Importance of an efficient tide-surge interaction model for the coast of Bangladesh: a case study with the tropical cyclone Roanu

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. M₂ (principal lunar semidiurnal), S₂ (principal solar semidiurnal), K₁ (lunisolar diurnal), and O₁ (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.

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

$$ \frac{\partial \zeta }{\partial t}=\frac{\zeta_{i,j}^{k+1}-{\zeta}_{i,j}^k}{\Delta t},\frac{\partial \zeta }{\partial x}=\frac{\zeta_{i+1,j}^k-{\zeta}_{i-1,j}^k}{2\Delta x},\mathrm{and}\kern0.24em \frac{\partial \zeta }{\partial y}=\frac{\zeta_{i,j+1}^k-{\zeta}_{i,j-1}^k}{2\Delta y}. $$

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

$$ {\overline{U_{i,j}^k}}^x=\frac{U_{i+1,j}^k+{U}_{i-1,j}^k}{2},{\overline{U_{i,j}^k}}^y=\frac{U_{i,j+1}^k+{U}_{i,j-1}^k}{2},{\overline{U_{i,j}^k}}^{xy}={\overline{{\overline{U_{i,j}^k}}^x}}^y. $$

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

$$ {\zeta}_{i,j}^{k+1}={\zeta}_{i,j}^k-\Delta t\left[ TL1+ TL2\right], $$

(14)

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

$$ {\tilde{u}}_{i,j}^{k+1}={\tilde{u}}_{i,j}^k\left(1-\Delta t. FUR\kern0.28em 3\right)-\Delta t.\left( UL1+ UL2+ UL3\right)+\Delta t.\left( UR\kern0.28em 1+ UR\kern0.28em 2+ UR\kern0.28em 3\right), $$

(15)

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

$$ FUR3=\frac{C_f\sqrt{{\left({U}_{i,j}^k\right)}^2+{\left({\overline{V_{i,j}^k}}^{xy}\right)}^2}}{\zeta_{i,j}^{k+1}+{h}_{i,j}}, $$

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:

$$ {\tilde{v}}_{i,j}^{k+1}={\tilde{v}}_{i,j}^k\left(1-\Delta t. FVR3\right)-\Delta t.\left( VL1+ VL2+ VL3\right)+\Delta t.\left( VR1+ VR2+ VR3\right), $$

(16)

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

$$ FVR3=\frac{C_f\sqrt{{\left({\overline{U_{i,j}^k}}^{xy}\right)}^2+{\left({V}_{i,j}^k\right)}^2}}{\zeta_{i,j}^{k+1}+{h}_{i,j}}, $$

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:

$$ {\zeta}_{i,1}^{k+1}=-{\zeta}_{i,3}^{k+1}-2\sqrt{\left({h}_{i,2}/g\right)}\kern1em {V}_{i,2}^k, $$

(17)

$$ {\zeta}_{i,N}^{k+1}=-{\zeta}_{i,N-2}^{k+1}+2\sqrt{\left({h}_{i,N-1}/g\right)}\kern1em {V}_{i,N-1}^k, $$

(18)

$$ {\zeta}_{M,j}^{k+1}=-{\zeta}_{M-2,j}^{k+1}+2\sqrt{\left({h}_{M-1,j}/g\right)}\kern1em {U}_{M-1,j}^k\kern0.5em +4\sum \limits_{p=1}^4{a}_p\sin \left(2\pi \kern0.28em k\Delta t/{T}_p+{\phi}_p\right), $$

(19)

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:

$$ {\left({U}_b\right)}_{1,j}^{k+1}={\left({U}_b\right)}_{3,j}^{k+1}+\frac{Q}{\left({\zeta}_{1,j}^{k+1}+{h}_{1,j}\right)B}. $$

(20)

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 UL1, UL2, UL3, UR1,UR2, FUR3 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 VL1, VL2, VL3, VR1,VR2, FVR3 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 M₂, S₂, K₁, and O₁ 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).