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**Ocean Science**
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- Abstract
- Introduction
- The numerical model
- Numerical methods for estimating the resonant frequency
- Influence of resonance in the main area of the South China Sea on the Gulf of Thailand
- A theoretical model
- Conclusions
- Code availability
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- References

**Research article**
29 Mar 2019

**Research article** | 29 Mar 2019

Tidal resonance in the Gulf of Thailand

^{1}First Institute of Oceanography, Ministry of Natural Resources, Qingdao, 266061, China^{2}Laboratory for Regional Oceanography and Numerical Modelling, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, China

^{1}First Institute of Oceanography, Ministry of Natural Resources, Qingdao, 266061, China^{2}Laboratory for Regional Oceanography and Numerical Modelling, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, China

**Correspondence**: Guohong Fang (fanggh@fio.org.cn)

**Correspondence**: Guohong Fang (fanggh@fio.org.cn)

Abstract

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The Gulf of Thailand is dominated by diurnal tides, which might be taken to indicate that the resonant frequency of the gulf is close to one cycle per day. However, when applied to the gulf, the classic quarter-wavelength resonance theory fails to yield a diurnal resonant frequency. In this study, we first perform a series of numerical experiments showing that the gulf has a strong response near one cycle per day and that the resonance of the South China Sea main area has a critical impact on the resonance of the gulf. In contrast, the Gulf of Thailand has little influence on the resonance of the South China Sea main area. An idealized two-channel model that can reasonably explain the dynamics of the resonance affecting the Gulf of Thailand is then established in this study. We find that the resonant frequency around one cycle per day in the main area of the South China Sea can be explained with the quarter-wavelength resonance theory, and the large-amplitude response at this frequency in the Gulf of Thailand is basically a passive response of the gulf to the increased amplitude of the wave in the southern portion of the main area of the South China Sea.

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Cui, X., Fang, G., and Wu, D.: Tidal resonance in the Gulf of Thailand, Ocean Sci., 15, 321–331, https://doi.org/10.5194/os-15-321-2019, 2019.

1 Introduction

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The Gulf of Thailand (GOT) is an arm of the South China Sea (SCS), the
largest marginal sea of the western Pacific Ocean (Fig. 1). The width of the
GOT is approximately 500 km, and the length of the GOT from the top to the
mouth of the gulf, as indicated by the blue B line in Fig. 1, is
approximately 660 km. The mean depth in this area is 36 m according to the
ETOPO1 depth data set (from the US National Geophysical Center). The mean
depth of the SCS from the southern opening of the Taiwan Strait to the cross
section at 1.5^{∘} N is 1323 m. If the GOT is excluded, the mean
depth of the rest of the SCS (herein called the SCS body and abbreviated as
SCSB) is 1457 m. Tidal waves propagate into the SCS from the Pacific Ocean
through the Luzon Strait (LS) and mainly propagate in the southwest
direction towards the Karimata Strait, with two branches that propagate
northwestward and enter the Gulf of Tonkin and the GOT. The energy fluxes
through the Mindoro and Balabac straits are negligible (Fang et al., 1999;
Zu et al., 2008; Teng et al., 2013). The GOT is dominated by diurnal tides,
and the strongest tidal constituent is *K*_{1} (Aungsakul et al., 2011; Wu
et al., 2015).

The resonant responses of the GOT to tidal and storm forcing have attracted extensive research interest. However, the previous results have been diverse. Yanagi and Takao (1998) simplified the GOT and Sunda Shelf as an L-shaped basin and concluded that this basin has a resonant frequency near the semi-diurnal tidal frequency. Sirisup and Kitamoto (2012) applied a normal mode decomposition solver to the GOT and Sunda Shelf area and obtained four eigenmodes with modal frequencies of 0.42, 1.20, 1.54 and 1.76 cpd (cycles per day). Tomkratoke et al. (2015) further studied the characteristics of these modes and concluded that the mode with a frequency of 1.20 cpd is the most important. Cui et al. (2015) used a numerical method to estimate the resonant frequencies of the seas adjacent to China, including the SCS and the GOT, and found that the GOT has a major resonant frequency of 1.01 cpd and a minor peak response frequency of 0.42 cpd. In these studies, except for Yanagi and Takao (1998), no effort was made to establish a theoretical model of the GOT, and the resonant period estimated by Yanagi and Takao (1998) cannot be used to explain the resonance of the diurnal tides.

