Theoretical understanding of the growth of wind-driven surface water waves has been based on two distinct mechanisms: growth due to random atmospheric pressure fluctuations unrelated to wave amplitude and growth due to wave coherent atmospheric pressure fluctuations proportional to wave amplitude. Wave-independent random pressure forcing produces wave growth linear in time, while coherent forcing proportional to wave amplitude produces exponential growth. While observed wave development can be parameterized to fit these functional forms and despite broad agreement on the underlying physical process of momentum transfer from the atmospheric boundary layer shear flow to the water waves by atmospheric pressure fluctuations, quantitative agreement between theory and field observations of wave growth has proved elusive. Notably, wave growth rates are observed to exceed laminar instability predictions under gusty conditions. In this work, a mechanism is described that produces the observed enhancement of growth rates in gusty conditions while reducing to laminar instability growth rates as gustiness vanishes. This stochastic parametric instability mechanism is an example of the universal process of destabilization of nearly all time-dependent flows.

Asymptotic linear stability of time-dependent flows is examined by extending to nonautonomous systems 
methods of nonnormal analysis that were recently developed for studying the stability of autonomous systems. 
In the case of either an autonomous or a nonautonomous operator, singular value decomposition (SVD) analysis 
of the propagator leads to identification of a complete set of optimal perturbations ordered according to the 
extent of growth over a chosen time interval as measured in a chosen inner product generated norm. The long-
time asymptotic structure in the case of an autonomous operator is the norm-independent, most rapidly growing 
normal mode while in the case of the nonautonomous operator it is the first Lyapunov vector that grows at the 
norm independent mean rate of the first Lyapunov exponent. While information about the first normal mode 
such as its structure, energetics, vorticity budget, and growth rate are easily accessible through eigenanalysis of 
the dynamical operator, analogous information about the first Lyapunov vector is less easily obtained. In this 
work the stability of time-dependent deterministic and stochastic dynamical operators is examined in order to 
obtain a better understanding of the asymptotic stability of time-dependent systems and the nature of the first 
Lyapunov vector. Among the results are a mechanistic physical understanding of the time-dependent instability 
process, necessary conditions on the time dependence of an operator in order for destabilization to occur, 
understanding of why the Rayleigh theorem does not constrain the stability of time-dependent flows, the de-
pendence of the first Lyapunov exponent on quantities characterizing the dynamical system, and identification 
of dynamical processes determining the time-dependent structure of the first Lyapunov vector.