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

Trapping, Tangles, and Transport

• transport and Hamiltonian mechanics
• periodic perturbations: traditional Hamiltonian chaos
• last ten years: aperiodic flows
• generalization: similar behavior
• qualitative vs. quantitative

Jeffrey B. Weiss University of Colorado, Boulder

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

• meandering jet or

large amplitude wave

structures have closed streamlines

• isolated vortex

(Bower, 1991) (Dewar and Flierl, 1985)

• 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, dψ/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
• trajectories cannot cross preserved torii
• would violate uniqueness of solutions
• torii are barriers to transport
• KAM for aperiodic perturbations?

ω

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

• f 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

• chaotic trajectories alternately behave like each of the regions
• Episodes of trapping and flights

(Weiss and Knobloch, 1989) (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

no merger 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 distance nearest neighbor identity

speed

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
• ne 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.