, Section 5, and Section 6, respectively. In Appendix A, the generation of the nodal factor for K₁ tide is detailed.

2. Study Domain and Data

2.1. Study Domain

The Gulf of Tonkin is a semi-enclosed gulf located in the northwest of the South China Sea (SCS). The average water depth of the Gulf of Tonkin is ~42 m while the deepest part can reach ~100 m. Due to resonance, the Gulf of Tonkin has the strongest diurnal tides in the SCS [16]. As shown in Figure 1, the largest amplitudes of K₁ and O₁ tides can reach 90 cm and 100 cm in the Gulf of Tonkin based on the EOT20 ocean tidal model which is one of the most accurate tidal models [17]. The EOT20 model is derived based on residual tidal analysis of multi-satellite altimeter records, including T/P-Jason, ERS, Envisat, and T/P-Jason interleaved. Compared to diurnal tides, semi-diurnal tides are much smaller, the largest amplitudes of M₂ and S₂ tides are near 45 cm and 10 cm in the Gulf of Tonkin.

2.2. Water Level Observations

The X-TRACK sea level anomaly (SLA) records are obtained from AVISO (https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/coastal-tide-xtrack.html) (accessed on 10 January 2022) over a 27-year period (from March 1993 to May 2020), including T/P data (March 1993–January 2002), Jason-1 data (January 2002–July 2008), Jason-2 data (July 2008–February 2016), and Jason-3 data (February 2016–May 2020). These satellites share the same orbit with a sampling period of 9.915642 days. Compared to other altimeter missions, T/P-Jason satellite observations have a shorter sampling period and longer LOR. Thus, T/P-Jason altimeter data are widely used in the research of tidal dynamics. Other satellites, such as Envisat which has a sampling interval of 35 days and relatively short LOR (8 years), cannot resolve nodal satellite constituents. Figure 2 displays the ground tracks of T/P-Jason satellite altimetry in the Gulf of Tonkin. There are 126 observation points in the Gulf of Tonkin, but to ensure the reliability of the results, we use 114 points which are selected based on the LOR (more than 18.61 years) and data completeness (more than 75%). Hourly water level records from three tide gauges in the Gulf of Tonkin (Figure 2) are downloaded from the University of Hawaii Sea Level Center (https://uhslc.soest.hawaii.edu/data/?fd) (accessed on 21 June 2020). These tide gauges are also selected according to the same criteria as satellite data. The missing values of three tide gauges are all smaller than 0.17%. Although water level observations at Beihai, Haikou, and Dongfang are outdated (~1975 to 1997), they are still analyzed considering the scarcity of long-term tide gauges in the Gulf of Tonkin.

3. Methodology

3.1. Processing Tide Gauge Data

Hourly tide gauge observations were harmonically analyzed without nodal corrections in yearly windows of 8767 h, using the S_TIDE MATLAB toolbox (Qingdao, China) [18] at monthly (720 h) time steps. The S_TIDE toolbox is used to realize classical harmonic analysis as based on the T_TIDE software package [19,20]. We used 8767 h used because this is the minimum length which can resolve MA₂ and MB₂ tides which represent the seasonal variation of M₂ tide. Yearly windows are long enough for resolving P₁ (K₂) from K₁ (S₂). A total of 67 tidal constituents (including 6 long-period constituents and 22 shallow water constituents) can be resolved according to LOR and Rayleigh criterion. Traditionally, the actual nodal cycle of tidal amplitudes as well as the linear trend are estimated by the least square model [3,4,5,21,22,23], which can be expressed as:

$$A(t)=C_0+C_{1}t+H_N\cos (\frac{2\pi }{18.61}t+G_N)$$

(1)

where A(t) is the estimated value of tidal amplitudes or phases at time t. C₀ is a constant, C₁ is the linear trend. H_N and G_N are the amplitude and phase of the nodal cycle, respectively. To decrease the potential effects of background noise on tidal estimation and increase the signal-to-noise ratio (SNR), we use an iteratively reweighted least squares (IRLS) regression [23,24] in the harmonic analysis model instead of ordinary least squares (OLS) regression [20]. The ratio of the calculated nodal amplitude H_N over the mean value of tidal amplitude is estimated as the actual nodal modulation and compared with the theoretical nodal factor f in

Table 1. Nodal factor f for the four major tidal constituents as derived from equilibrium tidal theory. N is the longitude of the lunar ascending node, which was 0° in November 1987 and June 2006, and was 180° in March 1997 and October 2015 [1,19].

