On regularizing the strongly nonlinear model for two-dimensional internal waves

2013 ◽  
Vol 264 ◽  
pp. 27-34 ◽  
Author(s):  
Ricardo Barros ◽  
Wooyoung Choi
Keyword(s):  
Internal Waves ◽  
Nonlinear Model ◽  
Two Dimensional ◽  
Strongly Nonlinear

Internal waves in a viscous atmosphere

1975 ◽  
Vol 72 (4) ◽  
pp. 773-786 ◽  
Author(s):  
W. L. Chang ◽  
T. N. Stevenson
Keyword(s):  
Energy Source ◽  
Incompressible Fluid ◽  
Internal Waves ◽  
Infinite Number ◽  
Similarity Solution ◽  
Small Amplitude ◽  
Horizontal Plane ◽  
Two Dimensional ◽  
The Waves ◽  
Isothermal Atmosphere

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas & Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.

scholarly journals Models for strongly-nonlinear evolution of long internal waves in a two-layer stratification

2007 ◽  
Vol 14 (1) ◽  
pp. 31-47 ◽  
Author(s):  
T. Sakai ◽  
L. G. Redekopp
Keyword(s):  
Internal Waves ◽  
Nonlinear Evolution ◽  
Phase Speed ◽  
Environmental Parameters ◽  
Exact Expression ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Polynomial Fit ◽  
Nonlinear Relation ◽  
Nonlinear Extensions

Abstract. Models describing the evolution of long internal waves are proposed that are based on different polynomial approximations of the exact expression for the phase speed of uni-directional, fully-nonlinear, infinitely-long waves in the two-layer model of a density stratified environment. It is argued that a quartic KdV model, one that employs a cubic polynomial fit of the separately-derived, nonlinear relation for the phase speed, is capable of describing the evolution of strongly-nonlinear waves with a high degree of fidelity. The marginal gains obtained by generating higher-order, weakly-nonlinear extensions to describe strongly-nonlinear evolution are clearly demonstrated, and the limitations of the quite widely-used quadratic-cubic KdV evolution model obtained via a second-order, weakly-nonlinear analysis are assessed. Data are presented allowing a discriminating comparison of evolution characteristics as a function of wave amplitude and environmental parameters for several evolution models.

scholarly journals A Possible Three-Dimensional Mechanism for Oscillating Wobbles in Tropical Cyclone–Like Vortices with Concentric Eyewalls

2018 ◽  
Vol 75 (7) ◽  
pp. 2157-2174 ◽  
Author(s):  
Konstantinos Menelaou ◽  
M. K. Yau ◽  
Tsz-Kin Lai
Keyword(s):  
Tropical Cyclone ◽  
Nonlinear Model ◽  
Three Dimensional ◽  
The Other ◽  
Sensitivity Analyses ◽  
Two Dimensional ◽  
Triggering Mechanism ◽  
Eigenmode Analysis ◽  
Concentric Eyewalls ◽  
Third Dimension

Abstract It is known that concentric eyewalls can influence tropical cyclone (TC) intensity. However, they can also influence TC track. Observations indicate that TCs with concentric eyewalls are often accompanied by wobbling of the inner eyewall, a motion that gives rise to cycloidal tracks. Currently, there is no general consensus of what might constitute the dominant triggering mechanism of these wobbles. In this paper we revisit the fundamentals. The control case constitutes a TC with symmetric concentric eyewalls embedded in a quiescent environment. Two sets of experiments are conducted: one using a two-dimensional nondivergent nonlinear model and the other using a three-dimensional nonlinear model. It is found that when the system is two-dimensional, no wobbling of the inner eyewall is triggered. On the other hand, when the third dimension is introduced, an amplifying wobble is evident. This result contradicts the previous suggestion that wobbles occur only in asymmetric concentric eyewalls. It also contradicts the suggestion that environmental wind shear can be the main trigger. Examination of the dynamics along with complementary linear eigenmode analysis revealed the triggering mechanism to be the excitation of a three-dimensional exponentially growing azimuthal wavenumber-1 instability. This instability is induced by the coupling of two baroclinic vortex Rossby waves across the moat region. Additional sensitivity analyses involving reasonable modifications to vortex shape parameters, perturbation vertical length scale, and Rossby number reveal that this instability can be systematically the most excited. The growth rates are shown to peak for perturbations characterized by realistic vertical length scales, suggesting that this mechanism can be potentially relevant to actual TCs.

Richtmyer–Meshkov instability on two-dimensional multi-mode interfaces

2021 ◽  
Vol 928 ◽  
Author(s):  
Yu Liang ◽  
Lili Liu ◽  
Zhigang Zhai ◽  
Juchun Ding ◽  
Ting Si ◽  
...  
Keyword(s):  
Nonlinear Model ◽  
Initial Conditions ◽  
Superposition Principle ◽  
Mode Competition ◽  
Two Dimensional ◽  
Competition Effect ◽  
Density Ratios ◽  
Multi Mode ◽  
Amplitude Growth ◽  
Perturbation Growth

Shock-tube experiments on eight kinds of two-dimensional multi-mode air–SF $_6$ interface with controllable initial conditions are performed to examine the dependence of perturbation growth on initial spectra. We deduce and demonstrate experimentally that the amplitude development of each mode is influenced by the mode-competition effect from quasi-linear stages. It is confirmed that the mode-competition effect is closely related to initial spectra, including the wavenumber, the phase and the initial amplitude of constituent modes. By considering both the mode-competition effect and the high-order harmonics effect, a nonlinear model is established based on initial spectra to predict the amplitude growth of each individual mode. The nonlinear model is validated by the present experiments and data in the literature by considering diverse initial spectra, shock intensities and density ratios. Moreover, the nonlinear model is successfully extended based on the superposition principle to predict the growths of the total perturbation width and the bubble/spike width from quasi-linear to nonlinear stages.

Instabilities of three-dimensional viscous falling films

1992 ◽  
Vol 242 ◽  
pp. 529-547 ◽  
Author(s):  
S. W. Joo ◽  
S. H. Davis
Keyword(s):  
Evolution Equation ◽  
Three Dimensional ◽  
Falling Film ◽  
Two Dimensional ◽  
Wave Evolution ◽  
Strongly Nonlinear ◽  
Long Wave ◽  
Threshold Amplitude ◽  
Permanent Wave ◽  
Dimensional Wave

A long-wave evolution equation is used to study a falling film on a vertical plate. For certain wavenumbers there exists a two-dimensional strongly nonlinear permanent wave. A new secondary instability is identified in which the three-dimensional disturbance is spatially synchronous with the two-dimensional wave. The instability grows for sufficiently small cross-stream wavenumbers and does not require a threshold amplitude; the two-dimensional wave is always unstable. In addition, the nonlinear evolution of three-dimensional layers is studied by posing various initial-value problems and numerically integrating the long-wave evolution equation.

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