Kelvin-Helmholtz instability as a boundary-value problem

The Kelvin-Helmholtz (KH) instability is traditionally viewed as an initial-value problem, wherein wave perturbations of a two-layer shear flow grow over time into billows and eventually generate vertical mixing. Yet, the instability can also be viewed as a boundary-value problem. In such a framework, there exists an upstream condition where a lighter fluid flows over a denser fluid, wave perturbations grow downstream to eventually overturn some distance away from the point of origin. As the reverse of the traditional problem, this flow is periodic in time and exhibits instability in space. A natural application is the mixing of a warmer river emptying into a colder lake or reservoir, or the salt-wedge estuary. This study of the KH instability from the perspective of a boundary-value problem is divided into two parts. Firstly, the instability theory is conducted with a real frequency and complex horizontal wavenumber, and the main result is that the critical wavelength at the instability threshold is longer in the boundary-value than in the initial-value situation. Secondly, mass, momentum and energy budgets are performed between the upstream, unmixed state on one side, and the downstream, mixed state on the other, to determine under which condition mixing is energetically possible. Cases with a rigid lid and free surface are treated separately. And, although the algebra is somewhat complicated, both end results are identical to the criterion for complete mixing in the initial-value problem.