  We consider a class of steady solutions of the semi-geostrophic equations on
$\mathbb{R}^3$ and derive the linearised dynamics around those solutions. The
linear PDE which governs perturbations around those steady states is a
transport equation featuring a pseudo-differential operator of order 0. We
study well-posedness of this equation in $L^2(\mathbb{R}^3;\mathbb{R}^3)$
introducing a representation formula for the solutions, and extend the result
to the space of tempered distributions on $\mathbb{R}^{3}$. We investigate
stability of the steady solutions by looking at plane wave solutions of the
linearised problem, and discuss differences in the case of the
quasi-geostrophic equations.
