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## Coherent Structures in Dynamical Systems |

Self-consistent
transport is the transport of a field F in which there is a feedback between F
and the advection velocity. Two examples of interest to this talk are: vortex
dynamics in shear flows, and charged particles in plasmas. In the first case, F
is the vorticity, the advection equation is the 2-D
Euler's equation, and the self-consistent constrain is the vorticity-streamfunction
relation. In plasmas, F is the electron distribution function, the advection
equation is the Vlasov equation, and the
self-consistent constrain is the Poisson equation. We show that near marginal
stability, the weakly nonlinear dynamics of these two systems can de described
by a universal mean-field Hamiltonian model known as the Single Wave Model
(SWM). We present numerical and analytical results on the SWM in the
N-->infinity kinetic limit and in the finite-N limit, where N is the number
of degrees of freedom. Using this model, we study the formation and persistence
of coherent structures in the presence of self-consistent chaos. Starting from
the finite-N SWM we construct mean-field coupled symplectic
map models of self-consistent transport. We discuss how these maps open
the possibility of studying chaos in Hamiltonian systems with a large number of
degrees-of-freedom.
The formation and interaction
of coherent structures that geometrically determine the transport properties of
laminar 3D flow will be discussed. The impact of these structures on 3D laminar
mixing will be demonstrated numerically and experimentally. Key result is the
role of fluid inertia that induces partial disintegration of coherent
structures of the non-inertial limit into chaotic regions and merger of
surviving parts into intricate 3D structures. The response follows a universal
scenario and reflects an essentially 3D route to chaotic mixing.
The bimodal behavior of the Kuroshio Extension (KE) in the North Pacific has fascinated
physical oceanographers since indications of this phenomenon were found. Why
would a western boundary current switch between coherent states of a
large-meander and a small- meander in a few years time? Why does this
phenomenon not appear in other western boundary currents, such as the Gulf
Stream? For the Kuroshio, both large- and small
meander states can persist over a period ranging from a few years to a decade.
With the analysis of satellite data and those of in situ measurements, a quite
detailed description of the different states and their transition behavior is
now available. There is still, however,
no consensus on which processes cause the low-frequency variability in the
KE. It appears that direct interpretation of the
observations often has been based on
mechanisms involving external causes (such as atmospheric forcing) and linear ocean dynamics while ocean modelers
have tended to suggest mechanisms
which involve elements of nonlinear
ocean dynamics. A theory of the bimodality of the Kuroshio
Extension should include a few essential explanations. It should explain (i) why the KE can be in different states and the origin of
the spatial patterns of these states, (ii) the interannual
time scale of the transition between the two states, and (iii) why there is
much irregularity in the large-meander state while spatial and temporal
variability are relatively low in the small-meander state. In this presentation a nonlinear theory is
proposed which is able to provide answers to the issues
above.
Fractional monodromy is a property of the geometry of the fibration of Liouville integrable Hamiltonian systems. Unlike standard monodromy which describes only the regular part of the fibration, fractional monodromy
naturally appears when we look at the complete fibration,
including singular fibres. In this talk I discuss
fractional monodromy for n_1 :
(-n_2) resonant Hamiltonian systems with n_1, n_2 coprime
natural numbers. The geometry of the fibration is
simplified by passing to an appropriate covering space where the same type of
standard 1: (-1) monodromy is obtained independently
of n_1, n_2. Pushing the results down to the original space gives fractional monodromy.
It is well known that a body
oscillating at high frequency in a viscous fluid will create a steady streaming
flow in its vicinity, due to the nonlinear interactions in the thin Stokes
layer surrounding the body. Fluid
particles in this streaming flow travel, in a time-averaged sense, along closed
orbits about streaming cells adjacent to the body. Less well understood, however, are the behavior of inertial particles of finite size in
such a flow, as well as the interacting streaming flows generated by multiple
oscillating bodies in close proximity.
In this talk, we will present high-fidelity numerical simulations of
streaming flows, generated by single and multiple oscillating bodies. The simulations are carried out with the
viscous vortex particle method, which has several advantages for such flows,
particularly a useful computational efficiency for simulating many bodies
separated by arbitrarily large distances.