The GOT is a semi-enclosed gulf with an amphidromic point in the basin for
each *K*_{1} and *O*_{1} constituent. Wu et al. (2013) reproduced the tidal
system well with superimposed incident and reflected Kelvin waves and a
series of Poincare modes. This result raises the question of whether the
quarter-wavelength resonant theory can explain the tidal resonance in the
gulf, as is the case with other areas (Miles and Munk, 1961; Garrett, 1972;
Sutherland et al., 2005). According to quarter-wavelength theory, because
the distance from the head of the gulf to the mouth is approximately 660 km
and the mean depth is approximately 36 m, the resonant frequency should be
0.61 cpd, which is much lower than the estimates by Cui et al. (2015) and
Tomkratoke et al. (2015). Thus, it is clear that the tidal resonance
phenomenon in the GOT cannot be reasonably explained by the
quarter-wavelength theory.

According to the theories of Garrett (1972) and Miles and Munk (1961), tidal oscillations are limited to a specific area. In contrast, Godin (1993) proposed that tides are a global phenomenon that cannot be separated into independent subdomains. The GOT is an auxiliary area of the SCS and is connected to the SCSB. We thus believe that Godin's theory is applicable to the GOT. In this paper, by considering the bathymetry of the SCSB in numerical experiments and theoretical analyses, we investigate the reasons for the GOT to have a strong response around the frequency of one cycle per day and how the physical properties of the SCSB primarily determine the resonances of both the SCSB and the GOT.

2 The numerical model

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In this paper, we use the Princeton Ocean Model (POM) for numerical investigation, but we partially modify its code to meet the needs of the present study. This study is limited to the main mechanism of diurnal tidal resonance; accordingly, the elimination of tide-generating forces and nonlinear terms and the linearization of bottom friction in the control equation will not affect the problem we are studying, and the two-dimensional model effectively suits our purpose, as shown in a number of previous studies (e.g., Garrett, 1972; Godin, 1993; Webb, 2014; Cui et al., 2015). The general forms of the equation of continuity and the equation of motion used in this study are as follows:

$$\begin{array}{}\text{(1)}& {\displaystyle}& {\displaystyle \frac{\partial \stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}{\partial t}}=-{\displaystyle \frac{\mathrm{1}}{R\mathrm{cos}\mathit{\phi}}}\left[{\displaystyle \frac{\partial \left(H\stackrel{\mathrm{\u0303}}{u}\right)}{\partial \mathit{\lambda}}}+{\displaystyle \frac{\partial \left(H\stackrel{\mathrm{\u0303}}{v}\mathrm{cos}\mathit{\phi}\right)}{\partial \mathit{\phi}}}\right],\text{(2)}& {\displaystyle}& {\displaystyle \frac{\partial \stackrel{\mathrm{\u0303}}{u}}{\partial t}}=\mathrm{2}\mathrm{\Omega}\stackrel{\mathrm{\u0303}}{v}\mathrm{sin}\mathit{\varphi}-{\displaystyle \frac{g}{R\mathrm{cos}\mathit{\phi}}}{\displaystyle \frac{\partial \stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}{\partial \mathit{\lambda}}}-{\displaystyle \frac{\mathit{\tau}\stackrel{\mathrm{\u0303}}{u}}{H}},\text{(3)}& {\displaystyle}& {\displaystyle \frac{\partial \stackrel{\mathrm{\u0303}}{v}}{\partial t}}=-\mathrm{2}\mathrm{\Omega}\stackrel{\mathrm{\u0303}}{u}\mathrm{sin}\mathit{\varphi}-{\displaystyle \frac{g}{R}}{\displaystyle \frac{\partial \stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}{\partial \mathit{\phi}}}-{\displaystyle \frac{\mathit{\tau}\stackrel{\mathrm{\u0303}}{v}}{H}},\end{array}$$

where *t* denotes time; *λ* and *ϕ*, respectively, refer to the
east longitude and north latitude; $\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}$ is the surface height
above the undisturbed sea level; $\stackrel{\mathrm{\u0303}}{u}$ and $\stackrel{\mathrm{\u0303}}{v}$ represent
the east and north components of the fluid velocity, respectively; *R*
indicates the Earth's radius, Ω its angular velocity, *g*
gravity, *H* the undisturbed water depth and *τ* the linearized bottom
friction coefficient.

