Three-Dimensional Modeling of Storm Surge and Inundation Including the Effects of Coastal Vegetation

Abstract

Two-dimensional (2D) and three-dimensional (3D) hydrodynamic models are used to simulate the hurricane-induced storm surge and coastal inundation in regions with vegetation. Typically, 2D storm surge models use an enhanced Manning coefficient while 3D storm surge models use a roughness height to represent the effects of coastal vegetation on flow. This paper presents a 3D storm surge model which accurately resolves the effects of vegetation on the flow and turbulence. First, a vegetation-resolving 1DV Turbulent Kinetic Energy model (TKEM) is introduced and validated with laboratory data. This model is both robust enough to accurately model flows in complex canopies, while compact and efficient enough for incorporation into a 3D storm surge-wave modeling system: Curvilinear Hydrodynamics in 3D-Surface WAves Nearshore (CH3D-SWAN). Using the 3D vegetation-resolving model, three numerical experiments are conducted. In the first experiment, comparisons are made between the 2D Manning coefficient approach and the 3D vegetation-resolving approach for simple wind-driven flow. In a second experiment, 2D and 3D representations of vegetation produce similar inundations from the same hurricane forcing, but differences in momentum are found. In a final experiment, varying inundation between seemingly analogous 2D and 3D representations of vegetation are demonstrated, pointing to a significant scientific need for data within wetlands during storm surge events. This study shows that the complex flow structures within vegetation canopies can be accurately simulated using a vegetation-resolving 3D storm surge model, which can be used to assess the feasibility for future wetland restoration projects.

Appendix

This is an extended derivation of the TKE model from the original RSM (Lewellen and Peter Sheng 1980). In the limit of no vegetation, the RSM from which this TKE model is derived is equivalent to the RSM for vegetation-free flow described in Sheng and Villaret (1989).

For simplicity, the RANS equations for the RSM are written in tensor form:

Mean Continuity Equation

$$ \frac{\partial {u}_i}{\partial {x}_i}=0 $$

(A1)

Mean Momentum Eqn.

$$ \begin{array}{l}\frac{\partial {u}_i}{\partial t}+{u}_j\frac{\partial {u}_i}{\partial {x}_j}=\left[{C}_f{A}_w+{C}_p{A}_f{\left(1+\frac{u_j^2}{q^2}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]q{u}_i-\frac{1}{p}\frac{\partial P}{\partial {x}_i}-2{\varepsilon}_{ijk}{\varOmega}_j{u}_k-\frac{\overline{\partial {u}_i^{\prime }{u}_j^{\prime }}}{\partial x}\hfill \\ {}+\frac{\partial }{\partial {x}_j}\left(v\frac{\partial {u}_{{}_i}}{\partial {x}_j}\right)\hfill \end{array} $$

(A2)

Reynolds Stress Equation

$$ \begin{array}{l}\frac{\overline{\partial {u}_i^{\prime }{u}_j^{\prime }}}{\partial t}+{u}_k\frac{\overline{\partial {u}_i^{\prime }{u}_j^{\prime }}}{\partial {x}_k}\hfill \\ {}=2{C}_p\left({u}_k^2+{q}^2\right){\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{A}_f{u}_i{u}_j{\delta}_{ij}-2{C}_f{A}_w\overline{q{u}_i^{\prime }{u}_j^{\prime }}-\overline{u_i^{\prime }{u}_j^{\prime }}\frac{\partial {u}_j}{\partial {x}_k}\hfill \\ {}-\overline{u_j^{\prime }{u}_k^{\prime }}\frac{\partial {u}_i}{\partial {x}_k}-2{\varepsilon}_{ikl}{\Omega}_k\overline{u_l^{\prime }{u}_j^{\prime }}-2{\varepsilon}_{jlk}{\Omega}_l\overline{u_k^{\prime }{u}_i^{\prime }}+0.3\frac{\partial }{\partial x}\left( q\varLambda \frac{\overline{\partial {u}_i^{\prime }{u}_j^{\prime }}}{\partial {x}_k}\right)\hfill \\ {}-\frac{q}{\varLambda}\left(\overline{u_i^{\prime }{u}_j^{\prime }}-{\delta}_{ij}\frac{q^2}{12\varLambda}\right)-{\delta}_{ij}\frac{q^3}{12\varLambda}\hfill \end{array} $$

(A3)

where (i, j, k) = (1, 2, 3), x _i are coordinate axes, t is time, (u _i , u _j , u _k ) are the mean velocity components, (u _i ’ , u _j ’, u _k ’ ) are the fluctuating velocity components, the overbar represents the Reynolds averaging, g is gravitational acceleration, ε _ijk is alternating tensor, Ω is the Earth’s rotation, δ _ij is the Kronecker delta, q is the total rms fluctuating velocity, κ is molecular diffusivity, and λ is the turbulence macroscale which is a measure of the average turbulent eddy size. The right hand side of Eqn. 7, which represents the dynamic equation for the Reynolds stresses, contains two vegetation terms, two shear production terms, two rotation terms, one diffusion term, one tendency towards isotropy term, and one dissipation term (Sheng and Villaret 1989). While the general RSM and TKEM include the buoyancy effects associated with temperature and salinity variation in the flow, those terms are neglected here for simplicity. In the 1DV model for vegetation, only vertical gradients in the momentum equation are considered.

In the Reynolds stress equations, both a source and a sink term have been added:

$$ 2{C}_p\left({u}_k^2+{q}^2\right){\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{A}_f{u}_i{u}_j{\delta}_{ij}-2{C}_f{A}_wq\overline{u_i^{\prime }{u}_j^{\prime }} $$

(A4)

where the first term represents the creation of wake turbulence due to the profile drag, and the second term recognizes the dissipation of Reynolds stress by the skin friction.

Eqn. A3 can be simplified as:

$$ \begin{array}{l}0=2{C}_p\left({u}_k^2+{q}^2\right){\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{A}_f{u}_{{}_i}{u}_j{\delta}_{ij}-2{C}_f{A}_wq\overline{u_i^{\prime }{u}_j^{\prime }}-\overline{u_j^{\prime }{u}_k^{\prime }}\frac{\partial {u}_j}{\partial {x}_k}\hfill \\ {}-\overline{u_j^{\prime }{u}_k^{\prime }}\frac{\partial {u}_i}{\partial {x}_k}-\frac{q}{\varLambda}\left(\overline{u_i^{\prime }{u}_j^{\prime }}-{\delta}_{ij}\frac{q^2}{3}\right)-{\delta}_{ij}\frac{q^3}{12\varLambda}\hfill \end{array} $$

(A5)

and the dynamics of turbulence are represented by the following equation for q ²:

$$ \begin{array}{l}\frac{\partial {q}^2}{\partial t}+{u}_k\frac{\partial {q}^2}{\partial {u}_k}=2{C}_p\left({e}^2+{q}^2\right){\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{A}_f{e}^2-2{C}_f{A}_w{q}^3\hfill \\ {}+2{A}_v\left[{\left(\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}\right)}^2\right]+0.3\frac{\partial }{\partial z}\left( q\varLambda \frac{\partial {q}^2}{\partial z}\right)-\frac{q^3}{4\varLambda}\hfill \end{array} $$

(A6)