Tides represent periodic fluctuations in the level of ocean waters, occurring at distinct frequencies, due to the combined gravitational effects of the Moon and the Sun. Given the continual variation in the relative positions of the Earth, Moon, and Sun, the tidal forces exhibit periodic changes, leading to the manifestation of two tidal events daily. Furthermore, observational data collected from different geographical locations on Earth will give distinct tidal profiles, reflecting the complexity and variability of tidal patterns.

In the conducted experiment, the mangrove population and soil samples were sourced from the Qi'ao-Dangan Island Nature Reserve located in Zhuhai, Guangdong Province (longitude: 113.635706°, latitude: 22.414662°), and an observation station with tidal and other hydrological data was established near this location. Consequently, the proposed model takes this specific location as a representative case and uses least squares approximation fitting methodology to conduct a harmonic analysis of the tidal parameters, and find the solution of the amplitude and the tidal epoch of each tidal constituent.

For a specific location or monitoring station, the water level at a given time can be mathematically approximated as a superposition of several individual simple harmonic vibrational components, known as tidal constituent, which are characterized by the cosine function. The water level at this point can be formulated using the following mathematical expression: $$ Z(t) = S_0 + \sum_{j=1}^{N}[H_jcos(\sigma_jt-g_j)] \tag{1}$$

where $Z(t)$ represents the water level recorded at the observation point at time $t$ , \(S_0\) represents the mean sea level height under the initial state, $H_j$ represents the amplitude of the tidal constituent, $sigma_j$ represents the angular velocity of the tidal constituent, $g_j$ represents the tidal epoch of the tidal constituent, and $N$ represents the total number of tidal constituents.

The expression is collated with the custom variables $a_j$ and $b_j$ : $$Z(t) = S_0 + \sum_{j=1}^{N}(a_jcos\sigma_jt+b_jsin\sigma_jt) \tag{2}$$

However, since the location we analyzed is close to the Pearl River Estuary, it is greatly affected by the runoff of the Pearl River. Consequently, pertinent parameters, including the amplitude and phase lag of each constituent wavelet, exhibit temporal variability in response to the fluctuating discharge. Therefore, when incorporating the impact of the runoff, the modified expression for water level dynamics is delineated as follows: $$Z(t) = S(t) + \sum_{j=1}^{N}(a_j(t)cos\sigma_jt+b_j(t)sin\sigma_jt) \tag{3}$$

In the process of solving equation $(3)$ , we adopt an independent point scheme. For the $j$ -th tidal constituent, several independent points are uniformly selected in the known time series, in which the $i$-th independent point is denoted as ($S_i$ , $a_{i,j}$ , $b_{i,j}$). These independent point values are ascertained through the application of interpolation techniques. The variation of mean sea level height and harmonic variable over time follows this expression: $$S(t) = \sum_{i=1}^{N_1}f_{i,t}S_i, \space a_j(t) = \sum_{i=1}^{N_2}f_{i,t}a_{i,j}, \space b_j(t) = \sum_{i=1}^{N_2}f_{i,t}b_{i,j} \tag{4}$$

where $f_{i,t}$ represents the interpolation weight of the selected $i$-th independent point at time $t$, $N_1$ represents the number of independent points selected for mean sea level, and $N_2$ represents the number of independent points selected for tidal constituent. By substituting equation (4) into equation (3), we get: $$Z(t)=\sum_{i=1}^{N_1}f_{i,t} S_i + \sum_{j=1}^N(\sum_{i=1}^{N_2}f_{i,t} a_{i,j} cos\sigma_j t+\sum_{i=1}^{N_2}f_{i,t} b_{i,j} sin\sigma_j t) \tag{5}$$

For ease of calculation, given T different time points of water level observations, the series of equations in (5) can be represented in the form of elementary transformations of a matrix, that is: $$\textbf{Z} = \textbf{KU} \tag{6} $$

