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 of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.

我们考虑\（{{mathbb {R}} ^ 3 \）上一类半地转方程的稳定解，并得出围绕这些解的线性化动力学。控制那些稳态周围扰动的线性PDE是一个输运方程，具有零阶伪微分算子。我们在\（L ^ 2（{\ mathbb {R}} ^ 3，{ \ mathbb {R}} ^ 3）\）引入解决方案的表示公式，并将结果扩展到\（{\ mathbb {R}} ^ {3} \）上的缓和分布空间。通过研究相关线性化问题的平面波解，我们研究了半地转方程稳定解的稳定性，并讨论了准地转方程情况下的差异。