Magistère des Sciences de la Matière Ecole Normale Supérieure de Lyon Université Claude Bernard Lyon 1.
Training Program 2004-2005 Mercier Matthieu Master M1, Physics.
ECOLE NORMALE SUPERIEURE DE LYON
Dynamical Systems Theory Applied to a Kinematic Model of a Meandering Jet
Abstract: In order to bring a new point of view on dynamical studies of models of jet, a kinematic model of a meandering jet is analyzed in a fixed frame. Two different approaches are being developed to understand and quantify Lagrangian transport. They are both based on geometrical considerations, and have been tested through computations. The statistical quantities obtained can help discussing the links between studies of models of jet in a moving frame and in a fixed frame.
Keywords: Dynamical systems, meandering jet, lobe dynamics, incoming/exit set, Gulf Stream.
Department of Mathematics University of Bristol University Walk Bristol, BS8 1TW England
Supervisor: Professor Stephen Wiggins
From May
to July
Contents 1
Introduction
2
2
Presentation of the Model and Preliminary Results 2.1 In the Moving Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Stagnation Points (SPs) . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Stable and Unstable Manifolds Associated to the Hyperbolic Points 2.2 Bower’s Model in the Fixed Frame . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Concept of Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Hyperbolic trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stable and Unstable Manifolds of a Hyperbolic Trajectory . . . . .
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3 3 4 5 6 6 7 8
Study of Exchange in a Perturbed Version of Bower’s Model 3.1 A General Geometric Approach to Transport . . . . . . . . . . . . . . 3.1.1 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Exit Set and Incoming Set . . . . . . . . . . . . . . . . . . . 3.1.3 Exit Time Decomposition of the Entrance Set and Transit Set 3.1.4 Statistical Quantities for Particle Transport . . . . . . . . . . 3.1.5 Application to Perturbed Version of Bower’s Model . . . . . . 3.2 Lobe Dynamics Approach . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Definitions and Description of the Turnstile Mechanism . . . 3.2.2 Application to the Perturbed Version of Bower’s Model . . . .
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9 9 9 10 10 11 12 14 14 15
3
4
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Conclusion
20
A Numerical Methods A.1 Integration of ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Computation of Hyperbolic Trajectories and their Local Stable and Unstable Manifolds A.3 Evolution of Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . . . . . . A.4 Extracting Lobes and Computing Their Areas . . . . . . . . . . . . . . . . . . . . . .
1
22 22 22 23 23
Chapter 1
Introduction In geophysical fluid mechanics, the study and undestanding of a meandering jet is a lively domain of research since this structure can be applied to real flows such as the Gulf Stream for instance. The progress in this field is well summarized by S.Wiggins in [1]. During the last ten years, dynamical systems theory has been applied to the study of this type of flows, and has reaveled itself very practical to define and to quantify Lagrangian transport. One must understand by Lagrangian transport, the effective transport of fluid particles studied using Lagrangian fluid mechanics, opposed to an Eulerian point of view. Different models of jet have been developped [15, 16], but here we will consider the one introduced by A.S.Bower in 1991 [2], and studied also by R.M.Samelson in 1992 [4] and F.Raynal in 2005 [5]. The work is mainly based on dynamical systems theory and thus require some knowledge that can be found in books and articles. It must be noticed that the mathematical background is introduced with the idea more to explain the mathematical tools than to give proofs of theorems. Although each new ideas is given with references on it, one can cite [6, 8, 9, 10, 11]. The computations made are based on these mathematical tools, and are detailled in the appendix. The references for them are [7, 12, 13]. We finally give two references that help to understand the general background of the study. The first one, an article from F.R.Hama [3] of 1962 presents a warning on studies of time-dependent flows in fluid mechanics; the second one is from S.Lozier (1997)[14], and draws comparison between experimental results in a fixed frame and in a moving frame, a matter of interest for what will be developed next. A lot of research works have been made on models of jet, but it has always been done in a frame moving with the jet, in order to study a steady flow, or quasi-steady flow. The problem raised by this approach is that it is not clear how one can compare the results obtained to experimental observations on geophysical flows that are always made in a fixed frame. So the motivation of this study is to try to draw a comparison between results obtained in the fixed frame and results already known in the moving frame, for the case of a model of a meandering jet. In the following, we will first present Bower’s model and give some preliminary results, then the study of exchange in a perturbed version of Bower’s model will introduce two different approaches, in order to quantify some statistical quantities.
2
Chapter 2
Presentation of the Model and Preliminary Results The simple kinematic model introduced by A.Bower [2] has been an attempt to understand and describe geophysical flows such as the Gulf Stream. Deduced from observations of drifters, the twodimensional model consists in a jet of uniform width deformed by a propagating sinusoidal meander, and the streamfunction used to describe the flow is: !#"%$'&)(*,+.-/+102"%3)4 6 565 798 :=,$?=2*A@CB = +1-D+102"%3)4 6 5A5FEHG This streamfunction has already been studied (see [5]) in the moving frame following the meander defined by: I
JLK 0 M
NO
and we will give some of the important results.