Where $\textbf{Z}$ represents the matrix of water level observations, $\textbf{K}$ represents the coefficient matrix, and $\textbf{U}$ represents the parameter matrix of substitution. The specific expression of each matrix is as follows: $$ \textbf{Z} = \left\lbrack Z\left( t_{1} \right),Z\left( t_{2} \right),\cdots,Z\left( t_{T} \right) \right\rbrack^{T} \tag{7} $$ $$ \textbf{K} = \begin{pmatrix} f_{1,t_1} & \cdots & f_{N_1,t_1} & f_{1,t_1}cos\sigma_1t_1 & \cdots & f_{1,N_2}sin\sigma_Nt_1 \\ f_{1,t_2} & \cdots & f_{N_1,t_2} & f_{1,t_2}cos\sigma_1t_2 & \cdots & f_{1,N_2}sin\sigma_Nt_2 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ f_{1,t_T} & \cdots & f_{N_1,t_T} & f_{1,t_T}cos\sigma_1t_T & \cdots & f_{1,N_2}sin\sigma_Nt_T \\ \tag{8} \end{pmatrix} $$ $$ \scriptsize \textbf{U} = \begin{pmatrix} S_1 & \cdots & S_{N_1} & a_{1,1} & \cdots & a_{N_1,1} & \cdots & a_{1,N} & \cdots & a_{N_1,N} & b_{1,1} & \cdots & b_{N_1,1} & \cdots b_{1,N} & \cdots & b_{N_1,N} \end{pmatrix}^T \tag{9} $$

The optimal solution of the matrix obtained by least square method is: $$ \textbf{U} = (\textbf{K}^T\textbf{K})^{-1}\textbf{K}^T\textbf{Z} \tag{10}$$

The tide table data from 2024/06/01 to 2024/06/30 of Qi'ao Island Observation Station in Zhuhai City (data source: National Marine Science Data Center) were collected and counted, and a mathematical model was established for the harmonic analysis of tides. The following table shows the prediction results of the height and phase of each tidal component on Qi'ao Island under 95% confidence interval:

Based on tidal harmonic analysis, we have a visual understanding of the tidal conditions in our environment. In addition to the laboratory, we also hope that PAO1 and rhodopseudomonas will continue to work effectively in the real environment. However, due to the complexity and variability of the actual Marine environment, the tides will also affect the distribution of microorganisms, so that they can not be completely fixed in the mangrove roots and work, thus affecting their degradation efficiency of microplastics to a certain extent.

The model analysis is based on the following assumptions:

(1) Tides and currents are regarded as incompressible fluids;

(2) For ease of solution, microorganisms are regarded as rods of equal length. (1μm×1μm× 3.25μm)

The local scour flow is three-dimensional unsteady flow, which can be simulated well by continuity equation and Navier-Stokes equation: $$ \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho V)=0 \tag{11}$$ $$\rho(\frac{\partial V}{\partial t}+V\cdot\nabla V)=-\nabla P+\rho g+\nabla\mathrm{T} \tag{12}$$

Where $\rho$ represents the density of the fluid, $V$ represents the velocity field of the fluid, $P$ represents the pressure of the fluid, $g$ represents the acceleration of gravity, and $T$ represents the viscous force.

The equation is converted into integral form as follows: $$\frac{\partial}{\partial t}\int_{V}\rho\mathrm{dV}+\oint_{\partial V}\rho(\boldsymbol{v}\cdot\boldsymbol{n})\mathrm{dS}=0 \tag{13}$$ $$\begin{aligned}\frac{\partial}{\partial t}\int_{V}\rho\boldsymbol{v}\mathrm{dV}+ \oint_{\partial V}\rho\boldsymbol{v}(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{n})\mathrm{dS}= \int_{V}\rho\boldsymbol{f}_{e}\mathrm{dV}-\ \oint_{\partial V}p\boldsymbol{\cdot}\boldsymbol{n}\mathrm{dS}+ \oint_{\partial V}(\overline{\tau}^{\prime}\boldsymbol{\cdot}\boldsymbol{n})\mathrm{dS} \end{aligned} \tag{14}$$

The three-dimensional finite volume method is used to segment and solve the tides, and the results are shown in the figure: