The instability of a horizontal five-layer jet

The instability of a symmetric jet moving horizontally, in which two shear layers with opposite shear of the same strength are separated by a central irrotational layer and are adjoined by unbounded, irrotational outer layers, is studied.First, the fluid is assumed to be homogeneous. Two unstable modes are found, the central wave one-quarter wave length out of phase with the outer wave. Mode I consists of central waves being in phase and outer waves being in phase. Mode II consists of central waves being in opposite phase and outer waves being in opposite phase. For a given width of the jet, the thicker the central irrotational layer, the stronger the shear of the shear layers, the stronger the instability. For a fixed ratio of the thickness of central layer to that of the shear layers, mode I is more unstable than mode II.Next, a density jump across the outer interface levels and another density jump across the central interface levels are introduced. The effect of these density jumps on mode I is to reduce the growth of the wave. The wave with equal density jump across every interface level grows somewhat slower than the waves with the entire density jump across outer or central interface levels. For an idealized velocity profile with isentropic layers with an overall Richardson number of 4.9, the linear theory predicts that the amplitude of the wave doubles in about 5 min and the wave-length is 241 m, which compares favorably with 320m obtained in the boundary layer by Gossard et al. (1970). For atmospheric parameters with an overall Richardson number of unity, linear theory predicts that the amplitude of the wave doubles in about % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaale% aaleaacaaIXaaabaGaaG4maaaaaaa!383C!\2{\textstyle{1 \over 3}}\] min and the wave-length is about 510 m, which is only slightly larger than the width of the jet.A physical argument is invoked to explain the evolution of finite-amplitude waves.