10,637 research outputs found
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
A phenomenological model of weakly damped Faraday waves and the associated mean flow
A phenomenological model of parametric surface waves (Faraday waves) is
introduced in the limit of small viscous dissipation that accounts for the
coupling between surface motion and slowly varying streaming and large scale
flows (mean flow). The primary bifurcation of the model is to a set of standing
waves (stripes, given the functional form of the model nonlinearities chosen
here). Our results for the secondary instabilities of the primary wave show
that the mean flow leads to a weak destabilization of the base state against
Eckhaus and Transverse Amplitude Modulation instabilities, and introduces a new
longitudinal oscillatory instability which is absent without the coupling. We
compare our results with recent one dimensional amplitude equations for this
system systematically derived from the governing hydrodynamic equations.Comment: Complete paper with embedded figures (PostScript, 3 Mb)
http://www.csit.fsu.edu/~vinals/mss/jmv1.p
Pole dynamics for the Flierl-Petviashvili equation and zonal flow
We use a systematic method which allows us to identify a class of exact
solutions of the Flierl-Petvishvili equation. The solutions are periodic and
have one dimensional geometry. We examine the physical properties and find that
these structures can have a significant effect on the zonal flow generation.Comment: Latex 40 pages, seven figures eps included. Effect of variation of
g_3 is studied. New references adde
On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation
We study the stability of a four parameter family of spatially periodic
traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to
two classes of perturbations: periodic perturbations with the same periodic
structure as the underlying wave, and long-wavelength localized perturbations.
In particular, we derive necessary conditions for spectral instability to
perturbations to both classes of perturbations by deriving appropriate
asymptotic expansions of the periodic Evans function, and we outline a
nonlinear stability theory to periodic perturbations based on variational
methods which effectively extends our periodic spectral stability results.Comment: 27 pages, 3 figure
Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane
We present an overview of pattern formation analysis for an analogue of the
Swift-Hohenberg equation posed on the real hyperbolic space of dimension two,
which we identify with the Poincar\'e disc D. Different types of patterns are
considered: spatially periodic stationary solutions, radial solutions and
traveling waves, however there are significant differences in the results with
the Euclidean case. We apply equivariant bifurcation theory to the study of
spatially periodic solutions on a given lattice of D also called H-planforms in
reference with the "planforms" introduced for pattern formation in Euclidean
space. We consider in details the case of the regular octagonal lattice and
give a complete descriptions of all H-planforms bifurcating in this case. For
radial solutions (in geodesic polar coordinates), we present a result of
existence for stationary localized radial solutions, which we have adapted from
techniques on the Euclidean plane. Finally, we show that unlike the Euclidean
case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf
bifurcation to traveling waves which are invariant along horocycles of D and
periodic in the "transverse" direction. We highlight our theoretical results
with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne
An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns
This paper presents an introduction to phase transitions and critical
phenomena on the one hand, and nonequilibrium patterns on the other, using the
Ginzburg-Landau theory as a unified language. In the first part, mean-field
theory is presented, for both statics and dynamics, and its validity tested
self-consistently. As is well known, the mean-field approximation breaks down
below four spatial dimensions, where it can be replaced by a scaling
phenomenology. The Ginzburg-Landau formalism can then be used to justify the
phenomenological theory using the renormalization group, which elucidates the
physical and mathematical mechanism for universality. In the second part of the
paper it is shown how near pattern forming linear instabilities of dynamical
systems, a formally similar Ginzburg-Landau theory can be derived for
nonequilibrium macroscopic phenomena. The real and complex Ginzburg-Landau
equations thus obtained yield nontrivial solutions of the original dynamical
system, valid near the linear instability. Examples of such solutions are plane
waves, defects such as dislocations or spirals, and states of temporal or
spatiotemporal (extensive) chaos
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