A new shallow water model with linear dependence on depth

Abstract

In this paper, we study the Euler equations in a domain with small depth. With this aim, we introduce a small adimensional parameter ε related to the depth and we use asymptotic analysis to study what happens when ε becomes small. Usually, when asymptotics are used to analyze fluids, they are used in the original domain or the surface is supposed to be constant. We, however, shall use the asymptotic technique making a change of variable to a reference domain independent of the parameter ε and the time. In this way we obtain a model for ε small that, after coming back to the original domain, gives us a shallow water model that considers the possibility of a non-constant bottom and the horizontal velocity components depend on z if the vorticity is not zero. This represents an interesting novelty with respect to shallow water models found in the literature. We stand out that we do not need to make a priori assumptions about velocity or pressure behavior to obtain the model. The new model is able to calculate exactly the solutions of Euler equations that are linear in z , whereas the classic model just obtains the averaged velocities.