Finite-amplitude internal wavepacket dispersion and breaking

2001 ◽  
Vol 429 ◽  
pp. 343-380 ◽  
Author(s):  
BRUCE R. SUTHERLAND
Keyword(s):  
Numerical Simulations ◽  
Internal Waves ◽  
Large Amplitude ◽  
Finite Amplitude ◽  
Critical Amplitude ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Stability ◽  
The Waves ◽  
Horizontal Wavelength

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

Fully nonlinear internal waves in a two-fluid system

1999 ◽  
Vol 396 ◽  
pp. 1-36 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
ROBERTO CAMASSA
Keyword(s):  
Solitary Wave ◽  
Euler Equations ◽  
Deep Water ◽  
Large Amplitude ◽  
Nonlinear Models ◽  
Weak Nonlinearity ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Waves ◽  
Two Fluid

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

Internal wave excitation from stratified flow over a thin barrier

1998 ◽  
Vol 377 ◽  
pp. 223-252 ◽  
Author(s):  
BRUCE R. SUTHERLAND ◽  
PAUL F. LINDEN
Keyword(s):  
Internal Wave ◽  
Numerical Simulations ◽  
Internal Waves ◽  
Wavelet Transforms ◽  
Large Amplitude ◽  
Internal Gravity Waves ◽  
Wave Excitation ◽  
Mean Flow ◽  
Propagating Waves ◽  
The Waves

We perform laboratory experiments in a recirculating shear flow tank of non-uniform salt-stratified water to examine the excitation of internal gravity waves (IGW) in the wake of a tall, thin vertical barrier. The purpose of this study is to characterize and quantify the coupling between coherent structures shed in the wake and internal waves that radiate from the mixing region into the deep, stationary fluid. In agreement with numerical simulations, large-amplitude internal waves are generated when the mixing region is weakly stratified and the deep fluid is sufficiently strongly stratified. If the mixing region is unstratified, weak but continuous internal wave excitation occurs. In all cases, the tilt of the phase lines of propagating waves lies within a narrow range. Assuming the waves are spanwise uniform, their amplitude in space and time is measured non-intrusively using a recently developed ‘synthetic schlieren’ technique. Using wavelet transforms to measure consistently the width and duration of the observed wavepackets, the Reynolds stress is measured and, in particular, we estimate that when large-amplitude internal wave excitation occurs, approximately 7% of the average momentum across the shear depth and over the extent of the wavepacket is lost due to transport away from the mixing region by the waves.We propose that internal waves may act back upon the mean flow modifying it so that the excitation of waves of that frequency is enhanced. A narrow frequency spectrum of large-amplitude waves is observed because the feedback is largest for waves with phase tilt in a range near 45°. Numerical simulations and analytic theories are presented to further quantify this theory.

Trapped internal waves over undular topography

1991 ◽  
Vol 226 ◽  
pp. 205-217 ◽  
Author(s):  
C. Kranenburg ◽  
J. D. Pietrzak ◽  
G. Abraham
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Large Amplitude ◽  
Bottom Topography ◽  
Stratified Flow ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Trapped Waves ◽  
Acoustic Images ◽  
The Waves

We describe observations of slowly decelerating stratified flow over undular bottom topography in an estuary. The flow, which initially was supercritical with respect to the first internal wave mode, approached a resonance after it had become subcritical. A series of acoustic images showed large-amplitude first-mode trapped waves during this phase of the tide. We derive a criterion for quasi-steady response, and present an extension of Yih's class II linear finite-amplitude solutions that accounts for the waves observed.

scholarly journals Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas

2004 ◽  
Vol 11 (2) ◽  
pp. 219-228 ◽  
Author(s):  
S. S. Ghosh ◽  
G. S. Lakhina
Keyword(s):  
Solitary Waves ◽  
Nonlinear Theory ◽  
Acoustic Waves ◽  
Large Amplitude ◽  
Magnetized Plasma ◽  
Amplitude Variation ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

Abstract. The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.

Finite-Amplitude Baroclinic Instability of Time-Varying Abyssal Currents

2006 ◽  
Vol 36 (1) ◽  
pp. 122-139 ◽  
Author(s):  
Seung-Ji Ha ◽  
Gordon E. Swaters
Keyword(s):  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Zonal Flow ◽  
Finite Amplitude ◽  
Marginal Stability ◽  
Time Variability ◽  
Time Varying ◽  
Weakly Nonlinear ◽  
Abyssal Current ◽  
The Stability

Abstract The weakly nonlinear baroclinic instability characteristics of time-varying grounded abyssal flow on sloping topography with dissipation are described. Specifically, the finite-amplitude evolution of marginally unstable or stable abyssal flow both at and removed from the point of marginal stability (i.e., the minimum shear required for instability) is determined. The equations governing the evolution of time-varying dissipative abyssal flow not at the point of marginal stability are identical to those previously obtained for the Phillips model for zonal flow on a β plane. The stability problem at the point of marginally stability is fully nonlinear at leading order. A wave packet model is introduced to examine the role of dissipation and time variability in the background abyssal current. This model is a generalization of one introduced for the baroclinic instability of zonal flow on a β plane. A spectral decomposition and truncation leads, in the absence of time variability in the background flow and dissipation, to the sine–Gordon solitary wave equation that has grounded abyssal soliton solutions. The modulation characteristics of the soliton are determined when the underlying abyssal current is marginally stable or unstable and possesses time variability and/or dissipation. The theory is illustrated with examples.

