Assessing Relative Dispersion

We assess the four point method of relative dispersion proposed by Jones and Winkler to identify the hyperbolic trajectories of a system. We sample a discrete analog to a quasi-geostrophic, single layered flow field and perform a comparison of the dispersion of neighboring points after iteration. We evaluate our method by performing a transformation to ( ) , x y space and comparing the trajectories corresponding to maximum dispersion with the ( ) , x y values of trajectories of the Standard map, derived from traditional techniques. We perform a similar evaluation using a 2D Ross by wave. We show that the method of relative dispersion is able to generate 2 order accurate (on the scale of the discretization) hyperbolic trajectories.


Introduction
Identifying the manifolds of finite time geophysical structures, which includes atmospheric and oceanographic circulations, is integral to understanding the mechanisms by which they evolve and move in both space and time. The geometry of the manifolds provide insight into intrinsic properties of the flows such as the transport and mixing within the system [1]- [7]. Due to the scale of some geophysical data (both oceanic and atmospheric) obtaining data can be extremely costly from a time, computational, and economic aspect. Traditionally, this information has been derived from sampled data taken from floats, weather balloons or even satellite imagery [8]. Unfortunately, the data sets available for effective investigations are often either too sparse in time or too large in space to be used to efficiently model and analyze with traditional numerical methods and high resolution in space, time, or both, limiting the research investigations [9].
The expenses related to data acquisition and processing dictate that an effective method should be used for sensor deployment to ensure that the maximum amount of data is obtained from the investigation, while minimizing associated costs [10].
For chaotic and sub-chaotic systems, the existence of finite-time, invariant manifolds help to distinguish regions of uniform transport or chaotic motion from neighboring regions of transport [11]. The stable and unstable manifolds within a region act as barriers to transport and contain similar circulations.
Depending upon the desired data, deploying sensors within, external to, or along the manifolds will be ideal for generating investigation appropriate data [10].
Data regarding manifold location is a major component to many studies, as they can help guide the investigation or assess the accuracy of the model itself [12].
For maps and computational velocity fields, the hyperbolic trajectories associated with given manifolds can be numerically and sometimes analytically derived from their Hamiltonian representation. In the case of observed data, a closed form representation of the system can only be assumed or approximated, leaving a great deal of room for errors. Other methods must be employed to efficiently locate trajectories of interest and boundaries to flow in more complex systems.
The problem is not trivial and additional methods must be developed, applied, and assessed for use in locating features of various systems.
Relative dispersion is a method that has been used to provide insight into the locations of key features within flows. We perform an assessment of how well relative dispersion locates trajectories of sample systems. As a first step to identifying the underlying mechanics of a system, the homoclinic and heteroclinic orbits of some well known 2D systems are used to evaluate relative dispersion.
An early strain-based method for identifying stable and unstable manifolds in the context of two-dimensional, incompressible atmospheric flows involved initializing two particles in the vicinity of a hyperbolic point within a flow. By iterating forward in time, the distance between two particles straddling a manifold will grow as the particles followed their associated trajectories. The distance between the two initialized points after iteration, divided by the distance between them after iteration formed the basis for the assessment of finite strain [13]. An important aspect of the method is that it requires no a priori knowledge of the location of hyperbolic points within the flow, rather it is a tool that can be effective in locating the stagnation points.
The two point method proposed by Bowman effectively detects the local extrema of strain, but is not sufficient to guarantee the existence of hyperbolic sets in the vicinity. The method is dependent upon the system running for an appropriate termination time, which may be unknown. For an appropriately chosen observational time interval however, the method is effective for identifying invariant manifolds at a given scale and provides a good approximation in test  [16]. We assess the four point method [16] to balance accuracy with computational efficiency. While the method of relative dispersion has been used in many studies to approximate the locations of manifolds, the accuracy of the method has not been as widely explored. To better assess the method, we focus on a single parameter map with known stagnation points and velocity field and then expand its application to the time independent Rossby wave.

