Asymptotic shallow water models with non smooth topographies

Abstract

We present new models to describe shallow water flows over non smooth topographies. The water waves problem is formulated as a system of two equations on surface quantities in which the topography is involved in a Dirichlet-Neumann operator. Starting from this formulation and using the joint analyticity of this operator with respect to the surface and the bottom parametrizations, we derive a nonlocal shallow water model which only includes smoothing contributions of the bottom. Under additional small amplitude assumptions, Boussinesq-type systems are also derived. Using these alternative shallow water models as references, we finally present numerical tests to assess the precision of the classical shallow water approximations over rough bottoms. In the case of a polygonal bottom, we show numerically that our new model is consistent with the approach developed by Nachbin.

Appendix: The case of polygonal topographies

In the case of two dimensional motions, Hamilton [22] and Nachbin [29] used a conformal mapping technique to derive long wave models. This conformal mapping technique can be adapted to derive shallow water models with polygonal topography. The idea is to use Schwarz-Christoffel mapping theory (see e.g. [31]) to find a conformal map from a strip to the fluid domain at rest (see [20, 29]). From a numerical point of view, the main interest of this technique is that such a mapping can be efficiently computed using, for instance, the Schwarz-Christoffel Toolbox [17] (see [20, Appendix A] for an application to the conformal mapping of a fluid domain with polygonal bottom). This particular conformal mapping can then be used to approximate the Dirichlet-Neumann operator. Broadly speaking, the derivation of this approximation proceeds via the following steps:

• 1. Transform the Laplace Eq. (1) into an elliptic boundary value problem defined on the flat strip.

• 2. Express the Dirichlet-Neumann operator in terms of the solution of this new problem (the so-called transformed potential).

• 3. Approximate the transformed potential using a BKW procedure.

• 4. Use this approximate solution in the expression of () to deduce an approximation of ${\mathcal G}_{\mu }[\varepsilon \zeta ,\beta b]\psi$.

Denoting by $\Sigma$ the Schwarz-Christoffel mapping function and setting $(\sigma (x),\rho (x))=\Sigma ^{-1}(x,\varepsilon \zeta (x))$ (the transformed free surface), the resulting approximation is

$$\begin{aligned} {\mathcal G}_{\mu }[\varepsilon \zeta ,\beta b]\psi =-\partial _{x} \biggl (\frac{1+\rho }{ \frac{\mathop {}\mathopen {}\mathrm {d}\sigma }{\mathop {}\mathopen {}\mathrm {d}x}} \partial _{x} \psi \biggr ) + O(\mu ), \end{aligned}$$

giving rise to the following nonlinear shallow water system with polygonal topography

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t} \zeta + \partial _{x} \biggl (\frac{1+\rho }{ \frac{\mathop {}\mathopen {}\mathrm {d}\sigma }{\mathop {}\mathopen {}\mathrm {d}x}} v_\mathrm{s} \biggr ) =0, \\ \partial _{t} v_\mathrm{s} + \partial _{x} \zeta + \varepsilon v_\mathrm{s} \partial _{x} v_\mathrm{s} =0. \end{array}\right. } \end{aligned}$$

(46)

To evaluate the behavior of the nonlocal shallow water model (27) when the bottom has polygonal shape, we compare the solutions produced by both systems (40) and (46) in the particular case of a rectangular bottom: $b=0$ on (0, 30) and $b=1$ on (30, 60).

The initial condition, defined in (44), generates a unidirectional wave propagating to the right and the amplitude parameters are set to $\varepsilon =0.1$ and $\beta =0.6$. Time histories of the surface elevation computed by both models are shown in Fig. 10. The simulation was performed using $N=$ 1,024 points and $\Delta t=10^{-1}$. As the wave passes over the step, both models produce similar results.

Fig. 10

Elevation for a wave passing over a rectangular hump: conformal mapping approach (dashed line) and nonlocal model (solid line)