The numerical model covers the ocean lying between 99, 131^{∘} E, 1.5 and 42^{∘} N. The northern and
eastern open boundaries are located far beyond the area of the SCS to
prevent the numerical values at the open boundaries from influencing the
results in the study area. The southern open boundary is set along a zonal
section of 1.5^{∘} N, which meets the southernmost tip of the Malay
Peninsula. The grid resolution is 1∕12^{∘}. The water depths are
basically taken from the ETOPO1 data set and are modified using depth data
extracted from navigational charts.

To examine the applicability of the modified POM model to the study area,
the model is first used to simulate the *K*_{1} tide in the SCS and its
neighboring area. The amplitudes and phase lags along the open boundary are
taken from the global tidal model TPXO9, which is based on satellite
altimeter observations (Egbert and Erofeeva, 2002). The model-produced
*K*_{1} tidal system is shown in Fig. 2a, and a comparison of the model
results with observations at 31 tidal gauge stations is shown in Table 1.
The locations of these tidal stations are shown in Fig. 1, from which it
can be seen that the stations are basically evenly distributed along the
coast of the SCSB and GOT.

From Table 1, we can see that the deviations in amplitudes are mostly within
0.05 m, while those in phase lags are mostly within 20^{∘}.
Considering that the governing equations are greatly simplified, the
linearized bottom friction is used to replace the more accurate quadratic
form, the agreement between model results and observations can be regarded
as satisfactory. Therefore, the modified model is applicable to the SCS and
GOT tidal study.

To show the characteristics of the wave propagation process, we calculate
the tidal energy flux density distribution, as given in Fig. 2b. One can
see that the *K*_{1} tidal wave mainly enters the SCS through the LS and
spreads southward, partially moves to the GOT and partially exits the SCS
through the Karimata Strait. Figure 2a shows that the *K*_{1} tidal
amplitudes are large in the Gulf of Tonkin and that there is an amphidromic
point at the mouth of the gulf caused by one-quarter-wavelength resonance
(Fang et al., 1999). In the GOT, there is also an amphidromic point, but
away from the mouth section, indicating that the amplified *K*_{1} tide
cannot be attributed to the quarter-wavelength resonance.

3 Numerical methods for estimating the resonant frequency

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The open boundary condition for the tidal resonant study can be written in the following form:

$$\begin{array}{}\text{(4)}& \stackrel{\mathrm{\u0303}}{\mathit{\zeta}}(i,j,t)=\sum _{n={N}_{i}}^{{N}_{\mathrm{2}}}{\stackrel{\mathrm{\u0303}}{Z}}_{n}(i,j)\mathrm{cos}\left[\mathrm{2}\mathit{\pi}{f}_{n}t-{\stackrel{\mathrm{\u0303}}{\mathit{\theta}}}_{n}(i,j)\right],\end{array}$$

where (*i*,*j*) indicates grid points on the open boundary; *f*_{n}=*n*Δ*f*
refers to the frequency of the *n*th wave of interest, with Δ*f*
referring to the spectrum resolution; and for $n={N}_{\mathrm{1}},{N}_{\mathrm{1}}+\mathrm{1},\mathrm{\dots},{N}_{\mathrm{2}}$, ${\stackrel{\mathrm{\u0303}}{Z}}_{n}$ and ${\stackrel{\mathrm{\u0303}}{\mathit{\theta}}}_{n}$ represent the amplitude
and phase lag of the *n*th wave, respectively. In this study, we choose
$\mathrm{\Delta}f=\mathrm{1}/\mathrm{1024}$ cph (cycles per hour) for the following two reasons.
First, the value 1024 is equal to 2^{10}, enabling us to efficiently
calculate spectra from model-produced time series by using a fast Fourier
transform (FFT). Second, because the minimum frequency difference between the
main tidal constituents (*Q*_{1}, *O*_{1}, *K*_{1}, *N*_{2}, *M*_{2} and
*S*_{2}) is equal to $\mathrm{1.51}\times {\mathrm{10}}^{-\mathrm{3}}$ cph, the resolution of $\mathrm{\Delta}f=\mathrm{1}/\mathrm{1024}$ cph is sufficient for separating these constituents.

