Particular analytical solutions of the hydrodinamic equations

Some particular solutions of the hydrodynamic equations are presented. In the solutions the velocity is a complex exponential on space variables, i.e. it is a real exponential multiplied by sine and cosine functions. The amplitude of the exponential is a time dependent complex vector. These particular solutions have been obtained using the approximation of a constant density fluid lying in the Earth’s tangent plane. The equations in eulerian approach contain the advective part which is balanced by two pressure terms. The first term corresponds to the kinetic energy, the second term to a rotational energy. The existence of this last energy implies rather strict conditions on the possible solutions. The exponential solutions are obtained neatly under the assumption of a vanishing Coriolis force. However the solutions obtained under this approximation may still be a good initial solution of the hydrodynamic equations for a sea like the Adriatic. The friction has been represented by a symmetric tensor with vanishing trace and constant coefficients, which operates linearly on the components of the velocity. The tensor is not assumed to be necessarily isotropic in the vertical direction.