Abstract
Using the Hamiltonian formulation, the Lagrangian formulation or Green’s formula, six kinds of the mild-slope equations are developed, depending upon how variable ambient currents and bottom topography behave, that is, two- or three-dimensional slowly-varying currents, and mildly-, rapidly- or even uniformly-varing bottom topography. What is the most important is a new type of conservative quantity, the product of phase velocity and group velocity arising from the general mild-slope equation for wave-current interactions, which itself holds true for a universal conservation law of wave action. In addition, an operator representing the wide wave-current interactions is introduced to develop a hierarchy of the mild-slope equations for linear surface gravity waves with two-dimensional slowly-varying currents over uniformly-varying bottom topography.
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