Energy Migration and Groundwater Response to Irregular Wave Forcing in Coastal Aquifers: A Spectral and Wavelet Analysis

Abstract

In recent years, the irregular wave characteristics of ocean dynamics have often been overlooked in the study of the driving mechanism of groundwater movement in coastal aquifers. To clarify the propagation mechanisms of groundwater fluctuations driven by irregular waves in beach aquifers, we employed spectral analysis based on numerical simulations to examine the energy migration processes and evolution characteristics of wave signals at different frequencies. It elucidates the response mechanism of groundwater movement characteristics (head, velocity) to irregular waves in the sea. The energy density in the low-frequency region is enhanced compared to the incident wave and continuously increases in the direction away from the sea within the aquifer. The wavelet power corresponding to the 1/2 spectral peak frequency is significantly enhanced. The energy density in the high-frequency region is generally weaker than that of the incident waves, and the wavelet power corresponding to double spectral peak frequency is enhanced. The correlation between incident waves and groundwater fluctuations is highest near the spectral peak period. This study addresses some problems in modeling surface water–groundwater interactions under irregular wave conditions and provides a theoretical reference for investigating the impacts of extreme climate events (such as typhoon waves and low-frequency offshore oscillations generated by storm surges) on seawater intrusion into coastal groundwater systems.

1. Introduction

Periodic oscillations on the ocean surface directly lead to periodic fluctuations in the groundwater level of aquifers [1,2]. Previous studies have focused on the effects of tides on the mass flux and groundwater level of beach aquifers through field observations, indoor experiments, and numerical simulations [3,4,5,6]. Hegge and Masselink [7] found that the groundwater level changes caused by waves exhibit asymmetry similar to those caused by tides, both of which reflect the characteristics of rapid rise and gradual decline. Some studies have also begun to observe the effects of infragravity waves and related free long waves on other oceanic long-period oscillations. Cartwright et al. [8] chose a typical long wave of 348 s as the ocean dynamic condition and investigated the wave evolution process inside the beach through experiments. It was found that due to the nonlinear filtering effect of the beach’s tilted boundary, low-frequency oscillations generated higher harmonics in the aquifer. This is consistent with the conclusions drawn by Nielsen [5] and Raubenheimer et al. [9] through analytical derivation of tidal effects. The low-frequency peaks in the energy spectrum of groundwater fluctuations have been proven to be the result of nonlinear interactions between ocean long or short waves at beach surface, exhibiting low-frequency dynamics within aquifers (Nielsen [5]). Similarly, Holman [10] and Elfrink and Baldock [11] pointed out that the interaction between continuous flows leads to a merging of swash periods on the beach surface and the enhancement of low-frequency energy in the aquifer. Swash due to larger incident waves may catch the swash from preceding smaller waves, or the backwash from a large uprush event may prevent the uprush of later smaller waves [12]. In addition, Waddell [13] observed standing waves at the bottom of the beach, with a correlation between their spectral peak frequencies and those in the aquifer. Low-frequency waves directly affect the groundwater level, and the position of the groundwater exit point changes the energy distribution along the coast in turn. When the exit point is close to the open sea, the peak energy of the spectrum is relatively small [14]. Cartwright et al. [14] also found that the magnitudes of pressure waves driven by swash exhibit exponential decay when they propagate from the beach boundary into the aquifer. Zheng et al. [15] and Zheng et al. [16], respectively, demonstrated through solitary wave and periodic wave experiments that zero-phase lag occurs for head changes at different cross-shore positions, leading to a standing wave oscillation mode. This can be attributed to changes in the seepage face and exit point during the backwash process.