Through the simulation of the *K*_{1} tide, it is shown that the modified
model can be applied to tidal study. The model setting is consistent with the
setting of the simulation of the *K*_{1} tide except that the water level
values at the open boundaries are changed as follows. The amplitudes
${\stackrel{\mathrm{\u0303}}{Z}}_{n}$ at the open boundaries are specified as a constant, 2 cm;
the phase lags ${\stackrel{\mathrm{\u0303}}{\mathit{\theta}}}_{n}$ are given as random numbers that are
evenly distributed in the interval (0,2*π*) and generated
using a normal random number generator. The purpose of using random phase
lags is to avoid all or some of the waves to have the same phase at a certain
time, which can lead to simultaneous unreasonably high or low sea levels. The
selected *N*_{1} and *N*_{2} values are 1 and 107, respectively. Thus, the
frequencies of the waves studied range from 1∕1024 to 107∕1024 cph or
0.0234–2.5078 cpd. The corresponding periods range from 10 to 1024 h (approximately 0.4–42.7 days), which
covers all main tidal constituents.

The model is run for 3×1024 h (see Cui et al., 2015, Fig. 2).
In the last cycle of 1024 h, the hourly results at each grid point are
preserved, and FFT analysis is performed to yield amplitude *Z*_{n} and
phase lag *θ*_{n}. The amplitude ratio is defined as follows:

$$\begin{array}{}\text{(5)}& {G}_{n}={Z}_{n}/{\stackrel{\mathrm{\u0303}}{Z}}_{n},\end{array}$$

and the phase lag difference is given by the following equation:

$$\begin{array}{}\text{(6)}& {\mathit{\psi}}_{n}={\mathit{\theta}}_{n}-{\stackrel{\mathrm{\u0303}}{\mathit{\theta}}}_{n}.\end{array}$$

According to Munk and Cartwright (1966), *G**e*^{−iψ} is the admittance. Specifically, as Munk and Cartwright (1966)
stated, *G*_{n} is the amplitude response and represents the amplification
factor of the *n*th wave in response to forcing. In the present study, we
call *G*_{n} and *ψ*_{n} the amplitude gain and phase change,
respectively, in accordance with Sutherland et al. (2005) and Roos et
al. (2011).

4 Influence of resonance in the main area of the South China Sea on the Gulf of Thailand

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Cui et al. (2015) revealed that both the SCS and GOT have strong response peaks around one cycle per day and suggested that the resonance of the SCS could be explained by one-quarter-wavelength resonance theory, but the authors did not provide the reasons for the strong response of the GOT around this frequency. As an arm of the SCS, the tidal energy of the GOT comes mainly from the SCSB (Fig. 2b), so we speculate that the strong response of the GOT around one cycle per day may be related to the SCS resonance.

To examine the influence of SCSB resonance on GOT, we conduct six numerical experiments. In Exp. 1, we use real bottom topography. In Exp. 2, we artificially make the depths in the SCSB equal to half of the real depths and retain the depths in the GOT. In Exp. 3, the depths in the SCSB are artificially doubled and the depths in the GOT remain unchanged. In Exps. 4 and 5, we change the water depths of the GOT by factors of 1∕2 and 2, respectively, and the SCSB depths remain unchanged. In Exp. 6, the mouth boundary of the GOT (indicated by the blue B line in Fig. 1) is artificially closed and the SCSB retains the real depths. The results of these six experiments are shown in Fig. 3, in which the area-mean values of the top 20 % amplitude gains are used to represent the response of the corresponding area. The resonant frequencies corresponding to peak responses are listed in Table 2.

Table 2 and Fig. 3 show that when real water depths are used, the
frequency corresponding to peak amplitude in the SCSB appears at 0.99 cpd,
while that in the GOT is 1.01 cpd. The frequencies corresponding to peak
amplitudes in the two areas are basically the same, and both are very close
to that of the diurnal tide *K*_{1}, whose frequency is equal to 1.00 cpd
(or more precisely, 1.0027 cpd). The spatial patterns of the amplitude gain
and phase change for the frequency 0.99 cpd are displayed in Fig. 4a, with
corresponding energy flux density vectors shown in Fig. 4b (the
corresponding figures for 1.01 cpd are almost the same and are thus not
shown). We can find that the patterns shown in Fig. 4 are quite similar to
the simulated *K*_{1} patterns shown in Fig. 2. Minor differences are
caused by the use of different open boundary conditions.

