The instability to longitudinal vortices of two-dimensional density-stratified temporally evolving wavy shear flow is considered. The problem is posited in the context
of Langmuir circulations, LCs, beneath wind-driven surface waves and the instability
mechanism is generalized Craik–Leibovich, either CLg or CL2. Of interest is the
influence of non-stationary base flows on the instability according to linear theory. It
is found that the instability is described by a family of similarity solutions and that the
growth rate of the instability, in non-stationary base flows, is doubly exponential in
time, although the growth rate reduces to exponential when the base flow is stationary.
An example is given for weakly sheared wind-driven flow evolving in the presence of
growing irrotational surface waves. Waves aligned both with the wind and counter
to it are considered, as is the role of stratification. Antecedent to the example is
an initial value problem posed by Leibovich & Paolucci (1981) for neutral waves in
slowly evolving shear. Here, however, the waves and shear may grow (or decay) at
rates comparable with the LCs. Furthermore the current here has two components: a
wind-driven portion due to the wind stress applied at the free surface and a second
due to the diffusion of momentum due to the wave-amplitude-squared free-surface
stress condition. Using the case for neutral waves in non-stratified uniform shear for
reference, it is found, in general, that growing waves are stabilizing while decaying
waves are destabilizing to the formation of LCs, although the latter applies only for
sufficiently large spanwise spacings and is subject to a globally stable lower bound.
Decaying waves in the absence of wind can also be destabilizing to LCs. When the
wind is counter to the waves, however, only decaying waves are unstable to LCs.
Furthermore, while growing waves are stable to the formation of LCs in the presence
of stable stratification, decaying waves are unstable in both aligned and opposed
wind-wave conditions. Unstable stratification on the other hand, is destabilizing to
LCs for all temporal waves in both aligned and opposed wind-wave conditions.
Waves’ Numerical Dispersion and Damping due to Discrete Dispersion Relation