In post-processing of these flows, we analyze the fluid and inertial
particle transport, as well as the finite-time Lyapunov
exponent field, in order to reveal the coherent structures.. It will be shown that inertial particles tend
to focus at the centers of the streaming cells.
Furthermore, adjacent oscillating bodies can exchange particles under
certain conditions. Finally, future
directions will be discussed.
Previous research has shown
that chemical reactions in a compressible fluid flow interact strongly with the
surrounding turbulence both in quantitative measures and qualitative character.
In the case of a flame propagating through homogeneous isotropic turbulence,
the rapid flow expansion generated in the reaction zone causes a significant attenuation
in the vorticity. This suppression of the vorticity magnitude complicates the tracking of individual
coherent structures using Eulerian methods,
therefore we use Lagrangian coherent structures to
study the nature of the vortex dynamics, focusing on structure creation,
destruction, and reorientation.
In this talk we present the
geometric and analytic framework underlying the methodology introduced in A. Haro's talk. First we present a parametric KAM result for
the existence of the invariant Lagrangian
deformations. The required non-degeneracy condition is very weak,
indeed this is always satisfied in the in close-to-integrable
case. Hereby, the infinite dimensional problem of finding invariant tori is reduced to the finite-dimensional problem of
finding zeros of a smooth function. Using symplectic
properties of invariant Lagrangian deformations, we
show that the latter problem can be transformed to a problem of finding
critical points of a real-valued function -- which we call the potential of the
Lagrangian deformation. It is also shown that
non-twist tori correspond to degenerate > critical
points of the potential. This naturally leads to apply Singularity Theory to
the critical points of the potential. This work is in collaboration with R. de
la Llave and A. Haro.
Lagrangian Coherent Structures (LCS) are
dynamically evolving surfaces that govern the evolution of complex material
patterns in moving fluids and solids. Examples of such patterns include oil
spills, ash clouds, plankton populations, schools of fish and moving crowds.
Because of their finite lifetime and aperiodic
nature, LCS have been challenging to locate, predict
or control. At the same time, LCS promise to play a key role in the real-time
assessment and short-term forecasting of a number of environmental conditions,
including hazardous winds over airports and the spread of contaminants in
coastal waters. In this talk, I describe a mathematical theory that enables a rigorous extraction of LCS from observational flow data.
In this approach, LCS are defined as invariant
surfaces that extremize an appropriate finite-time
normal repulsion or attraction measure in the governing dynamical system. Solving this variational problem leads to computable sufficient and
necessary criteria for LCS. I will show recent applications to large scale
oceanic and atmospheric flow problems.
Let us consider a
quasi-periodic time-dependent perturbation of an autonomous Hamiltonian system
and focus on an elliptic equilibrium point of the unperturbed Hamiltonian.
Under suitable hypotheses, this equilibrium point becomes an invariant torus
under the effect of the perturbation, with the same internal frequencies as the
perturbation. The normal frequencies of this torus are a perturbation of the
normal frequencies around the initial equilibrium point. In this work I consider the
interaction between these frequencies. Varying a parameter of the system in
such a way that the internal frequencies remain fixed and the normal ones move
with the parameter, two normal frequencies may enter into 1:-1 resonance with
the internal frequencies, thus violating the second Mel'nikov
condition. In fact, this typically occurs for a dense subset of parameters. In
the talk I will study the Hamiltonian Hopf
bifurcations that take place when the normal and internal frequencies are in resonance.
I will also discuss the consequences for the dynamics of the system, in
particular concerning the geometry of the bundle of Lagrangean
tori.
We present a novel method to
find KAM tori with fixed frequency. The method
enables us to prove existence of invariant tori with
fixed frequency in degenerate cases, including persistence of invariant tori for perturbations of isochronous systems. Our method
also leads to a natural classification of KAM tori which
is based on Singularity Theory. This talk aims to illustrate the main ideas of
our approach. Motivating examples are given in the integrable
and near-integrable cases --where the interpretation
of our method is immediate-- and also in the far from integrable
case which highlight the generality and the strength of our approach. This work
is in collaboration with R. de la Llave and A.
Gonzalez.
