scholarly journals Nonlinear dynamics induced by linear wave interactions in multilayered flows

2017 ◽  
Vol 816 ◽  
pp. 412-427 ◽  
Author(s):  
Anirban Guha ◽  
Firdaus E. Udwadia
Keyword(s):  
Dynamical System ◽  
Normal Mode ◽  
Nonlinear Phenomena ◽  
Linear Wave ◽  
Neutral Stability ◽  
Wave Problem ◽  
Wave Interactions ◽  
Fluid System ◽  
Normal Mode Theory ◽  
The Waves

Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two-dimensional (2D), inviscid, multilayered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar ‘normal-mode’ type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be nonlinear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symmetry is explored. Fixed points of the system for different ranges of a Froude number like control parameter $\unicode[STIX]{x1D6FE}$ are derived, and their stability evaluated. Depending upon the initial condition and $\unicode[STIX]{x1D6FE}$, the dynamical system may reveal transient growth, weakly positive Lyapunov exponents, as well as different nonlinear phenomena such as the formation of periodic and pseudo-periodic orbits. All these occur for those ranges of $\unicode[STIX]{x1D6FE}$ where normal-mode theory predicts neutral stability. Such rich nonlinear phenomena are not observed in a 2D dynamical system resulting from the two-wave problem, which reveals only stable and unstable nodes.

Discrete Normal Mode Decompositions in Quasigeostrophic Theory

2021 ◽  
Author(s):  
Houssam Yassin ◽  
Stephen Griffies
Keyword(s):  
Normal Mode ◽  
Rossby Wave ◽  
Degrees Of Freedom ◽  
Wave Interaction ◽  
Linear Wave ◽  
Wave Problem ◽  
Dirac Delta ◽  
Complete Basis ◽  
Geostrophic Turbulence ◽  
Baroclinic Modes

<p>The baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple equivalent barotropic modes, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state. Using this complete basis, we are able to obtain a series expansion to the potential vorticity of Bretherton (with Dirac delta contributions). We compare the convergence and differentiability properties of these complete modes with various other modes in the literature. We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. In the process, we introduce the concept of the quasigeostrophic phase space which we define to be the space of all possible quasigeostrophic states. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes prescribed boundary buoyancy gradients into account.</p>

scholarly journals APPLICATIONS ON NONLINEAR WAVE COMBINATION

1986 ◽  
Vol 1 (20) ◽  
pp. 26
Author(s):  
J.T. Juang
Keyword(s):  
Wave Height ◽  
Taiwan Strait ◽  
Western Coast ◽  
Linear Wave ◽  
Wave System ◽  
Height Distribution ◽  
Wave Height Distribution ◽  
Local Wave ◽  
Source Wave ◽  
The Waves

Due to the special bathymetry in Taiwan Strait, the waves off the western coast of Taiwan are considered to be composed of two-source wave system. One propagates from the central part of the Strait named main wave, and the other is generated by the local wind known as local wave which occurs along the shore. After the combination and the transformation procedure from these two-nonlinear-source wave system, the wave height distribution in Taiwan Strait should be modified. A comparison of the wave height distributions based on the present proposed method with the field data indicates that the present method yields a better result than other theorems. Furthermore, the result of application of two non-linear wave theorem to wave prediction are also presented.

scholarly journals A normal‐mode theory of shallow‐water reverberation

1993 ◽  
Vol 94 (3) ◽  
pp. 1785-1785 ◽  
Author(s):  
Renhe Zhang ◽  
Guoliang Jin ◽  
Dinghua Guan ◽  
Wenhua Li ◽  
Xinfang Qiu
Keyword(s):  
Shallow Water ◽  
Normal Mode ◽  
Mode Theory ◽  
Normal Mode Theory

scholarly journals Multi-spacecraft determination of wave characteristics near the proton gyrofrequency in high-altitude cusp

2005 ◽  
Vol 23 (3) ◽  
pp. 983-995 ◽  
Author(s):  
D. Sundkvist ◽  
A. Vaivads ◽  
M. André ◽  
J.-E. Wahlund ◽  
Y. Hobara ◽  
...  
Keyword(s):  
Magnetic Field ◽  
High Altitude ◽  
Nonlinear Effects ◽  
Linear Wave ◽  
Wave Growth ◽  
Ion Cyclotron Waves ◽  
Background Magnetic Field ◽  
Ion Cyclotron ◽  
The Waves ◽  
Proton Gyrofrequency

Abstract. We present a detailed study of waves with frequencies near the proton gyrofrequency in the high-altitude cusp for northward IMF as observed by the Cluster spacecraft. Waves in this regime can be important for energization of ions and electrons and for energy transfer between different plasma populations. These waves are present in the entire cusp with the highest amplitudes being associated with localized regions of downward precipitating ions, most probably originating from the reconnection site at the magnetopause. The Poynting flux carried by these waves is downward/upward at frequencies below/above the proton gyrofrequency, which is consistent with the waves being generated near the local proton gyrofrequency in an extended region along the flux tube. We suggest that the waves can be generated by the precipitating ions that show shell-like distributions. There is no clear polarization of the perpendicular wave components with respect to the background magnetic field, while the waves are polarized in a parallel-perpendicular plane. The coherence length is of the order of one ion-gyroradius in the direction perpendicular to the ambient magnetic field and a few times larger or more in the parallel direction. The perpendicular phase velocity was found to be of the order of 100km/s, an order of magnitude lower than the local Alfvén speed. The perpendicular wavelength is of the order of a few proton gyroradius or less. Based on our multi-spacecraft observations we conclude that the waves cannot be ion-whistlers, while we suggest that the waves can belong to the kinetic Alfvén branch below the proton gyrofrequency fcp and be described as non-potential ion-cyclotron waves (electromagnetic ion-Bernstein waves) above. Linear wave growth calculations using kinetic code show considerable wave growth of non-potential ion cyclotron waves at wavelengths agreeing with observations. Inhomogeneities in the plasma on the order of the ion-gyroradius suggests that inhomogeneous (drift) or nonlinear effects or both of these should be taken into account.

Normal mode theory for electron-beam plasma amplification

Microwaves ◽  
1963 ◽  
pp. 333-337
Author(s):  
M. Chodorow ◽  
J. C. Eidson ◽  
G. S. Kino
Keyword(s):  
Electron Beam ◽  
Normal Mode ◽  
Mode Theory ◽  
Beam Plasma ◽  
Normal Mode Theory ◽  
Electron Beam Plasma
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