From Fig. 3a, b, we can see that the peak amplitude gains in the GOT and SCSB are reduced when the depths of the SCSB are changed to half of the real depths. This reduction in amplitude gain occurs because friction increases as depth decreases (see Eqs. 2 and 3). Moreover, the amplitude peak frequencies in the GOT and SCSB both change to 0.75 cpd (Fig. 3a, b and Table 2), indicating that the resonant frequency in the GOT is determined by that of the SCSB. When the depths of the SCSB are doubled, the amplitude peak frequencies of the SCSB and GOT are increased to 1.49 and 1.50, respectively (Fig. 3a, b and Table 2), again indicating that the peak frequency of the GOT is determined by that of the SCSB. It is worth noting that when the depths in the SCSB are artificially changed by factors of 1∕2 and 2, the resonant frequencies are roughly changed by factors of $\sqrt{\mathrm{1}/\mathrm{2}}$ and $\sqrt{\mathrm{2}}$, respectively. This indirectly indicates that the quarter-wavelength resonance theory is applicable to the SCSB.

In Exp. 3, there is another weaker peak in the SCSB at the frequency of approximately 1.15 cpd (Fig. 3a). The peak frequency response may also have an effect on the GOT (Fig. 4b, Exp. 3), which results in a plateau peak of GOT between 0.5 and 1.2 cpd. This may be due to the deepening of the SCSB, resulting in discontinuity of topographic data at the junction with the GOT. Moreover, the amplitude gains in the GOT are significantly increased by increasing the depth, which results in reduced friction (see Eqs. 2 and 3).

Experiments 1–3 suggest that the peak response frequency of the GOT is strongly affected by the SCSB. Here, we conduct further experiments (Exps. 4–6) to investigate whether the GOT can also influence the resonance of the SCSB. From the results of Exps. 4–5 (Table 2, Fig. 3c, d), it can be seen that changing the depths of the GOT has little effect on the peak response frequencies of the SCSB and GOT. That is, the resonant frequencies of both areas are still close to one cycle per day.

In Exp. 6, the mouth boundary of the GOT (indicated by the blue B line in
Fig. 1) is artificially closed. The results of this experiment show that
the GOT has a small influence on the response of the SCSB, as shown in Fig. 3c.
When boundary B is closed, the resonant frequency of the SCSB becomes
slightly higher than the frequency of the *K*_{1} tide, and the response
amplitude of the SCSB in the vicinity of the resonant frequency is slightly
reduced. As indicated by Arbic et al. (2009), when a shallow basin is
connected to a deep basin, the impact of the shallow basin on the tidal
response in the deep basin is determined by the depth ratio, width ratio
and length ratio of these two basins as well as the friction in the shallow
basin. In the present case, the depth and width ratios of the GOT against
the SCSB are small, and tides in the GOT are strongly damped, so the impact
of the GOT on the SCSB is not significant. In addition, there is a weak
response peak at the frequency 0.45 cpd in the GOT (Exp. 1 in Fig. 3b).
Since the GOT has a length of 660 km and a mean depth of 36 m, the
quarter-wavelength theory gives a resonant frequency of 0.61 cpd. It seems
that the peak at 0.45 cpd is associated with the local regional resonance.

In summary, the resonant periods or frequencies change when the depth of the SCSB varies, and the trends in the two sea areas are consistent (Fig. 3). The resonant frequency decreases when the SCSB is shallower and increases when the SCSB is deeper. Additionally, the resonant frequencies of the SCSB and GOT remain almost identical (Table 2). The experimental results show that the SCSB has a critical impact on the tidal response of the GOT and that the GOT is not an independent sea area in terms of tidal resonance.

5 A theoretical model

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The quarter-wavelength resonant theory is based on the wave behavior in a
single channel. As discussed above, this theory does not explain the
enhanced tides around one cycle per day in the GOT. Here, we establish a
two-channel model and examine its applicability to the SCSB-GOT system.
Tidal waves from the Pacific Ocean propagate through the LS, pass the SCSB
and finally enter the GOT. The tidal waves in the Karimata Strait are very
weak (Fig. 2a and Wei et al., 2016) and are not able to propagate into the
GOT. Thus, the tidal energy in the GOT is mainly from the SCSB. Therefore,
we use a two-channel model to represent the SCSB-GOT system, as shown in
Fig. 5. This model is quite similar to Webb's (2011) 1-D model, except that
we add a forcing at the entrance of the deep channel. In the figure, *H*_{1}
is the depth of channel 1 (the deep channel), *H*_{2} is the depth of
channel 2 (the shallow channel), and *L*_{1} and *L*_{2} are the lengths of
channels 1 and 2, respectively. The tidal waves enter channel 1 through the
opening at *x*=*L*_{1}, enter channel 2 through the junction point at *x*=0
and finally reach the top of channel 2 at $x=-{L}_{\mathrm{2}}$. The equations
governing the tidal motion in channels can be expressed as follows:

