Beyond Linear Decomposition: a Nonlinear Eigenspace Decomposition for a Moist Atmosphere with Clouds

Abstract: A linear decomposition of states underpins many classical systems. This is the case of the Helmholtz decomposition, used to split vector fields into divergence-free and potential components, and of the dry Boussinesq system in atmospheric dynamics, where identifying the slow and fast components of the flow can be viewed as a decomposition. The dry Boussinesq system incorporates two leading ingredients of mid-latitude atmospheric motion: rotation and stratification. In both cases the leading order dynamics are linear so we can rely on an eigendecomposition to decompose states. Here we study the extension of dry Boussinesq to incorporate another important ingredient in the atmosphere: moisture and clouds. The key challenge with this system is that nonlinearities are present at leading order due to phase boundaries at cloud edge. Therefore standard tools of linear algebra, relying on eigenvalues and eigenvectors, are not applicable. The question we address in this paper is this: in spite of the nonlinearities, can we find a decomposition for this moist Boussinesq system? We identify such a decomposition adapted to the nonlinear balances arising from water phase boundaries. This decomposition combines perspectives from partial differential equations (PDEs), the geometry, and the conserved energy. Moreover it sheds light on two aspects of previous work. First, this decomposition shows that the nonlinear elliptic PDE used for potential vorticity and moisture inversion can be used outside the limiting system where it was first derived. Second, we are able to rigorously justify, and interpret geometrically, an existing numerical method for this elliptic PDE. This decomposition may be important in applications because, like its linear counterparts, it may be used to analyze observational data. Moreover, by contrast with previous decompositions, it may be used even in the presence of clouds.