2.1 In the Moving Frame In the moving frame following the meander, and after nondimensionalization, the streamfunction is: P
QRST U V %
P with
K_S W " XY&)(*Z+.[>\ 565 8 \ 5 E^]
I
a` K_SdeQRST f hg " : Ni _kjl mg " :n poq mr gJ ksu t kKv mgT K 0 xwy z g " s
`cb ]
The streamlines and stagnation points (zero of the velocity field) in the moving frame are given at figure 2.1. The flow is steady in this frame and one can define three different regimes: the central jet (flow is from west to east, or left to right), exterior retrograde motion, and intermediate closed circulations above meander troughs and below crests. No exchange occurs between these regimes because the flow is steady. A brief study using dynamical systems theory will introduce the main ideas for more complicated cases to come. We will study dynamical systems of the type:
| { f }~ | _ }~ _ is the velocity field considered. where bold notation is for vectors, and |
It must be noted that to draw some comparisons with the articles [4] and [5], we will take the following values for the applications unless other values are specified:
jl V2
st V> k [ 3
K zR
Figure 2.1: Streamlines and stagnation points in the moving frame.
2.1.1 Stagnation Points (SPs)
Search for SPs. The system of equations for fluid particles in this steady flow are given by:
W
\
W
] \
: HL s W " X 3 +1[>\ 5 8 :\ 5 EG ] 8 :\ 5 : jio oTQc Y % s W " X 3 +.[>\ 5 hX [ o j v jo s " : 8 LK# = g . !ZK>o j jo s " : 8 K> = ] Q S Q S s are two hyperbolic fixed points since the eigenvalues : : s We can conclude that and Q S Q S of the linearized system of equations are real and of opposite signs. . . and 0 0 are two elliptic
fixed points since the two eigenvalues are purely imaginary and conjugated.
2.1.2 Stable and Unstable Manifolds Associated to the Hyperbolic Points For a two-dimensional linear velocity field, the unstable (resp. stable) manifold is the onedimensional subspace associated to the negative (resp. positive) eigenvalue of the Jacobian at the hyperbolic point. For non-linear velocity fields, this subspace associated to the linearized part of the flow is only tangent to the true unstable (resp. stable) manifold near the hyperbolic point. A more s general definition of the unstable (resp. stable) manifold is the invariant set of whose elements tend
a
to the hyperbolic point as (resp. ] ). We have an approximated form of the manifold near the hyperbolic point. Furthermore, if one
has an idea of the unstable (resp. stable) manifold at time , by making it evolve to some later time
(resp. earlier time) under the flow, one gets a part of the unstable (resp. stable) manifold at time since stable and unstable manifolds are invariant sets. Finally, the stable and unstable manifolds associated to the hyperbolic points define exactly the different regimes of the flow, since these invariant sets are material curves so fluid trajectories cannot cross them. There are different ways to compute them (see [6], [7]). The same ideas are developed in both articles, but implemented differently. To obtain figure 2.2, the evolution in time of the local manifold near the hyperbolic point has been used (each manifold contains points). : 8 v5000 R j L # K is also the unstable " It turns out that the stable manifold ] : 8 LK# of point v j vj " manifold of point ; and respectively the stable manifold of point ] : : " 8 HLK# is the unstable manifold of point Rij " 8 vLK# . Thus points initially] ] RUj % " : 8 HLK/ on its unstable manifold tend j " : 8 H K/ as close to to point ] ] time tend to infinity. So the stable and unstable manifolds define heteroclinic trajectories, which can be parameterized by the equations: P S zS : , by for hyperbolic points located at Q _S U V K j " : 8 LK# 8 LK , ] ]
,
P
,
S zS s , by Q _S U VvLK j " : 8 LK# 8 LK , ] ]
for hyperbolic points located at
5
Figure 2.2: Streamlines(black), stable(blue) and unstable(red) manifolds in the moving frame.