The Lagrangian
description of fluid transport has been largely enriched with the introduction
of stretching quantifiers such as the different types of Lyapunov
exponents. Lagrangian Coherent Structures can be
readily identified from them. Here we will focus on the finite-size Lyapunov exponent case, as applied to horizontal ocean
flows estimated from satellite altimetry. The influence on biological organisms
of the structures revealed by the Lyapunov analysis
will be illustrated with examples from the bottom and from the top of marine
ecosystems (phytoplankton and marine birds, respectively). Full understanding
of the results requires discussion of the robustness of the approach, as well
as the consideration of the vertical dimension. With this aim, preliminary
results for three-dimensional flows will be presented.
This talk is the result of
several years of collaboration with oceanographers at the University of Miami's
Rosenstiel School of Marine and Atmospheric Science
(Francisco Beron-Vera, Michael Brown, and María Olascoaga) and will deal with Lagrangian
Coherent Structures (LCS) of elliptic type in incompressible two-dimensional
fluid flows with special attention to geophysical fluid flows. Unlike
hyperbolic LCS, which are locally the strongest normally attracting or
repelling material fluid curves over a finite-time interval, elliptic LCS are material
fluid curves which do not experience exponential stretching and folding over a
finite-time interval. Thus while hyperbolic LCS can be regarded as finite-time
generalizations of invariant manifolds of hyperbolic trajectories, elliptic LCS
can be regarded as finite-time generalizations of invariant tori.
As such, rather than facilitating mixing as hyperbolic LCS, elliptic LCS
inhibit mixing. A vivid example of an elliptic LCS in a geophysical fluid flow
system is that one at the perimeter of the austral stratospheric polar vortex.
Such an elliptic LCS prevents ozone-depleted air from spreading toward lower latitudes,
leading to the formation of the so-called ozone hole. The occurrence of
elliptic LCS in various geophysical fluid flow systems will be illustrated.
Also, a partial theory that provides support for their occurrence will be
discussed. Such a partial theory relies on recent results relating to the Kolmogorov--Arnold--Moser theory for time-quasiperiodic one-degree-of-freedom Hamiltonian systems for
which the frequency mapping is degenerate in the Kolmogorov
sense.
We introduce and discuss new Lagrangian tools which are successful in achieving a
detailed description of transport in aperiodic time
dependent dynamical systems. We illustrate applications for purely advective transport events in general aperiodic
geophysical flows. First is discussed the concept of Distinguished Trajectory
which generalizes the concept of fixed point for aperiodic
dynamical systems. It is built on a function that detects simultaneously,
invariant manifolds, hyperbolic and non-hyperbolic flow regions, thus
insinuating the active transport routes in the flow. Once these are recognized,
the transport description is completed by means of the direct computation of
the stable and unstable manifolds of the DHTs.
The term coherent structure
is usually interpreted to mean some kind of dynamical object that is localized
and persistent in both space and time. Nonlinear systems, however, also
typically have nontrivial spectral properties, since nonlinearities in real
space lead to nonlocalities in Fourier space. Here,
we extend the idea of a coherent structure to include the spectral properties
of the dynamical system. Using a recently developed filtering technique, we
study the flux of energy and enstrophy between scales
as a function of space and time in an experimental quasi-two-dimensional flow
that is weakly turbulent. We show that these fluxes are localized in space,
time, and scale, and discuss the implications of this observation.
The language of coherent
structures provides a new means for discussion of transport and mixing of
atmospheric pathogens, paving the way for new modeling and management
strategies for the spread of infectious diseases affecting plants, domestic
animals, and humans. Atmospheric dynamical structures have an influence on aeroecology, namely the population structure of airborne
microbe species. We report on a recent integration of experimental biology and
applied mathematics uncovering how Lagrangian
coherent structures and their phase space complements, strongly connected
(almost-invariant) regions, provide a framework for understanding how airborne
microbe populations are dispersed and mixed. Applications to identification of
frontiers between qualitatively different kinds of behavior in other areas will
also be discussed.
