The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

Abstract

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.

Evolution of interface singularities in shallow water equations with variable bottom topography

“…The problem of predicting the motion of a shoreline in the presence of waves has a long history. When the bottom has a constant slope, so that b xx = 0, the SWE can be linearized by the so-called Carrier-Greenspan transform [7,23]. Several closed-form solutions can be constructed in this case and the corresponding shoreline motion can be described exactly, offering an elegant global perspective in the study of the breaking of waves.…”

Evolution of interface singularities in shallow water equations with variable bottom topography

“…The problem of predicting the motion of a shoreline in the presence of waves has a long history. When the bottom has a constant slope, so that $$b_{xx}=0$$, the SWE can be linearized by the so‐called Carrier–Greenspan transform 1,12,32 . Several closed‐form solutions can be constructed in this case and the corresponding shoreline motion can be described exactly, offering an elegant global perspective in the study of the breaking of waves.…”

“…When the bottom has a constant slope, so that 𝑏 𝑥𝑥 = 0, the SWE can be linearized by the socalled Carrier-Greenspan transform. 1,12,32 Several closed-form solutions can be constructed in this case and the corresponding shoreline motion can be described exactly, offering an elegant global perspective in the study of the breaking of waves. However, this hodograph-like transformation is no longer available for general bathymetry.…”

Extreme Wave Runup on a Steep Coastal Profile

“…In the presence of bathymetry, it is somewhat more difficult to find the requisite change of variables than in the case of constant coefficients. Nevertheless, an appropriate hodograph transformation was found by Carrier and Greenspan 15 , and there have been a number of works seeking to extend and generalize that idea (see [16][17][18][19][20][21] and references therein).…”