The deep-atmosphere Euler equations: Explicit solutions and perturbation analysis

We present a method to construct a family of explicit solutions to the deep-atmosphere Euler equations in spherical coordinates. These solutions represent steady purely–azimuthal flows accommodating general stratification that depends on the height and latitude. After prescribing the stratification we derive explicit formulas for the velocity field and for the pressure. From the gas law the temperature is determined explicitly and the heat sources are identified as well from the first law of thermodynamics. By expressing the solution in spherical coordinates, we avoid making any approximations that would simplify the geometry in the governing equations. Following the construction of our explicit solution, we address the suitability of these solutions to describe physically realistic atmospheric flows. By analyzing different choices of the density profile, we demonstrate that the resulting temperature distribution may either decrease (cf. Proposition 3.1, formula (3.27)) or increase (cf. Proposition 3.3, formula (3.38)) with altitude. Both behaviors are physically relevant and correspond to distinct atmospheric regimes. Our study concludes with a short-wavelength stability analysis, revealing that certain flows are stable for a wide range of density distributions that depend on the depth, and on the depth and latitude, respectively.