The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our
understanding of the dynamics and their numerical approximation. This paper presents a preliminary
numerical study of one type of phase averaging applied to mean flows in the 2d Boussinesq equations
that also has application to numerical ...
The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our
understanding of the dynamics and their numerical approximation. This paper presents a preliminary
numerical study of one type of phase averaging applied to mean flows in the 2d Boussinesq equations
that also has application to numerical methods. The phase averaging technique, well-known in
dynamical systems theory, relies on a mapping using the exponential operator, and then averaging
over the phase. The exponential operator has connections to the Craya-Herring basis pioneered by
Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper we
perform numerical experiments to study the effect of this averaging technique on the time evolution
of the solution. We explore its potential as a definition for mean-flows. We also show that, as expected
from theory, the phase averaging method can reduce the magnitude of the time rate of change of the
PDEs making them potentially suitable for time stepping methods.