Travelling waves appearing in
the thermal convection of a pure fluid contained in a spherical shell with the
boundaries at different temperatures are studied. They are computed, by using
continuation methods, as steady solutions of a system for the waves, in the
frame of reference of the spheres. Navier-Stokes
equations are written in terms of two scalar potentials for the velocity, which
are expanded, as the temperature, in spherical harmonics, and collocation is
employed in the radius. The special block-tridiagonal
structure of the linear part of the equations provides a preconditioner,
which allows an efficient calculation of the waves. Their stability is also
studied, and the secondary bifurcations to subharmonic
or modulated waves are detected.
Mikhail
B. Sevryuk Following mainly our paper [1], we will present a general theory of the birth of quasi-periodic solutions with n+N basic frequencies in perturbations of families of autonomous dynamical systems, under the assumption that the unperturbed systems admit quasi-periodic solutions with n basic frequencies whereas the perturbation is quasi-periodic in time with N Diophantine basic frequencies. The first n frequencies of the perturbed solutions are close to the unperturbed frequencies whereas the last N frequencies of the perturbed solutions coincide with the frequencies of the perturbation itself. In the particular case n=0 one speaks of a quasi-periodic response to the perturbation. The theory is `structured' and treats systems belonging to various symmetry classes (dissipative, volume preserving, Hamiltonian, and reversible) separately but in a unified manner. The nondegeneracy conditions to be imposed on the smooth families of the unperturbed solutions are very weak (of the so-called Ruessmann type). The perturbed quasi-periodic solutions are organized into Whitney-smooth families. We will also consider some results concerning more conventional nondegeneracy conditions (following, e.g., the fundamental paper [2] devoted to Hamiltonian systems) as well as the situations where the perturbation is quasi-periodic not only in time but also in spatial variables (the recent paper [3] examining planar Hamiltonian and reversible systems provides a key example). References [1] M. B. Sevryuk, Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method, Discrete Contin. Dyn. Syst. 18 (2007) 569-595 [2] A. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci. 7 (1997) 427-473 [3] H. Hanssmann and J. Si, Quasi-periodic solutions and stability of the equilibrium for quasi-periodically forced planar reversible and Hamiltonian systems under the Bruno condition, Nonlinearity 23 (2010) 555-577
In this talk we explore the
connection between the variability of decaying turbulent diffusion-reaction and
Lagrangian Coherent Structures (LCS). In particular
the reaction processes work against homogenization by turbulent diffusion.
Consequently they are highly dependent on the local flow information, provided
by stretching and folding of the chaotic flow, characterized by LCS. The
particular example we study addresses nutrient uptake advantage of motile
micro-organisms over non-motile species. We find that the uptake advantage,
quantified as the difference in total uptake over time and space between the
species, can be associated with attracting LCS. We will discuss the
implications of our approach to large-scale geophysical fluid dynamics
problems, particularly establishment of parameterization on eddy diffusion
based on LCS.
Jean-Luc
Thiffeault
In contrast to its
counterpart in the 3D world, turbulence in 2D is characterized by an inverse
energy cascade. The presence of this inverse cascade in 2D turbulence is visible
in the so-called self-organization of such flows: larger coherent vortex
structures are observed to emerge from initially random flow fields. The lecture will address the
evolution of 2D turbulent flows on a finite domain with no-slip walls. The
organized state consists of a large, domain-filling cell. Results of both
laboratory experiments in rotating / stratified fluids and numerical
simulations have revealed the crucial role played by the unsteady boundary
layers: the domain boundaries act as important sources of large-amplitude vorticity filaments that may influence the motion in the
interior. Attention will be given to global flow quantities like the kinetic
energy, the enstrophy, and the total angular
momentum. In the case of forced 2D turbulence, the latter quantity may show a
remarkable flip-flopping behaviour, associated with a
collapse of the organized flow state followed by its re-organization. In the 2D
turbulent flow, vortex structures may undergo interactions and deformations due
to shear and strain imposed by the ambient flow field. Some theoretical models
and laboratory experiments will be discussed in which the effects of merger and
time-periodic shear upon individual vortex structures are investigated.
An endless vortex is an eddy
whose axis of rotation is a continuous curve with neither beginning nor end.