Previous numerical simulations and experiments on wave-driven groundwater fluctuation characteristics have typically focused on limited locations near the beach surface [17,18]. Some studies only consider the pore pressure gradient near the beach surface to illustrate its impact on the morphological changes of the beach surface [19,20], making it impossible to explore the response characteristics of groundwater fluctuations to waves on the broader scale of beach aquifers. Yang et al. [21] investigated the spatiotemporal distribution characteristics of total head and local water surface elevation throughout the entire beach aquifer through indoor experiments, but the wave conditions were limited to solitary waves and did not reflect the wave characteristics of irregular waves. In recent years, some studies have begun to use Fourier transform and other methods for spectral analysis of the fluctuation characteristics of groundwater movement characteristics (water head, velocity, discharge, salinity, etc.) with tidal oscillations [22]. Yu et al. [22] found through their study on the impact of different tidal constituent combinations on seawater intrusion that the interaction of high-frequency tidal components in aquifers produces low-frequency oscillations, thereby enhancing seawater intrusion. Will the interaction of high-frequency waves in aquifers also produce low-frequency oscillations? In addition, Robinson et al. [23] displayed that under the strong wave action caused by storms, seawater intrusion is significantly enhanced, and chemical substances flood into the aquifer in large quantities. Waves enhance the mixing of salt and fresh water, and also alter the geochemical conditions of aquifers. The migration of wave energy (such as the strengthening of low-frequency energy) may lead to further enhancement or weakening of these phenomena and may have long-term impacts on the groundwater environment near the coast. Therefore, this study uses FUNWAVE-TVD and spectral analysis to explore the energy migration process and evolution characteristics of wave signals at different frequencies after irregular waves are introduced into a beach aquifer. The response mechanism of groundwater movement characteristics (water head, velocity) to irregular waves is elucidated. The study of the propagation mechanism of irregular waves in aquifers provides a theoretical reference for investigating the impacts of extreme climate events (such as typhoon waves and low-frequency offshore oscillations generated by storm surges) on seawater intrusion into coastal groundwater systems and is significant in that it will be able to guide the layout of groundwater extraction wells (such as avoiding strong energy migration zones).

Firstly, the calculation results of FUNWAVE-TVD are used as the ocean boundary conditions for SUTRA-MS, and the groundwater movement characteristics (water head, velocity) at each measurement point are calculated. Afterwards, using Fourier transform, the differences between the characteristics of beach groundwater movement and the energy density spectrum distribution characteristics of irregular waves in the sea are explained. Through wavelet transform, we reveal the differences and correlations of the temporal distribution of wavelet power between groundwater movement and the incident waves.

2. Method

2.1. Governing Equation

FUNWAVE-TVD is a total variation decreasing (TVD) version developed by Shi et al. [24] based on the fully nonlinear Boussinesq wave model (FUNWAVE). The conservation form of the FUNWAVE-TVD governing equation is given by the following equation:

$${\eta }_t+\nabla ·\mathit{M}=0$$

(1)

$${\mathit{u}}_{\alpha ,t}+\left({\mathit{u}}_{\alpha }·\nabla \right){\mathit{u}}_{\alpha }+g\nabla \eta +{\mathit{V}}_1+{\mathit{V}}_2+{\mathit{V}}_3+\mathit{R}=0$$

(2)

where

$$\mathit{M}=\left(h+\eta \right)\left({\mathit{u}}_{\alpha }+{\overline{\mathit{u}}}_2\right)$$

(3)

$\mathit{M}$ is the mass flux; $h$ is the still water depth; η is the free surface water level; ${\mathit{u}}_{\alpha }$ is the horizontal velocity vector at reference elevation $z_{\alpha }$; the subscript t represents the partial derivative of time; and ${\overline{\mathit{u}}}_2$ is the vertical velocity correction term after averaging the water depth, which can be expressed as:

$${\overline{\mathit{u}}}_2=\left[{\displaystyle \frac{z_{\alpha }^2}{2}}-{\displaystyle \frac{1}{6}}\left(h^2-h\eta +{\eta }^2\right)\right]\nabla B+\left[z_{\alpha }+{\displaystyle \frac{1}{2}}\left(h-\eta \right)\right]\nabla A$$

(4)

where

$$A=\nabla ·\left(h{\mathit{u}}_{\alpha }\right)$$

(5)

$$B=\nabla ·{\mathit{u}}_{\alpha }$$

(6)

g is the gravitational acceleration; $\mathit{R}$ is the dissipation term, including bottom friction and sub-grid turbulent mixing; and ${\mathit{V}}_1$ and ${\mathit{V}}_2$ are the dispersion terms, which can be written as:

$${\mathit{V}}_1={\left({\displaystyle \frac{z_{\alpha }^2}{2}}\nabla B+z_{\alpha }\nabla A\right)}_t-\nabla \left({\displaystyle \frac{{\eta }^2}{2}}{B}_t+\eta A_t\right)$$

(7)

$${\mathit{V}}_2=\nabla \left[\left(z_{\alpha }-\eta \right)\left({\mathit{u}}_{\alpha }·\nabla \right)A+{\displaystyle \frac{1}{2}}\left(z_{\alpha }^2-{\eta }^2\right)\left({\mathit{u}}_{\alpha }·\nabla \right)B+{\displaystyle \frac{1}{2}}{\left(A+\eta B\right)}^2\right]$$

(8)

${\mathit{V}}_3$ can be written as:

$${\mathit{V}}_3={\omega }_0{\mathit{i}}^z\times {\overline{\mathit{u}}}_2+{\omega }_2{\mathit{i}}^z\times {\mathit{u}}_{\alpha }$$

(9)

where

$${\omega }_0=\left(\nabla \times {\mathit{u}}_{\alpha }\right)·{\mathit{i}}^z$$

(10)

$${\omega }_2=\left(\nabla \times {\mathit{u}}_2\right)·{\mathit{i}}^z$$

(11)

and where ${\mathit{i}}^z$ is the unit vector in the z-direction.

