Semi-Lagrangian numerical simulation method for tides in coastal regions

Abstract

In this paper, a numerical computation method is proposed to simulate tides in coastal regions. The proposed method is based on the hyperbolic form of governing equations and employs a semi-Lagrangian scheme to ensure the accuracy and stability of numerical computations. Open and wall boundary conditions can be treated universally by combining them with the semi-Lagrangian scheme. Furthermore, the method is applied to some benchmark problems of shallow water to examine its performances in wave propagation, wave transparency through open boundaries, and tides in semi-enclosed bays. The results obtained demonstrate that the proposed method can be utilized as a practical tool to investigate tidal dynamics in coastal regions.

Appendices

Appendix 1

The mathematical treatment for obtaining the analytical solution in [17] is briefly described here.

By applying the perturbation technique, the solution of the 1D shallow water equation can be mathematically obtained under the assumption that the Froude number $F$ is small.

The initial condition is given as;

$$ \left\{ {\begin{array}{*{20}l} {h\left( {x,{\mkern 1mu} {\mkern 1mu} t = 0} \right) = H\left( x \right),} \hfill \\ {u\left( {x,\;t = 0} \right) = 0,} \hfill \\ \end{array} } \right. $$

where $H\left( x \right)$ is the water depth in the calm state. The boundary condition is;

$$ \left\{ {\begin{array}{*{20}l} {h\left( {x = 0,{\mkern 1mu} {\mkern 1mu} t} \right) = \varphi \left( t \right),} \hfill \\ {u\left( {x = L,{\mkern 1mu} {\mkern 1mu} t} \right) = 0.} \hfill \\ \end{array} } \right. $$

The variables (water column height and flow velocity) are asymptotically expanded as power series of the bookkeeping parameter F in the following manner,

$$ \left\{ \begin{aligned} h = h_{0} + Fh_{1} + F^{2} h_{2} + \cdots , \hfill \\ u = u_{0} + Fu_{1} + F^{2} u_{2} + \cdots . \hfill \\ \end{aligned} \right. $$

Substituting these perturbation expansions into the governing equations, equating the coefficients with the same powers of $F$, we have the partial differential equations with respect to the expansion coefficients $\left( {h_{0} ,h_{1} ,h_{2} , \ldots ,u_{0} ,u_{1} ,u_{2} , \ldots } \right)$. Further, the solution of the zeroth order is given by;

$$ \left\{ {\begin{array}{*{20}l} {h_{0} \left( {x,t} \right) = \varphi \left( t \right) + H\left( x \right),} \hfill \\ {u_{0} \left( {x,t} \right) = \frac{1}{{h_{0} \left( {x,t} \right)}}\left\{ { - \left( {x - L} \right)\varphi^{\prime } \left( t \right)} \right\}.} \hfill \\ \end{array} } \right. $$

The benchmark problem in this study assumes the incidence of a sinusoidal wave at x = 0 m,

$$ \varphi \left( t \right) = H\left( 0 \right) + dH\left\{ {1 - \sin \left( {\frac{2\pi }{T}t + \frac{\pi }{2}} \right)} \right\}, $$

where the amplitude of the incident wave is $dH = 4.00\;{\text{m}}$, and its period is $T = 43200.0\;{\text{s}}$.

Another assumption of the benchmark is the topographic undulation expressed by the sinusoidal function as,

$$ H\left( x \right) = A - B\frac{x}{L} - C\sin \left( {\frac{4\pi }{L}x - \frac{\pi }{2}} \right), $$

where $\left( {A,B,C} \right) = \left( {50.5\,{\text{m,}}\,\,40.0\,{\text{m,}}\,\,10.0\,{\text{m}}} \right)$.

Appendix 2

The linearized shallow water equation is solvable by mathematical techniques (e.g., [16, 19]). In this study, the solution was obtained assuming a constant water depth, denoted by $h_{0}$. The modeled bay (Fig. 10) is rectangular with an open boundary along $x = 0$, and coasts along $x = L$, $y = 0$, and $y = B$. The variables are assumed to have a common frequency, $\sigma$, as expressed by the equation: $\left( {\eta ,u,v} \right) = {\text{Re}}\left\{ {\left( {\hat{\eta },\hat{u},\hat{v}} \right)e^{i\sigma t} } \right\}$, where $\left( {\hat{\eta },\hat{u},\hat{v}} \right)$ are the complex amplitudes.

The applications of the variable separation and eigenfunction expansion methods yield the solution comprising four modes: positive Kelvin and Poincaré modes, and negative Kelvin and Poincaré modes. Here, the terms “positive” and “negative” indicate wave propagations in positive and negative x-directions, respectively. Though the theoretically exact solution requires the superimposition of infinite numbers of Poincaré modes, the computations in this study truncate the number maximally at N.