$$\begin{array}{}\text{(7)}& {\displaystyle}\left\{\begin{array}{l}{\displaystyle \frac{\partial {\stackrel{\mathrm{\u0303}}{u}}_{m}}{\partial t}}=-g{\displaystyle \frac{\partial {\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{m}}{\partial x}}-{\mathit{\gamma}}_{m}{\stackrel{\mathrm{\u0303}}{u}}_{m}\\ {\displaystyle \frac{\partial {\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{m}}{\partial t}}=-{H}_{m}{\displaystyle \frac{\partial {\stackrel{\mathrm{\u0303}}{u}}_{m}}{\partial x}},\end{array}\right.,\end{array}$$

where ${\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{m}(x,t)$ and ${\stackrel{\mathrm{\u0303}}{u}}_{m}(x,t)$ represent the
elevation and velocity, respectively; *H*_{m} is the depth; *γ*_{m} is
the friction parameter (equivalent to *τ*∕*H* in Eqs. 2 and 3); *g* is the
acceleration due to gravity; and $m=\mathrm{1},\mathrm{2}$ represents the different channel
segments. Here, ${\stackrel{\mathrm{\u0303}}{u}}_{m}(x,t)$ and ${\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{m}(x,t)$ can be
expressed in the forms of ${\stackrel{\mathrm{\u0303}}{u}}_{m}\left(x,t\right)={R}_{e}\left({u}_{m}\left(x\right){e}^{-i\mathit{\omega}t}\right)$ and
${\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{m}\left(x,t\right)={R}_{e}({\mathit{\zeta}}_{m}\left(x\right){e}^{-i\mathit{\omega}t}$), respectively, where *ω* is the angular
frequency, *t* is time, and *ζ*_{m}(x) and *u*_{m}(x) represent the complex amplitudes of the elevation and velocity,
respectively. By eliminating the common factor *e*^{−iωt}, Eq. (7) can
be reduced to ordinary differential equations as follows:

$$\begin{array}{}\text{(8)}& \left\{\begin{array}{l}-i\mathit{\omega}{u}_{m}=-g{\displaystyle \frac{\mathrm{d}{\mathit{\zeta}}_{m}}{\mathrm{d}x}}-{\mathit{\gamma}}_{m}{u}_{m}\\ -i\mathit{\omega}{\mathit{\zeta}}_{m}=-{H}_{m}{\displaystyle \frac{\mathrm{d}{u}_{m}}{\mathrm{d}x}}.\end{array}\right.\end{array}$$

The boundary and matching conditions are as follows:

$$\begin{array}{}\text{(9)}& {\displaystyle}& {\displaystyle}{\mathit{\zeta}}_{\mathrm{1}}\left({L}_{\mathrm{1}}\right)={a}_{\mathrm{0}},{\mathit{\zeta}}_{\mathrm{1}}\left(\mathrm{0}\right)={\mathit{\zeta}}_{\mathrm{2}}\left(\mathrm{0}\right),\text{(10)}& {\displaystyle}& {\displaystyle}{H}_{\mathrm{1}}{u}_{\mathrm{1}}\left(\mathrm{0}\right)={H}_{\mathrm{2}}{u}_{\mathrm{2}}\left(\mathrm{0}\right),\end{array}$$

and

$$\begin{array}{}\text{(11)}& {\displaystyle}{\displaystyle}{u}_{\mathrm{2}}(-{L}_{\mathrm{2}})=\mathrm{0}.\end{array}$$

From the governing equations and boundary/matching conditions, the complex amplitudes of the elevations of the two channels can be obtained as follows (see the Appendix for a detailed derivation):

$$\begin{array}{}\text{(12)}& {\displaystyle}{\displaystyle}{\mathit{\zeta}}_{\mathrm{1}}\left(x\right)={a}_{\mathrm{0}}{R}_{\mathrm{0}}Q\left(x\right),\end{array}$$

and

$$\begin{array}{}\text{(13)}& {\displaystyle}{\displaystyle}{\mathit{\zeta}}_{\mathrm{2}}\left(x\right)={a}_{\mathrm{0}}{R}_{\mathrm{0}}\mathrm{cos}{\mathit{\beta}}_{\mathrm{2}}\left(x+{L}_{\mathrm{2}}\right).\end{array}$$

