A new shallow water model with polynomial dependence on depth

In this paper, we study two-dimensional Euler equations in a domain with small depth. With this aim, we introduce a small nondimensional parameter (epsilon)related to the depth and we use asymptotic analysis to study what happens when (epsilon)becomes small. We obtain a model for (epsilon)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 has a dependence on z introduced by the vorticity when it 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 behaviour to obtain the model. The new model is able to approximate the solutions to Euler equations with dependence on z (reobtaining the same velocities profile), whereas the classic model just obtains the average velocity.