The Stability of Radiatively Cooling Jets. II. Nonlinear Evolution

1997 ◽  
Vol 483 (1) ◽  
pp. 136-147 ◽  
Author(s):  
James M. Stone ◽  
Jianjun Xu ◽  
Philip E. Hardee
Keyword(s):  
Nonlinear Evolution ◽  
The Stability

The Rayleigh–Taylor instability in magnetic fluids and nonlinear interfacial waves

1992 ◽  
Vol 70 (8) ◽  
pp. 603-609 ◽  
Author(s):  
Abdel Raouf F. Elhefnawy
Keyword(s):  
Nonlinear Evolution ◽  
Multiple Scale ◽  
Magnetic Fluids ◽  
Wave Number ◽  
Perturbed System ◽  
Stability Diagrams ◽  
Rayleigh Taylor Instability ◽  
The Stability ◽  
Normal Magnetic Field ◽  
Horizontal Interface

The nonlinear evolution of a horizontal interface separating two magnetic fluids of different densities, including surface tension effects, is investigated. The fluids are considered incompressible and inviscid, being stressed by the force of gravity, the normal magnetic field, and a constant acceleration in a direction normal to the interface. The method of multiple-scale perturbations is used to obtain two nonlinear Schrödinger equations describing the behavior of the perturbed system. The stability of the perturbed system is discussed both analytically and numerically, and the stability diagrams are obtained. We also obtain the nonlinear cutoff wave number, which separates the region of stability from that of instability.

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

STABILITY OF SPATIALLY PERIODIC SOLUTIONS IN COUPLED MAP LATTICES

2003 ◽  
Vol 13 (02) ◽  
pp. 343-356 ◽  
Author(s):  
M. DUBCOVÁ ◽  
A. KLÍČ ◽  
P. POKORNÝ ◽  
D. TURZÍK
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Nonlinear Evolution ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Evolution Operators ◽  
Spatially Periodic ◽  
The Stability ◽  
Steady State Solutions ◽  
Theoretical Results

Stability of steady-state solutions of 1-dim coupled map lattices is studied. The stability is determined by the spectrum of linear operators on two-sided sequences of vectors in [Formula: see text] arising as a linearization of the corresponding nonlinear evolution operators. Theoretical results are applied to several examples.

Zonal flow in toroidal ion temperature gradient mode turbulence

2007 ◽  
Vol 73 (4) ◽  
pp. 565-573 ◽  
Author(s):  
INGMAR SANDBERG ◽  
V. P. PAVLENKO
Keyword(s):  
Temperature Gradient ◽  
Nonlinear Evolution ◽  
Zonal Flow ◽  
Shear Layers ◽  
Ion Temperature ◽  
Ion Temperature Gradient ◽  
Zonal Flows ◽  
Characteristic Oscillation ◽  
The Stability ◽  
Gradient Mode

AbstractThe properties of zonal flows in the toroidal ion temperature gradient mode turbulence are investigated taking into account the polarization drift effects. The stability criterion and the characteristic oscillation frequency of the zonal flow are determined in terms of the spectra of turbulent fluctuations. The nonlinear evolution of zonal flows may lead to the formation of stationary long-lived coherent structures supporting stationary shear layers. These results indicate the existence of regions with reduced levels of anomalous transport attributed to zonal flows generalizing previous findings regarding zonal flows in electron drift turbulence.

Modulational Instability in JONSWAP Sea States Using the Alber Equation

2017 ◽  
Author(s):  
Odin Gramstad
Keyword(s):  
Stability Analysis ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wave Spectrum ◽  
Rogue Waves ◽  
Systems Of Nonlinear Equations ◽  
Optimal Parameters ◽  
Weakly Nonlinear ◽  
Marquardt Algorithm ◽  
The Stability

An investigation of the instability of JONSWAP wave fields is carried out within the framework of the Alber equation [1]. The Alber equation describes the weakly nonlinear evolution of an inhomogeneous wave spectrum, and by linear stability analysis of this equation [1–3] the instability of an arbitrary wave spectrum subject to inhomogeneous perturbation is investigated. We are solving the equations for instability using a numerical method based on the Levenberg-Marquardt algorithm for solving systems of nonlinear equations, as implemented in the FORTRAN library MINPACK. Results from previous works addressing related topics [4, 5] are verified and refined, providing new results for the stability of JONSWAP wave spectra. Based on the results of the instability analysis we propose more optimal parameters for parameterizing the effects of modulational instability and probability of rogue waves in JONSWAP sea states. The results from the stability analysis of the Alber equation as well as the proposed parameters for parameterizing the effect of modulational instability are verified and tested by performing phase-resolving numerical simulations with the Higher Order Spectral Method [6, 7].