Constituents	f
M2/N2	1.0004 − 0.0373cos(N) + 0.0003cos(2N)
S2	1
K1	1.0060 + 0.1150cos(N) − 0.0088cos(2N) + 0.0006cos(3N)
K2	1.0241 + 0.2863cos(N) + 0.0083cos(2N) − 0.0015cos(3N)
O1/Q1	1.0089 + 0.1871cos(N) − 0.0147cos(2N) + 0.0014cos(3N)

, showing that major tidal constituents have significant 18.61-year cycles as well as weak 9.3-year cycles and 6.2-year cycles [19].

3.2. Processing Satellite Altimeter Data

The analysis of altimeter data requires consideration of the aliasing effect. The periods of the diurnal and semi-diurnal constituents are shorter than twice the T/P repeat period; thus, aliasing is induced according to the Nyquist sampling theorem.

Table 2. Major tidal constituents resolved in classical harmonic analysis, showing frequency, Doodson numbers, and alias periods. The first Doodson number represents tidal species (2 means semidiurnal tide, 1 means diurnal tide, 0 means long-period tide).

Tidal Constituent	Doodson Numbers	Frequency (h−1)	Alias Period (Days)
Mn	0 0 0 0 1 0	0.000006129	-
Sa	0 0 1 0 0-1	0.000114074	-
Ssa	0 0 2 0 0 0	0.000228159	-
Mf	0 2 0 0 0 0	0.003050092	36.168
Q1n	1-2 0 1-1 0	0.037212374	68.682
Q1	1-2 0 1 0 0	0.037218503	69.383
O1n	1-1 0 0-1 0	0.038724526	46.015
O1	1-1 0 0 0 0	0.038730654	45.706
P1	1 1-2 0 0 0	0.041552587	88.925
K1	1 1 0 0 0 0	0.041780746	173.322
K1n	1 1 0 0 1 0	0.041786875	177.856
J1	1 2 0-1 0 0	0.043292898	32.763
N2n	2-1 0 1-1 0	0.078993120	49.190
N2	2-1 0 1 0 0	0.078999249	49.548
NU2	2-2 2 0 0 0	0.079201620	65.251
M2n	2 0 0 0-1 0	0.080505272	62.648
M2	2 0 0 0 0 0	0.080511401	62.076
MKS2	2 0 2 0 0 0	0.080739560	46.328
S2	2 2-2 0 0 0	0.083333333	58.772
K2	2 2 0 0 0 0	0.083561492	86.661
K2n	2 2 0 0 1 0	0.083567624	87.780
SK3	3 3-2 0 0 0	0.125114080	43.889
M4	4 0 0 0 0 0	0.161022801	31.038
2MS6	6 2-2 0 0 0	0.244356135	65.775
2MK6	6 2 0 0 0 0	0.244584294	48.358

displays the tidal periods and T/P alias periods of major tidal constituents. Note that for the long-period constituents Ssa and Sa, no aliasing is involved. To fully separate two constituents of alias periods T_i and T_j, the LOR must satisfy Equation (2) (the Rayleigh criterion):

$$LOR\ge \left|\frac{T_i{T}_j}{T_j-T_i}\right|$$

(2)

As displayed in

Table 3. Minimum length (years) for resolving each pair of constituents from T/P-Jason altimetry observations [30].

	Ssa	Q1	O1	P1	K1	N2	M2	S2	K2
Sa	1.00	0.23	0.14	0.32	0.90	0.16	0.20	0.19	0.31
Ssa		0.31	0.17	0.47	9.18	0.19	0.26	0.24	0.45
Q1			0.37	0.86	0.32	0.47	1.63	1.05	0.95
O1				0.26	0.17	1.63	0.47	0.56	0.27
P1					0.50	0.31	0.56	0.47	9.18
K1						0.19	0.27	0.24	0.47
N2							0.67	0.86	0.32
M2								2.97	0.60
S2									0.50

