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Regular version of the site

Article

The Generalized Carrier-Greenspan Transform for the shallow water system with arbitrary initial and boundary conditions

Water Waves. 2021. Vol. 3. No. 1. P. 267-296.
Rybkin A., Nikolsky D., Pelinovsky E., Buckel M.

We put forward a solution to the initial boundary value (IBV) problem for the nonlinear
shallow water system in inclined channels of arbitrary cross section by means of
the generalized Carrier–Greenspan hodograph transform (Rybkin et al. in J Fluid
Mech, 748:416–432, 2014). Since the Carrier–Greenspan transform, while linearizing
the shallow water system, seriously entangles the IBV in the hodograph plane, all
previous solutions required some restrictive assumptions on the IBV conditions, e.g.,
zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking
initial conditions in the physical space, we present an explicit formula for equivalent
IBV conditions in the hodograph plane, which can readily be treated by conventional
methods. Our procedure, which we call the method of data projection, is based on the
Taylor formula and allows us to reduce the transformed IBV data given on curves in
the hodograph plane to the equivalent data on lines. Our method works equally well
for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical
for U-shaped bays. Numerical simulations show that our method is very robust and
can be used to give express forecasting of tsunami wave inundation in narrow bays
and fjords.