| Gravity currents, produced by the instantaneous release of a finite volume of dense fluid beneath a layer of lighter fluid and overlying a spatially-varying rigid bottom boundary, are modelled as discontinuous solutions to the systems of nonlinear hyperbolic conservation laws arising from a shallow-water model.;Equations of motion for two stably-stratified fluids of constant density are derived for the incompressible Navier-Stokes Equations for small aspect ratio flow in an Eulerian fluid, and the equations are nondimensionalized using a gravity current scaling so that they may be stated as a first order system of partial differential equations. The model equations neglect the effects of turbulence, entrainment, density stratification, and viscosity, but include the Coriolis force, variable topography, and bottom friction. Special cases are stated for one-layer three-dimensional axisymmetric flow, and in the two-dimensional case for flow with a free surface, rigid lid, thin upper or lower layer, or small density differences. These equations are then stated as a nonlinear system of conservation laws.;The model equations are classified as hyperbolic, with defined regions of hyperbolicity stated where possible. When in conservation form, discontinuous solutions are considered, and the Rankine-Hugoniot jump conditions derived for solutions which are trivial on one side of the shock. The initial release problem is shown to be well-posed by the method of localization.;By approximating a gravity current front as a vertical discontinuity, the initial release problem is solved numerically by use of a relaxation method designed for systems of hyperbolic conservation laws and adapted to include boundary conditions and forcing terms. The usefulness of this method is demonstrated by several diagrams which show the effects of bottom slope and friction in the two-dimensional case, and of bottom slope and rotation in the three-dimensional one.;Since the relaxation method is applicable to systems in conservation form, a result is proved showing that an infinite number of polynomial conservation laws do not exist for the two-layer shallow-water equations in one spatial dimension, and it is conjectured that this is the case for one layer in two dimensions. The conservation laws which are known to exist are described, and correspond to the conserved quantities of mass, momentum, and energy. |