2.2 Bower’s Model in the Fixed Frame Studying Bower’s Model in a fixed frame requires a new mathematical approach. As the flow was steady in the moving frame, classic dynamical systems theory was applicable and did not present much difficulty. But now the flow is time-dependent. In order to be able to add some non-periodic time-dependent perturbations to the model, the periodicity of the time-dependent flow will not be considered and no Poincaré map will be used. The streamfunction of the flow in the fixed frame is given by: P
QRS O
W " XY&)(* _ ]
This define a trajectory in time that we call : _ # Q _S _ #
Q _S _ We want to demonstrate that this trajectory is an hyperbolic one. The Jacobian,
system evaluated at is: K2vLK# = Q _S _ a vKMjio s Ljio s " : 8 vLK# Q2 f K_ M
of the
which is a time-independent matrix, already studied before. It must be noticed that the case of a time-independent Jacobian of a trajectory in a time-dependent velocity field is special. Thus the linear ODE associated to the hyperbolic trajectory is an ordinary differential equation with time-independant coefficients. The Lyapunov exponents are effectively : the eigenvalues of , that is to say two reals of g J g
K o j v L i j o % H L / K s " 8 = ). with opposite signs (
So the trajectory is a hyperbolic trajectory.
!
7
2.2.3 Stable and Unstable Manifolds of a Hyperbolic Trajectory The computation of stable and unstable manifolds of a hyperbolic trajectory for a time-dependent velocity field is usually complicated. It is based on evolving in time approximations of these manifolds near the trajectory, and these approximations are obtained by studying the linearized system. Here the linearized system is time-independent, hence the manifolds at each time can be calculated much more quickly. A natural behaviour for the manifolds, as time evolves, would be to translate, looking the same than what has been drawn in the moving frame. We can observe that it is effectively the case with figure 2.3 (each manifold contains 2000 points).
4
4
2
2
eta
0
0 0
2
4
6
0
10
8
2
4
6
8
10
xi -2
-2
-4
-4
(a) t=0
(b) t=15
Figure 2.3: stable(blue) and unstable(red) manifolds evolving in the fixed frame.
Q
In the fixed frame, we consider a window of observation whose width along the -axis is equal S to one meander, and its height along the -axis is at least two times larger than the meander amplitude (and centered on the mean position of the jet). With the numerical values considered for computa'>>D>R#~>> . At each time, one can find two(or three) points belonging to tions, the window is two different hyperbolic trajectories. The stable and unstable manifolds associated to these hyperbolic trajectories can be considered as (time-dependent) boundaries of the jet in this flow if one chooses the Q contour always following the manifold the farthest from the -axis (see figure 2.4). With these bound-
'
*
(a) t=0
(b) t=15
Figure 2.4: Boundaries (in black) defined thanks to the manifolds (red and blue) at different times. aries defined, then we can say that no fluid particule can cross the jet in the window of observation in the fixed frame since the boundaries are stable and unstable manifolds of hyperbolic trajectories. Bower’s model cannot explain with her simple model the observations showing crossing and particles exchange in the Gulf Stream. One can also define boundaries between the three different regimes of the flow. There is no exchange possible between these regimes neither. 8
Chapter 3
Study of Exchange in a Perturbed Version of Bower’s Model In 1992, Samelson presented a perturbed version of Bower’s model in order to take into account the observed phenomenon of transport across the Gulf Stream [4]. He considered three types of perturbations: a time-dependent meander amplitude, a time-dependent spatially uniform meridional velocity superimposed on the basic flow, and a propagating plane wave superimposed on the basic flow. Expected from these perturbations is a radical change in the behaviour of the manifolds associated to these trajectories, and the loss of the heteroclinic trajectories already observed. This would lead to the breaking of the barriers to transport and allow exchange in the Gulf Stream. The study of this kind of perturbation has been done in the moving frame [4, 5] using Melnikov’s function or lobe dynamics. To tackle the problem in the fixed frame, we will consider a more general approach dealing with the notion of exit and incoming set of a defined domain, that will be defined in next section. More Details on the Perturbations. For a reason of time, only the first perturbation has been studj ied. The time-dependent meander amplitude can be expressed by the change of in the streamfunction by:
P
j
QRSd
,
j zj
] :
D :
We will call : the streamfunction of the flow defined previously, but now with the new definij j tion of . Note that this first expression for allows one to generalize later to a time-dependent non periodic amplitude, by using its Fourier decomposition.
3.1 A General Geometric Approach to Transport To deal with transport in aperiodically time-dependent velocity fields, a general approach has been developped by J.D.Meiss and co-workers [9], and it has been synthesized by S.Wiggins [10]. We present here these ideas applied to our special case, which is a aperiodically time-dependent velocity field known analytically for the time interval needed. We now define the general background to describe the transport in and out of a chosen domain named (the subscript shows the time-dependence of the domain).
3.1.1 Time Evolution Time Slice We will need to observe the flow at different times, so we need to define the notion of time slice as follow:
f b
9
wR
By noting
M
#
the trajectory passing through the point at time , then we have
z
w the point of this trajectory in the time slice corresponding to .
(Forward and Backward) Exit Time If we consider a point time as:
; b M ]
#
,
#
b
;
b
b , we define the forward exit
It represents the time when the particle located at at is not in b ; . Similarly, we define the backward exit time as:
" b M b "
; d/
" / )
If b (resp. b
for all ).
