Uniqueness of solutions and conservation laws for the quasigeostrophic model

In this study we investigate which conditions are needed to assure uniqueness of solutions of the barotropic QG model, and how these conditions are related to the conservation laws of mass, vorticity and energy. Uniqueness and conservation laws are analyzed for a simply connected domain (a closed basin) and for a doubly connected domain (a periodic or re-entrant channel). For the multiply connected domain we find, besides the model proposed by McWilliams in 1977, another consistent model whose solutions satisfy the conservation laws. The additional conditions in our model are: (a) the sum of the circulations around all closed solid walls is time-independent; (b) the value of the QG streamfunction (psi) is the same at all closed walls. McWilliams' model and ours are not equivalent. As a simple application we study the free Rossby normal modes in a channel. For a non-zonal channel on a beta-plane there are solutions (modes) that are independent of the coordinate along the channel. These are used to compare the modes and frequencies obtained from three different, but well posed, models, Solutions that are independent of the along-channel coordinate do not exist for the planetary geostrophic and for the shallow water equations on a beta-plane.