Relative periodic orbits in point vortex systems

We give a method to determine relative periodic orbits in point vortex systems: it consists mainly into perform a symplectic reduction on a fixed point submanifold in order to obtain a two-dimensional reduced phase space. The method is applied to point vortices systems on a sphere and on the plane, but works for other surfaces with isotropy (cylinder, ellipsoid, ...). The method permits also to determine some relative equilibria and heteroclinic cycles connecting these relative equilibria.Comment: 27 pages, 17 figure

Flow reversals in turbulent convection via vortex reconnections

We employ detailed numerical simulations to probe the mechanism of flow reversals in two-dimensional turbulent convection. We show that the reversals occur via vortex reconnection of two attracting corner rolls having same sign of vorticity, thus leading to major restructuring of the flow. Large fluctuations in heat transport are observed during the reversal due to this flow reconfiguration. The flow configurations during the reversals have been analyzed quantitatively using large-scale modes. Using these tools, we also show why flow reversals occur for a restricted range of Rayleigh and Prandt numbers

Interaction of point sources and vortices for incompressible planar fluids

We consider a new system of differential equations which is at the same time gradient and locally Hamiltonian. It is obtained by just replacing a factor in the equations of interaction for N point vortices, and it is interpreted as an interaction of N point sources. Because of the local Hamiltonian structure and the symmetries it obeys, it does possess some of the first integrals that appear in the N vortex problem. We will show that binary collisions are easily blown up in this case since the equations of motion are of first order. This method may be easily generalized to the blow up of higher order collisions. We then generalize the model further to interactions of sources and vortices.Comment: 9 page

Rain, power laws, and advection

Localized rain events have been found to follow power-law size and duration distributions over several decades, suggesting parallels between precipitation and seismic activity [O. Peters et al., PRL 88, 018701 (2002)]. Similar power laws are generated by treating rain as a passive tracer undergoing advection in a velocity field generated by a two-dimensional system of point vortices.Comment: 7 pages, 4 figure

Viscous evolution of point vortex equilibria: The collinear state

When point vortex equilibria of the 2D Euler equations are used as initial conditions for the corre- sponding Navier-Stokes equations (viscous), typically an interesting dynamical process unfolds at short and intermediate time scales, before the long time single peaked, self-similar Oseen vortex state dom- inates. In this paper, we describe the viscous evolution of a collinear three vortex structure that cor- responds to an inviscid point vortex fixed equilibrium. Using a multi-Gaussian 'core-growth' type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a 'viscously induced' instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the long-time asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns.Comment: 17 pages, 15 figure

Euler configurations and quasi-polynomial systems

In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3 point vortices in the plane, there are at most three collinear relative equilibria. The "at most three" part is common to both statements, but the respective arguments for it are usually so different that one could think of a casual coincidence. By proving a statement on a quasi-polynomial system, we show that the "at most three" holds in a general context which includes both cases. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.Comment: 21 pages, 6 figure

Chaos in Shear Flows

Almost 25 years ago Lorenz published his seminal study on the existence of a strange attractor in the phase space of a severely truncated model system arising from the hydrodynamical equations describing two-dimensional convection. Nearly a century ago Poincare published his famous treatise Les Methodes Noovelles de la Mecaniaue Celeste (1892) in which the possible complexity of behavior in nonintegrable, conservative systems was first envisioned. Both these works address an age old puzzle: How do apparently stochastic outputs arise from an entirely deterministic system subject to non-stochastic inputs

THE ONSET, CESSATION, AND RATE OF GROWTH OF LOBLOLLY PINES IN THE FACE EXPERIMENT

The Duke Forest FACE experiment was set up to investigate the impact of elevated CO2 levels on a larger eco system. One of the studies dealt with the impact of elevated CO2 levels on the onset and cessation of growth of loblolly pine trees (Pinus taeda L.). In this study the times of these events were determined for each year, 1996 - 2002. The rate of growth, the growth duration, and actual growth were determined from the models of onset and cessation of growth. Adjusted for initial basal area, the rate of growth, the actual growth, and the current basal area were slightly greater for elevated CO2 levels. There was no difference between the two CO2 levels for any of the time variables, onset, cessation, and growth period