Stability, Bifurcations and Explicit Solutions in Geophysical Fluid Models with Simplified Backscatter

Abstract

Analytical investigation methods are used in this work in order to study a simplified form of a numerical scheme, the so-called kinetic energy backscatter scheme, which models the effects of unresolved small scales on the resolved large scales in the simulations of geophysical flows in atmosphere and oceans. This thesis is divided into three main parts, an additional introduction in the beginning about the thesis and the topics it contains as well as a short outlook in the end about further ideas for research topics based on the results of this work. In the first main part certain traveling wave functions are presented, which can be used in order to explicitly solve nonlinear fluid equations. Their properties are studied, in particular possible superposition among these functions, which also solves the underlying fluid equations. The occurrence of such explicit solutions is studied in the incompressible Euler, Navier-Stokes and rotating Boussinesq equations, as well as in more general fluid equations with certain forcing terms. In the end of this part the new contributions of the presented solutions are pointed out and a comparison of them with already known ones is provided. In the second main part the rotating shallow water and Boussinesq equations with simplified kinetic energy backscatter and hyperviscosity are analyzed in a continuous setting. For these investigations the functions presented in the first part are used and numerous solutions that grow exponentially and unboundedly in time are found, which indicates the possibility of undesired energy concentration into specific modes due to the backscatter. The stability of trivial as well as certain nontrivial steady solutions are studied and the occurrence of unboundedly growing unstable perturbations shown in certain cases. In the third main part the rotating shallow water equations with backscatter, hyperdiffusion and linear as well as non-smooth quadratic bottom drag terms are studied. The provided stability analysis shows, that by decreasing the linear bottom drag the trivial flow becomes unstable after a certain threshold, which generates nonlinear flows. These nonlinear flows are investigated in more details. For isotropic backscatter and hyperdiffusion the simultaneous supercritical bifurcation of (steady) Rossby waves and (temporally oscillating) inertia-gravity waves is proved, while in the anisotropic case only Rossby waves primarily bifurcate. The bifurcation results are illustrated by numerical computations and branches are extended in parameter space beyond the analytical investigations. Furthermore, it is shown that purely smooth bottom drag cannot completely suppress the occurrence of explicit solutions as presented in the second part, so that steady and unboundedly growing explicit flows can exist in this case as well.