]
#
, it simply means that the point is in
3.1.2 Exit Set and Incoming Set
We define the set of points in b that are not in ; b ; ,
b b b ; b
#
b b ; is called the t-exit set on the time slice
, is similarly defined as: b " ,
b b "
;
for all
, as:
. The set of points that are in
# b " b
b b "
(resp. in
b
"
b that were in
is called the t-entrance set on the time slice b . Figure 3.1 shows examples of these definitions, where we have used the notation for the application that give the image of a point belonging to the time slice on time slice .
3.1.3 Exit Time Decomposition of the Entrance Set and Transit Set We can describe the t-entrance set on the time slice time from the domain. The subsets are defined as:
b b " ; b b b "
#
b by the subsets which have the same exit
f zw w
R
Thus, according to the definition, we have:
b b "
#
b b
"
#
# ' '
Furthermore, we define the sets:
A b b " R w b ; b b " ]
f . Therefore, the set of points: It is to be noted that b b " is the image of b b at time ] b b " with b b " b b
b ; visited by points that are in b " and not in b ; , ie that enter
~ L and leave it at time w .1 the domain at time ] / b , With all these definitions established, we can defined the t-transit set, denoted b b "
; to be the set of points that are in b " and will not be in b for some by: b 1b " b b " exit times. the sets $ # ! $" # are disjoint since they have different is the set of points in
1
!
10
(a) t
(b) t+t’
(c) t-t’
(d) t
Figure 3.1: The exit set on time slice in (a) is the part of that is not black. The incoming set on time slice in (d) is the part of that is not black. (b) and (c) are representing ; and " respectively.
3.1.4 Statistical Quantities for Particle Transport The geometrical structures defined in the previous sections are very practical to deduce some statistical results for particle transport. First, we define the notions needed to obtain averages. The Lebesgues measure of a set will be used, but it is used in order to keep a general approach although in the case of fluid dynamics, it simply represents the volume. over a subset S is defined as: The average of a scalar valued function
f
Average Exit Time We consider the subset b b " b b " of the points that do exit . Then the
since time M/ c is given by: average exit time for points entering at time
;
; b
%w b b " w ] b ! b b b "
; w
b b " w b b ! b b b " b b " b b " In the case of incompressible velocity field, it is obvious that the same quantity is considered, but only at a different time. Hence we have: ; b /b " b b " w E ;
w b b " w
or simply, the average elapsed time is:
11
since
/
Finally we have an interesting formula for the average elapsed time:
b
b b " b ! b b b "
3.1.5 Application to Perturbed Version of Bower’s Model Now that the tools for a general approach have been introduced, we can try to apply them to the case of Bower’s perturbed model. The idea is to find numerical results (in the fixed frame of course) for quantities such as the mean residence time within the jet, the mean crossing time of the jet, etc. We expect to be able to compare these results with those obtained thanks to the lobe dynamics approach explained later, and also with results found in the moving frame and given in [5]. Choice of the Domain of Study. In geophysical studies, people are generally interested in exchanges between the jet and its surroundings [2, 14, 15]. So an interesting choice for the boundaries should be delimitating a domain representing the jet. The mathematical definition of its boundaries would be the following:
? f " : 8 HLK/ D oqLKM
!
jq
#
This choice corresponds to a meandering jet of uniform width with its boundaries containing the hyperbolic trajectories found when studying the unperturbed model in the fixed frame. Furthermore, we will restrain the window of observation to one meander, that is to say we will search for the incoming set of the time-dependent domain defined as one meander and represented at different times in figure 3.2.
(a) t=0
(b) t=12.5
(c) t=25
(d) t=37.5
Figure 3.2: Meandering boundaries of the jet (black lines) and different hyperbolic trajectories (red symbols).
12
(a) T-incoming set
(b) Time decomposition of the incoming set, zoom on the left part.
Figure 3.3: T-incoming set and its time decomposition at time
z
for
R
.
Average Residence Time in the Jet In order to compute the average residence time, we need to find
and . To be able the t-incoming set b b " for the domain defined and thus we need to choose
s t to compare with the lobe dynamics approach, we will set where is the frequency of
O
but the time of observation should note the perturbation considered; and we will usually take matter. Thus, we will compute:
"
# Y " f
with being the domain defined as the meandering jet. The flow is perturbed with the values for the different parameters being:
j O
2 K zR : zR and R zR
hR (resp. R ), using an integrator Runge-Kutta of fifth order with adaptative step on For : T f l " . " . has provided an incoming set with > each point of the grid of precision
points (resp. ), and its transit time decomposition, allows the computation of the mean residence
R ). We took " since almost all the points of the time in the jet (see figure 3.3 for : " w R 2 (resp. w R ), only incoming set left the jet ( b b " have been computed for y w
2 y w
three points were remaining in the jet till at least )). Hence we found: j (resp. z
R
" T : zR " :
# *
/
# *
*
*
And since the T-incoming set represents the points entered in the jet since one period of the perturbation
), we must add it so that the mean residence time in the defined meandering jet is: (
R
: R : R
: zR : zR
Conclusion. This very general approach that can be directly applied to observations made in the Gulf Stream gives interesting results but tackles a burning issue. Is the exchange phenomenon observed here is associated to Lagrangian transport? Indeed, the choice of the boundaries was influenced by the spatial structure of the flow, which is an Eulerian point of view, different form the Lagrangian point of view in time-dependent velocity fields. In order to answer this question, we will consider another approach using lobe dynamics.
