This document includes a variety of stochastic systems of equations that describe the flow of shallow water. Though we typically do not consider the oceans to be shallow, when comparing the miles of depth to the thousands of miles of breadth, the oceans can in fact be assumed to be shallow in a relative sense. In the first chapter, we consider the barotropic quasigeostrophic equations used to model the circulation of currents in the ocean. In the remaining two chapters, we consider the one and two layer shallow water equations. With modest assumptions on the initial data and assuming that the stochastic terms satisfy certain Lipschitz conditions, we obtain the existence and uniqueness of a maximal pathwise solution to all of these problems. In the barotropic quasigeostrophic model, the maximal solution is actually global in time. In order to obtain global solutions for the single and double layer shallow water models, we must utilize the classical small initial data assumption. The proofs rely on the Galerkin scheme, Cauchy estimates, the Skorohod representation theorem, the Gyongy-Krylov theorem, stopping time arguments, and anisotropic estimates.
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