Benjamin-Feir Wave Instability on the Opposite Current

2016 ◽  
Author(s):  
Igor V. Shugan ◽  
Hwung-Hweng Hwung ◽  
Ray-Yeng Yang
Keyword(s):  
Wave Interaction ◽  
Stokes Wave ◽  
Initial State ◽  
Weakly Nonlinear ◽  
Initial Wave ◽  
Steady Current ◽  
Cubic Schrödinger Equation ◽  
Energy Peak ◽  
Waves Interaction ◽  
Weakly Nonlinear Model

An analytical weakly nonlinear model of Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly-varying steady current. In contrast to the models based on versions of the cubic Schrodinger equation the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, results in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current.

scholarly journals An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

2015 ◽  
Vol 22 (3) ◽  
pp. 313-324 ◽  
Author(s):  
I. V. Shugan ◽  
H. H. Hwung ◽  
R. Y. Yang
Keyword(s):  
Schrödinger Equation ◽  
Spatial Scale ◽  
Schrodinger Equation ◽  
Wave Interaction ◽  
Experimental Results ◽  
Stokes Wave ◽  
Initial State ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Cubic Schrödinger Equation

Abstract. An analytical weakly nonlinear model of the Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly varying steady current. In contrast to the models based on versions of the cubic Schrödinger equation, the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. The spatial scale of the current variation is assumed to have the same order as the spatial scale of the Benjamin–Feir (BF) instability. The model includes wave action conservation law and nonlinear dispersion relation for each of the wave's triad. The effect of nonuniform current, apart from linear transformation, is in the detuning of the resonant interactions, which strongly affects the nonlinear evolution of the system. The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, resulting in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the waves accompanied by energy income from sufficiently strong opposing current. We employ the model for the estimation of the maximum amplification of wave amplitudes as a function of opposing current value and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation.

A weakly nonlinear model of long internal waves in closed basins

2002 ◽  
Vol 467 ◽  
pp. 269-287 ◽  
Author(s):  
D. A. HORN ◽  
J. IMBERGER ◽  
G. N. IVEY ◽  
L. G. REDEKOPP
Keyword(s):  
Internal Waves ◽  
Kdv Equation ◽  
Wave Interaction ◽  
Second Order ◽  
Weakly Nonlinear ◽  
Closed Basin ◽  
Kdv Equations ◽  
Main Effect ◽  
Weakly Nonlinear Model ◽  
Oscillatory Waves

A simple model is developed, based on an approximation of the Boussinesq equation, that considers the weakly nonlinear evolution of an initial interface disturbance in a closed basin. The solution consists of the sum of the solutions of two independent Korteweg–de Vries (KdV) equations (one along each characteristic) and a second-order wave–wave interaction term. It is demonstrated that the solutions of the two independent KdV equations over the basin length [0, L] can be obtained by the integration of a single KdV equation over the extended reflected domain [0, 2L]. The main effect of the second-order correction is to introduce a phase shift to the sum of the KdV solutions where they overlap. The results of model simulations are shown to compare qualitatively well with laboratory experiments. It is shown that, provided the damping timescale is slower than the steepening timescale, any initial displacement of the interface in a closed basin will generate three types of internal waves: a packet of solitary waves, a dispersive long wave and a train of dispersive oscillatory waves.

scholarly journals Benjamin–Feir instability of waves in the presence of current

2014 ◽  
Vol 1 (2) ◽  
pp. 1803-1832
Author(s):  
I. V. Shugan ◽  
H. H. Hwung ◽  
R. Y. Yang
Keyword(s):  
Nonlinear Waves ◽  
Wave Interaction ◽  
Experimental Results ◽  
Initial State ◽  
Resonance Model ◽  
Freak Waves ◽  
Energy Peak ◽  
Stokes Waves ◽  
Moving Media ◽  
Nonlinear Focusing

Abstract. The development of Benjamin–Feir instability of Stokes waves in the presence of variable current is presented. We employ a model of a resonance system having three coexisting nonlinear waves and nonuniform current. The model is free from the narrow-band approximation for surface waves and relatively weak adverse current. The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, results in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the resonant waves accompanied by energy income from sufficiently strong opposing current. We employ the resonance model for the estimation of the maximum amplification of wave amplitudes as a function of gradually increasing opposing current and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation.

scholarly journals Precessional instability of a fluid cylinder

2011 ◽  
Vol 666 ◽  
pp. 104-145 ◽  
Author(s):  
ROMAIN LAGRANGE ◽  
PATRICE MEUNIER ◽  
FRANÇOIS NADAL ◽  
CHRISTOPHE ELOY
Keyword(s):  
Reynolds Number ◽  
Nonlinear Effects ◽  
Intermittent Flow ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Aspect Ratios ◽  
Image Velocimetry ◽  
Precessional Motion ◽  
Weakly Nonlinear Model