The best-known example is the smoke ring, which is a vortex whose axis of
rotation is a circle. Here we study the dynamics and transport properties of
more complex examples, namely knotted and linked toroidal
vortices. These are thin tubular vortices uniformly coiled on an immaterial
torus in an otherwise quiescent fluid. Their evolution depends on the number of
vortices in the fluid, the winding number of the vortices (the ratio of the
number of coils round the symmetry axis to the number of coils round the centreline), and the aspect ratio of the supporting torus
(the ratio of the cross-section radius to the centreline
radius). The evolution was computed using the Rosenhead¬Moore
approximation to the Biot¬Savart law to evaluate the
velocity field and a fourth-order Runge-Kutta scheme
to advance in time. It was found that when the supporting torus is thin and the
number of vortices is small the system progresses along and rotates around the
torus symmetry axis in an almost steady manner, with each vortex approximately
preserving its shape. In these cases the velocity field, observed in the comoving frame, has two stagnation points. The stream tube
starting at the front stagnation point transversely intersects the stream tube
ending at the rear stagnation point along a finite number of streamlines,
creating a three-dimensional chaotic tangle. An analysis of the geometry of the
tangle showed that more vortices coiled on tori of
equal aspect ratio carry more fluid, whereas the same number
of vortices coiled on thicker tori carry less
fluid.
Coherent Structures are known
to drive biological dynamics, from plankton to top predators, thus it is very
important to be able to characterize them in realistic three dimensional flows.
The Finite-Size Lyapunov Exponent (FSLE) is a measure
of particle dispersion in fluid flows and the ridges of this scalar field
locate regions of the velocity field where strong exponential separation
between particles occur. These regions are referred to as Lagrangian
Coherent Structures (LCS). We have identified LCS in two different 3D flows: a
canonical turbulent velocity field, that is the
turbulent flow between two parallel stationary plates, driven by a pressure
gradient in the mean flow direction and a primitive equation model (ROMS)
simulation of the oceanic flow in the Benguela
region.
For the abstract, click here.
Normally hyperbolic invariant
manifolds (NHIM's) are a generalization to hyperbolic fixed points. The fixed
point is replaced by a whole invariant manifold with corresponding generalized hyperbolicity criteria. Similar to a hyperbolic fixed
point, a NHIM has (un)stable manifolds and persists
under small perturbations. The classical theorems [1,2] on persistence assume compactness of the invariant
manifold. We formulate a persistence theorem for general noncompact
NHIM's. The results include C^{k,\alpha} smoothness
with Hölder continuity of the highest derivatives. The proof is based on the Perron method and uses ideas of Henry [3] and Van Gils and Vanderbauwhede [4]. To properly generalize to arbitrary
manifolds, e.g. with non-trivial normal bundle, we require the concept of a
Riemannian manifold of bounded geometry. We illustrate that the
smoothness of the perturbed manifold is governed by the spectral gap condition
and that it is optimal. Using further examples we show some of the issues
specific to the noncompact setting and how bounded
geometry comes into play. References: [1] Neil Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math.
J. 21 (1971/1972), 193–226. [2] M. W. Hirsch, C. C. Pugh,
and M. Shub, Invariant manifolds, Lecture Notes
in Mathematics, vol. 583, Springer-Verlag,
Berlin, 1977. [3] Daniel Henry, Geometric
theory of semilinear parabolic equations, Lecture Notes
in Mathematics, vol. 840, Springer-Verlag,
Berlin, 1981. [4] A. Vanderbauwhede
and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), no. 2, 209–224.
