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