13
3.2 Lobe Dynamics Approach When one has little knowledge concerning a flow, looking at the incoming and exit sets of a domain defined arbitrary to quantify Lagrangian transport is subtle, but it is the only possibility. If one can search for, and find, hyperbolic trajectories, then lobe dynamics and the turnstile mechanism are extremely powerful products of dynamical systems theory (see [5, 6, 16, 1] for a complete explanation of the turnstile mechanism).
3.2.1 Definitions and Description of the Turnstile Mechanism
The following paragraphs explain the mathematical background of lobe dynamics with a will to describe more than to show rigorously the following facts. Hyperbolic trajectories are denoted
and associated stable and unstable manifolds, and .
Intersection Points. Stable and unstable manifolds of hyperbolic trajectories can intersect each other, whereas two stable (resp. unstable) manifolds cannot 2 . This is simply a consequence of unicity of the infinite time limit for fluid particle trajectory. Since a trajectory on a stable (resp. unstable)
), if two stable (resp. unstable) (resp. manifold tends to the hyperbolic trajectory as tends to ]
) intersect, then the intersection s : and s (resp. : and manifolds point has two different limits as tends to ] (resp. ), which is not possible. This will explain the behaviour of lobes near hyperbolic trajectories later.
: Lobe on a Time Slice. Consider two following intersection points and between
s and is simply the sets of points within the domain defined by the segments of : and . A lobe s contained between and as shown in figure 3.4.(a).
Figure 3.4: A lobe, stable manifolds in blue, unstable manifolds in red. Evolution of Points belonging to Stable and Unstable Manifolds. To understand lobe dynamics, two properties of points belonging to stable and/or unstable manifolds must be given. They are consequences of uniqueness of solutions and of invariance of the manifolds. First of all, the order of points on a manifold is maintained since it is an invariant one-dimensional
and of a manifold associated to can be orset. On any time slice , two points than in the sense of arclength of the manifold dered by saying if is closer to (or ) with considered. Hence, for and belonging to , then ; ; for any . This is due to uniqueness of solutions of ODE, since a change in the order along this one dimensional subspace would imply a crossing of two trajectories. Second of all, if one can find intersection points on a time slice, then they can be found on any other time slice since manifolds are invariant sets.
2
There are different types of intersection points. Even if there is no need here to differentiate them, it can be noted that lobes will be defined thanks to primary intersection points
14
The evolution of manifolds, and thus of lobes, is ruled by these properties as shown in figure 3.4.(b). Finally, a last property is to be noticed: the area of a lobe must be constant (in time) since it represents a volume of fluid and the flow is incompresible.
Turnstile Lobes and Mechanism. The turnstile lobes are special lobes with respect to a boundary : and s intersect. On defined thanks to stable and unstable manifolds. Suppose that s each time slice, a point is chosen to define a boundary as the union of the segment of to and of the segment of : from to : (the order given describes the sense from s of evolution of points in forward time). will be called boundary intersection points (BIP) and it is w , is farther chosen on each time slice such that the image under the flow of on time slice ; , w% than 3. These BIPs are used to define the time-dependent boundaries, and the turnstile of : lobes. The turnstile mechanis is represented in figure 3.5, with:
,
,
being the application that evolves a point under the flow from to , and
(a)
" :
(b)
is its inverse.
:
Figure 3.5: The turnstile mechanism ( > ).
and s represent the two domains defined thanks to the time dependent boundary. The BIPs are chosen so that: : s : and : s s s : s and s : : : s and s : represent the turnstile lobes on the time slice . The only points that cross the boundary : defined, from to s (resp. s to : ) are the points in : s (resp. s : ). This is due to the fact that the boundary are invariant sets, and to the way points evolve on a manifold. One can remark that the turnstile lobes play the part of the incoming and exit sets described in the previous approach.