(derived from Equation (2) and Table 2), full resolution of the M₂ and S₂ tide from T/P-Jason satellite altimeter data requires at least 2.97 years of observations. To fully separate K₂ from P₁, and K₁ from Ssa, at least 9.18 years of data are needed (Table 3). The data we use are long enough (27-year) to sufficiently separate aliased pairs, such as S_sa-K₁ and P₁-K₂, and to fully resolve the 18.61-year nodal satellites (such as M_2n and K_1n) and their parent constituents (such as M₂ and K₁). The nodal satellites such as M_2n and K_1n are denoted with subscript n to represent the nodal modulations of major constituents (e.g., M₂ and K₁). M_n is the long-period constituent with a period of 18.61-years. Shallow water constituents such as SK₃, 2MK₆, and MKS₂ should resolved because their alias periods are close to the alias period of O_1n tide (Table 2). The ratios of the amplitudes of the nodal satellites over their parent constituents are calculated as the actual nodal modulation and compared with the theoretical values derived from the equilibrium tidal theory. The theoretical values for M_2n/M₂, K_1n/K₁, O_1n/O₁, and Q_1n/Q₁ are 0.0373, 0.1356, 0.1885, and 0.1884, respectively [13]. There is an obvious difference between the K_1n/K₁ ratio of 0.1356 while the K₁ amplitude is nodally modified by about 11.6%; a detailed explanation of this phenomenon is provided in Appendix A.

Note that the SCS is known to exhibit significant mesoscale eddy activity [25]. The presence of such strong mesoscale activity can influence the accuracy of tidal estimation from altimetry time series [26]. Ray and Byrne (2010) [27] used multi-satellite mapped sea level anomaly (SLA) fields as a prior correction for the mesoscale ocean variability before tidal harmonic analysis and found that this method can significantly improve the along track altimeter tidal estimates. To ensure obtained tidal amplitudes are purely tidal, we compiled satellite data into small overlapping geographical blocks and the sea levels in each circular block with radius of 0.25° were independently subjected to harmonic analysis [28]. To account for the distinct behavior of sea levels, collected sea level data were weighted using a Gaussian function based on the distance to the analyzed point [16,29]. We selected 0.25° because it can effectively decrease the errors of tidal amplitudes.

We mainly focused on nodal variability of M₂, O₁, K₁, and Q₁ constituents because they are strongest constituents in the Gulf of Tonkin. To ensure the reliability of the results of harmonic analysis and minimize the potential effects of aliasing and non-tidal signals, two editing criteria are adopted: (1) Only satellite altimeter records which are at least 75% complete are used for harmonic analysis to extract tidal amplitudes. (2) The SNR (defined as the square of the ratio of the estimated amplitude to its error) should be larger than two. Applying both of these criteria can eliminate most spurious results.

4. Results

4.1. 18.61-Year Nodal Variability from Tide Gauges

Table 4. Actual 18.61-year nodal modulations and their errors of tidal amplitudes of main constituents at three tide gauges in the Gulf of Tonkin. Errors are calculated using 95% confidence intervals.

Constituents	M2	N2	K2	K1	O1	Q1
Theoretical	3.73	3.73	28.63	11.5	18.7	18.7
Beihai	10.4 ± 0.2	13.6 ± 1.4	14.5 ± 0.4	8.2 ± 0.1	16.0 ± 0.2	17.4 ± 0.8
Dongfang	8.5 ± 0.1	9.7 ± 1.1	16.6 ± 0.5	8.0 ± 0.1	15.9 ± 0.2	17.5 ± 0.7
Haikou	11.1 ± 0.2	10.5 ± 2.4	18.4 ± 0.4	9.9 ± 0.1	17.2 ± 0.2	18.3 ± 0.6