We established a groundwater movement model using a variable saturation, density, and multi-component solute transport model SUTRA-MS [25], with the governing equation consistent with Chen et al. [26].

2.2. Boundary Conditions

The slope of the beach is 1:10. The size and range of the model are consistent with the linear beach in Chen et al. [26], as shown in Figure 1. With no precipitation or evapotranspiration considered, the permanently exposed surface BC was set to be no-flow [27,28]. DE was also set to be no-flow [27,28,29,30]. AB was set as the inland water head boundary. Three simulations were conducted with varied inland water heads—i.e., low water head = 9 m, middle water head = 10 m, and high water head = 11 m. CDE was assigned a specified pressure boundary based on wave signal. In Figure 1, WSI represents wave-driven saltwater infiltration.

This article uses the JONSWAP (Joint North Sea Wave Project) spectrum to drive wave generation. The model domain was divided into 480 grids along the x-direction (AD) (Figure 2), each with a size of 0.25 m. Point B is the foot of the slope, corresponding to point D in Figure 1. Point C is the top of the inclined beach surface, corresponding to point C in Figure 2. Point A is the position of the wavemaker, located 50 m from the foot of the slope (AB). The water depth at the flat bottom is 4 m, which is greater than that of Xin et al. [31] and Robinson et al. [23]. Significant wave height $H_{1/3}$ was set to 0.5 m, and the low-frequency cutoff and high-frequency cutoff were set to 0.008 Hz and 1 Hz [32]. The CFL (Courant–Friedrichs–Lewy number) value was set to 0.1. The peak period of the spectrum was set to 10 s. Many previous studies using the JONSWAP spectrum for nearshore waves have observed or set the spectral peak period to 10 s or close to 10 s [32,33,34,35,36]. The peak enhancement parameter $\gamma$ was set to 3.3, which is the mean value of the JONSWAP spectrum [37]. The model simulated water level fluctuations for 1000 s under quasi-steady-state conditions, and output instantaneous water level results for all grid points every 0.5 s (close to the output step size of 0.1 s in Xin et al. [31]). Afterwards, the water level results at each time point in 1000 s were cyclically input into SUTRA-MS for groundwater simulation. The numerical model of the groundwater continues to read the instantaneous water level calculated by FUNWAVE-TVD as a boundary condition after fluctuations have started and are operating in a quasi-steady state.

In this section, 50 measurement points were set in the trapezoidal area of CDEF (FE = 70 m) (Figure 3). The distance between different measurement points is 6 m. The distance between the first measurement point and CF is 5 m, and the distance between measurement points in the same column is equal. In order to reduce workload, 15 representative measurement points were selected in the following analysis (blue box in Figure 3). Three wave gauges were set up at equal intervals, with the first station located 10 m away from the shoreline. The interval between each gauge is 10 m. (Figure 3). All the parameter values used in this section are listed in

Table 1. Parameters for the generalized models.

Parameter	Symbol	Unit	Value
Hydraulic conductivity	$K_{\mathrm{s}}$	m/s	$1.16\times {10}^{-4}$
Porosity	$\varphi$	-	0.4
Longitudinal dispersivity	${\alpha }_{\mathrm{L}}$	m2	0.5
Transverse dispersivity	${\alpha }_{\mathrm{T}}$	m2	0.05
Molecular diffusion	$D_0$	m2/s	$1\times {10}^{-9}$
Residual water saturation	$S_{wres}$	-	0.1
Pore size distribution index	n	-	2.68
Inverse of air entry suction parameter	a	1/m	14.5
Seawater density	${\rho }_{\mathrm{s}}$	kg/m3	1025
Freshwater density	${\rho }_0$	kg/m3	1000
Seawater concentration	$C_{\mathrm{s}}$	ppt	35
Freshwater concentration	$C_{\mathrm{f}}$	ppt	0.01
Beach slope	Sb	-	0.1

, and the values of porous media parameters are basically the same as Robinson et al. [29,30].