3.2.2 Application to the Perturbed Version of Bower’s Model The stable and unstable manifolds already described along this study define barriers for fluid particles. With the introduction of the perturbation, the heteroclinic trajectories are “destroyed” and stable and unstable manifolds become infinite curves that intersect each other an infinite number of time. 3
It is clear that the choice is not unique
15
Hyperbolic Trajectories and their Manifolds. First thing to be done is to find the new hyperbolic trajectories in the window of observation, and their stable and unstable manifolds. One can first consider an approximation of the hyperbolic trajectory and its invariant manifolds. When we studied the
s z ), we found an hyperbolic trajectory: not-perturbed case (equivalent to :
_ #
Q _S _ # : (small compared to j ), keeping s
If we use small value of , we can use perturbation theory to approximate the real hyperbolic trajectory (its existence is prooved by general dynamic systems theory, see for example [11] (p253-254). At zero order in : , the hyperbolic trajectory is:
Q S
KM
: (3.1) ] j " : LK# : (3.2) ] ] The Jacobian of the system developped in order of : also gives the result found in the previous part: K v K> = : _ K j #o s j#o s " : 8 HLK/ ]
R , but is not This approximation is good enough for values of the perturbation up to
kR for instance. No further order can be obtained easily since to obtain the precise enough for
first order, one must solve an non-linear, time-dependent ODE which is as difficult as the original problem. Boundaries and Turnstile Lobes. To study transport in our case, we define a domain using segments of the stable and unstable manifolds of hyperbolic trajectories, starting at the hyperbolic trajectories untill a chosen intersection point, as shown in figure 3.6.(a) where we also named the different domain defined: : and represent regions of recirculation above and under the jet, and s is the jet, and gave the turnstile lobes in figure 3.6.(b).
.
(a) Boundaries of the jet using stable(blue) and unstable(red) manifolds
(b) Turnstile lobes
Figure 3.6: Boundaries and turnstile lobes at t=0. The choice of the BIPs through time is made in order to have a domain that “looks” like a meandering jet, although we can only use segments of the stable and unstable manifolds of hyperbolic trajectories in our window of observation. The choice is not unique, but the one made for the boundary 16
between : and s is shown in figure 3.7, for to . The BIPs are plotted in green, the
R
hyperbolic trajectories in black. For , the BIP at time , denoted , is the image under the flow H
V
H
of the BIP chosen for ; and at time , a new BIP is chosen so that the turnstile mechanism takes place at that moment. As the perturbation considered is periodic, this choice can be repeated for
#* '
(a) t=1
(b) t=3
(c) t=5
(d) t=T
Figure 3.7: Upper boundary of the jet from
' r _Fr
*r # %
z
to
f
.
$# %
, and we have the boundary for all time. The exchange between all time intervals ] ,
zr , r the defined jet and the exterior takes place at each time .
Remark. Numerical computation of the manifolds around the hyperbolic trajectory have been made j V2 , : R , R , Kv R and oq ps: t - by evolving forward an for the following values to f , and by evolving backward an initial approximation of the unstable manifold from '
to ' , with up to 80. The principle initial approximation of the stable manifold from ] of the algorithm is given in the appendix.
Mean residence Time in the jet ( s ). To compute this statistical quantity and be able to compare it
with the result obtained with the previous approach, the T-incoming set and the T-transit at some time are needed. Here the turnstile lobes show their importance since all the points that were in the domain
(resp. that were not in the domain at time ) are contained in ": s and "s (resp. at time ] s : s if we keep the notations of the previous section. Figure 3.8 shows the T-incoming set at and
f z in )orange, time the hyperbolic trajectory are in black and the chosen BIPs are in green. > " ). As the We now want to find its time decomposition (supposing again that " turnstile mechanism is responsible for exchange, the only way points in the T-incoming set can leave the jet is to belong also to a lobe that will play the role of a turnstile lobe at some later time. Once a lobe found, it also gives us the exit time since the choice of our boundaries allows exchange only at
k that recover the incoming set, then some specific times. So we need to find the lobes at time
.
.
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Figure 3.8: T-incoming set at t=0. compute the area of each intersection and find at what time each lobe is playing the part of a turnstile lobe to get the time distribution and the mean residence time. Because of the symmetries, we can study only the part of the T-incoming set in the lobe of the upper boundary, the result will be the same for the other part. A general view of the stable and unstable manifolds for different hyperbolic trajectories is given
f z in figure 3.9. As usual, hyperbolic trajectories are the black dots, the stable manifolds are at time the blue lines and the unstable manifolds the red lines (each type of line is associated to a hyperbolic trajectory), and the studied part of the incoming set is in orange. With this view, one quickly notice
Figure 3.9: General structure of manifolds at t=0. two main points. First of all, the lobes from the north boundary never intersect the ones from the south
R ). Second of all, the lobes mainly composed of boundary since the perturbation is too small ( :
stable (resp. unstable) manifold are meandering up (resp. down) the jet, where to go up the jet is to go in the opposite direction of the flow. So in order to find the lobes covering the incoming set at
f z , one must look for the lobes mainly composed of stable manifolds associated to hyperbolic times trajectories of the north boundary, located farther in the -direction. 2 Unfortunately, by this method, computations have permitted to cover only up to of the incoming set (see figure3.10.(a) where the incoming set is covered by lobes playing the turnstile lobes
, ), knowing about the remaining that it is points leaving the defined at times domain after more than seven periods of the perturbation. To have an idea of the average residence time in the defined domain, we consider that the remaining area is covered by lobes being turnstile
~ , with ' , for . We take so that the lobes at times b b " = following serie converges:
#
#
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b b "
b b "
and is chosen to fit the first term of each series, that is to say s = figure 3.10.(b).