displays the actual nodal modulations of the tidal amplitudes of the six major constituents in the Gulf of Tonkin. At all tide gauges in the SCS, M₂ and N₂ nodal modulations are anomalously larger than theoretical values while K₂, K₁, and O₁ nodal modulations are noticeably smaller than theoretical values. Q₁ nodal modulations are only slightly smaller than the theoretical value. The temporal variations of M₂, K₁, O₁, and Q₁ amplitudes at Haikou are highly consistent with theory in terms of phase: they all reach extreme values in November 1987 and M₂ amplitudes vary reversely to K₁, O₁, and Q₁ amplitudes (Figure 3 and Table 1). The results at Dongfang and Beihai stations (not shown) are consistent with those of Haikou. In fact, extreme nodal modulations at Beihai, Dongfang, and Haikou have been observed by Feng et al. (2015) [1]. A possible factor that may affect M₂ nodal variability involves resonant triads, also known as three-wave resonances [31], which are nonlinear interactions between the M₂ and K₁/O₁ tides which may transfer energy to each other, and in cases may decrease the K₁/O₁ nodal modulations and increase the M₂ nodal modulation [1]. Similarly, nonlinear interactions between the N₂ and K₁/Q₁ tides may decrease the K₁/Q₁ nodal modulations and increase the N₂ nodal modulation. The K₂ tide can be generated by the nonlinear interaction of K₁ tide with itself which should increase K₂ nodal modulation; however, actual K₂ nodal modulations are anomalously small in the Gulf of Tonkin (Table 4) which needs further study. It is obvious that only three coastal tide gauges cannot represent the whole gulf. There are 114 valid T/P-Jason observation points in the Gulf of Tonkin, which can provide an opportunity to explore whether the whole gulf shows similar characteristics.

4.2. Main Constituents Extracted from Satellite Altimeter

Figure 4 shows the amplitudes of M₂, S₂, K₁, and O₁ tides in the Gulf of Tonkin from T/P-Jason observations, which are generally consistent with Figure 1. The results from satellite altimeter data indicate that the largest amplitude of M₂, S₂, K₁, and O₁ tides can reach 25.10 cm, 8.05 cm, 81.22 cm, and 90.12 cm, respectively. For diurnal tides, extreme small amplitudes occur in the southern boundary of the gulf because the effect of diurnal resonance is negligible. The theoretical ratio for P₁ to K₁ amplitude is 0.3309 based on the equilibrium tidal theory [30] and after considering the nearby diurnal free wobble resonance, the P₁/K₁ ratio reduces to ~0.3180 [32]. As displayed in Figure 5a, satellite-derived P₁/K₁ ratios are basically consistent with the theory and the spatially averaged P₁/K₁ ratio is 0.3237. The theoretical ratio for O₁ to K₁ amplitude is 0.711 while actual O₁ amplitudes are slightly larger than K₁ amplitudes in the Gulf of Tonkin (Figure 5b). It is believed that the O₁ period is closer to the resonance period of the gulf, thus, the amplification of O₁ amplitude is more significant than K₁ [33].

4.3. 18.61-Year Nodal Variability from Satellite Altimeter

As displayed in Figure 6a, M_2n amplitudes in the Gulf of Tonkin range from 0.6 cm (in the central part of the Gulf of Tonkin) to 1.8 cm (in the northern part of the Gulf of Tonkin). In the central of the Guf of Tonkin, M_2n tides are very weak and their amplitudes are insignificant. The results from satellite data indicate that the M₂ nodal modulation (namely, M_2n/M₂) in the Gulf of Tonkin ranges from 3.59% to 8.01% while the spatially averaged value is 5.90%, which is significantly larger than the theoretical value (3.73%). There are indeed extremely large M₂ nodal modulations in the northeast of the Gulf of Tonkin (Figure 6b). However, in the southwest of the Gulf of Tonkin, M₂ nodal modulations are generally consistent with the theoretical value. Thus, the findings on M₂ nodal modulations from three tide gauges (Beihai, Haikou and Dongfang) are not universal in the Gulf of Tonkin.

K_1n amplitudes in the Gulf of Tonkin (Figure 6c), ranging from 3.31 cm (in the southern part of the Gulf of Tonkin) to 10.20 cm (in the northern part of the Gulf of Tonkin), are much larger than M_2n amplitudes. Satellite results indicate that the K₁ nodal modulation in the Gulf of Tonkin ranges from 11.05% to 14.76% while the spatially averaged value is 12.07% which is clearly smaller than theoretical value (13.56%). Note that large K₁ nodal modulations only occur near the southeastern boundary of the Gulf of Tonkin (Figure 6d). In general, the findings of small K₁ nodal modulation from tide gauges are universal in the Gulf of Tonkin.

O_1n amplitudes are largest among nodal tides in the Gulf of Tonkin (Figure 7a), ranging from 4.59 cm (in the southeastern part of the Gulf of Tonkin) to 16.60 cm (in the northern part of the Gulf of Tonkin). Satellite results indicate that the O₁ nodal modulation in the Gulf of Tonkin ranges from 15.30% to 18.42% while the spatially averaged value is 17.38% which is clearly smaller than the theoretical value (18.85%). Extremely small O₁ nodal modulations occur near the southeastern boundary of the Gulf of Tonkin (Figure 7b). At all T/P-Jason observation points, O₁ nodal modulations are smaller than the theoretical value, which is generally consistent with tide gauges.

