## Abstract

In this study, shallow water equations (SWEs) in Cartesian coordinates were used to foresee water levels (WLs) on account of the nonlinear tide-surge interaction (TSI) associated with the recent cyclonic storm Roanu that hit the eastern coast of Bangladesh. A fully explicit finite-difference method (FDM) was implemented to solve the SWEs. A one-way nesting approach was implemented for incorporating coastal complexity with the lowest cost. The land-sea interface of every nested scheme was made favorable for employing the FDM via the stair-step technique. To incorporate river dynamics, freshwater Meghna River discharge was taken into consideration in the innermost scheme. An appropriate tidal regime in the domain was established after forcing the sea-level to be oscillatory applying the four effective tidal constituents, viz. *M*_{2} (principal lunar semidiurnal), *S*_{2} (principal solar semidiurnal), *K*_{1} (lunisolar diurnal), and *O*_{1} (principal lunar diurnal) on the southern boundary of the outermost scheme. Numerical experiments were performed by the model to estimate WLs due to tide, surge, and TSI associated with tropical storm Roanu in the coastal areas of Bangladesh. Our estimated WLs due to TSI were in good agreement with the observed data procured from the Bangladesh Inland Water Transport Authority (BIWTA) and some reported results. The model was found to perform well on the basis of the root mean square error (RMSE) values and computational cost.

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## Acknowledgments

This research is partially supported by a grant from the University Grants Commission of Bangladesh with No. 6(74)/UGC/RSP/ST/PHY(4)/2016/3530. The first author acknowledges this support. We would like to express our sincere thanks to the Editor-in-Chief, Associate Editor, and the anonymous referee for thoughtful suggestions and detailed comments that helped us to substantially improve our original manuscript. The authors are thankful to Mr. Md. Mahabub Alam, an M.Sc. thesis student, Department of Mathematics, University of Rajshahi, Bangladesh for his help in editing a figure.

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## Appendices

### Appendix 1-Discretization of model equations

The basic Eqs. (1)–(3) and the BCs prescribed by Eqs. (4)–(6) are discretized adopting the Arakawa C-grid technique using an explicit FD approximation method, where the forward difference approximation is used for discretizing time derivatives and central difference approximation for spatial derivative.

We represent any variable, say *ζ*, at any spatial grid point (*i*, *j*) at any time *t*_{k} by

\( \zeta \left({x}_i,{y}_j,{t}_k\right)={\zeta}_{1,j}^k \), FD operators are defined as

We define the averaging operations for any variable (say *U*) as

Thus, the FD approximation to Eq. (1) is

where

\( TL1=\left(\frac{{\tilde{u}}_{i+1,j}^k-{\tilde{u}}_{i-1,j}^{`k}}{2\Delta x}\right) \) and \( TL2=\left(\frac{{\tilde{v}}_{i,j+1}^k-{\tilde{v}}_{i,j-1}^k}{2\Delta y}\right) \),

\( {\zeta}_{i,j}^{k+1} \) in Eq. (14) are computed at the points (*i*, *j*), where *i* = 2, 4, 6, …, *M* − 2 and *j* = 3, 5, 7, …, *N* − 2.

Similarly, one can see Eq. (2) as

where

\( UL1=\left(\frac{{\tilde{u}}_{i+2,j}^k{U}_{i+2,j}^k-{\tilde{u}}_{i-2,j}^k{U}_{i-2,j}^k}{4\Delta x}\right) \), \( UL2=\left(\frac{{\overline{{\tilde{u}}_{i,j+1}^k}}^y{\overline{V_{i,j+1}^k}}^x-{\overline{{\tilde{u}}_{i,j-1}^k}}^y{\overline{V_{i,j-1}^k}}^x}{2\kern1em \Delta y}\right) \),

\( UL3=-{f}_i{\overline{{\tilde{v}}_{i,j}^k}}^{xy} \), \( UR\;1=-g\left({\zeta}_{i,j}^{k+1}+{h}_{i,j}\right)\left\{\frac{\zeta_{i+1,j}^{k+1}-{\zeta}_{i-1,j}^{k+1}}{2\Delta x}\right\} \),

\( UR2=\frac{\tau_x}{\rho } \), \( UR3=\frac{P_x}{\rho } \), where *P*_{x} is the Right hand side of Eq. (11) and

where the term \( {\tilde{u}}_{i,j}^{k+1} \) in Eq. (15) is computed at the points (*i*, *j*), where *i* = 3, 5, 7, …, *M* − 1 and *j* = 3, 5, 7, …, *N* − 2.

