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Distinguished Hyperbolic Trajectories

Hyperbolic or ``saddle type'' stagnation points for two-dimensional, incompressible, steady flows are often of great significance in a flow because they tend to be the ``origin'' of qualitatively different fluid particle motions. On the boundaries of a fluid flow they may be manifested as ``separation points'' or ``convergence points'' of the flow. Moreover, associated with them are curves; the stable and unstable manifolds of the hyperbolic stagnation point. Fluid particle trajectories cannot cross these curves (they are invariant manifolds). Thus they divide the flow into regions of qualitatively different fluid particle trajectories. We want to generalize the notion of hyperbolic stagnation point to two-dimensional, incompressible, unsteady flows. We want our notion of ``moving hyperbolic stagnation point'' to have two characteristics in common with the steady case.

It must be a fluid particle trajectory.

This is certainly necessary because the significance of the saddle point structure is that it ``organizes'' fluid particle trajectories that behave in qualitatively different ways. While this requirement may seem ``obvious'', we will see that if we consider ``frozen time'' velocity fields then there are some (generally unfamiliar to most) subtleties that arise when we consider unsteady flows.

The ``moving hyperbolic stagnation point'' must have one dimensional (invariant) stable and unstable manifolds (which, generally, will also be time-dependent).

In this way the ``moving hyperbolic stagnation point'' will give rise to one-dimensional ``flow barriers''. First, we establish some notation. We consider two-dimensional, incompressible¹ velocity fields of the form:

$\begin{displaymath} \dot{x} = v(x, t), \qquad x \in {\rm I\hskip -0.2em R}^2, \quad t \in {\rm I\hskip -0.2em R}, \end{displaymath}$

(1)

Now we consider two examples that illustrate in a concrete manner the issues that we will face. The examples are one (space) dimensional. This may seem far removed from the fluid mechanical applications of interest. However, this is not the case since in many applications the boundary conditions may be free slip and then the issue of saddle type trajectories on one dimensional boundaries becomes of interest.

Example 1. Consider the following example from Szeri et al. [1991]:

$\begin{displaymath} \dot{x} = -x + t, \end{displaymath}$

(2)

The solution through the point

is given by

$\begin{displaymath} x(t) = t-1 + e^{-t} \left( x_0 + 1 \right). \end{displaymath}$

(3)

A typical way to visualize the trajectories of time dependent vector fields is to consider the ``frozen time'' setting. That is, one fixes time and then considers the resulting instantaneous direction field, instantaneous streamline contours, instantaneous stagnation points (henceforth, ISPs), etc. However, such information can be very misleading if one uses it to try to understand Lagrangian transport issues. Consider the ISPs for (2). These are given by

$\begin{displaymath} x=t. \end{displaymath}$

(4)

At a fixed

, this is the unique point where the velocity is zero. However,

is not a solution of (2). This is very different from the case of a steady flow where a stagnation point is a solution of the velocity field. Now let us return to the issue of a hyperbolic trajectory. We will define this more formally later. Now we will content ourselves with a less mathematically formal description in order to motivate the ideas. A trajectory is said to be hyperbolic if the associated linear equations (linearized about the trajectory in question) have

linearly independent exponentially growing and decaying solutions (as $t \rightarrow \infty$ ). That is, all solutions of the linearized equations exhibit exponential growth and decay. The linearization of (2) is given by

$\begin{displaymath} \dot{\xi} = -\xi, \end{displaymath}$

(5)

that is, the linearization of (2) is the same for any trajectory. Clearly, all trajectories of (2) are hyperbolic². This brings us to the notion of a distinguished hyperbolic trajectory (henceforth, DHT). Despite the fact that all trajectories of (2) are hyperbolic, upon examining the form of the general solution given in (3) we see that all trajectories decay at an exponential rate to the trajectory

$\begin{displaymath} x(t) = t-1. \end{displaymath}$

(6)

This trajectory is our DHT. Note also that it is the only trajectory that does not exhibit exponential growth or decay, which can be clearly seen for the repelling situation as $t\rightarrow -\infty$ . It remains for us to give it a precise mathematical definition in such a way that it lends itself to numerical computation. However, before doing that let's return to the issue of ISPs and their relationship to DHT's. In Fig. 1 we plot some of the trajectories of (2). In particular, we plot the DHT, we show some trajectories converging to it, and we plot the curve of instantaneous stagnation points.