When the denominator of *R*_{0} reaches its minimum, the amplitudes of the
two-sea area become the largest and resonance occurs. The condition is as
follows (see expression of *R*_{0} in the Appendix, Eq. A10):

$$\begin{array}{}\text{(14)}& \mathrm{|}\mathrm{cos}{\mathit{\beta}}_{\mathrm{1}}{L}_{\mathrm{1}}\mathrm{cos}{\mathit{\beta}}_{\mathrm{2}}{L}_{\mathrm{2}}-\left(r{p}_{\mathrm{1}}/{p}_{\mathrm{2}}\right)\mathrm{sin}{\mathit{\beta}}_{\mathrm{1}}{L}_{\mathrm{1}}\mathrm{sin}{\mathit{\beta}}_{\mathrm{2}}{L}_{\mathrm{2}}\mathrm{|}=min,\end{array}$$

where $r=\sqrt{{H}_{\mathrm{2}}/{H}_{\mathrm{1}}}$ and ${p}_{m}=\sqrt{\mathrm{1}+i{\mathit{\mu}}_{m}}$.

Based on the configurations of the SCSB and GOT, we take *L*_{1}=2600,
*L*_{2}=660 km, *H*_{1}=1457, *H*_{2}=36 m, and
${\mathit{\gamma}}_{m}=\frac{\mathrm{0.0001}}{{H}_{m}}$. By substituting these values into
Eqs. (12) and (13), we can obtain the solutions of these equations. The
amplitude gains at locations *x*=0 and −*L*_{2} are functions of *ω* or
functions of the frequency *f* ($f=\mathit{\omega}/\mathrm{2}\mathit{\pi})$ and can be used to
represent the response properties of channels 1 and 2, respectively. The
results are shown with red curves in Fig. 6. For comparison, we also
calculate the corresponding functions in the absence of friction, as shown by
the blue curves in Fig. 6. From these curves, the resonant frequencies can be
readily obtained, as given in Table 3.

Figure 6a displays the response function at $x=-{L}_{\mathrm{2}}$, which represents
the response of channel 2. This figure shows that the results in the
presence of friction are more realistic than those in the absence of
friction. The former case has a maximum peak at a frequency of 1.040 cpd.
The corresponding resonant period is 23.08 h, and this value is similar to
the result of the numerical experiment involving the natural basin (Table 2,
Exp. 1). The secondary response peak appears at a frequency of 0.558 cpd,
which is also fairly consistent with the results for the natural basin, as
shown in Fig. 3b (red curve). The third peak is very small and appears at a
frequency of 1.848 cpd. If we carefully examine the red curve in Fig. 3b, we
can also find a small peak near a frequency of 1.9 cpd. The response
function is worse when friction is neglected than when friction is retained,
but the obtained resonant frequencies are almost unchanged, as shown by the
blue curve in Fig. 6a. Figure 6b shows the response function at *x*=0,
which represents the response of channel 1. The red curve has only one peak
at a frequency of 1.040 cpd, which is also similar to the results of the
numerical experiment applied to the natural basin (Table 2, Exp. 1). When
the friction is neglected, the frequency of the main peak is unchanged. In
addition, there are two other peaks that are very narrow, indicating that
these two peaks are relatively insignificant.

If we apply the quarter-wavelength resonance theory to channel 1, we can
obtain resonant frequencies of 0.99 cpd. If we apply the quarter-wavelength
and three-quarter-wavelength resonance theories to channel 2, we can obtain
resonant frequencies of 0.61 and 1.84 cpd, respectively. Therefore, we can
conclude that the major peaks around the frequency of 1.04 cpd in Fig. 6
are caused by resonance in channel 1. This indicates that channel 1 plays a
determinative role in the two-channel system. Similarly, we can also
conclude that the secondary and tertiary peaks around the frequencies of 0.55
and 1.85 cpd in Fig. 6 are caused by resonances in channel 2, associated
with the quarter-wavelength and three-quarter-wavelength resonances.
Although the frequencies of the peaks shown in Fig. 6 correspond well with
those estimated based on the quarter-wavelength and three-quarter-wavelength
theories, there are small discrepancies. This is due to the connection of
the two channels. In fact, the resonant frequencies of the two-channel
system also depend on the depth ratio of two channels, as shown in Eq. (14).
In comparison to channel 2, the secondary, especially the tertiary peak, in
channel 1 is much more less significant. This can be explained as follows:
The tidal incident wave from the channel 1 partially enters channel 2 across
the steep topography at *x*=0, and here, the rest of the wave is reflected.
The reflected wave is superimposed with the incident wave, and tidal
resonance occurs around the frequency of 1.04 cpd. That is, the steep
topography at *x*=0 acts as a wall for channel 1, which causes the
quarter-wavelength resonance to occur in the channel. Furthermore, the steep
topography can also block most energy of the wave in channel 2 from entering
channel 1. Therefore, the relatively large amplitudes in channel 2 at
frequencies around 0.55 and 1.85 cpd are not obvious in channel 1 under
the action of friction.

