In a pair of recent papers, Tabak, Jacobson, and Milewski present an alternative closure for shocks in two-layer shallow water flow. This closure replaces conservation of mass with conservation of energy, allowing breaking waves to move fluid mass across density surfaces. This thesis extends this type of closure to shallow water flows with arbitrarily many layers, as well as to continuously stratified hydrostatic flows in isopycnal coordinates. The optimization closures introduced here enforce minimal production across shocks of some quantities (such as layer mass) subject to exact conservation of some others (such as energy). After introducing and motivating optimization closures, Chapter 1 develops the analogue of the classical jump conditions for these closures, and develops finite volume methods which enforce optimization closures. These methods are used to test the new jump conditions numerically. Chapter 2 studies shocks and optimization closures in continuously stratified hydrostatic flow in isopycnal coordinates, which emerges as the limit of layered shallow water with many layers. The simple wave problem for this system is reduced to a linear eigenproblem amenable to efficient numerical solution. The simple waves, exact solutions which break nonlinearly, are used as a test bed for studying shocks. It is then shown that optimization closures for this system are equivalent to a nonlocal vertical forcing term. The effects of this forcing term are given an interpretation involving the Bernoulli polynomials, and it is argued that this forcing term indeed mixes the fluid according to two diagnostic criteria for irreversible diapycnal mixing, background potential energy and mixing entropy. Finally, a numerical integration of a breaking simple wave in continuously stratified flow with a optimization closure is presented.
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