Instability of marginally stable streamwise varying shear flow to long Rossby. (English) Zbl 1218.76013

Hodyss, Daniel; Nathan, Terrence R.

Commun. Math. Sci. 5, No. 4, 833-849 (2007).

Summary: The instability of streamwise varying shear flow that is marginally stable to long Rossby waves is examined. Both Hamiltonian and non-Hamiltonian flows are considered within the framework of coupled wave instability (CWI). The CWI is shown to be mediated by a ‘physical’ wave and a ‘virtual’ wave. The physical/virtual wave model concisely describes the differences between the instabilities that develop on locally supercritical flow and those that develop on globally subcritical flow. Globally subcritical Hamiltonian flows are proven to be stable. In contrast, non-Hamiltonian flows may be unstable in both locally supercritical and globally subcritical regimes. In locally supercritical flow, the CWI grows via wave-resonance between the physical/virtual wave pair and is highly localized to the supercritical region. In contrast, in globally subcritical flow, the CWI grows via pseudomomentum extraction from the background flow and either radiates away or remains attached to the streamwise variation in the flow.

MSC:

76B65	Rossby waves (MSC2010)
76E09	Stability and instability of nonparallel flows in hydrodynamic stability
76E15	Absolute and convective instability and stability in hydrodynamic stability
76E20	Stability and instability of geophysical and astrophysical flows

Keywords:

linear instability; coupled wave; long Rossby waves; streamwise varying