Figure 7 displays the amplitude gains along the channels when channel 2 is in a resonant state. The solution in presence/absence of friction is shown in Fig. 7a, b. The figure shows that when the forcing frequency is equal to 1.040 cpd, which is the major resonant frequency, the amplitude gain gradually increases from the mouth towards the head in channel 1. In channel 2, the amplitude gain decreases first and reaches a trough before increasing again towards the head. The trough corresponds with the amphidromic zone of the diurnal tides in the GOT (see Fig. 2a or the cotidal charts in Wu et al., 2015). For a frequency of 0.558 cpd, the amplitude gain is nearly constant in channel 1 and increases with a relatively high rate towards the head in channel 2. For a frequency of 1.848 cpd, the amplitude gain is also nearly constant in channel 1, and in channel 2, there are two antinodes and one node. This result is similar to the distribution of semi-diurnal tides in the GOT. When friction is neglected, the basic characteristics are the same but the amplitude gain significantly increases.

6 Conclusions

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The GOT is dominated by diurnal tides, indicating that the response near the diurnal tide frequency in the GOT is stronger than that at other frequencies. However, when applied to the GOT, the classical quarter-wavelength resonant theory fails to yield a diurnal resonant period. Changing the water depths in the SCSB in our numerical experiments further shows that the resonance of the SCSB has a critical impact on the resonance of the GOT. An idealized two-channel model that can reasonably explain the resonance in the GOT is established. Through the numerical experiments and two-channel model, we found that the resonant frequency around one cycle per day in the South China Sea main area can be explained with the quarter-wavelength resonance theory, and the large-amplitude response at this frequency in the GOT is basically a passive response of the gulf to the increased amplitude of the wave in the southern portion of the main area of the South China Sea. However, there are still some problems that require further exploration, such as the effects of the length, width, depth, Coriolis force and friction of the SCSB on the GOT, which will be the focus of subsequent studies.

Code availability

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Code availability.

In this paper, we use the Princeton Ocean Model (POM), which is available online at http://www.ccpo.odu.edu/POMWEB/ (Mellor, 2002).

Data availability

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Data availability.

The ETOPO1 data (https://doi.org/10.7289/V5C8276M) came from NGDC, NOAA (https://www.ngdc.noaa.gov/mgg/global/, Amante and Eakins, 2009). The tidal data at the open boundaries came from the Oregon State University (ftp://ftp.oce.orst.edu/dist/tides, Egbert and Erofeeva, 2002.)

Author contributions

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Author contributions.

XC modified and ran the POM model, solved the analytical model, and prepared the draft of the manuscript. GF designed the method for numerically computing response functions and the configuration of the analytical model, and modified the manuscript. DW examined the derivation of the analytical solution.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This study was supported by the National Key Research and Development Program
of China (2017YFC1404200), the NSFC-Shandong Joint Fund for Marine Science
Research Centers (grant no. U1406404) and the Basic Scientific Fund for
National Public Research Institutes of China (grant no. 2015G02). The authors
sincerely thank the topical editor, Neil Wells, for handling our manuscript.
We also sincerely thank two referees for reviewing our manuscript. In
particular, David Webb thoroughly reviewed our manuscript and provided many
useful comments and suggestions, which were of great help in improving our
study.

Edited by: Neil Wells

Reviewed by: David Webb and one anonymous referee

References

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Short summary

The resonant period of the Gulf of Thailand is potentially close to 1 day. However, the classic quarter-wavelength resonant theory fails to give a diurnal resonant period. In this study, we first perform a series of numerical experiments showing that the resonance of the South China Sea body has a critical impact on the resonance of the gulf. An idealised two-channel model that can reasonably explain the dynamics of the tidal resonance in the Gulf of Thailand is then established in this study.

The resonant period of the Gulf of Thailand is potentially close to 1 day. However, the classic...

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