Trapping, Tangles, and Transport Jeffrey B. Weiss University of - PowerPoint PPT Presentation

Trapping, Tangles, and Transport Jeffrey B. Weiss University of Colorado, Boulder • transport and Hamiltonian mechanics • periodic perturbations: traditional Hamiltonian chaos • last ten years: aperiodic flows  generalization: similar behavior • qualitative vs. quantitative

Geophysical Flows • strong environmental influences:  rotation: Rossby number Ro = U/fL << 1  stratification: Froude number F = U/NH << 1 • leads to anisotropy  small vertical velocity w << u, v • asymptotic limit is quasigeostrophy • inverse turbulent cascade gives coherent structures  vortices  jets

Pictures of Coherent Structures

2d advection and Hamiltonian dynamics • advection in 2d and 3d qg is 2d and incompressible  flow described by Eulerian streamfunction ψ (x,y,z,t) • passive tracers move with fluid velocity (diffusion = 0)  Lagrangian tracer position (x,y) • same as Hamilton’s equations of motion  (u, v, ψ ) (q, p, H)  physical space equals phase space

structures have closed streamlines • meandering jet or • isolated vortex large amplitude wave (Dewar and Flierl, 1985) (Bower, 1991) • steady flows • comoving frame • stagnation point = fixed point • separatrix divides flow regions

Integrable Hamiltonians • H(x,y) independent of time, autonomous, steady flow • along trajectories, dH/dt = 0, d ψ /dt = 0 • H(x(t),y(t)) = H(x(0),y(0)) = E • y(t) = f(x(t),E) • equations of motion • integrate • invert • flow is integrable • trajectories are circles (topologically), invariant torii

Non-integrable Hamiltonians • what happens when an integrable system is perturbed?  time dependence typically destroys integrability  historically, arises from planetary motion, Earth, Sun, and Jupiter  simplest case: perturbation is periodic in time • well understood  torii described by their frequency of rotation relative to perturbation • three aspects  KAM • which invariant torii remain  Poincare-Birkhoff • fractal structure  tangles • chaos

KAM Theory • naive perturbation theory gives terms ~  problem of “small divisors” • KAM says irrational torii remain • as perturbation grows, less irrational torii are destroyed  degree of irrationality based on continued fraction expansion ω • trajectories cannot cross preserved torii  would violate uniqueness of solutions • torii are barriers to transport • KAM for aperiodic perturbations?

Poincare-Birkhof • KAM says irrational torii preserved, what about rational ones? • stroboscopic map at period of perturbation • go into frame rotating with torus • deduce that there is an infinte hierarchy of stable and unstable periodic points

Tangle • expect perturbation to break separatrices of unstable fixed points • typically the manifolds intersect each other • intersect once implies intersect infinite number of times • cannot self-intersect • area between intersections constant • result, called a tangle, is the backbone of chaos

It may be futile to even try to represent the figure formed by these two curves and their infinite intersections, each of which corresponds to a doubly asymptotic solution [to the equations of motion]. The intersections form a kind of trellis, a tissue, an infinitely tight lattice; each of the curves must never self- intersect, but it must fold itself in a very complex way, so as to return and cut the lattice an infinite number of times. The complexity of this figure is so astonishing that I cannot even attempt to draw it. Nothing is more appropriate to give us an idea of the complexity of the three-body problem and in general of all the problems in Dynamics where there is no uniform integral. -Poincare, 1899

Cantori • as perturbation grows, less irrational tori are destroyed • they dissolve by developing a fractal Cantor set of holes result called a Cantorus • preserved tori are barriers to transport • Cantori are leaky barriers, and can leak very slowly

standard map • Illustrate Hamiltonian chaos with standard map • Program by Jim Meiss  http://amath.colorado.edu/faculty/jdm/stdmap.html

trapping and flights • separatrix between qualitatively different fluid regions becomes chaotic region (Weiss and Knobloch, 1989) • chaotic trajectories alternately behave like each of the regions  Episodes of trapping and flights (Solomon, et al, 1993)

Long episodes lead to anomalous diffusion • Probability of episodes decays algebraically • standard diffusion • anomalous diffusion

Finite-time manifolds • For flows with aperiodic time dependence, structures don’t live forever • Fixed points and their manfolds come and go via bifurcations • Finite-time invariant manifolds similar to periodic case (Haller and Poje, 1998; Poje and Haller, 1999)

Manifolds in the Gulf of Mexico (Kuznetsov, et al, 2002) • Used to analyze Lagrangian transport • need Eulerian and Lagrangian data  not predictive  perhaps a good summary of large datasets

Manifolds and Vortex Merger • 2d vortices merge when closer than some critical distance • critical distance is when manifolds enter the vortex  causes mixing of vorticity of two vortices (Velasco Fuentes, 2001) • compare with statistical mechanics  mixing causes merger, but says nothing about when mixing occurs merger no merger

Merger experiment (Velasco Fuentes, 2001)

Trapping by Vortices (Weiss, et al, 1998) • model 2d turbulence as collection of point vortices • point vortices are a chaotic Hamiltonian system • H has chaotic time dependence due to motion of vortices • vortices trap other vortices and passive particles nearest neighbor speed identity nearest neighbor distance time time

Trapping Mechanism • Warning: rampant speculation • two possibilities 1. Cantori as in periodic perturbation 2. boundary of regular region around vortex moves to envelope particle • trapping and release when no close vortices probably similar to Cantori • during vortex close approach, boundary of chaos changes significantly

Model as stochastic process • model transport as random jumps between regions with qualitatively different behavior, Markov partitions (Meiss and Ott, 1985) • algebraic trapping distributions, Levy processes, lead to anomalous diffusion • meandering jet example, works for some mixing properties (Cencini, et al, 1999) • could include in a pde by using fractional derivatives (del Castillo-Negrete, et al, 2003)  difficult, lots of technical details

Summary • transport in 2d incompressible and 3d QG fluids is equivalent to one degree-of-freedom Hamiltonian dynamics • coherent structures can be defined by manifolds • qualitatively similar behavior for complex and periodic time- dependence • turbulent background is a tangle of manifolds • Overall: phase space structures give generic and qualitative description of transport.