Dynamical Systems Tools for Ocean Studies: Chaotic Advection versus Turbulence Reza Malek-Madani

Monterey Bay Surface Currents - August 1999

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)

Chesapeake Bay (Tom Gross- NOAA)

Close-up of VF

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

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 Institute’s $ 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?

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 ……

Lagrangian Perspective

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.

A “toy” problem

Duffing with eps = 0

Stream lines versus Particle Paths, eps= 0

Stream lines versus Particle Paths, eps= 0.01

Stream lines versus Particle Paths, eps= 0.1

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

Can’t 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)

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.

Exponential Dichotomy

Double Gyre Flow

Wiggins, Small and Mancho

Double Gyre with Large (turbulent) Wind Stress

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.

Dynamical Systems and Data Assimilation Chris 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.

Point vortex flows

Two point vortices

Stream function in the co-rotating frame

Two vortices, N=2, one tracer, L=1