Stationary Solutions and Oscillatory Dynamics in a Mathematical Model of a Saturated Moist Atmosphere

Abstract. This paper investigates the mathematical properties of a simplified atmospheric system describing vertical air flows with water condensation. We provide a rigorous proof for the existence and uniqueness of the stationary solution under specific technical conditions. Numerical simulations reveal damped oscillations in the air flow intensity and liquid water content, interpreted as a self-regulating physical cycle driven by latent heat and droplet accumulation. By establishing a qualitative analogy with a Volterra integro-differential equation, we demonstrate that these oscillations are governed by memory effects. The convergence of the evolutionary variables toward the stationary values provides a robust validation of the model's consistency.