2.3. Model Validation

In order to verify the reliability of the combined use of the wave model FUNWAVE-TVD and the groundwater model SUTRA-MS, this paper cites the experimental results of Yang et al. [21] on the impact of solitary waves on groundwater movement for verification. The flume is 35 m long, 0.8 m high, and 0.2 m wide. A piston-type wavemaker is installed at one end of the flume to generate irregular waves (Figure 4a). The beach slope is 1:10, with the toe of the beach slope located 14 m from the initial position of the wavemaker. The initial water depth was set to 30 cm. The origin is defined as the intersection of the still water level and the beach surface, with the x-axis positive landward and the z-axis positive upward. Unshaded circles in the beach aquifer indicate the positions of pressure sensors (Figure 4b).

In groundwater dynamics, the total head is the sum of the positional head and the pore-water pressure head. Compared to groundwater level, total head can better reflect the characteristics of groundwater dynamics. From Figure 5, it can be seen that the calculated values during the rising stage of the total head are in good agreement with the experimental values. The maximum value of the total head is almost the same, while the calculated values during the falling stage of the total head are different from the experimental values. This may be mainly due to certain differences between the selection of soil V-G param [38] and the experimental soil and hysteresis effects, resulting in a delay in the withdrawal process.

The eight measuring points in Figure 5 are all located at different horizontal positions 5 cm above the bottom of the flume. It can be clearly observed from the distribution of experimental and calculated values that the response characteristics of the total head at different horizontal positions of measurement points are different when solitary waves are introduced into the beach aquifer. Firstly, it can be observed that the total head fluctuation period gradually increases in the direction away from the sea, and the magnitude of the increase becomes significant. In response to this phenomenon, this article set the measurement point closest to the ocean boundary (x = −90 cm, z = 5 cm, 5 cm refers to the distance from the bottom) as the origin, and set the distance between other measurement points and the origin as L. The L of each measurement point was fitted with the total head fluctuation period to analyze the relationship. The results indicate that L has an exponential relationship with the total head fluctuation period, and the goodness of fit R² reached 0.99975 (Figure 6). The farther away from the sea, the greater the fluctuation period of the total head, and the growth rate also increases. In addition, as L increases, the asymmetry of total head fluctuation also strengthens. At points (x = 110 cm, z = 5 cm) and (x = 130 cm, z = 5 cm), the duration of the head retreat is much longer than the duration of the rise.

2.4. Wavelet Transform

In wavelet transform, the approximation of the original function using wavelet functions is employed to identify localized frequency at specific time instants in a time series [39]. Similar to Fourier transform, the continuous wavelet transform of a time series x(t) can be expressed as:

$$W_x\left(a,\tau \right)={\displaystyle \frac{1}{\sqrt{a}}}{\int }_{-\infty }^{+\infty }x\left(t\right){\psi }_{a,\tau }^{*}\left(t\right)dt$$

(12)

$${\psi }_{a,\tau }={\displaystyle \frac{1}{\sqrt{a}}}\psi \left({\displaystyle \frac{t-\tau }{a}}\right)$$

(13)

where ${\psi }_{a,\tau }^{*}$ denotes a set of wavelet functions and is the complex conjugate of ${\psi }_{a,\tau }$. a is the scaling parameter. When a increases, the wavelet broadens the signal window it covers, reflecting the overall characteristics (low-frequency) of the time series. Conversely, when a decreases, the signal window narrows, allowing the study of local variations (high-frequency) in the time series. τ is the translation parameter.

Cross-wavelet analysis, built upon continuous wavelet analysis, is a tool used to analyze significant periods and phase relationships between two time series. The cross-wavelet transform of two time series can be defined as the wavelet coefficient matrix $W_x$ of time series $x\left(t\right)$ multiplied by the complex conjugate matrix $W_y^{\ast }$ of the wavelet coefficient matrix $W_y$ of time series $y\left(t\right)$, yielding the cross-wavelet coefficient matrix W_xy for series $x\left(t\right)$ and $y\left(t\right)$. Its specific form is:

$$W_{xy}\left(\tau ,a\right)=\left(W_x\left(\tau ,a\right)W_y^{\ast }\left(\tau ,a\right)\right)$$

(14)

The cross-wavelet transform corresponding to time series $x\left(t\right)$ and $y\left(t\right)$ is defined as:

$$D\left({\displaystyle \frac{\left|W_x(\tau ,a)W_y^{\ast }(\tau ,a)\right|}{{\sigma }^X{\sigma }^Y}}<p\right)={\displaystyle \frac{Z_v\left(p\right)}{v}}\sqrt{P_k^X{P}_k^Y}$$

