 
Summary: Under consideration for publication in J. Fluid Mech. 1
Nonnormal stability analysis
of a shear current
under surface gravity waves
D. AMBROSI1 and M. ONORATO2
1
Dip. di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
2
Dip. di Fisica Generale, Universit`a di Torino, via Pietro Giuria 1, 10125 Torino, Italy
(Received 13 May 2008)
The stability of a horizontal shear current under surface gravity waves is investigated on
the basis of the Rayleigh equation. As the differential operator is nonnormal, a standard
modal analysis is not effective in capturing the transient growth of a perturbation. The
representation of the stream function by a suitable basis of biorthogonal eigenfunctions
allows one to determine the maximum growth rate of a perturbation. It turns out that, in
the considered range of parameters, such a growth rate can be two orders of magnitude
larger than the maximum eigenvalue obtained by standard modal analysis.
1. Introduction
A longstanding problem in fluid mechanics is the stability of a shear flow bounded
by longcrested gravity waves. After Burns (1953), this topic has received wide attention
