A variational formulation of geophysical fluid motion in non-Eulerian coordinates

Systematic methods to derive geophysical equations of motion possessing conservation laws for energy, momentum and potential vorticity have recently been developed. One approach is based on Hamilton's least action principle in Eulerian curvilinear coordinates and the other is based on the covariance upon time-dependent changes of spatial coordinates. The variational approach unifies, facilitates and generalizes the formulation of a wide range of approximations. The covariant approach makes it straightforward to rewrite approximate equations of motion derived in a particular coordinate system in another coordinate system. The variational approach encompasses models that, like the β-plane and non-traditional shallow-atmosphere approximations, lack a global inertial frame, so that the resulting motion cannot be interpreted as Newtonian motion observed in a non-inertial frame. It has been previously suggested that such models are not covariant. In this work a covariant variational formulation is developed. As a consequence, it is shown that all models previously obtained by the variational approach are covariant upon time-dependent changes of spatial coordinates.