Two-dimensionality of gravity water flows governed by the equatorial \(f\)-plane approximation (Q1681839)

The purpose of this paper is to study the system of governing equations for gravity wave trains at the free surface, governed by the equatorial \(f\)-plane approximation. This system consists of seven equations. Choose a rotating coordinate system with origin at a point on the Earth's surface, with the \(x\)-axis pointing to the east, the \(y\)-axis pointing to the north, and the \(z\)-axis pointing upward. The main theorem states that the water flow over this flat bed governed by the first five equations has a constant vorticity only if the flow is two-dimensional. The proof by contradiction, which uses the Phragmen-Lindelöf method, is made by showing a lack of \(y\)-dependence for the velocity field.