J. Vanneste, EXPLOSIVE RESONANT INTERACTION OF BAROCLINIC ROSSBY WAVES AND STABILITY OF MULTILAYER QUASI-GEOSTROPHIC FLOW, Journal of Fluid Mechanics, 291, 1995, pp. 83-107
The amplitude equations governing the nonlinear interaction among norm
al modes are derived for a multilayer quasi-geostrophic channel. The s
et of normal modes can represent any wavy disturbance to a parallel sh
ear flow, which may be stable or unstable. Orthogonality in the sense
of pseudomomentum or pseudoenergy is used to obtain the amplitude equa
tions in a direct fashion, and pseudoenergy and pseudomomentum conserv
ation laws permit the properties of the interaction coefficients to be
deduced. Particular attention is paid to triads exhibiting explosive
resonant interaction, as they lead to nonlinear instability of the bas
ic flow. The relationship between this mechanism and the most recently
discovered nonlinear stability conditions is discussed. Situations in
which the basic velocity is constant in each layer are treated in det
ail. A particular formulation of the stability condition is given that
emphasizes the close connection between linear and nonlinear stabilit
y. It is established that this stability condition is also a necessary
condition: when it is not satisfied, and when the flow is linearly st
able, explosive resonant interaction of baroclinic Rossby waves acts a
s a destabilizing mechanism. Two- and three-layer models are specifica
lly considered; their stability features are presented in the form of
stability diagrams, and interaction coefficients are calculated in par
ticular cases.