which is linear in and has simple sinusoidal solutions of the form:
where and are constant coefficients and is given by:
It is interesting to note that eq.(71) is identical to the equation governing stationary, barotropic Rossby waves under the -plane assumption where, in that case, -- the tangent-plane approximation to the meridional gradient of the Coriolis parameter. Eq.(70) is then equivalent to the conservation of absolute vorticity. In the context of stratified flow, eq.(73) is the dispersion relation for stationary gravity
waves in a uniform flow with constant stability. It is identical to the dispersion relation arising in the linearized gravity wave problem yet here the solution is exact within the context of the AQB equations.
The Moncrieff/Green conservation law eq.(69) is closely related to the conservation law derived by Long (1953) for an incompressible fluid and further refined by Yih (1965). Since Long's equation is of fundamental significance in the dynamics of incompressible stratified flow over two-dimensional mountains, it will reviewed here. The derivation that follows is that of Yih (1965) since his form of conservation principle which resembles
eq.(69).
Yih begins with the unapproximated Euler equations of motions for a steady, incompressible fluid i.e.
(the equation of state need not concern us here).
Eq.(75) (incompressibility relation) implies that the velocity is non-divergent for steady flow i.e.