(15)

where $Z_v\left(p\right)$ is the confidence level associated with the probability p with the chi-square probability density function with degree of freedom $v$, i.e., $f_v\left(z\right)$. In other words, $p={\int }_0^{Z_v\left(p\right)}{f}_v\left(z\right)dz$ [40]. $P_k^X$ and $P_k^Y$ are the Fourier spectra for time series $x\left(t\right)$ and $y\left(t\right)$ at scale k. ${\sigma }^X$ and ${\sigma }^Y$ are the standard deviations of respective time series.

Cross-wavelet phase angle can be used to estimate the phase difference between two time series (which can be understood as the lag of event occurrence), including:

$$A^{XY}=\mathrm{a}\mathrm{r}\mathrm{c}\tan \left({\displaystyle \frac{i_{\mathrm{m}\mathrm{a}\mathrm{g}}\left\{W_{xy}\right\}}{r_{\mathrm{e}\mathrm{a}\mathrm{l}}\left\{W_{xy}\right\}}}\right)$$

(16)

where $i_{mag}\left\{W_{xy}\right\}$ and $r_{eal}\left\{W_{xy}\right\}$ represent the imaginary and real parts, respectively. The average phase angle can be expressed as:

$$\overline{A^{XY}}=\mathrm{a}\mathrm{r}\mathrm{g}\left[\sum _1^n\cos a_i,\sum _1^n\sin a_i\right]$$

(17)

where $a_i$ is the phase angle.

3. Results

3.1. Variations in Energy Density Distribution

The energy density is obtained by dividing the square of the magnitude obtained by Fourier transform by half of the total sample size and by the sampling frequency. In this paper, the total sample size is 2000 and the sampling frequency is 2 Hz. This section distinguishes high-frequency (f > 0.05 Hz) and low-frequency (f < 0.05 Hz) fluctuations by 0.05 Hz [41]. From the results (Figure 7, Figures S1 and S2 of the Supplementary Materials (SM)), it can be observed that the energy density distribution curves of groundwater fluctuation characteristics (water level, velocity) in three cases are basically overlapping. This indicates that different inland water heads do not alter the response mechanism of groundwater fluctuation to irregular waves. Therefore, we proceed to investigate the middle water head (10 m) boundary condition.

Fourier transform was introduced to reflect the response of groundwater head fluctuations in beach aquifers to wave action, as shown in Figure 8. It can be seen that there is a difference in the energy density distribution between the incident wave (water level at gauge 1) and the water head at each measuring point. From Figure 8, it is evident that compared to incident waves, the energy density of groundwater head in the low-frequency region (as shown in the light-green area in Figure 8(a1)) at each measuring point is generally enhanced. Specifically, first of all, from measuring points near the inland boundary to measuring points near the ocean boundary, the increment of energy density in the low-frequency region gradually decreases. When the position of measuring point 41 is reached, the increment is basically zero, and the phenomenon of increased energy density compared to incident waves effectively disappears. The enhancement of energy density in the low-frequency region is more sensitive to changes in the x-direction and less sensitive to changes in the z-direction. From a quantitative perspective, the energy density corresponding to the spectral peak period (10 s) is always the highest at all measurement points. The energy density corresponding to the 20 s period of water head from point 1 to point 25 significantly increases, only second to the peak period.

In addition, compared to the incident wave, the energy density of water head in the high-frequency region (as shown in Figure 8(a1), flesh-colored area) at each measuring point has generally weakened, especially in areas above 0.1 Hz. In the z-direction, the weakening of energy density of measurement points far from the beach surface is more pronounced. It can be seen that the magnitude and fluctuation of the energy density of the water head at point 41 are weakened to a very small extent between 0.4 and 1 Hz (light-gray area in Figure 8(e1)), indicating that the head fluctuation in this frequency range has almost disappeared. This can be explained by short-wave energy being more easily filtered by sand than long-wave energy [42]. The attenuation of high-frequency energy near inland measurement points may also be due to energy transfer caused by the evolution of high-frequency fluctuations towards low-frequency. Yu et al. [22] found through their study on the impact of different tidal component combinations on seawater intrusion that the interaction of high-frequency tidal components in aquifers produces low-frequency oscillations. This study indicates that the interaction between high-frequency waves also generates low-frequency oscillations. Tides and waves are essentially periodic fluctuations in the ocean, and they may follow the same principle. From the perspective of energy conservation, a portion of the attenuated high-frequency energy is converted into other forms of energy and dissipated into the aquifer, while a portion is transferred to the low frequency to promote an increase in low-frequency energy.