Q_1n amplitudes in the Gulf of Tonkin are much smaller than O_1n and K_1n amplitudes, but noticeably larger than M_2n amplitudes (Figure 7c), ranging from 1.10 cm (in the southwestern part of the Gulf of Tonkin) to 3.28 cm (in the northern part of the Gulf of Tonkin). Satellite results indicate that the Q₁ nodal modulation in the Gulf of Tonkin ranges from 13.02% to 25.58% while the spatially averaged value is 17.66%, which is slightly smaller than the theoretical value (18.84%). Extremely large Q₁ nodal modulations occur in the southeastern part of the Gulf of Tonkin while extremely small Q₁ nodal modulations occur near the southwestern boundary of the Gulf of Tonkin (Figure 7d). Except for 27 observation points located in the southeastern part of the Gulf of Tonkin, the findings on Q₁ nodal modulations from satellite data are consistent with tide gauges.

The maximum amplitude of N₂ tide in the Gulf of Tonkin is 4.9 cm (Figure 8a), thus, the maximum amplitude of N_2n tide should be 0.184 cm based on the equilibrium tidal theory. However, satellite-derived N_2n amplitudes significantly exceed 0.184 cm (Figure 8b). In the central part of the Gulf of Tonkin, N₂ tides are very weak while N₂ nodal modulations are abnormally large (more than 80%). Such large N_2n amplitudes should not be real, but induced by background noise (e.g., mesoscale ocean variability). Although overlapping geographical blocks are used to eliminate the effect of tidal alias originating from long-period sampling intervals, the estimation of nodal cycles of minor constituents are still questionable (Figure 8). K_2n shows similar problematic features as N_2n (not displayed).

5. Discussion

5.1. Systematic Nomenclature of Nodal Satellite Tides

As displayed in Appendix A, the nodal cycle of K₁ tide is the resultant of two nodal satellite constituents whose frequencies are w_K1 − w_Mn and w_K1 + w_Mn, where w_K1 and w_Mn represent the frequencies of K₁ and M_n constituents. The constituent whose frequency is w_K1 + w_Mn, has been named as K_1n (Table 2) while the other can be named as K_−1n. K_−1n tides are very small in theory (

Table A1. Constituents near K1 tide. Their frequency, Doodson numbers, and theoretical amplitude ratios are listed. The theoretical values for ${\mathrm{K}}_1^{\mathrm{N}}$/K₁ and ${\mathrm{K}}_1^{-\mathrm{N}}$/K₁ are 0.1356 and 0.0198, respectively [32].

Tidal Constituent	Doodson Numbers	Frequency (h−1)	Amplitude Ratio
${\text{K}}_1^{-\text{2N}}$	1 1 0 0-2 0	0.041768488	0.0001
${\mathrm{K}}_1^{-\mathrm{N}}$	1 1 0 0-1 0	0.041774617	0.0198
K1	1 1 0 0 0 0	0.041780746	1
${\text{K}}_1^{\text{N}}$	1 1 0 0 1 0	0.041786875	0.1356
${\mathrm{K}}_1^{2\mathrm{N}}$	1 1 0 0 2 0	0.041793004	0.0029

), thus, they are not analyzed in this paper. Like K₁, the nodal cycle of M₂ tide is derived from M_2n and M_−2n tides, but the frequency of M_2n is w_M2 − w_Mn, not w_M2 + w_Mn. The present nomenclature of nodal satellite tides may cause potential confusion on their frequencies. In addition, except for the 18.61-year nodal cycle, main constituents are also modulated by 9.3-year and 6.2-year cycles (Table 1). The constituents that generate 9.3-year and 6.2-year cycles are not named yet. Here, we systematically name these nodal satellite tides. Taking K₁ for example, constituents whose frequencies are w_K1 − w_Mn, w_K1 – 2 × w_Mn, and w_K1 – 3 × w_Mn are named as ${\mathrm{K}}_1^{-\mathrm{N}}$, ${\text{K}}_1^{-2\text{N}}$, and ${\mathrm{K}}_1^{-3\mathrm{N}}$ where N is the longitude of the lunar ascending node (Table A1). Using this naming rule, K_1n, O_1n, K_2n, and M_2n tides can be renamed as ${\text{K}}_1^{\text{N}}$, ${\mathrm{O}}_1^{-\mathrm{N}}$, ${\mathrm{K}}_2^{\mathrm{N}}$, and ${\mathrm{M}}_2^{-\mathrm{N}}$ which are more intuitive in terms of frequencies.

