The Obukhov–Charney Equation; Rossby Waves

Abstract

The equations of rotating shallow-water (6.21) and (6.22) should not be considered as a completed next step in the expansion of the original equations in parameter ε with respect to the initial approximation (6.18). In fact they exceed the precision O(ε), which was basic for this expansion, as can be seen already from the fact that these equations describe the propagation of long gravitational-gyroscopic waves for which the smallness condition for the Mach and Rossby–Kibel numbers is not valid. For the Earth’s atmosphere, for instance, the group velocity of their propagation nearly coincides with the speed of sound, which corresponds to ε≈1 already for the wave length of one and a half thousand kilometers (see Sect. 7.4 below).

For the further reduction according to Sect. 6.1 we need to turn to the property (III) of geophysical flows, i.e., the conservation equation for the potential vorticity, which singles out the vorticity component of the motion. This will allow us to get rid of “extra” fast processes (e.g., gravitational-gyroscopic waves), for which the potential vorticity vanishes and concentrate on slow geophysical flows, which are naturally vortical (we discuss this question in more detail in Sect. 7.4).