baroclinic instability in a pizza box
I describe the domain as a "pizza box" because it is a shallow box.
Two simulations are shown: a lowresolution simulation with 41×41×11 grid points
and a highresolution simulation with 161×161×21 grid points.
In distance, it is much more shallow: 2000×2000×1.
So we can think of the domain as a box 20,000 km on a side and 10 km deep, draped over the
north pole ... but we keep the geometry Cartesian ... an fplane.

Computation is with Python and numpy.

In the simulations shown here, the Rossby radius of deformation is 500.

The initial state is stably stratified,
with the dense fluid gathered toward the center, in thermal wind balance.

The initial polar vortex is set slightly offcenter so that the perturbation is not purely
a wave number 4.

The boundaries are freeslip.

We initialize with zero thermal wind at the surface, and so put
a "jet stream" aloft. The makes all the weather at the surface very quiet, until the baroclinic instability
develops.

The Boussinesq approximation is applied.
Description of the plots
Upper left: horizontal crosssection at the surface
 White contours: pressure anomaly at the surface.
 Color contours: fluid density at the surface.
 Green contours: pressure anomaly at midheight.
 Black vectors: wind at the surface.
Lower left: vertical crosssection at mid y
 Color contours: fluid density
 Grey contours: wind. dashed: coming at you; solid: going away.
Right: time trace
 red: total kinetic energy in domain
 blue: total potential energy anomaly in domain
 green: sum of red and blue
Hires: 161x161x21
The highresolution simulation.
If you want to see it: the lowresolution simulation