The complex amplitudes of the positive Kelvin and Poincaré modes are

$$ \left\{ {\begin{array}{*{20}l} {\hat{\eta } = \frac{{h_{0} k}}{\sigma }a,} \hfill \\ {\hat{u} = - a{\mkern 1mu} e^{\alpha y} e^{ikx} ,} \hfill \\ {\hat{v} = 0,} \hfill \\ \end{array} } \right. $$

and

$$ \left\{ {\begin{array}{*{20}l} {\hat{\eta } = \sum\limits_{n = 1}^{N} {\kappa_{n} \left\{ {C_{n} { \cos }\left( {\gamma_{n} y} \right) + D_{n} { \sin }\left( {\gamma_{n} y} \right)} \right\}e^{{il_{n} x}} } ,} \hfill \\ {\hat{u} = \sum\limits_{n = 1}^{N} {\kappa_{n} \left\{ {A_{n} { \cos }\left( {\gamma_{n} y} \right) + B_{n} { \sin }\left( {\gamma_{n} y} \right)} \right\}e^{{il_{n} x}} } ,} \hfill \\ {\hat{v} = \sum\limits_{n = 1}^{N} {\kappa_{n} { \sin }\left( {\gamma_{n} y} \right)e^{{il_{n} x}} } ,} \hfill \\ \end{array} } \right. $$

respectively.

The complex amplitudes of the negative Kelvin and Poincaré modes are

$$ \left\{ {\begin{array}{*{20}l} {\hat{\eta } = \frac{{h_{0} k}}{\sigma }b{\mkern 1mu} e^{ - \alpha y} e^{ - ikx} ,} \hfill \\ {\hat{u} = be^{ - \alpha y} e^{ - ikx} ,} \hfill \\ {\hat{v} = 0,} \hfill \\ \end{array} } \right. $$

and

$$ \left\{ {\begin{array}{*{20}l} {\hat{\eta } = \sum\limits_{n = 1}^{N} {\lambda_{n} \left\{ {C_{n} { \cos }\left( {\gamma_{n} y} \right) - D_{n} { \sin }\left( {\gamma_{n} y} \right)} \right\}e^{{ - il_{n} x}} } ,} \hfill \\ {\hat{u} = \sum\limits_{n = 1}^{N} {\lambda_{n} \left\{ { - A_{n} { \cos }\left( {\gamma_{n} y} \right) + B_{n} { \sin }\left( {\gamma_{n} y} \right)} \right\}e^{{ - il_{n} x}} } ,} \hfill \\ {\hat{v} = \sum\limits_{n = 1}^{N} {\lambda_{n} { \sin }\left( {\gamma_{n} y} \right)e^{{ - il_{n} x}} } .} \hfill \\ \end{array} } \right. $$

The notations $\left( {A_{n} ,B_{n} ,C_{n} ,D_{n} } \right)$ in the Poincaré modes are defined as

$$ \begin{aligned} A_{n} & = \frac{{ - \gamma_{n} l_{n} \left( { - 1 + f^{\prime 2} } \right)}}{{i\left( {\gamma_{n}^{2} + l_{n}^{2} f^{\prime 2} } \right)}}, \\ B_{n} & = \frac{{f^{\prime}\left( {l_{n}^{2} + \gamma_{n}^{2} } \right)}}{{i\left( {\gamma_{n}^{2} + l_{n}^{2} f^{\prime 2} } \right)}}, \\ C_{n} & = \frac{{ih_{0} }}{\sigma }\frac{{\left( {l_{n}^{2} + \gamma_{n}^{2} } \right)p}}{{\gamma_{n}^{2} + l_{n}^{2} f^{\prime 2} }}, \\ D_{n} & = \frac{{ih_{0} }}{\sigma }\frac{{\left( {l_{n}^{2} + \gamma_{n}^{2} } \right)f^{\prime}\,l_{n} }}{{\gamma_{n}^{2} + l_{n}^{2} f^{\prime 2} }}, \\ \end{aligned} $$

where $f^{\prime} \equiv {f \mathord{\left/ {\vphantom {f \sigma }} \right. \kern-0pt} \sigma }$ is a dimensionless Coriolis parameter normalized by the frequency σ.

The y-component of the wavenumber of Poincaré modes is forced to have discrete values by the condition that the bay is bounded along $y = 0$ and $y = B$, written as,

$$ \gamma_{n} = \frac{n}{B}\pi . $$

The x- and y-components of the wavenumber $\left( {l_{n} ,\gamma_{n} } \right)$ of the Poincaré modes are related to the frequency (dispersion relation) as follows;

$$ \begin{aligned} & \gamma_{n}^{2} = - l_{n}^{2} + k^{2} \left( {1 - f^{\prime 2} } \right), \\ & k \equiv \frac{\sigma }{{\sqrt {gh_{0} } }}. \\ \end{aligned} $$

Here, $\alpha$ defined as

$$ \alpha \equiv \frac{f}{{\sqrt {gh_{0} } }} = k{\kern 1pt} f^{\prime}, $$

is the inverse of the Rossby radius of deformation.