Also, one can easily see Eq. (3) in the following form:

where

\( VL1=\frac{{\overline{U_{i+1,j}^k}}^y\;{\overline{{\tilde{v}}_{i+1,j}^k}}^x-{\overline{U_{i-1,j}^k}}^y{\overline{{\tilde{v}}_{i-1,j}^k}}^x}{2\Delta x} \), \( VL2=\frac{V_{i,j+2}^k{\tilde{v}}_{i,j+2}^k-{V}_{i,j-2}^k{\tilde{v}}_{i,j-2}^k}{4\kern1em \Delta y} \),

\( VL3={f}_i{\overline{{\tilde{u}}_{i,j}^{k+1}}}^{xy} \), \( VR1=-g\left({\zeta}_{i,j}^{k+1}+{h}_{i,j}\right)\left\{\frac{\zeta_{i,j+1}^{k+1}-{\zeta}_{i,j-1}^{k+1}}{2\Delta y}\right\} \),

\( VR2=\frac{\tau_y}{\rho } \), \( VR3=\frac{P_y}{\rho } \), where *P*_{y} is the Right hand side of Eq. (12) and

where \( {\tilde{v}}_{i,j}^{k+1} \) in Eq. (16) is computed at (*i*, *j*) at (*k* + 1)th time level, where *i* = 2, 4, 6, …, *M* − 2 and *j* = 2, 4, 6, …, *N* − 1.

The BCs specified by Eqs. (4) - (6), the elevations at *j* = 1 (eastarn boundar *y*), *j* = *N* (western boundary ), and

*i* = *M* (southern boundary) are computed in the following manner, respectively:

where *i* = 2, 4, 6, …, *M* − 2 and *j* = 1, 3, 5, 7, …, *N.*

The freshwater Meghna River discharge is incorporated through Eq. (7), where the velocity component *U*_{b} is calculated at (1, *j*), *j* = 7, 9, 11, …, 19, in the following manner:

In the case of a semi-implicit scheme, the last term on the right hand side of each of Eqs. (2) and (3) is discretized in a technique so that the scheme becomes semi-implicit in nature. For example, from Eq. (2), the term \( \tilde{u}\sqrt{\left({u}^2+{v}^2\right)} \) is discretized as \( {\tilde{u}}^{K+1}\sqrt{\left({u}^{k^2}+{v}^{k2}\right)} \), where the descriptions about the superscripts are aforementioned. Therefore, the FD form of Eq. (2) in the case of semi-implicit scheme can be put as

\( {\tilde{u}}_{i,j}^{k+1}=\frac{{\tilde{u}}_{i,j}^k-\varDelta t\left( UL1+ UL2+ UL3\right)+\varDelta t\left( UR1+ UR2\right)}{\left(1+\varDelta t. FUR3\right)}, \) where *UL*1, *UL*2, *UL*3, *UR*1,*UR*2, *FUR*3 are defined above.

Similarly, the FD form of Eq. (3) in the case of a semi-implicit scheme can be set as

\( {\tilde{v}}_{i,j}^{k+1}=\frac{{\tilde{v}}_{i,j}^k-\varDelta t\left( VL1+ VL2+ VL3\right)+\varDelta t\left( VR1+ VR2+ VR3\right)}{\left(1+\varDelta t. FVR3\right)}, \) where *VL*1, *VL*2, *VL*3, *VR*1,*VR*2, *FVR*3 are defined above. The discretized boundary conditions mentioned by Eqs. (17) - (19) are the same for the semi-implicit scheme along with Eq. (20).

### Appendix 2-Generation of stable tidal condition

To generate a fully tidal oscillatory motion in the area of choice, the influences of the four effective tidal constituents, namely *M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1} are taken into consideration along the southern OB of the CMS as per Eq. (19) (see Appendix 1). The tidal constants (tidal period, amplitude, and phase) for the chosen tidal constituents were those used in Paul et al. (2016). After inserting the values of the tidal constituent constants, the integration was started from an initial condition of rest with the nonappearance of wind stress and atmospheric pressure gradient force. The integration was continued until a fully tidal oscillatory motion got established in the area of our choice. In our investigation, it was attained after four tidal cycles of integration. A detail of the procedure can be found in Roy (1995), Paul and Ismail (2012), and Paul et al. (2016, 2018).

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Paul, G.C., Khatun, R., Ali, E. *et al.* Importance of an efficient tide-surge interaction model for the coast of Bangladesh: a case study with the tropical cyclone Roanu.
*J Coast Conserv* **25, **12 (2021). https://doi.org/10.1007/s11852-020-00787-z

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### Keywords

- Non-linear interaction
- Roanu
- Explicit finite difference method
- Storm surge
- Bay of Bengal