A recent variational
theory of Lagrangian Coherent Structures (LCS) [1]
provides a sufficient and necessary criterion for ridges of the Finite-time Lyapunov Exponent (FTLE) field to mark LCS. The criterion
requires the computation of the eigenvalues and
eigenvectors of the Cauchy–Green strain tensor field over a grid of initial
conditions. We propose a numerical method to accurately calculate these eigenvalues and eigenvectors without high computational
cost. We then apply this numerical technique to study LCS in experimental data
on swarming bacterial colonies [2]. We show that only a
subset of the ridges of this FTLE field correspond to hyperbolic LCS. A
careful study of the velocity field suggests that the non-hyperbolic ridges
result from jet-like structures. Finally, the hyperbolic LCS are
carefully advected under the flow map in order to
visualize the temporal motion of these bacterial colonies. References: [1] G. Haller, Variational theory of hyperbolic Lagrangian
Coherent Structures, Physica D (2010), doi:10.1016/j.physd.2010.11.010 [2] H. P. Zhang et al, EPL, 87(2009), 48011, doi:
10.1209/0295-5075/87/48011
In certain two-dimensional
time-dependent flows, the braiding of periodic orbits provides a way to analyze
chaos in the system through application of the Thurston-Nielsen classification
theorem (TNCT). We apply the TNCT to braiding of almost-cyclic sets, which are
closely related to almost-invariant sets (AIS). AIS in a fluid flow are regions
with high local residence time that can act as stirrers or ghost rods. We also
discuss the break up of the AIS as a parameter value
is changed, which results in a sequence of topologically distinct braids. We
show that, for Stokes flow in a lid-driven cavity, these various braids give
good lower bounds on the topological entropy over the respective parameter
regimes. Hence we make the case that even in the absence of periodic orbits,
almost-cyclic regions identified using a transfer-operator approach can reveal
the underlying structure that enables the topological analysis of chaos in the
domain.
We focus on the transport
dynamics of coastal marine larvae in an idealized ROMS Eastern Boundary current
model. Coherent structures control two main processes of ecological interest.
First, nearshore to far offshore transport is
dominated by filamentation correlating with
attracting LCS. Second, settlement (transport back to the coast) is moderated
by a meandering upwelling jet and an associated upwelling front coincident with
a separatrix. We also explore the relationship
between mixing efficiency, filamentation and wind
forcing.
Horizontal mixing and the
distribution of coherent structures in the global ocean are analyzed using
Finite-Size Lyapunov Exponents (FSLE), computed for a
surface layer of a velocity field obtained from the Ocean general circulation
model For the Earth Simulator (OFES). FSLEs measure horizontal stirring and
dispersion, and horizontal structures organizing the oceanic flow can be
identified as ridges of the FSLE field, for which we have performed a
comparative study between hemispheres and different oceans. The computed
Probability Distributions Functions (PDFs) of FSLE, obtained for all the oceans
and for the main currents, are broad and symmetric. Horizontal mixing is
generally more active in the northern hemisphere than in the southern one.
Nevertheless the Southern Ocean is the most active ocean, being the Pacific the
less active one. A striking result is that the main currents aggregate into two
groups of activity: Western Boundary Currents, which have broad PDFs with large
FSLE values, and Eastern Boundary Currents with
narrower ranges and lower FSLE values. Both groups of activity are also found
when we correlate FSLE fields with Eddy Kinetic Energy (EKE) and vorticity (w), finding dispersion relations.These
relations reveal the existence of dynamical and dispersion relations which
characterize the dynamics of different ocean regions, which can be used for
instance to explain differences in the distribution of quantities of biological
interest.
For the abstract, click here.
In this poster, an algorithm
for efficiently computing Lagrangian coherent
structures (LCS) in three dimensions based on the finite time Lyapunov exponent field (FTLE) is presented. This algorithm
relies on a process of surface tracking which computes the FTLE values only
near ridges in the FTLE field. This offers a reduction in computational order
from about O(dx3) to O(dx2). The ridge points are then
used to build a surface triangulation to represent the LCS. This algorithm
offers significant speed ups for LCS computations and offers several other
benefits including ease of LCS visualization and lobe extraction for further
analysis. The properties of the algorithm are discussed in the context of
several test cases and corresponding results are presented. Examples include a
double gyre flow, Arnold-Beltrami-Childress flow, and a hurricane simulation.
Abstract: Discontinuous
vector fields find many applications in several fields including mechanical and
electrical engineering. In this work we consider a planar Filippov
vector field which consists of two parts separated by a discontinuity boundary.
On one side a vector field goes through a Hopf
bifurcation, and on the other side a constant vector field transversally
crossing the boundary. A generic three parameter family of this Hopf-transversal vector field is studied and all the
bifurcations and the phase portraits of the system are determined. Furthermore,
this bifurcation scenario persists under small perturbations.
