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

Monterey Bay Surface Currents - August 1999

Observed Eulerian FieldsVector field is known at discrete points at discrete times interpolation becomes a major mathematical issueVF 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 DataVery expensive (CPU, personnel)Data may not be on rectangular grid (from FEM code)But No gaps in data, either in time or space

Nonlinear PDEsDo 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?

Typical SettingA 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 EulerianTo understand transport, mixing, exchange of fluids, we need to solve the set of differential equations

Lagrangian Perspective

Steady FlowStagnation (Fixed) Points Hyperbolic (saddle) pointsDirections 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

Streamlines versus Particle Paths,eps= 0

Streamlines versus Particle Paths,eps= 0.01

Streamlines versus Particle Paths,eps= 0.1

Unsteady FlowsThe 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

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)

New Methodologies for Unsteady FlowsChaotic 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 flowsThese 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 dichotomyThe 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 DHTIn 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

SummaryDynamical 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 ModelMajor 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 methodsKey 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

Start talking about Poincare cross section and its implications about quantifying chaos Talk about Poincare SectionWe use complex coordinates: z vortices, \zeta: tracers. \Gamma: vortex circulations.\eta: system noise, assumed to be white Gaussian with covariance Q.The system is modeled by the deterministic part of the equations, \eta=0.Superscripts: t for full system (truth), f for model (forecast). The error is assumed unbiased, =0 (an approximation: in nonlinear models there is a bias)We take vortices of equal circulation, by rescaling time and coordinates we can put z=-+1, \Gamma=2\pi. Due to the noise, the deterministic model (without data assimilation) looses track of the full system after certain time. This time can be estimated by approximating the shear (induced by another vortex) at vortex location with linear shear, then the average distance between vortices in the full system and model vortices grows as t^(3/2), when it reaches ~1 the model breaks down.For our range of model noise (sigma = 0.01 0.05) it happens in 1-3 T (T=4\pi motion period)

Purple circle #1 tracer launch location for the animations (next two slides)Vortices: z(0)=+-1, tracer \zeta(0)=0.6-0.3i.Tracer position is assimilated every 1 time unit (i.e. approx 12 times per period).Triangles: full systemStars: model that assimilates tracer position Arrows: update steps