A Tropical Atmosphere Model with Moisture: Global Well-posedness and Relaxation Limit

Abstract

In this paper, we consider a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture; that was derived in [Frierson, D.M.W.; Majda, A.J.; Pauluis, O.M.: Dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit, Commum. Math. Sci., 2 (2004), 591-626.] We establish the global existence and uniqueness of strong solutions to this system, with initial data in $H^1$, for each fixed convective adjustment relaxation time parameter $\epsilon >0$. Moreover, if the initial data enjoy slightly more regularity than $H^1$, then the unique strong solution depends continuously on the initial data. Furthermore, by establishing several appropriate $\epsilon$-independent estimates, we prove that the system converges to a limiting system, as the relaxation time parameter $\epsilon$ tends to zero, with convergence rate of the order $O(\sqrt{\epsilon })$. Moreover, the limiting system has a unique global strong solution, for any initial data in $H^1$, and such unique strong solution depends continuously on the initial data if the the initial data posses slightly more regularity than $H^1$. Notably, this solves the VISCOUS VERSION of an open problem proposed in the above mentioned paper of Frierson, Majda and Pauluis.

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• The primitive equations of the atmosphere in presence of vapor saturation
• Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion
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Cited by in corpus (8)

• Global well-posedness for passively transported nonlinear moisture dynamics with phase changes
• The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: rigorous justification of the hydrostatic approximation
• Hamilton's Principle with Phase Changes and Conservation Principles for Moist Potential Vorticity
• Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ initial data
• Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities
• Moisture dynamics with phase changes coupled to heat-conducting, compressible fluids
• Global regularity for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion
• Global regularity of 2D tropical climate model with zero thermal diffusion