The Hamilton Principle for the extended GN Equations

The three-dimensional Euler equations for a rotating stratified incompressible fluid in the Boussinesq approximation follow from an action principle , with   where is twice the rotation vector, which we note has a horizontal component when S has a nonvanishing gradient. Also, , where is the Jacobian matrix for the map from Eulerian coordinates to Lagrangian fluid labels, , on which the buoyancy stratification depends. These Lagrangian labels specify the fluid particle currently occupying Eulerian position . They satisfy the advection law, , thereby determining the velocity components in the action principle, as

where, as usual, we sum on repeated indices. Variations in () with respect to l^A at fixed and t yield the Euler equations for a rotating stratified incompressible fluid, with appropriate kinematic boundary conditions. The constraint D=1 imposed by the Lagrange multiplier p (the pressure) implies incompressibility. For more details, see Holm [27] and Miles and Salmon [26].

Following Miles and Salmon [26], we find an action principle for the Green-Naghdi equations (extended to account for stratification and rotation) by restricting the action principle () to variations of the form,

from which () implies   This is the columnar motion Ansatz. For restricted variations of this type, we choose a buoyancy stratification profile which is linear in ,  so the bottom and top surfaces of the domain of flow are taken to be level surfaces of the buoyancy, . After performing the vertical integrations, the action () reduces to

From incompressibility we have   so the depth is also the two-dimensional Jacobian for this class of flows. From advection of the fluid labels, we have   Thus, and the components of are expressed in terms of derivatives of , with A = 1,2. Advection of the fluid labels also implies the continuity equation for ,  The extended Green-Naghdi motion equations then result from stationarity of the action in () under variations with respect to Lagrangian fluid labels at fixed Eulerian position and time. These motion equations arise in the following form,   which implies Kelvin's circulation theorem,

where we have used the transport theorem for the contour moving with the fluid velocity in the first line and substituted equation () in the second.

As a consequence of equation () and the continuity equation for , potential vorticity is convected; namely, we have   Equation () is the vortex stretching relation for the extended GN equations: when the fluid depth changes locally, the ``vorticity" quantity changes in proportion. In the GN theory, this quantity - the scalar curl of the total specific horizontal momentum of the flow - is a linear homogeneous functional of the three dimensional vertical vorticity (R. Camassa, private communication). This means, in particular, that potential flows in three dimensions have q=0 in the GN theory. Advection of the potential vorticity q, combined with the continuity relation for and tangential boundary conditions on the mean velocity , yields an infinity of conserved quantities,   for any function . The results in equations () through () apply generally for all vertically averaged fluid action principles in the Eulerian description and do not depend on the specific form of the action .

Evaluating the variational derivatives of appearing in equation () for the GN casefor the GN case yields

where F and G are given by (cf. equation ())

Consequently equation () yields the following explicit form of the extended GN motion equation, cf. equations () and (),

In this equation, the quantities F and G are defined in (), nonzero indicates stratification, is horizontal component of the rotation vector and is its vertical component. When the quantities and S are absent, F and G reduce to A and B, respectively, in equation (). When the quantities , , S and are all absent, equation () for the extended GN theory reduces to the GN motion equation given in ().