Abstract
In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate "double stagnation" point. Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.
Highlights
The world’s oceans are stratified in the sense that the fluid density increases with depth
In many settings there are two regions with nearly constant density separated by a thin layer, called the pycnocline, where density gradients are large
This permits the system to be modeled as two constant density fluids with different densities, divided by a sharp interface along which waves can propagate
Summary
The world’s oceans are stratified in the sense that the fluid density increases with depth. In many settings there are two regions with nearly constant density separated by a thin layer, called the pycnocline, where density gradients are large This permits the system to be modeled as two constant density fluids with different densities, divided by a sharp interface along which waves can propagate. Unlike surface waves in a homogeneous density fluid, these internal waves can take the form of fronts or (smooth) hydrodynamical bores These are steady solutions where the internal interface is asymptotically flat both upstream and downstream of the wave but with different heights. Alternative proofs have subsequently been given using different methods by Mielke [28], Makarenko [25], and the authors [7] In this announcement, we report the first construction of genuinely large-amplitude bores. One can no longer expect to base such an analysis on a well-chosen model equation, and instead we rely on a new abstract global bifurcation theory tailored to front-type solutions of elliptic equations in cylindrical domains [8]
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