Raphael Stuhlmeier: Instability and evolution of inhomogeneous, broad-banded seas

Submitted by Jeremy Upsal on

Speaker: Raphael Stuhlmeier, University of Plymouth

Date: December 6, 2018

Title: Instability and evolution of inhomogeneous, broad-banded seas

Abstract: Nonlinear interaction, along with wind input and dissipation, is one of the three drivers of wave evolution at sea, and is included in every modern wave-forecast model. The mechanism behind the nonlinear interaction terms in such models is based on the kinetic equation for wave spectra derived by Hasselmann. This does not allow, for example, for statistically inhomogeneous wave fields, nor for the modulational instability which depends on such inhomogeneity, and which has been implicated in the appearance of exceptionally high rogue waves.

Beginning with the basics of third-order wave theory, we sketch the derivation of a discretized equation for the evolution of random, inhomogeneous surface wave fields on deep water from Zakharov's equation, along lines first laid out by Crawford, Saffman, and Yuen. This allows for a general treatment of the stability and long-time behaviour of broad-banded sea states. It is investigated for the simple case of degenerate four-wave interaction, and the instability of statistically homogeneous states to small inhomogeneous disturbances is demonstrated. Furthermore, the long-time evolution is studied for several cases and shown to lead to a complex spatio-temporal energy distribution. The possible impact of this evolution on the statistics of rogue wave occurrence is explored within the framework of this simplified example.