Abstract
This chapter discusses the stability of steady, finite amplitude interfacial waves in the presence of a current jump to infinitesimal three-dimensional disturbances. The flow is assumed to be inviscid, incompressible, and irrotational; and the effect of surface tension is neglected. The presence of a current jump at the interface causes the flat surface to be unstable to disturbances of sufficiently short wavelength. This short wave instability is the “Kelvin-Helmholtz instability,” which can be suppressed by the inclusion of surface tension. The chapter focuses on the instabilities of longer scales, which are Kelvin-Helmholtz stable. The present study establishes the presence of long wave instabilities different from the classical Kelvin-Helmholtz type, which might be dominant when the unperturbed waves are of finite amplitude. Stable eigenvalues appear in pairs, whereas unstable eigenvalues appear in quartets. The transition from stability to instability is marked by the coalescence of pairs of eigenvalues, which can also be interpreted as a resonance between two modes that have the same frequency in the frame of reference moving with the unperturbed wave.
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