Theory of Rossby Waves

Abstract

We shall now consider adiabatic motions without external forces. Then in the quasistatic and Boussinesq approximations the equations of motion, mass conservation, and entropy evolution have the form (see Kamenkovich and Monin, 1978, §§2 and 5):

$$\frac{{du}}{{dt}} - \frac{{uv}}{a}\tan \varphi - fv = - \frac{1}{{{\varrho _0}}}\frac{{\partial p}}{{a{\text{ cos }}\varphi {\text{ }}\partial \lambda }};$$

((1.1))

$$\frac{{dv}}{{dt}} + \frac{{{u^2}}}{a}\tan \varphi + fu = - \frac{1}{{{\varrho _0}}}\frac{{\partial p}}{{a\partial \varphi }};$$

((1.2))

$$\frac{{\partial p}}{{\partial z}} = - g\varrho ;$$

((1.3))

$$\frac{{\partial u}}{{a\cos \varphi {\text{ }}\partial \lambda }} + \frac{1}{{\cos {\text{ }}\varphi }}\frac{\partial }{{a{\text{ }}\partial \varphi }}\left( {v\cos {\text{ }}\varphi } \right) + \frac{{\partial w}}{{\partial z}} = 0;$$

((1.4))

$$\frac{{{\text{d}}\varrho }}{{{\text{d}}t}} = \frac{1}{{{c^2}}}\frac{{{\text{d}}p}}{{{\text{d}}t}},$$

((1.5))

where

$$\frac{{\text{d}}}{{{\text{d}}t}} = \frac{\partial }{{\partial t}} + \frac{u}{{a\cos {\text{ }}\varphi {\text{ }}}}\frac{\partial }{{\partial \lambda }} + \frac{v}{a}\frac{\partial }{{\partial \varphi }} + w\frac{\partial }{{\partial z}};{\text{ }}f = 2{\omega _e}\sin \;\varphi {\text{.}}$$

((1.6))