For the x- and z-direction flow velocities at each measurement point of the aquifer, the distribution and pattern of change of energy density at different frequencies are similar to that of water head (Figure 9 and Figure 10). The difference is that the energy densities of the x- and z-direction velocities are significantly weaker than the incident waves in the low-frequency region at the measurement points near the ocean boundary. This may be due to the fact that the wave-generated groundwater fluctuations are gradually weakened over the entire frequency range when they travel from the swash zone to the aquifer interior, but the merging of the swash and the prolongation of the internal beach drainage process result in an increase in the low-frequency energy of the groundwater fluctuations at the points far from the offshore boundary. The increase in the low-frequency energy is sufficient to offset the weakening caused by the sand filtration. For sites close to the ocean boundary, x- and z-direction velocity fluctuations at these sites show reduced energy in the low-frequency region because the extended drainage process contributes little to the energy enhancement in the low-frequency region, and the swash merging does not contribute enough to the energy enhancement in the low-frequency region to offset attenuation due to sand filtration. In addition, at the points in bottom and middle layers near the offshore boundary (point 31, point 33, point 41, point 43), there is an enhancement of the energy density between 0.1 and 0.3 Hz (e.g., gray area in Figure 9(e2)). The reason for this is partly due to the interaction of spectral peak period fluctuations in the propagation process in the aquifer, generating multiplicative fluctuations (0.2 Hz), and partly due to the interaction of higher-frequency fluctuations (0.4~1 Hz), generating low-frequency fluctuations.

3.2. Variations in Wavelet Power Distribution

Continuous wavelet analysis was used in Figures S3 and S5 of the Supplementary Materials to reflect the time–frequency distribution differences between the water head and flow velocity at various measurement points and the signals of incident waves. It can be observed that the results of the wavelet power spectrum from points 1 to 15 near the inland boundary are very similar, and the head and z-direction flow velocity of each measurement point are also very similar (Figure S3 of the Supplementary Materials). The incident wave exhibits a coherent and stable bright band (deep red) near the spectral peak period of 10 s, located within a solid black line. This is the region with the highest wavelet power. The solid black line represents the area that has passed the significance test with a 95% confidence level. High stability is demonstrated by the fact that the bright band does not exhibit discontinuities, deletions, or significant changes in width over time. In the results for water head and flow velocity from point 1 to point 15, there are stable bright bands around the period of 10 s and 20 s, and the color of the bright band near 20 s is darker than that near 10 s, indicating the presence of higher wavelet power near 20 s. In addition, there are relatively continuous dark regions around the period of 125 s, but their stability is not as good as the bright bands of 10 s and 20 s, and they did not pass the significance test. As a whole, it can be observed that the low-frequency region of water head and flow velocity (with a period greater than 20 s) has a high-power region connected to the incident wave, indicating an overall enhancement of low-frequency energy from point 1 to point 15. On the contrary, for regions with a period less than 10 s, there is a high-power region in the incident wave that has passed the significance test near the period of 1 s to 6 s, exhibiting a temporal alternating distribution. In the water head and velocity spectrum, this area has significantly weakened and almost disappeared, indicating a decrease in the overall high-frequency energy at points 1 to 15.

The results of the power spectrum from points 21 to 35 are much more chaotic than those near inland measurement points (Figure S4 of the Supplementary Materials). It can be seen that there are significant differences in the time–frequency distribution of wavelet power among different measurement points, and the results of water head and flow velocity at the same measurement point are also significantly different. Compared to measuring points closer to land, the fragmentation degree of the bright area in the velocity spectrum is greatly deepened, while the stability and coherence of the head spectrum are relatively good. Another significant change is that the stable bright band around the period of 20 s has basically disappeared. Overall, in the results from points 21 to 25, bright areas can still be found in the low-frequency region. However, in the column further away from the inland boundary from points 31 to 35, the bright areas in the low-frequency region completely disappear. This indicates that the phenomenon of low-frequency energy enhancement does not exist anywhere in the beach, but rather has the characteristic of gradually weakening and disappearing from inland to offshore, which is consistent with the results of Fourier transform. In addition, it can be observed that there is a higher-power region with a denser distribution and wider range in the high-frequency region near the surface measurement points, because the high-frequency energy of the surface measurement points is relatively less weakened compared to the bottom layer. For points 41 to 45, which are farthest from land, except for point 45 on the surface, there are still stable and continuous bands in the head power spectrum at points 41 and 43 around 10 s. There are bright bands in the velocity power spectrum around 10 s, but they do not pass the significance test (Figure S5 of the Supplementary Materials). Overall, compared to incident waves, the wavelet power in the low-frequency region with a period of more than 20 s is close to zero. The continuity of the high-power region in the period of 10 s to 20 s is greatly reduced, and the degree of fragmentation is deepened. The range and density of the high-power area in the high-frequency region within the period of 10 s in the flow velocity power spectrum have increased.

