156 research outputs found
Amplitude equations for coupled electrostatic waves in the limit of weak instability
We consider the simplest instabilities involving multiple unstable
electrostatic plasma waves corresponding to four-dimensional systems of mode
amplitude equations. In each case the coupled amplitude equations are derived
up to third order terms. The nonlinear coefficients are singular in the limit
in which the linear growth rates vanish together. These singularities are
analyzed using techniques developed in previous studies of a single unstable
wave. In addition to the singularities familiar from the one mode problem,
there are new singularities in coefficients coupling the modes. The new
singularities are most severe when the two waves have the same linear phase
velocity and satisfy the spatial resonance condition . As a result
the short wave mode saturates at a dramatically smaller amplitude than that
predicted for the weak growth rate regime on the basis of single mode theory.
In contrast the long wave mode retains the single mode scaling. If these
resonance conditions are not satisfied both modes retain their single mode
scaling and saturate at comparable amplitudes.Comment: 34 pages (Latex), no figure
Spatially localized magnetoconvection
Numerical continuation is used to compute branches of time-independent, spatially localized convectons in an imposed vertical magnetic field focusing on values of the Chandrasekhar number Q in the range 10 < Q < 103. The calculations reveal that convectons initially grow by nucleating additional cells on either side, but with the build-up of field outside owing to flux expulsion, the convectons are able to transport more heat only by expanding the constituent cells. Thus, at large Q and large Rayleigh numbers, convectons consist of a small number of broad cells
Spatially localized binary fluid convection in a porous medium
The origin and properties of time-independent spatially localized binary fluid convection in a layer of porous material heated from below are studied. Different types of single and multipulse states are computed using numer- ical continuation and the results related to the presence of homoclinic snaking of single and multipulse states
Magnetohydrodynamic convectons
Numerical continuation is used to compute branches of spatially localized structures in convection in an imposed vertical magnetic field. In periodic domains with finite spatial period, these branches exhibit slanted snaking and consist of localized states of even and odd parity. The properties of these states are analysed and related to existing asymptotic approaches valid either at small amplitude (Cox and Matthews, Physica D, vol. 149, 2001, p. 210), or in the limit of small magnetic diffusivity (Dawes, J. Fluid Mech., vol. 570, 2007, p. 385). The transition to standard snaking with increasing domain size is explored
Nearly inviscid Faraday waves in containers with broken symmetry
In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for various types of drift instabilities of standing waves, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments
Stationary peaks in a multivariable reaction--diffusion system: Foliated snaking due to subcritical Turing instability
An activator-inhibitor-substrate model of side-branching used in the context
of pulmonary vascular and lung development is considered on the supposition
that spatially localized concentrations of the activator trigger local
side-branching. The model consists of four coupled reaction-diffusion equations
and its steady localized solutions therefore obey an eight-dimensional spatial
dynamical system in one dimension (1D). Stationary localized structures within
the model are found to be associated with a subcritical Turing instability and
organized within a distinct type of foliated snaking bifurcation structure.
This behavior is in turn associated with the presence of an exchange point in
parameter space at which the complex leading spatial eigenvalues of the uniform
concentration state are overtaken by a pair of real eigenvalues; this point
plays the role of a Belyakov-Devaney point in this system. The primary foliated
snaking structure consists of periodic spike or peak trains with identical
equidistant peaks, , together with cross-links consisting of
nonidentical, nonequidistant peaks. The structure is complicated by a multitude
of multipulse states, some of which are also computed, and spans the parameter
range from the primary Turing bifurcation all the way to the fold of the
state. These states form a complex template from which localized physical
structures develop in the transverse direction in 2D.Comment: 30 pages, 14 figure
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