Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2018 / 6 / a

Linearize about a uniform horizontal flow $U$ and a hydrostatic, vertically stratified basic state. The basic-flow material derivative is $D_0={\partial }_t+U{\partial }_x$. The first two equations are the horizontal and vertical perturbation momentum balances: pressure gradients accelerate the disturbance, while the vertical equation retains the buoyancy force $-g{\rho }^{\prime }/{\rho }_0$ after the basic hydrostatic balance is subtracted.

The Boussinesq approximation neglects fractional density variations in inertia and enforces incompressibility, $u_x+w_z=0$, but retains them in buoyancy. Advecting the stable background density gradient gives $D_0{\rho }^{\prime }+wd\bar{\rho }/dz=0$. With $N^2=-(g/{\rho }_0)d\bar{\rho }/dz>0$, this is the fourth supplied equation. A displaced parcel then has a restoring buoyancy tendency.

Small amplitude allows products of perturbations to be discarded; two-dimensional motion removes $y$ dependence and transverse flow. Rotation, viscosity, diffusion, diabatic heating and mean shear are neglected. ${\rho }_0$ is the reference inertial density, whereas the small background density gradient still sets $N$. In a compressible real atmosphere the appropriate stability measure involves potential temperature; this Boussinesq system is a local idealization that filters acoustic waves rather than a full compressible atmospheric model.