3.3. Variations in Cross-Wavelet Power Distribution

Cross-wavelet analysis is a tool used on the basis of continuous wavelet analysis to analyze the significant period, phase relationship, and lag time of two time series. The arrows in the cross-power spectrum represent the phase angle between two time series in cross-wavelet analysis, which can be divided into four quadrants, as shown in Figure 11. This article conducts cross-wavelet analysis on the time series of water head and flow velocity at each measuring point, respectively, with incident waves (Figure 12, Figure 13 and Figure 14). For the measurement points near the inland boundary (points 1–15), it can be clearly observed that there is a stable and continuous bright band around the 10 s spectral peak period, which passed the significance test. During the 3–8 s period, there were also regions that passed the significance test, but they were not continuous and sparsely distributed, only showing significant relationships at certain moments. There is a twisted continuous bright band near the 20 s period, but it did not pass the significance test. Compared with Figure 11, it can be observed that the phase angles in the bright bands at points 1 to 15 are all located in the second and fourth quadrants. The phase angle of the head and the z-direction flow velocity is located in the second quadrant. Therefore, it is negatively correlated with the incident wave and lags behind the incident wave. The phase angle of the x-direction flow velocity is located in the fourth quadrant. Therefore, it is positively correlated with the incident wave and lags behind the incident wave.

For measurement points far from the inland boundary (points 21–35), the phase angle is mostly located in the second and fourth quadrants. In some measurement point distribution results, the continuous bright bands around 10 s have become discontinuous and fragmented, including the flow velocity in the x-direction at point 23, the flow velocity in the z-direction at point 25, the flow velocity in the x-direction at point 31, the flow velocity in the x-direction at point 33, and the flow velocity in the x-direction at point 35. Another characteristic of these results is that the phase angle points randomly, alternating between the same and opposite directions. This means that although they have intermittent bright bands around 10 s and pass the significance test, there is actually no significant correlation. In the third column (points 21 to 25), the change in head is negatively correlated with the incident wave, while the z-direction change in flow velocity is positively correlated with the incident wave. On the contrary, in the fourth column (points 31 to 35), the change in head is positively correlated with the incident wave, while the z-direction change in flow velocity is negatively correlated with the incident wave. The phase angles in the fifth column (points 41 to 45) are all located in the second and fourth quadrants. The change in head is negatively correlated with the incident wave, while the change in flow velocity is positively correlated with the incident wave. In addition, at measurement points close to the ocean boundary (points 31 to 45), the high cross-power region with intermittent alternation has significant development in the high-frequency region (period 1–8 s).

