Andrew C. Poje

Finite time transport in aperiodic flows

Abstract We study the transport of particles in a general, two-dimensional, incompressible flow in the presence of a transient eddy, i.e., a bounded set of closed streamlines with a finite time of existence. Using quantities obtained from Eulerian observations, we provide explicit conditions for the existence of a hyperbolic structure in the flow, which induces mixing between the eddy and its environment. Our results can be used directly to study finite-time transport in numerically or experimentally generated vector fields with general time-dependence.

Geometry of Cross-Stream Mixing in a Double-Gyre Ocean Model

Abstract New dynamical systems techniques are used to analyze fluid particle paths in an eddy resolving, barotropic ocean model of the Gulf Stream. Specifically, the existence of finite-time invariant manifolds associated with transient, mesoscale events such as ring detachment and merger is proved based on computer-assisted analytic results. These “Lagrangian” invariant manifolds completely organize the dynamics and mark the pathways by which fluid parcels may be exchanged across stream. In this way, the Lagrangian flow geometry of a detaching ring or a ring–jet interaction event, as well as the exact associated particle flux, is obtained. The detaching ring geometry indicates that a significant amount of the fluid entrained by the ring originates in a long thin region on the far side of the jet and that this region extends as far upstream as the western boundary current. In the ring–stream interaction case, particle transport occurs both to and from the ring and is concentrated in thin regions on the ne...

THE GEOMETRY AND STATISTICS OF MIXING IN APERIODIC FLOWS

The relationship between statistical and geometric properties of particle motion in aperiodic, two-dimensional flows is examined. Finite-time-invariant manifolds associated with transient hyperbolic trajectories are shown to divide the flow into distinct regions with similar statistical behavior. In particular, numerical simulations of simple, eddy-resolving barotropic flows indicate that there exists a close correlation between such geometric structures and patchiness plots that describe the distribution of Lagrangian average velocity over initial conditions. For barotropic turbulence, we find that Eulerian velocity correlation time scales are significantly longer than their Lagrangian counterparts indicating the existence of well-defined Lagrangian structures. Identification of such structures shows a similar, close relationship between the invariant manifold geometry and patchiness calculations at intermediate time scales, where anomalous dispersion rates are found.

Universal properties of chaotic transport in the presence of diffusion

The combined, finite time effects of molecular diffusion and chaotic advection on a finite distribution of scalar are studied in the context of time periodic, recirculating flows with variable stirring frequency. Comparison of two disparate frequencies with identical advective fluxes indicates that diffusive effects are enhanced for slower oscillations. By examining the geometry of the chaotic advection in both high and low frequency limits, the flux function and the width of the stochastic zone are found to have a universal frequency dependence for a broad class of flows. Furthermore, such systems possess an adiabatic transport mechanism which results in the establishment of a “Lagrangian steady state,” where only the asymptotically invariant core remains after a single advective cycle. At higher frequencies, transport due to chaotic advection is confined to exchange along the perimeter of the recirculating region. The effects of molecular diffusion on the total transport are different in these two cases...

Low-dimensional models for flows with density fluctuations

A low-dimensional model, using the proper orthogonal, or Karhunen–Loeve decomposition, has been remarkably successful in representing the behavior of the wall region of a turbulent boundary layer. We briefly summarize this work. We may hope for similar success in other flows in which coherent structures play an important role, in particular flows with density fluctuations. We sketch an approach to such a decomposition for flows with density fluctuations, suggesting various alternatives which weigh the available information differently. In such a low-dimensional model, obtaining the empirical eigenfunctions poses a problem, since they can usually be determined only from extensive measurements or direct numerical simulations. However, recent work with energy method stability theory (modified by use of an anisotropic eddy viscosity and feedback to the mean profile) has been remarkably successful in predicting the form of the empirical eigenfunctions in the isothermal boundary layer. We present here prelimina...