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

Stability and Three-Wave Interaction of Disturbances in a Supersonic Boundary Layer With Cooling

2010 ◽  
Vol 5 (3) ◽  
pp. 52-62
Author(s):  
Sergey A. Gaponov ◽  
Natalya M. Terekhova
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Considerable Change ◽  
Supersonic Boundary Layer ◽  
Surface Cooling ◽  
Weakly Nonlinear ◽  
Linear Evolution ◽  
Weakly Nonlinear Theory ◽  
Compressed Gas

In linear and nonlinear approach (weakly nonlinear theory of stability) interaction of disturbances on a boundary layer of compressed gas is considered at surface cooling. The regimes of moderate (Max number М = 2) and high (М = 5.35) are considered at supersonic speeds. It is established that the surface cooling leads to considerable change of linear evolution of disturbances: the vortical disturbances of the first mode are stabilised, and the acoustic disturbances of the second mode are destabilised, the change degree is defined by the degree of change of the temperature factor. The nonlinear interaction in three-wave systems on high (М = 5.35) supersonic regimes on a boundary layer of compressed gas is carried out between waves of the different nature (acoustic and vortical) in a regime of a parametrical resonance. As a rating wave the flat acoustic wave which raises three-dimensional subharmonic components of the vortical modes. However, the similar interactions for vortical waves at М = 2 considerably weaken. It is possible to expect that surface cooling will lead to delay of a laminar regime at М = 2 and to accelerate of turbulization at М = 5.35

scholarly journals Dynamical Relationship between the Phase of North Atlantic Oscillations and the Meridional Excursion of a Preexisting Jet: An Analytical Study

2008 ◽  
Vol 65 (6) ◽  
pp. 1838-1858 ◽  
Author(s):  
Dehai Luo ◽  
Tingting Gong ◽  
Linhao Zhong
Keyword(s):  
North Atlantic ◽  
Stationary Wave ◽  
Negative Phase ◽  
Initial State ◽  
Weakly Nonlinear ◽  
The North ◽  
Induced Dipole ◽  
Dipole Component ◽  
Southward Shift ◽  
The Impact

Abstract In this paper, it is shown from an analytical solution that in the presence of a preexisting jet the interaction between the zonal jet and the topography of the land–sea contrast (LSC) in the Northern Hemisphere (NH) tends to induce a dipole component that depends crucially upon whether this zonal jet exhibits a north–south excursion. This phenomenon cannot be observed if the zonal jet has no north–south shift. When the preexisting jet is located more northward (southward), the induced dipole can have a low-over-high (high-over-low) structure and thus can make the center of the stationary wave anomaly shift southward (northward), which can be regarded as an initial state or embryo of a positive (negative) phase North Atlantic Oscillation (NAO). This dipole component can be amplified into a typical NAO event under the forcing of synoptic-scale eddies. To some extent, this result provides an explanation for why the positive (negative) phase of the NAO can be controlled by the northward (southward) shift of the zonal jet prior to the NAO. In addition, the impact of the jet shift on the occurrence of NAO is examined in a weakly nonlinear NAO model if the initial state of an NAO is prespecified. It is found that the northward (southward) shift of a zonal jet favors the occurrence of the subsequent positive (negative) phase NAO event and then results in a northward (southward)-intensified jet relative to the preexisting jet. In addition, during the decaying of the positive phase NAO, a strong blocking activity is easily observed over Europe as the jet is moved to the north.

Excitation of superharmonics by internal modes in non-uniformly stratified fluid

2016 ◽  
Vol 793 ◽  
pp. 335-352 ◽  
Author(s):  
Bruce R. Sutherland
Keyword(s):  
Single Mode ◽  
Stratified Fluid ◽  
Buoyancy Frequency ◽  
Nonlinear Interactions ◽  
Initial State ◽  
Weakly Nonlinear ◽  
Resonant Wave ◽  
Vertical Scale ◽  
Scale Structures ◽  
Parametric Subharmonic Instability

Theory and numerical simulations show that the nonlinear self-interaction of internal modes in non-uniform stratification results in energy being transferred to superharmonic disturbances forced at twice the horizontal wavenumber and frequency of the parent mode. These disturbances are not in themselves a single mode, but a superposition of modes such that the disturbance amplitude is largest where the change in the background buoyancy frequency with depth is largest. Through weakly nonlinear interactions with the parent mode, the disturbances evolve to develop vertical-scale structures that distort and modulate the parent mode. Because pure resonant wave triads do not exist in non-uniformly stratified fluid, parametric subharmonic instability (PSI) is not evident even though noise is superimposed upon the initial state. The results suggest a new mechanism for energy transfer to dissipative scales (from large to small vertical scale and with frequencies larger and smaller than that of the parent mode) through forcing superharmonic rather than subharmonic disturbances.

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