5.2. The Advantages and Disadvantages of X-TRACK Data

Although there are thousands of tide gauges around the world, many of them have only been established in recent decades and some are poorly maintained. Nearly all available long-term (more than 50 years) tide gauges are located on the coasts of Japan, North America, Australia, and Europe. For the tide gauges analyzed in this paper, both the location, number, and time span are highly limited which hinder us from determining basin-wide patterns of tidal variability in the Gulf of Tonkin. There are almost no continuous long-term (more than 18.61 years) tide gauges in the coasts of Vietnam although they have very long coastlines. Recent high-frequency (daily or hourly) water level observations are often not publicly available because of concerns of security and propriety, and only limited monthly or yearly averaged data are accessible for scientific research. Compared to this sparse distribution of tide gauges, T/P-Jason observations processed by X-TRACK software are abundant both in number and location, as well as freely available, which give us a unique opportunity to verify whether the findings from limited tide gauges are universal in the study area.

However, satellite altimeter observations also have their disadvantages. The satellite records mainly start in 1993, and long-period sampling intervals make the results of harmonic analysis more influenced by background noise such as mesoscale ocean variability, especially for minor constituents (e.g., N_2n and K_2n in this study). When tides are weak, especially near the amphidromic points, the ratios of nodal satellite constituents and their parent constituents may be abnormally large or small. Therefore, it is not recommended to use satellite altimeter observations to explore nodal modulations of weak tidal constituents.

6. Conclusions

Careful analysis of time-varying tidal characteristics is helpful and necessary for many practical purposes, such as navigation, coastal engineering, the utilization of tidal energy, and flood prevention. It is common knowledge that the actual nodal modulations of tidal amplitudes and phases have often been observed to be different from equilibrium theory in many coastal regions around the world based on long-term tide gauge observations. Less known, and less studied, is the nodal variability in the open sea where few tide gauges exist. In this paper, we analyzed the 18.61-year tidal variability in the Gulf of Tonkin using the combination of 27-year T/P-Jason satellite altimeter records and three coastal tide gauges. The main findings are summarized as follows:

1.

Both tide gauges and satellite altimeter observations indicate that 18.61-year nodal cycles in tidal amplitudes significantly deviate from the equilibrium theory in the Gulf of Tonkin. In general, M₂ and N₂ nodal modulations significantly exceed theoretical values while K₂, K₁, and O₁ nodal modulations are significantly lower than theoretical values.

2.

Compared to point-based tide gauges, satellite altimeter records can provide basin-wide features of nodal modulations of main constituents. It is found that M₂ nodal modulations are generally consistent with the theoretical value in the southwest of the Gulf of Tonkin. Furthermore, Q₁ nodal modulations are noticeably larger than the theoretical value in the southeastern part of the Gulf of Tonkin. Although overlapping geographical blocks are applied to eliminate the potential effect of tidal alias, the estimated N₂ and K₂ nodal modulations are still questionable.

Partial results obtained from satellite data are disturbed by tidal aliasing due to long-period sampling intervals. However, the methods presented here provide an important supplement to the observed variability seen at coastal tide gauges, which are limited in time and space in the study area. In the future, the accuracy of satellite altimeter records will be improved, and their length or record will increase, which will improve estimates. Additionally, next-generation altimetry missions, such as the recently launched Sentinel-6 mission from ESA (http://www.esa.int/Applications/Observing_the_Earth/Copernicus/Sentinel-6, (accessed on 21 June 2020)), and the upcoming Surface Water Ocean Topography (SWOT) platform from NASA (https://eospso.nasa.gov/missions/surface-water-ocean-topography, (accessed on 21 June 2020)) will bring new and improved observations of the coastal ocean, which will further help to resolve decadal variability behavior of sea level and tides and close the gap between coastal tide gauge observations and open-ocean altimetry observations.