Figure 1:The trajectories of (1) plotted in

space.The DHT is given by

and the curve of ISPs is plotted as a dashed line and given by

In Fig. 2 we plot the ``frozen time'' velocity field at some time $t=t^{*}$ . In this figure we see something that seems somewhat counterintuitive. Trajectories to the right of the DHT appear to be moving away from the DHT, towards the ISP. However, we know from (3) that all trajectories decay to the DHT at an exponential rate. What we are ``seeing'' in Fig. 2 is an artifact of drawing incorrect conclusions from instantaneous velocity fields. Trajectories to the immediate right of the DHT are indeed moving to the right (i.e., away from the DHT). However, the DHT is moving to the right at a faster speed and it eventually overtakes these trajectories. Fig. 2 might also lead us to believe that trajectories converge to the ISP. But we know this is not true since we have the exact solutions.

Figure 2: The ``frozen time'' velocity field at $t=t^{*}$ . The DHT and the ISP are indicated by diamond and circle, respectively

Example 2. The time dependent term (or ``forcing'') on the right hand side of (2) is unbounded as $t \rightarrow \infty$ . However, we give another example that shows the phenomena described above is not a consequence of this unboundedness. Consider the equation

$\begin{displaymath} \dot{x} = -x + \sin t. \end{displaymath}$

(7)

The general solution through any point

is given by

$\begin{displaymath} x(t) =\frac{1}{2} \left( \sin t - \cos t \right) + e^{-t} \left( x_0 + \frac{1}{2} \right). \end{displaymath}$

(8)

As in the previous example, all solutions are hyperbolic and any solution decays exponentially to the solution

$\begin{displaymath} x(t) =\frac{1}{2} \left( \sin t - \cos t \right), \end{displaymath}$

which is the DHT. One can also verify that the ISPs, $x = \sin t$ , is not a solution of (7).

To summarize, these simple examples illustrate the following points.

A given velocity field can contain an uncountable infinity of hyperbolic trajectories. Indeed, in these examples all trajectories are hyperbolic with the same decay rates.
Despite this fact, we see that there may be certain distinguished hyperbolic trajectories. In these examples, this was the one trajectory that all trajectories are attracted to exponentially as $t \rightarrow \infty$ .
Due to this abundance of hyperbolic trajectories, we see that a numerical method that is designed just to find hyperbolic trajectories may not be sufficiently refined for applications. For this reason one needs to precisely define the notion of a DHT for an analytically given velocity filed. Then one needs to develop a methodology for numerical identification of the DHT, according to the refined definition, when the velocity field is given as a discrete data set, rather than an analytical function.
ISPs are not necessarily trajectories of the velocity field. Moreover, viewing them in instantaneous velocity fields may lead to misleading information about fluid particle trajectories.
Numerical methods for locating hyperbolic trajectories that utilize the stretching and contraction properties to allow certain ``test regions'' to converge to the hyperbolic trajectory are not adequate for time-dependent velocity fields that are only known for a finite interval of time. In the process of convergence we ``lose'' much of the velocity field. Moreover, such methods require a good guess for the ``test regions'' that somehow bracket the hyperbolic trajectory as discussed earlier. We have seen that the instantaneous stagnation points do not necessarily provide us with a good guess for the location of the hyperbolic trajectory.

Motivated by the simple examples above, we define two classes of DHTs. The first class considers a velocity field whose linear part is independent and constant so that it closely relates to these examples. The second class considers a general velocity field as an extention of the first class.

