Atmospheric dynamics

Definition

Atmospheric dynamics is the study of the motion of the atmosphere of a rotating planet. The atmosphere is modelled as a compressible gas under gravity, in a frame rotating with angular velocity $\boldsymbol{\Omega}$. Its state is given by the velocity $\mathbf{u}$, pressure $p$, density $\rho$ and temperature $T$, which satisfy:

1. the momentum equation $\dfrac{D\mathbf{u}}{Dt} + 2 \boldsymbol{\Omega} \times \mathbf{u} = -\dfrac{1}{\rho} \nabla p + \mathbf{g} + \mathbf{F}$, where $D/Dt$ is the material derivative, $2\boldsymbol{\Omega} \times \mathbf{u}$ is the Coriolis term, $\mathbf{g}$ is the effective gravity and $\mathbf{F}$ is friction;
2. the continuity equation $\dfrac{D\rho}{Dt} + \rho \, \nabla \cdot \mathbf{u} = 0$;
3. the thermodynamic energy equation $c_p \dfrac{DT}{Dt} - \dfrac{1}{\rho} \dfrac{Dp}{Dt} = Q$, with $c_p$ the specific heat at constant pressure and $Q$ the heating rate per unit mass;
4. the ideal gas law $p = \rho R_d T$, with $R_d$ the gas constant of dry air.