+ = b b 5 + b !! b 5
b!
(b) Evolution of
(a) Incoming set covered by computed lobes
Figure 3.10: Time decomposition of the T-incoming set at time
as shown in
with
.
With the estimation of the remaining lobes, we have the following results (figure 3.11 shows this evolution, such as the result found with the general geometric approach (red line)):
, b
b
b ! !
0.1 124.7
0.2 63.2
0.3 43.3
0.4 33.7
0.5 28.2
0.6 24.7
0.7 22.4
0.8 20.7
0.9 19.5
1.0 18.7
16.9
8.6
5.9
4.6
3.8
3.3
3.0
2.8
2.6
2.5
Figure 3.11: Conclusion. The results really quantify Lagrangian transport but depend highly on the remaining part of the time decomposition that has not been obtained.
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Chapter 4
Conclusion The study of Bower’s kinematic model of a meandering jet in both, a fixed frame and a moving frame, has given the opportunity to understand transport between the different regimes of the flow according to different points of view. The study in the moving frame has introduced the main already known results on the model, whereas the one made in the fixed frame offers new results. The theory, such as the computations, developped here in the fixed frame where based on ideas already applied in dynamical systems theory, but never to this model studied in the fixed frame. Quantified results have been obtained in the fixed frame based on different approaches, and more concern has been granted to the average residence time in the jet. The general geometric approach is the closest to what could be simply applied when studying real flows well modeled by a meandering jet such as the Gulf Stream. It uses the “brute” power of computers. The results obtained give more an Eulerian point of view on transport than a Lagrangian one, since the way to define the jet is not based on barriers to particle paths but on streamlines behaviour. The lobe dynamics approach, exactly quantifying Lagrangian transport, has not been able to give an exact quantified value for the average residence time. But an estimation of what remains to be found shows that the results can possibly be of the same order than the one found with the general geometric approach. If it is to be verified, then it would help observations on real flows to estimate Lagrangian transport, with a simpler tool than lobe dynamics (but which gives exact results). More computations are needed to clearly answer the question of the similarity of the two approaches. Another important feature expected is to be able to compare with results based on the lobes dynamics approach too, and obtained in the moving frame. Thus one would be able to say if the studies made in the moving frame (usually much simpler than in the fixed frame) give results that are the same than the studies made in the fixed frame, as geophysicians do with observation of drifters.
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Bibliography [1] Wiggins S., The Dynamical Systems Approach to Lagrangian Transport in Oceanic Flows, Annu. Rev. Fluid Mech. 2005, 37, 295-328 (2005). [2] Bower A.S., A Simple Kinematic Mechanism for Mixing Fluid Parcels across a Meandering Jet, Journ. of Phys. Oceano. 21, 173-180 (1991). [3] Hama F.R., Streaklines in a Perturbed Flow, The Physics of Fluids, 5, 644-650 (1962). [4] Samelson R.M., Fluid Exchange across a Meandering Jet, Journ. of Phys. Oceano., 22, 431-440 (1992). [5] Raynal F., Wiggins S., Global Horseshoes and Lobe Dynamics in a Kinematic Model of a Meandering Jet, to be published (2005) [6] Malhotra N., Wiggins S., Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow, Journ. Nonlinear Science, 8, 401456 (1998). [7] Mancho A.M., Small D., Wiggins S., Ide K., Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields, Physica D, 182, 188-222 (2003). [8] Mancho A.M., Small D., Wiggins S., A Tutorial on Dynamical Systems Concepts Applied to Lagrangian Transport in Coastal and Oceanic Environments, to be published in a book (?). [9] Meiss J.D., Average Exit Time for Volume-Preseving Maps, Chaos, 7, 139-147 (1997). [10] Wiggins S., Transport in Finite-Time, Aperiodically Time-Dependent Velocity Field without Hyperbolicity or Lobe Dynamics: Application to Transport in Monterey Bay, internal paper (?). [11] Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag (1990). [12] Ide K., Small D., Wiggins S., Distinguished Hyperbolic Trajectories in Time Dependent Fluid Flows: Analytical and Computational Approach for Velocity Fields Defined as Data Sets, Nonlinear Processes in Geophysics, 9, 237-263 (2002). [13] Ju N., Small D., Wiggins S., Existence and Computation of Hyperbolic Trajectories of Aperiodically Time Dependent Vector Fields and their Approximations, International Journal of Bifurcation and Chaos, vol.13, No.13, 1449-1457 (2003). [14] Lozier S., Lawrence J.P., Rogerson A.M., Miller P.D., Exchange Geometry Revealed by Float Trajectories in the Gulf Stream, Journal of Physical Oceanography, 27, 2327-2341 (1997). [15] Meyers S.D., Cross-Frontal Mixing in a Meandering Jet, Journal of Physical Oceanography, 24, 16411646 (1994). [16] Coulliette C., Wiggins S., Intergyre Transport in a Wind-Driven, Quasigeostrophic Double Gyre: an Application of Lobe Dynamics, Nonlinear Processes in Geophysics, 8, 69-94 (2001). [17] Dieci L., Russell R.D., Van Vleck E.S., On the Computation of Lyapunov Exponents for Continuous Dynamical Systems, SIAM Journal on Numerical Analysis, vol.34, No.1, 402-423, (1997).