Model-based directed drifter launches in the Adriatic Sea: Results from the DART experiment

Abstract : A high-resolution numerical model of the Adriatic Sea is used to predict Lagrangian coherent structure boundaries, quantified by finite-size Lyapunov exponents (FSLE), for flow features in the region of the Gargano Peninsula during the course of the Dynamics of the Adriatic in Real Time (DART) observational program. FSLE fields computed from two-day model forecasts of the surface velocity indicate distinct regions of high relative drifter dispersion. Model predictions of such regions located on available ship-tracks were used to direct the launching of pairs of surface drifters on three days during March 2006, with the goal of maximizing coverage of the sampling area. For two of the three launches, the observed trajectories separated at locations and along directions closely approximated by those predicted from the model FSLE fields. The third case acted as an inadvertent control experiment. Model predictions at release-time showed minimal FSLE structure at the launch locations and the observed drifter pair advected in a coherent fashion for two days. While there are considerable differences between individual drifter observations and trajectory envelopes computed from ensembles of synthetic drifters, the experiment confirms the models ability to approximate the location and shape of energetic flow features controlling the near-time fate of quasi-Lagrangian particles. Overall, the combined use of FSLEs with realistic coastal circulation models appears to be a promising avenue to aid real-time-directed drifter launches in observational programs.

A model for large-scale structures in turbulent shear flows

A procedure based on energy stability arguments is presented as a method for extracting large-scale, coherent structures from fully turbulent shear flows. By means of two distinct averaging operators, the instantaneous flow field is decomposed into three components: a spatial mean, coherent field and random background fluctuations. The evolution equations for the coherent velocity, derived from the Navier-Stokes equations, are examined to determine the mode that maximizes the growth rate of volume-averaged coherent kinetic energy. Using a simple closure scheme to model the effects of the background turbulence, we find that the spatial form of the maximum energy growth modes compares well with the shape of the empirical eigenfunctions given by the proper orthogonal decomposition. The discrepancy between the eigenspectrum of the stability problem and the empirical eigenspectrum is explained by examining the role of the mean velocity field. A simple dynamic model which captures the energy exchange mechanisms between the different scales of motion is proposed. Analysis of this model shows that the modes which attain the maximum amplitude of coherent energy density in the model correspond to the empirical modes which possess the largest percentage of turbulent kinetic energy. The proposed method provides a means for extracting coherent structures which are similar to those produced by the proper orthogonal decomposition but which requires only modest statistical input.

Drifter launch strategies based on Lagrangian templates

Abstract A basin-scale, reduced-gravity model is used to study how drifter launch strategies affect the accuracy of Eulerian velocity fields reconstructed from limited Lagrangian data. Optimal dispersion launch sites are found by tracking strongly hyperbolic singular points in the flow field. Lagrangian data from drifters launched from such locations are found to provide significant improvement in the reconstruction accuracy over similar but randomly located initial deployments. The eigenvalues of the hyperbolic singular points in the flow field determine the intensity of the local particle dispersion and thereby provide a natural timescale for initializing subsequent launches. Aligning the initial drifter launch in each site along an outflowing manifold ensures both high initial particle dispersion and the eventual sampling of regions of high kinetic energy, two factors that substantially affect the accuracy of the Eulerian reconstruction. Reconstruction error is reduced by a factor of ∼2.5 by using a co...

Minimum time feedback control of autonomous underwater vehicles

We study the problem of steering a vehicle from its initial position in a 2D, time-varying, ocean flow field to a desired target position in minimum time. In particular, we focus on the case where the magnitude of the flow field sometimes exceeds the speed of the vehicle, and thus controllability is an issue. In order to obtain globally optimal, closed loop trajectories, one solves a dynamic Hamilton Jacobi Bellman equation for the optimal “time-to-go” and associated optimal feedback control law. We do this indirectly via a simple but powerful extremal field algorithm, which allows incremental refinement of the solution and is trivial to parallelize. We characterize solutions and the resulting closed loop optimal trajectories for a time-invariant double gyre flow field and for a numerically-defined, time-varying flow field from a real model of the Adriatic Sea.

Improved surface velocity and trajectory estimates in the Gulf of Mexico from Blended satellite altimetry and drifter data

AbstractThis study investigates the results of blending altimetry-based surface currents in the Gulf of Mexico with available drifter observations. Here, subsets of trajectories obtained from the near-simultaneous deployment of about 300 Coastal Ocean Dynamics Experiment (CODE) surface drifters provide both input and control data. The fidelity of surface velocity fields are measured in the Lagrangian frame by a skill score that compares the separation between observed and hindcast trajectories to the observed absolute dispersion. Trajectories estimated from altimetry-based velocities provide satisfactory average results (skill score > 0.4) in large (~100 km) open-ocean structures. However, the distribution of skill score values within these structures is quite variable. In the DeSoto Canyon and on the shelf where smaller-scale structures are present, the overall altimeter skill score is typically reduced to less than 0.2. After 3 days, the dataset-averaged distance between hindcast and drifter trajectorie...