Extreme internal waves: gravity currents and overturning fronts

Abstract

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. The velocity field in each layer is assumed to be incompressible and irrotational, and it limits to distinct laminar flows upstream and downstream. Rigid horizontal boundaries confine the fluids from above and below. A constant gravitational force acts on the waves, but surface tension is neglected. It was recently shown by the authors that there exist two large-amplitude families of hydrodynamic bores: a curve of depression bores and a curve of elevation bores. We now prove that in the limit along the elevation bore family, the solutions must overturn: the interface separating the layers develops a vertical tangent. This type of behavior was first observed over 45 years ago in numerical computations of internal gravity waves and gravity water waves with vorticity. Despite considerable progress over the past decade in constructing families of water waves that potentially overturn, a proof that overturning definitively occurs has been stubbornly elusive. We further show that in the limit along the depression bore family, either overturning occurs or the solutions converge to a gravity current: the free boundary contacts the upper wall and the relative velocity in the upper fluid is stagnant. We also determine the contact angle between the interface and the rigid barrier for the limiting gravity current, giving the first rigorous confirmation of a conjecture of von K\'arm\'an. The resolutions of these questions in the specific case of hydrodynamic bores is accomplished through the use of novel geometric analysis techniques, including bounds on the decay of the velocity field near a hypothetical double stagnation point. These ideas may have broader applications to bifurcation theoretic studies of large-amplitude waves.