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Appendix A
Numerical Methods Along with the theoritical work, numerical simulations have been done in order to compute mainfolds, exit and incoming set, statistical quantities, etc. Most of the simulations done for the brief study in the moving frame have been done with Maple c . Since the streamfunction was known analytically, it was helpful to use the symbolic and numeric computations. But it is too slow to do the computations associated to the perturbated version of the model, since the computation of the manifolds for the unperturbed case took already more than half an hour with Maple c . The rest of the computations has been done under Linux, using C language. The codes were compiled on different computers (depending on the availability) of the cluster of workstations of the Department of Mathematics of Bristol University. Typically, a workstation would be running a Linux sytem, processor 2.8GHz, 2G (or 4G) RAM (DDR ECC), cache 512kb.
A.1 Integration of ODE To obtain the manifolds and fluid particles trajectories, one need to integrate ordinary differential equations. Runge-Kutta algorithms were developped to do so, with increased precision, needed by the order with fixed step, of increase of the amplitude of perturbations. Runge-Kutta algorithms of order with fixed and adaptative step were thus implemented, inspired by Numerical Recipes and codes already developed in Python Language at the Department of Mathematics, with articles [7, 12, 13].
A.2 Computation of Hyperbolic Trajectories and their Local Stable and Unstable Manifolds To find a hyperbolic trajectory and its associated manifolds in a two-dimensional time-dependent velocity field, some numerical methods have been developed in [12, 13]. The outline of the numerical approach is the following. Given a velocity field (or a interpolated velocity field from datas), one first compute a hyperbolic trajectory, once found its stable and unstable manifolds at a certain time are obtained by evolving (forward or backward in time) a linear approximation of these manifolds. The code to find a hyperbolic trajectory gives the linear approximation of the manifolds. The general algorithm for finding hyperbolic trajectories has been implemented without success, even with some help from the developer of the codes of the articles (D.Small), but the zero-order estimation of the hyperbolic trajectory and manifolds was enough to obtain the real manifolds, based on the following algorithm.
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A.3 Evolution of Stable and Unstable Manifolds To obtain manifolds (of hyperbolic trajectories) bigger than local approximations is a tricky task since their behaviours under perturbation generate lobes that requires a high number of points to be represented accurately. In order to fill the gaps that appear by evolving an unstable (resp. stable)
(resp. ), , one need manifold forward (resp. backward) in time from time to time ]
to introduce points at time between the ones already known. It is not reasonable to simply introduce
), since it might requires an infinite number of points untill the gaps disappear at time ] (resp. points, and it does not take into account that some zones need less points than others. So one need to interpolate the behaviour of the manifolds in order to insert a lower number of points in the manifold. Comparisons between different interpolation and insertion techniques are given in [7] and we refer to it for the names of the insertion and interpolation technique used. For the purpose of this study, we considered the Dritschel-Ambaum criterion for points insertion and interpolation. To have a smooth curve two successive points and ;: , the interpolation between
V 2 2 uses four consecutive points , ] ] and the local curvature defined by a and ;: . The insertion criterion circle through " : , is based on the density of points along the interpolated manifold.
A.4 Extracting Lobes and Computing Their Areas Once the stable and unstable manifolds have been fully evolved, one can be interested in the lobes created by their intersections. Small codes have been implemented in order to find intersection points of curves defined as a set of points, to extract lobes from stable an unstable manifolds intersecting. Also using Green’s Theorem, one can easily compute the area of the lobes extracted, or more generally of any close curve.
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