Dynamical Systems Tools for Ocean Studies: Chaotic Advection versus Turbulence

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Dynamical Systems Tools for Ocean Studies
Chaotic Advection versus Turbulence

• Reza Malek-Madani

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Monterey Bay Surface Currents - August 1999
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Observed Eulerian Fields

• Vector field is known at discrete points at
  discrete times interpolation becomes a major
  mathematical issue
• VF is known for a finite time only How are we
  to prove theorems when invariance is defined
  w.r.t. continuous time and for all time?
• VF is often known in parts of the domain this
  knowledge may be inhomogeneous in time. How
  should we be filling the gaps?
• Normal Mode Analysis (NMA) is one way to fill in
  the gaps. (Kirwan, Lipphardt, Toner)

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Chesapeake Bay (Tom Gross- NOAA)
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Close-up of VF
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Modeled Eulerian Data

• Very expensive (CPU, personnel)
• Data may not be on rectangular grid (from FEM
  code)
• But
• No gaps in data, either in time or space

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Nonlinear PDEs

• Do not have adequate knowledge of the exact
  solution.
• Need to know if a solution exists if so, in
  which function space? (Clay Institutes 1M
  prize for the NS equations)
• Need to know if solution is unique. Otherwise why
  do numerics?
• How does one choose the approximating basis
  functions? Convergence?

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Typical Setting

• A velocity field is available, either
  analytically (only for toy problems), or from a
  model (Navier-Stokes) or from real data, or a
  combination VF is typically Eulerian
• To understand transport, mixing, exchange of
  fluids, we need to solve the set of differential
  equations

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Lagrangian Perspective
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Steady Flow

• Stagnation (Fixed) Points Hyperbolic (saddle)
  points
• Directions of stretching and compressions (
  stable and unstable manifolds)
• Linearization about fixed points spatial
  concept always done about a trajectory time
  enters nonlinearly in unsteady problems.
• Instantaneous stream functions are particle
  trajectories. Trajectories provide obstacle to
  transport and mixing.

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A toy problem
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Duffing with eps 0
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Streamlines versus Particle Paths,eps 0
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Streamlines versus Particle Paths,eps 0.01
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Streamlines versus Particle Paths,eps 0.1
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Unsteady Flows

• The basic concepts of stagnation points and
  poincare section as tools to quantify transport
  and mixing fail when the flow is aperiodic.
• How does one define stable and unstable manifolds
  of a solution in an unsteady flow? How does one
  compute these manifolds numerically?
• Mancho, Small, Wiggins, Ide, Physica D, 182,
  2003, pp. 188 -- 222

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Cant we just integrate the VF?

• Is it worth to simply
• integrate the velocity
• field to gain insight
• about the flow?
• Where are the coherent structures?
• (Kirwan, Toner,
• Lipphardt, 2003)

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New Methodologies for Unsteady Flows

• Chaotic oceanic systems seem to have stable
  coherent structures. However, Poincare map idea
  does not work for unsteady flows.
• Distinguished Hyperbolic Trajectories moving
  saddle points Their stable/unstable manifolds
  play the role of separatrices of saddle point
  stagnation points in steady flows
• These manifolds are material curves, made of
  fluid particles, so other fluid particles cannot
  cross them. It is often difficult to observe
  these curves by simply studying a sequence of
  Eulerian velocity snapshots.
• Wiggins group has devised an iterative algorithm
  that converges to a DHT.
• Exponential dichotomy
• The algorithm starts with identifying the
  Instantaneous Stagantion points (ISP), i.e.,
  solutions to v(x,t) 0 for a fixed t. Unlike
  steady flows, ISP are not generally solutions to
  the dynamical system.
• The algorithm then uses a set of integral
  equations to iterate to the next approximation of
  the DHT
• In real data sets (and, in general in unsteady
  flows) ISP may appear and disappear as time goes
  on
• Stable and unstable manifolds are then determined
  by (very careful) time integration of the vector
  field (using an algorithm by Dritschel and
  Ambaum)
• Have applied this method to the wind-driven
  quasigeostrophic double-gyre model.

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Exponential Dichotomy
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Double Gyre Flow
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Wiggins, Small and Mancho
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Double Gyre with Large (turbulent)Wind Stress
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Summary

• Dynamical Systems tools have been extended to
  discrete data.
• Concepts of stable and unstable manifolds have
  been tested on numerically generated aperiodic
  vector fields.
• What about stochasticity? Data Assimilation?
• Our goal is to determine the relevant manifolds
  for the Chesapeake Bay Model
• Major obstacle VF is given on a triangular
  grid.

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Dynamical Systems and Data AssimilationChris
Jones

• Computing stable and unstable manifolds requires
  knowing the Eulerian vector field backward and
  forward in time. But we lack that information in
  a typical operational setting. We do, however,
  have access to Lagrangian data (drifter, etc.)
• Integrate Dynamical Systems Theory into
  Lagrangian data assimilation (LaDA) strategy
  develop computationally efficient DA methods
• Key Idea The position data by a Lagrangian
  instrument is assimilated directly into the
  model, not through a velocity approximation.
• Behavior near chaotic saddle points in vortex
  models showed the need for patches for this
  technique. Ensemble Kalman Filtering.

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Point vortex flows
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Two point vortices
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Stream function in the co-rotating frame
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Two vortices, N2, one tracer, L1