The ocean and atmosphere are
full of vortices, i.e., locally recirculating flows
with approximately circular streamlines and trajectories. Most often the
recirculation is in horizontal planes, perpendicular to the gravitational acceleration
and rotation vectors in the vertical direction. Vortices are often referred to
as coherent structures, connoting their nearly universal circular flow pattern,
no matter what their size or intensity, and their longevity in a Lagrangian coordinate frame that moves with the
larger-scale, ambient flow [1]. We model the flow at the mouth of a river, with
different flow rates. Coherent vortex dipole is formed at the output of
channel. The formation and destruction are not considered by the potential theory.
We analyze the vortex formation stage with for different Reynolds numbers using
a numerical method. We compare numerical results with potential theory in the
case of shallow water, and we analyze the fluid particle trajectories and solid
particles trajectories, the later by solving an ordinary differential equation
[2]. The vortex causes a suction effect, which can be seen in the trajectories
of fluid particles. The simulation is done by solving the system of equations
in stream function ( References: [1] McWilliams, J. C. (2006)
Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press 105-107 [2] Mordant, N. (2001). Mesure lagrangienne en
turbulence: mise en ouvre et analyse. Ph. D. Thesis. ENS de Lyon, France. [3] Peyret,
R. (2002) Spectral methods for incompressible viscous flow. Applied
Mathematical Sciences 148 Springer, N. Y. USA.
For the abstract, click here.
It is argued that the
complexity of fluid particle trajectories provides the basis for a new method,
referred to as the Complexity Method (CM), for estimation of Lagrangian coherent structures in aperiodic
flows that are measured over finite time intervals. The basic principles of the
CM are explained and the CM is tested in a variety of examples, both idealized
and realistic, and in different reference frames. Two measures of complexity
are explored in detail: the correlation dimension of trajectory, and a new
measure -- the ergodicity defect. Both measures yield
structures that strongly resemble Lagrangian coherent
structures in all of the examples considered. Since the CM uses properties of
individual trajectories, and not separation rates between closely spaced
trajectories, it is well suited for the analysis of ocean float and drifter
data sets in which trajectories are typically widely and non-uniformly spaced.
Jetting swimmers, such as
squid or jellyfish, propel themselves by forming axisymmetric
vortex rings. It is known that vortex rings cannot grow indefinitely, but
rather 'pinch off' once they reach their physical limit, and that a decrease in
the efficiency of fluid transport is associated with pinch-off. Therefore, in
order to evaluate the performance of jetting swimmers, we must assess the
optimality of the vortex wakes they produce, which requires an understanding of
their stability. We consider the Norbury family of
vortices (Norbury 1973) as a model for the vortex
rings produced by jetting swimmers. Pozrikidis (1986)
has studied the stability of Hill's spherical vortex under axisymmetric
prolate and oblate shape perturbations. However, the
stability of other members of the Norbury family to axisymmetric perturbations of the type that might occur
during the vortex formation process in jetting swimmers is unknown. In order to
assess the stability of different members of the family, we introduce
physically pertinent shape perturbations and simulate their development in a
manner akin to Pozrikidis' analysis. As we progress
through the members of the family, we observe a change in the perturbation
response that mirrors the occurrence of pinch-off in jetting flows. We observe
the evolution of the Lagrangian coherent structures
(LCS) for the different perturbation responses, and compare them to
experimental results for pinched-off vortex rings (O'Farrell and Dabiri 2010).
Lake St Clair is a large
(SA~1100 km2) but shallow lake (maximum depth of 8 m) situated in between Lake
Huron and Lake Erie and carries the outflow of the 3 upper Great Lakes. Empirical evidence including significant
differences in residences times and water quality measurements in the east and
west basins of Lake St Clair suggests a barrier to mixing between these
regions. Using depth averaged 2D
velocity data produced from the NOAA finite volume coastal ocean model (FVCOM)
of Lake St Clair we extracted Lagrangian Coherent
Structures (LCSs) for July 2009. The
existence of a persistent LCS transport barrier between the east and west basin
is observed even with changes in wind speed and direction. Numerical drifter trajectories are compared
with 9 GPS track drifters released during the week of July 17th to July 23rd
2009. Comparison demonstrates fairly
good agreement between the field experiment and the numerical simulation and
the actual drifter release results correspond with the presence of the main
transport LCS barrier calculated from the model. [Back] |