Numerical Discretization

It is well known that the main features of non linear waves are controlled by the interplay of non linear effects, which steepen wavefronts counteracted by dispersive effects. Hence, it is important to consider a numerical method which minimizes the error introduced in the computation of derivatives. We use a spectral discretization (Canuto et al., 1987) employing sine/cosine-collocation points, following a 2/3 rule concerning the dealiasing of the nonlinear term. The physical domain, whose lower boundary represents realistic topography (see Figure 6), is mapped by means of a conformal map to a rectangular domain, using the technique described in Fornberg (1980). Time is advanced using a compact Runge-Kutta scheme.

The numerics was validated at several level. The overall correctness of the implementation was tested solving the initial value problem for a ${\rm sech}^2$ downward pulse. The treatment of the geometry was verified reproducing results available in the literaturre, such as the simulations of Lamb for a stratified flow across a bank edge. (JGR, 99, C1 843).

Though formally non hydrostatic, the model can be run either as hydrostatic or as non hydrostatic according to the numerical resolution adopted. To illustrate this point, consider the expression for the vorticity

$$\omega= -\left({\partial^2\psi\over \partial z^2}+{\partial^2\psi\over \partial x^2}\right).$$ (3)

If the horizontal resolution is kept low, the latter is equivalent to

$$\omega\simeq -{\partial^2\psi\over \partial z^2},$$ (4)

which is the hydrostatic (more precisely, the long wavelength) limit in the present formulation. Hence, even if formally our model is non hydrostatic, a low horizontal resolution makes it de facto. As the horizontal resolution increases, the dispersive effects become important (dispersion is a function of wavelength), which sets the stage for the occurence of solitary waves. If the resolution is further increased, we now begin to enter the region of wavelength that can become instable a la Kelvin-Helmholtz. This problem is well known to affect n-layer models and reflects a real instability of the Euler equations.

Some discussion about the Grue's experiments.

Looking at the observed data, the wavelength of solitary waves observed in the basin is about 160 m. Based on experience gained in solving spectrally 2-layer models (Scotti and Camassa, personal communication), a resolution of about 20 m should be enough to ensure proper treatment of dispersive effects without incurring in severe instabilities.