Disturbing vortices

2001 ◽  
Vol 426 ◽  
pp. 95-133 ◽  
Author(s):  
N. J. BALMFORTH ◽  
STEFAN G. LLEWELLYN SMITH ◽  
W. R. YOUNG
Keyword(s):  
Critical Radius ◽  
Nonlinear Effects ◽  
Rotation Frequency ◽  
Finite Amplitude ◽  
Matched Asymptotic Expansion ◽  
Critical Amplitude ◽  
Subsequent Evolution ◽  
Weakly Nonlinear ◽  
Vorticity Distribution ◽  
The Core

Inviscid spatially compact vortices (such as the Rankine vortex) have discrete Kelvin modes. For these modes, the critical radius, at which the rotation frequency of the wave matches the angular velocity of the fluid, lies outside the vortex core. When such a vortex is not perfectly compact, but has a weak vorticity distribution beyond the core, these Kelvin disturbances are singular at the critical radius and become ‘quasi-modes’. These are not true eigenmodes but have streamfunction perturbations that decay exponentially with time while the associated vorticity wraps up into a tight spiral without decay. We use a matched asymptotic expansion to derive a simplified description of weakly nonlinear, externally forced quasi-modes.We consider the excitation and subsequent evolution of finite-amplitude quasi- modes excited with azimuthal wavenumber 2. Provided the forcing amplitude is below a certain critical amplitude, the quasi-mode decays and the disturbed vortex returns to axisymmetry. If the amplitude of the forcing is above critical, then nonlinear effects arrest the decay and cat's eye patterns form. Thus the vortex is permanently deformed into a tripolar structure.

Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 2. The weak-beta case

2000 ◽  
Vol 412 ◽  
pp. 161-196 ◽  
Author(s):  
RICHARD H. KARSTEN ◽  
GORDON E. SWATERS
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Large Amplitude ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Steady Flows ◽  
Leading Order ◽  
Unstable Flow ◽  
Amplitude Analysis

This paper is a continuation of our study on nonlinear processes in large-amplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-β models. These models arise when the intrinsic length scale is large enough so that the dynamics is geostrophic to leading order but not so large that the β-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous analysis of strong-β LAG models in Part 1, we show that the weak-β models allow for vigorous linear baroclinic instability.For two-layer weak-β LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultraviolet catastrophe. We argue that it is not possible to establish conditions for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable flow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the development of large-amplitude meanders, squirts and eddies is correlated with the presence of these explosively unstable modes.For two-layer weak-β LAG models in which one of the two layers is substantially thinner than the other, the linear stability problem does not exhibit an ultraviolet catastrophe and it is possible to establish conditions for the nonlinear stability in the sense of Liapunov for steady flows. A finite-amplitude analysis for a marginally unstable flow suggests that nonlinearities act to stabilize eastward and enhance the instability of westward flows. Numerical simulations are presented to illustrate these processes.

Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity

1987 ◽  
Vol 178 ◽  
pp. 491-506 ◽  
Author(s):  
D. R. Jenkins
Keyword(s):  
Thermal Convection ◽  
Fluid Layer ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Temperature Dependent Viscosity ◽  
Weakly Nonlinear ◽  
Expansion Procedure ◽  
The Stability ◽  
Dependent Viscosity

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.

scholarly journals Propagation regimes of interfacial solitary waves in a three-layer fluid

2015 ◽  
Vol 2 (1) ◽  
pp. 1-41
Author(s):  
O. E. Kurkina ◽  
A. A. Kurkin ◽  
E. A. Rouvinskaya ◽  
T. Soomere
Keyword(s):  
Internal Waves ◽  
Fluid Layer ◽  
Soliton Solutions ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Stable Configuration ◽  
Weakly Nonlinear ◽  
Small Parameters ◽  
Specific Equation ◽  
Kdv Type Equations

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid immiscible layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most of near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equation (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

scholarly journals Propagation regimes of interfacial solitary waves in a three-layer fluid

2015 ◽  
Vol 22 (2) ◽  
pp. 117-132 ◽  
Author(s):  
O. E. Kurkina ◽  
A. A. Kurkin ◽  
E. A. Rouvinskaya ◽  
T. Soomere
Keyword(s):  
Internal Waves ◽  
Fluid Layer ◽  
Soliton Solutions ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Stable Configuration ◽  
Weakly Nonlinear ◽  
Small Parameters ◽  
Specific Equation ◽  
Kdv Type Equations

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equations (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown that both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

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