Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows
Shepherd, T. G. (1988) Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows. Journal Of Fluid Mechanics, 196. pp. 291-322. ISSN 0022-1120
To link to this article DOI: 10.1017/S002211208800271X
A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.