Green-Naghdi Equations

The GN equations are [10]

where is the local depth of the water, is the free surface height and is the fixed bottom boundary. The quantities A and B are given by

and is the material derivative following the horizontal velocity, . The GN equations provide a depth-averaged (barotropic) description of shallow water motion with a free surface under gravity, g, in nondominant asymptotics. That is, the GN equations are derived using shallow water scaling asymptotics for a domain of small aspect ratio, but no restriction is placed on the Froude number, or nonlinearity parameter of the flow, cf. reference [25].

The GN equations are often derived by making a solution Ansatz of columnar motion in which the horizontal fluid velocity is independent of the vertical coordinate. One then imposes three-dimensional incompressibility and certain symmetry requirements. The GN equations are derived in [10] by requiring the incompressible columnar motion to satisfy conservation of energy and invariance under rigid body translations. They are rediscovered in [11] by inserting the columnar motion solution Ansatz, , and incompressibility into the Euler equations, then averaging over depth. Finally, they are derived from a variational principle in [26] by inserting the columnar motion Ansatz into the variational principle for the Euler equations for an inviscid incompressible fluid, and explicitly performing the vertical integrations before varying. The Hamiltonian structure of these equations is discussed in [12].

The approach of Miles and Salmon [26] in obtaining the GN equations by restricting to columnar motion in Hamilton's principle for an incompressible fluid with a free surface affords a convenient starting point for extending the GN equations to include effects of a rotating frame and weak buoyancy stratification. Our derivation by Hamilton's principle also provides a systematic derivation of the Kelvin circulation theorem and potential vorticity conservation laws for the extended GN equations.