Onset of treelike patterns in negative streamers.

We present analytical and numerical studies of the initial stage of the branching process based on an interface dynamics streamer model in the fully three-dimensional case. This model follows from fundamental considerations on charge production by impact ionization and balance laws, and leads to an equation for the evolution of the interface between ionized and nonionized regions. We compare some experimental patterns with the numerically simulated ones, and give an explicit expression for the growth rate of harmonic modes associated with the perturbation of a symmetrically expanding discharge. By means of full numerical simulation, the splitting and formation of characteristic treelike patterns of electric discharges is observed and described.

Electric-discharge contour-dynamics model: the effects of curvature and finite conductivity.

In this paper we present the complete derivation of the effective contour model for electrical discharges which appears as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, when the electron diffusion is small. It consists of two integro-differential equations defined at the boundary of the plasma region: one for the motion and a second equation for the net charge density at the interface. We have computed explicit solutions with cylindrical symmetry and found the dispersion relation for small symmetry-breaking perturbations in the case of finite resistivity. We implement a numerical procedure to solve our model in general situations. As a result we compute the dispersion relation for the cylindrical case and compare it with the analytical predictions. Comparisons with experimental data for a 2D positive streamers discharge are provided and predictions confirmed.

Contour dynamics model for electric discharges.

We present an effective contour model for electrical discharges deduced as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, in the limit of small electron diffusion. The incorporation of curvature effects to the velocity propagation and not to the boundary conditions is a feature and makes it different from the classical Laplacian growth models. The dispersion relation for a nonplanar two-dimensional discharge is calculated. The development and propagation of fingerlike patterns are studied and their main features quantified.

Critical conditions for phytoplankton blooms.

We motivate and analyse a reaction-advection-diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton-water-light-system characterized by the same dimensionless parameters will show the same dynamics.

Strong enhancement of noise-induced escape by nonadiabatic periodic driving due to transient chaos.

We have found a mechanism by which a moderately weak nonadiabatic periodic driving may significantly facilitate noise-induced interwell transitions in an underdamped multiwell system. The mechanism is associated with the onset of a homoclinic tangle in the noise-free system: if the ratio of the driving amplitude A to the damping gamma exceeds a critical value approximately 1, then the basins of attraction of the linear responses related to different wells are mixed in a complex manner in some layer associated with the separatrix of the undriven nondissipative system, and the minimal energy in such layer is lower than the top of the barrier. Thus the energy to which the system needs to be activated by the noise, to be able to make a transition, is lower than the top of the barrier.

Noise-induced escape on time scales preceding quasistationarity: New developments in the Kramers problem.

Noise-induced escape from the metastable part of a potential is considered on time scales preceding the formation of quasiequilibrium within that part of the potential. It is shown that, counterintuitively, the escape flux may then depend exponentially strongly, and in a complicated manner, on time and friction. (c) 2001 American Institute of Physics.

Kramers problem for a multiwell potential

Fluctuational escape from a multiwell potential is shown to display new features, as compared to the conventional single-well case. The flux J may depend on friction gamma exponentially strongly, over an exponentially long period; for small enough temperatures, J(gamma) undergoes marked oscillations in the range of small gamma, and the time evolution of J changes drastically as gamma exceeds a critical value.

Ratchet driven by quasimonochromatic noise

The currents generated by noise-induced activation processes in a periodic potential are investigated analytically, by digital simulation and by performing analog experiments. The noise is taken to be quasimonochromatic and the potential to be a smoothed sawtooth. Two analytic approaches are studied. The first involves a perturbative expansion in inverse powers of the frequency characterizing quasimonochromatic noise and the second is a direct numerical integration of the deterministic differential equations obtained in the limit of weak noise. These results, together with the digital and analog experiments, show that the system does indeed give rise, in general, to a net transport of particles. All techniques also show that a current reversal exists for a particular value of the noise parameters.

Symmetry breaking of fluctuation dynamics by noise color

Activated escape is investigated for systems that are driven by noise whose power spectrum peaks at a finite frequency. Analytic theory and analog and digital experiments show that the system dynamics during escape exhibit a symmetry-breaking transition as the width of the fluctuational spectral peak is varied. For double-well potentials, even a small asymmetry may result in a parametrically large difference of the activation energies for escape from different wells.