Abstract:

Perturbation growth in uncertain systems is examined and related to previous work in which linear stability 
concepts were generalized from a perspective based on the nonnormality of the underlying linear operator. In 
this previous work the linear operator, subject to an initial perturbation or a stochastic forcing distributed in 
time, was either fixed or time varying, but in either case the operator was certain. However, in forecast and 
climate studies, complete knowledge of the dynamical system being perturbed is generally lacking; nevertheless, 
it is often the case that statistical properties characterizing the variability of the dynamical system are known. 
In the present work generalized stability theory is extended to such uncertain systems. The limits in which 
fluctuations about the mean of the operator are correlated over time intervals, short and long, compared to the 
timescale of the mean operator are examined and compared with the physically important transitional case of 
operator fluctuation on timescales comparable to the timescales of the mean operator. Exact and asymptotically 
valid equations for transient ensemble mean and moment growth in uncertain systems are derived and solved. 
In addition, exact and asymptotically valid equations for the ensemble mean response of a stable uncertain 
system to deterministic forcing are derived and solved. The ensemble mean response of the forced stable uncertain 
system obtained from this analysis is interpreted under the ergodic assumption as equal to the time mean of the 
state of the uncertain system as recorded by an averaging instrument. Optimal perturbations are obtained for 
the ensemble mean of an uncertain system in the case of harmonic forcing. Finally, it is shown that the remarkable 
systematic increase in asymptotic growth rate with moment in uncertain systems occurs only in the context of 
the ensemble.