Influence of nonlinear interactions on the development of instability in hydrodynamic wave systems (original) (raw)
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Instability and Evolution of Nonlinearly Interacting Water Waves
Physical Review Letters, 2006
We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schrödinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.
Physical Review Fluids, 2017
Both surface tension and buoyancy force in stable stratification act to restore perturbed interfaces back to their initial positions. Hence, both are intuitively considered as stabilizing agents. Nevertheless, the Taylor-Caulfield instability is a counterexample in which the presence of buoyancy forces in stable stratification destabilize shear flows. An explanation for this instability lies in the fact that stable stratification supports the existence of gravity waves. When two vertically separated gravity waves propagate horizontally against the shear, they may become phase locked and amplify each other to form a resonance instability. Surface tension is similar to buoyancy but its restoring mechanism is more efficient at small wavelengths. Here, we show how a modification of the Taylor-Caulfield configuration, including two interfaces between three stably stratified immiscible fluids, supports interfacial capillary gravity whose interaction yields resonance instability. Furthermore, when the three fluids have the same density, an instability arises solely due to a pure counterpropagating capillary wave resonance. The linear stability analysis predicts a maximum growth rate of the pure capillary wave instability for an intermediate value of surface tension corresponding to We −1 = 5, where We denotes the Weber number. We perform direct numerical nonlinear simulation of this flow and find nonlinear destabilization when 2 We −1 10, in good agreement with the linear stability analysis. The instability is present also when viscosity is introduced, although it is gradually damped and eventually quenched.
Shear instability for stratified hydrostatic flows
Communications on Pure and Applied Mathematics, 2009
Stratified flows in hydrostatic balance are studied in both their multilayer and continuous formulations. Novel stability criteria, both linear and nonlinear, are developed, both for general systems of mixed type and for the specific cases of contiuously stratified and two and three layer flows. These criteria go beyond the classic results concerning steady and planar flows, into the realm of flows that are non-uniform and unsteady. A nonlinear map is established between baroclynic and barotropic flows, as described by a two-layer system with rigid lids and a single layer with free surface respectively. The global emphasis is on the dynamics of stratified flows as they lead to wave breaking or shear instability. This sets the scenario for a future in-depth study of fluid mixing.
Strong dispersive effects on internal nonlinear waves in a sheared, stably stratified fluid layer
Wave Motion, 1999
Internal solitary waves are widely believed to propagate due to a balance between nonlinearity and dispersion. The expansion procedure introduced by Benney (J. Math. Phys. 45 (1966) 52-63) for weakly nonlinear, planar waves in sheared, stratified flows in shallow layers, approximates the motion by the Korteweg-de Vries equation (KdV) when the Ursell number Ur = ε/µ 2 ≈ 1, where ε is the ratio of the amplitude of the wave to the height of the waveguide and µ is the ratio of the same height to the wavelength. However, the scaling group Ri = N 2 /γ 2 which is the squared ratio of the buoyancy frequency to the shear rate, is left as a free parameter. In the limit of high relative shear as Ri ↓ 1/4, the leading order dispersion coefficient in the KdV equation becomes vanishingly small and the coefficient of nonlinearity becomes unbounded. Conversely, in relatively strong stratifications as Ri → ∞, the coefficient of nonlinearity becomes vanishingly small. Thus, higher order terms and other mechanisms need to be considered. This paper focuses on the role of higher order dispersion which permits consideration of shortwave disturbances. In the representative case of Couette shear and constant buoyancy frequency, estimates of the higher order dispersion coefficients are made in closed form, allowing the truncation of the nonlinear wave equation at the appropriate level for short nonlinear waves via the estimation of the radius of convergence of the phase velocity in wavenumber for linear waves.
Within the context of the well-known interpretation in terms of the wave interaction [P. G. Baines and H. Mitsudera, J. Fluid Mech. 276, 327 (1994); J. R. Carpenter et al., Phys. Fluids 22, 054104 (2010)], instability of sharply stratified (so that the vertical scale ' of density variation is much smaller than the scale K of velocity shear) flows with inflection-free velocity profiles should be treated as Holmboe's instability. In such flows with a relatively weak stratification (when the bulk Richardson number J < ('/K) 3/2 ), eigenoscillations (i.e., Holmboe waves) have much the same phase velocities in a broad spectral range. This creates favorable conditions for a wide variety of three-wave interactions, in contrast to the homogeneous boundary layers where subharmonic resonance is the only effective three-wave process. In the paper, evolution equations are derived which describe three-wave interactions of Holmboe waves and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively. Some possible relations of the results obtained with those of numerical simulations and laboratory experiments are briefly discussed.
Convectively induced shear instability in large internal solitary waves
An experimental laboratory study has been carried out to investigate the instability in- duced by an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign focuses on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification is varied and it is found that the onset, and intensity of breaking are dramatically affected by change in the background stratification. A combination of shear and convective insta- bility is seen on the leading face of the wave. It is shown that there is interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation of the upper boundary condition is also found to have an effect on stability. The implications for convective instability are profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.
Convectively induced shear instability in large amplitude internal solitary waves
Physics of Fluids, 2008
An experimental laboratory study has been carried out to investigate the instability induced by an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign focuses on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification is varied and it is found that the onset, and intensity of breaking are dramatically affected by change in the background stratification. A combination of shear and convective instability is seen on the leading face of the wave. It is shown that there is interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation of the upper boundary condition is also found to have an effect on stability. The implications for convective instability are profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.
Ordered and disordered dynamics in inertialess stratified three-layer shear flows
Physical Review Fluids, 2022
Unlike inertialess two-layer shear flows, three-layer ones can become unstable to long-wave interfacial instabilities due to a resonance mechanism between the interfaces. This interaction is codified in this paper through a set of coupled nonlinear evolution equations derived here in the limit of strong surface tension. A number of parameters are employed to cover a fairly general range of three-layer shear flows driven by a constant pressure gradient. The equations are analyzed using a combination of linear and computational techniques, identifying two linear instability mechanisms noted in the literature previously. The first is a kinematic instability due to the viscosity jumps across fluid phases and the second is a counterintuitive diffusion-derived instability, known in the literature as the Majda-Pego instability and mostly studied for second order diffusion. In the present work it is fourth order, due to surface tension, making the problem mathematically much more challenging. Three unstable parameter regimes of interest are identified linearly and are explored nonlinearly via pseudospectral numerical simulations. For thin middle layers we find steady-state traveling waves or states with asymptotically thinning regions leading to interfacial contact. However, for thin upper or lower layers, complex spatiotemporal dynamics emerge at large times that are characterized by fast time oscillations of the near-wall interface and slow oscillations of that farther away. Data analysis suggests that the dynamics is quasiperiodic in time and additionally coarsening phenomena are observed for large domain sizes leading to modulated traveling wave trains. The kinematic instability mechanism is shown to be triggered nonlinearly via the Majda-Pego mechanism. It can also be triggered by sufficiently large amplitude initial disturbances where linear instabilities are absent, although the transition is not necessarily self-sustaining in all cases.
Hydrodynamic instability of multiple four-wave mixing
Optics Letters, 2010
In the regime of normal dispersion and low-frequency detunings (or high powers), four-wave mixing is shown to undergo a hydrodynamic type of instability. Such instability involves the formation of shocks (steep fronts) from smooth initial data that are regularized through the appearance of trains of fast oscillations, which exhibit solitonlike behavior, colliding elastically.