4. Discussion

4.1. The Mechanism of Low-Frequency Energy Enhancement Phenomenon Through Fourier Analysis

The energy changes in the low-frequency region in Figure 8 are reflected in two phenomena: (1) the low-frequency energy of groundwater head fluctuations in the aquifer is stronger than that of incident waves; (2) the phenomenon of low-frequency energy enhancement at measuring points far from the ocean boundary is more significant than that close to the ocean boundary. The reason why the energy density of groundwater head increases in the low-frequency region compared to incident waves may first be that gauge 1 is located outside the wave breaking point, and the energy is transferred from high frequency to low frequency after wave breaking [42,43,44]. Other possible reasons are the merging of periods caused by the interaction between continuous waves, where the larger incident waves may catch up with the earlier smaller wave [11], or the backwash from a large uprush event may prevent the uprush of later smaller waves [12]. In Figure 15a, it can be observed from the normalized wave characteristics of time variation curves of the incident wave and swash that the swash lags behind the incident wave at gauge 1 by about 6 s. Here, the fluctuation process of swash is defined as the time series of the run-up elevation on the beach surface. In the dashed blue box of Figure 15a, the two peaks in the incident wave merge or will merge into one peak in the corresponding swash after 6 s, which may reflect the merging of the swash. The second phenomenon has similar characteristics to the horizontal evolution process of solitary waves in aquifers as described in Section 2.3. In the isolated wave experiment results, as the groundwater head gradually moves away from the ocean boundary in the horizontal direction, the fluctuation period of the total head shows an exponential growth trend (Figure 15), which also provides a reference for explaining the phenomenon that the low-frequency energy of water head becomes stronger when it moves away from the sea in irregular waves. Sous et al. [42] pointed out that, when there is swash generated by the incident wave uprush and downwash on the beach surface, the groundwater flow is mainly manifested in the displacement of the groundwater level in the saturated zone and the corresponding changes in the thickness of the capillary edge. The drainage process, i.e., the significant change in saturation of porous media, is a low-frequency oscillation. When the swash reaches its peak on the beach surface, a wetting front is formed on the upper part of the water table. During the drainage process, there is an aeration zone between the wetting front and the top of the capillary, forming a vertical “saturated–unsaturated–saturated” pattern that hinders the retreat of the water table [45]. During the movement of the wetting front, the area of the aeration zone may gradually increase, and the hindering effect on the retreat of the groundwater surface is gradually enhanced. The continuous extension of the drainage process directly leads to an increase in the period of water head fluctuation, evolving from high frequency to low frequency. Therefore, the low-frequency energy enhancement of measurement points closer to the inland boundary is more significant.

4.2. The Mechanism of Low-Frequency Energy Enhancement Phenomenon Through Wavelet Analysis

Figure 16 shows the time-averaged distribution of the wavelet power spectrum in Figure S3 of the Supplementary Materials to Figure S5 of the Supplementary Materials, which provides a more intuitive quantitative analysis of the wavelet power spectrum. Here, time-averaged refers to averaging the wavelet power at each moment in 1000 s. From the graph, it can be observed that the water head and flow velocity distribution in the column closest to the inland boundary (points 1 to 5) completely overlap, and there are three obvious wavelet power peaks around the periods of 10 s, 20 s, and 125 s. For the water head and flow velocity in the aquifer, 20 s exceed the spectral peak period of the incident wave (10 s) to become the peak period of the wavelet power. This can be explained by the interaction of the spectral peak period fluctuation (0.1 Hz) during the propagation process in the aquifer, generating a half frequency fluctuation (0.05 Hz). This is similar to the findings in Yu et al. [22]. The low-frequency fluctuation peak gradually increases landward and decreases seaward. The maximum truncation period set for the JONSWAP spectrum is 125 s, and the power is below the 95% confidence level. This enhancement of the extremely low-frequency signal can be explained as an extension of the drainage process. For the high-frequency range with a period less than 10 s, the average wavelet power of water head and flow velocity is generally smaller than that of incident waves at measurement points near the inland boundary (points 1–15). In points 1 to 25, the overall wavelet power in the extremely high-frequency range with a period less than 8 s is lower than the confidence level. In addition, at the bottom and middle measurement points far from land (points 31, 33, 41, 43), in the extremely high-frequency range, the wavelet power of the flow velocity exhibits a peak higher than the confidence level near a period of about 5 s. The period 5 s is exactly half of the peak period of the spectrum. This can be explained by the interaction of peak frequency fluctuations in the propagation process of the aquifer, generating doubling frequency fluctuations (0.2 Hz). The peak of high-frequency fluctuations gradually decreases in the direction away from the ocean boundary. The reduction is more remarkable at measurement points near the ocean boundary.

5. Conclusions

We provide complex boundary conditions for a groundwater numerical model through a nearshore wave numerical model, and use spectral analysis methods such as Fourier transform and wavelet analysis to explore the response characteristics and mechanisms of groundwater fluctuations to irregular wave signals. Due to the energy transfer caused by the period merging of continuous swash on the beach surface, the energy density of groundwater fluctuation characteristics (head, velocity) in the low-frequency region is enhanced compared to the incident wave. Due to the effect of the vadose zone, the drainage process is prolonged, and the characteristic of enhanced energy density in the low-frequency region is also continuously enhanced in the direction away from the sea. The energy density in the high-frequency region is generally weakened compared to the incident waves. The fluctuations of spectral peak frequency interact with each other during the propagation process within the aquifer, resulting in a significant increase in the energy of 1/2 frequency fluctuation at positions far from the sea, and a significant increase in the energy of doubling frequency fluctuation near the ocean boundary. The cross-wavelet power of incident waves and groundwater fluctuations reaches its maximum near the spectral peak period, which means a strong correlation.