Motion of unstable two interfaces in a three-layer fluid with a non-zero uniform current

Abstract

The nonlinear motion of two interfaces in a three-layer fluid with density stratification is investigated theoretically and numerically. We consider the situation such that a uniform current is present in one of the three layers. The linear dispersion relation is calculated by the Newton's method, from which the initial conditions for numerical computations are determined. When the uniform current is present in the upper (lower) layer, strong vorticity is induced on the upper (lower) interface, and it rolls up involving the other interface at the late stage of computations. When the current is present in the middle layer, a varicose wave appears at the initial stage, and it evolves into an asymmetric heart-shaped vortex sheet at the last computed stage. These phenomena are presented using the vortex sheet model (VSM) with and without regularizations.

Modulational Instability of Nonlinear Wave Packets within (2+4) Korteweg–de Vries Equation

“…Many studies have been devoted to the modulation instability of surface water waves to explain the rogue-wave formation [45][46][47][48][49][50], but in the present study we discuss waves in the interior of the fluid, where the nonlinear instabilities were also predicted [51][52][53][54]. Equation (1) was derived, in particular, for long lowest-mode interfacial waves in a three-layer fluid with an almost symmetric configuration [13,14] in which the lower-order nonlinear terms vanish and higher-order contributions govern the behavior of wave phenomena in the system.…”

“…The region in l is chosen to be limited taking into account the fact that equation ( 1) is valid only in the vicinity of the critical point l = h cr /H = 9/26 (for convenience, this critical value of the layer thickness is marked with a magenta dash-dotted line in Figure 3). The boundary of sign change for parameter q (or the boundary of instability region) is shown in Figure 3 by a white dotted line, instability zone q < 0 lies under this line, it consists of two sub-regions: + (l), in opposite, rises no higher than k+ (l) (53) at the same time. That is, for h < h cr for more intense waves the threshold of modulation instability shifts to the region of longer waves, and the amplitude here is limited from above by the value a 0* , and for h > h cr , on the contrary, the threshold of modulation instability shifts to the region of shorter waves, and the critical amplitude value a 0* must be exceeded for instability to occur.…”