Definition 1. Let is a trajectory of (1) that remains in a bounded region for all time. Then is said to be a distinguished hyperbolic trajectory if:

it is hyperbolic,
there exists a neighborhood ${\cal B}$ in the flow domain having the property that the DHT remains in ${\cal B}$ for all time, and all other trajectories starting in $\cal B$ leave $\cal B$ in finite time, as time evolves in either a positive or negative sense,
it is not a hyperbolic trajectory contained in the chaotic invariant set created by the intersection of the stable and unstable manifolds of another hyperbolic trajectory.

If the data spans only a finite time interval the DHT can not be determined uniquely. Instead, there is a small region in ${\cal B}$ where the DHT can exist. We will present a method to obtain an approximation to the DHT assuming that the time dependence of the velocity field persists outside the time-interval of the data set. The second part of this definition can be stated also in terms of the stable and unstable manifolds of the DHT. Points on the stable manifold can leave $\cal B$ in negative time, points on the unstable manifold can leave $\cal B$ in positive time, and points on neither manifold leave $\cal B$ in both positive and negative time. In the case where the DHT does not remain in a bounded region the definition is more tricky. This leads to the second class of DHTs.

Definition 2. Let us consider the general velocity field given by (1). Let us assume that there exists an invertible coodinate change from to , which is based on the movement of an Eulerian structure in , such as a path of an ISP. Let be a solution of the transformd velocity field that satisfies the three conditions given in Definition 1. Then the corresponding is said to be a distinguished hyperbolic trajectory.

If is a DHT in the coordinates, then the corresponding trajectory in the original velocity field is also a DHT, because DHTs are frame independent. Our task now will be to show that this definition does indeed satisfy the requirement of picking out the important hyperbolic trajectories for the application of Lagrangian transport theory using the method of lobe dynamics. This is the motivation for the third part of Definition 1. If the stable and unstable manifolds of a hyperbolic trajectory intersect transversely then there is an associated lobe dynamics that describes the motion of trajectories through the homoclinic tangle. A consequence of the transverse intersection of the stable and unstable manifold is the formation of an invariant Cantor set on which the dynamics is chaotic, with all trajectories in the Cantor set being hyperbolic. However, for our purposes, we would not call the hyperbolic trajectories in the Cantor set ``distinguished'' as the transport of trajectories through this Cantor set is governed by the lobe dynamics associated with the hyperbolic trajectory whose transversely intersecting stable and unstable manifolds give rise to the hyperbolic Cantor set.

References

Coppel, W. A. [1978] Dichotomies in Stability Theory. Springer Lecture Notes in Mathematics, vol. 629. Springer-Verlag: New York, Heidelberg, Berlin.
Ide, K., Small, D., Wiggins, S. [2001] Distinguished hyperbolic trajectories in time dependent flows: analytical and computational approach for velocity fields defined as data sets. accepted for publication in Nonlinear Processes in Geophysics. This paper, as well as the software used to produce the results presented in the paper, is available at http://lacms.maths.bris.ac.uk/publications/dht.
Ju, N., Small, D., Wiggins, S. [2001] Existence and Computation of Hyperbolic Trajectories of Aperiodically Time Dependent Vector Fields and Their Approximations. submitted to Journal of Mathematical Analysis & Applications. This paper, as well as the software used to produce the results presented in the paper, is available at http://lacms.maths.bris.ac.uk/publications/integral/.
Szeri, A., Leal, L.G., Wiggins, S. [1991] On the Dynamics of Suspended Microstructure in Unsteady, Spatially Inhomogeneous Two-Dimensional Fluid Flows. Journal of Fluid Mechanics, 228, 207-241.

Footnotes:

¹ Recall, incompressibility means that $\frac{\partial v_1}{\partial x_1} + \frac{\partial v_2}{\partial x_2}=0$ , where $v(x,t) \equiv (v_1 (x_1, x_2, t), v_2 (x_1, x_2, t))$ .
² A given trajectory is said to be hyperbolic if the linearly independent solutions of the velocity field linearized about the given trajectory exhibit exponential growth and/or decay in time. A detailed discussion for velocity fields with arbitrary time dependence can be found in Coppel [1978]