  A vertical slice model is developed for the Euler-Boussinesq equations with a
constant temperature gradient in the direction normal to the slice (the
Eady-Boussinesq model). The model is a solution of the full three-dimensional
equations with no variation normal to the slice, which is an idealized problem
used to study the formation and subsequent evolution of weather fronts. A
compatible finite element method is used to discretise the governing equations.
To extend the Charney-Phillips grid staggering in the compatible finite element
framework, we use the same node locations for buoyancy as the vertical part of
velocity and apply a transport scheme for a partially continuous finite element
space. For the time discretisation, we solve the semi-implicit equations
together with an explicit strong-stability-preserving Runge-Kutta scheme to all
of the advection terms. The model reproduces several quasi-periodic lifecycles
of fronts despite the presence of strong discontinuities. An asymptotic limit
analysis based on the semi-geostrophic theory shows that the model solutions
are converging to a solution in cross-front geostrophic balance. The results
are consistent with the previous results using finite difference methods,
indicating that the compatible finite element method is performing as well as
finite difference methods for this test problem. We observe dissipation of
kinetic energy of the cross-front velocity in the model due to the lack of
resolution at the fronts, even though the energy loss is not likely to account
for the large gap on the strength of the fronts between the model result and
the semi-geostrophic limit solution.