Instability of coupled geostrophic density fronts and its nonlinear evolution

2008 ◽  
Vol 613 ◽  
pp. 309-327 ◽  
Author(s):  
EMILIE SCHERER ◽  
VLADIMIR ZEITLIN
Keyword(s):  
Nonlinear Evolution ◽  
Wave Mode ◽  
Water Model ◽  
Shallow Water Model ◽  
Long Wave ◽  
Classical Stability ◽  
Rotating Shallow Water ◽  
Zero Potential ◽  
The Stability ◽  
Stability Results

Instability of coupled density fronts, and its fully nonlinear evolution are studied within the idealized reduced-gravity rotating shallow-water model. By using the collocation method, we benchmark the classical stability results on zero potential vorticity (PV) fronts and generalize them to non-zero PV fronts. In both cases, we find a series of instability zones intertwined with the stability regions along the along-front wavenumber axis, the most unstable modes being long wave. We then study the nonlinear evolution of the unstable modes with the help of a high-resolution well-balanced finite-volume numerical scheme by initializing it with the unstable modes found from the linear stability analysis. The most unstable long-wave mode evolves as follows: after a couple of inertial periods, the coupled fronts are pinched at some location and a series of weakly connected co-rotating elliptic anticyclonic vortices is formed, thus totally changing the character of the flow. The characteristics of these vortices are close to known rodon lens solutions. The shorter-wave unstable modes from the next instability zones are strongly concentrated in the frontal regions, have sharp gradients, and are saturated owing to dissipation without qualitatively changing the flow pattern.

Construction of the numerical and analytical wave solutions of the Joseph–Egri dynamical equation for the long waves in nonlinear dispersive systems

2020 ◽  
Vol 34 (30) ◽  
pp. 2050289
Author(s):  
Abdulghani R. Alharbi ◽  
M. B. Almatrafi ◽  
Aly R. Seadawy
Keyword(s):  
Solitary Wave ◽  
Numerical Solution ◽  
Evolution Equations ◽  
Dynamical Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations.

scholarly journals Kinetic effects on the stability properties of field-reversed configurations. II. Nonlinear evolution

2004 ◽  
Vol 11 (5) ◽  
pp. 2523-2531 ◽  
Author(s):  
Elena V. Belova ◽  
Ronald C. Davidson ◽  
Hantao Ji ◽  
Masaaki Yamada
Keyword(s):  
Nonlinear Evolution ◽  
Stability Properties ◽  
Kinetic Effects ◽  
The Stability ◽  
Field Reversed Configurations

scholarly journals The stability and nonlinear evolution of quasi-geostrophic toroidal vortices

2019 ◽  
Vol 863 ◽  
pp. 60-78 ◽  
Author(s):  
Jean N. Reinaud ◽  
David G. Dritschel
Keyword(s):  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Azimuthal Mode ◽  
Stability Properties ◽  
Central Vortex ◽  
Minor Radius ◽  
Toroidal Vortices ◽  
The Stability ◽  
Major Radius ◽  
Vortex Array

We investigate the linear stability and nonlinear evolution of a three-dimensional toroidal vortex of uniform potential vorticity under the quasi-geostrophic approximation. The torus can undergo a primary instability leading to the formation of a circular array of vortices, whose radius is approximately the same as the major radius of the torus. This occurs for azimuthal instability mode numbers $m\geqslant 3$, on sufficiently thin tori. The number of vortices corresponds to the azimuthal mode number of the most unstable mode growing on the torus. This value of $m$ depends on the ratio of the torus’ major radius to its minor radius, with thin tori favouring high mode $m$ values. The resulting array is stable when $m=4$ and $m=5$ and unstable when $m=3$ and $m\geqslant 6$. When $m=3$ the array has barely formed before it collapses towards its centre with the ejection of filamentary debris. When $m=6$ the vortices exhibit oscillatory staggering, and when $m\geqslant 7$ they exhibit irregular staggering followed by substantial vortex migration, e.g. of one vortex to the centre when $m=7$. We also investigate the effect of an additional vortex located at the centre of the torus. This vortex alters the stability properties of the torus as well as the stability properties of the circular vortex array formed from the primary toroidal instability. We show that a like-signed central vortex may stabilise a circular $m$-vortex array with $m\geqslant 6$.

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