Relative Dispersion in the Standard Map
We use the four point method [16] of identifying the hyperbolic trajectories of a system based on the relative motion of particles that are initialized in close proximity to one another. To assess the effectiveness of the method and simulate regions of bounded chaotic motion in close proximity, we consider the Chirikov Standard Map. The Standard map is a classic example of nonlinear motion. It is an autonomous, two dimensional, area preserving map, with the degree of chaos dependent upon a single parameter, K. Often used in physics and mathematics as an initial model of gyre motion or motion within a vortex, it is a prime example of the conventionally known "kicked rotor". The Standard map is a turbulent map that receives regular contributions to the motion of the next iteration from the previous iteration. It is well known for its distinct regions of chaotic motion, bound by invariant orbits, and existing at various scales [17] [18] [19]. The map is traditionally computed modulo 2π. In the case of 0 K = , the map is equivalent to constant level-set rotation on a cylinder (invariant circles).
For increasing values of K, the chaos inducing contribution is increased. The instance of 0.971635 K = is the traditionally accepted value for which the system becomes chaotic [20]. Increasing the parameter K will result in the , To identify the hyperbolic trajectories, we use a filter to apply a threshold to our phase space and highlight regions that undergo large dispersion. For a given grid point, ( ) , i j , and threshold value, τ , the filtered matrix of dispersion values is given by, Similarly, for points ( ) , the separation distance after advection is bound in the vertical direction by: For the Standard map: In other versions of this method, the dispersion value is defined relative to the initial dispersion of the particles in the x and y directions. To apply the method to the Standard map, we use a uniformly spaced, square grid to generate the initial particle locations. Setting the initial dispersion in the horizontal and vertical directions to 2 10 − , we can consider only the dispersion after iteration.
Using Equations 8 and 9 the relative dispersion response can be bound by We compare the R value at each location, ( ) , i j , with the upper bound of R values to generate our threshold as a percentage: In applying the method of relative dispersion to the Standard map, we see that points that are initialized within one of the chaotic islands of the Standard map ) remain entrained within the circulation that travels between the upper and lower halves of the phase space.
The particles that begin above the line y Kx = are trapped outside of the chaotic islands and travel more as a packet, undergoing less dispersion relative to each other.

T. C. Redd et al.
For initial positions near the hyperbolic trajectory as in Figure 3, the particles may follow different travel paths. Consider the particles on the horizontal axis, the particle on the left starts inside of the chaotic region, while the particle on the right is largely restricted to travel within the non-chaotic island. Similarly, in the vertical direction, the particle on the bottom will travel within the nonchaotic region while the corresponding particle at the top follows a path within the chaotic region. The result is a larger relative dispersion between points that begin in the vicinity of the central grid point (the lighter blue regions of Figure  4). In instances where the quartet of points is initialized within the chaotic regions (or between the chaotic regions) dispersion is minimized, resulting in lower relative dispersion values (the darker blue regions of Figure 4).

Relative Dispersion in a Rossby Wave
A system representing an autonomous Rossby wave is used to further assess how well relative dispersion identifies hyperbolic trajectories. The use of the Rossby wave helps provide insight into the method's impact and validity relative to dynamical systems. The general formulation of the Rossby wave under consideration: The equation is often used to simulate gyre motion. The system as used can be considered a simple model for a double gyre flow ( Figure 5). The Rossby wave is selected as an example of a physical, Hamiltonian system, providing some additional tools to aid in evaluation. Like the Standard map, the differential

Comparisons
We investigate the distribution of the relative dispersion values to determine optimal threshold values for isolating specific trajectories. The histograms in

Results
We performed a relative dispersion assessment for several values of K for the Standard map. Recall the parameter K controls the chaos in the map. Chaotic behavior in the map is traditionally associated with values of K above 0.971. We Examples of the intensity images returned from relative dispersion are presented in Figure 9. For comparison, the same images are overlaid with the corresponding hyperbolic trajectories that are used for validation (in red). Per initial observations, relative dispersion seems to provide very good alignment with the manifolds.
From our assessment of R distribution (Figure 7), we found that values of 12 R > , are the optimal focal points for efforts to delineate the key trajectories.
We also found that values of 16 R > , provided accurate but very sparse data for approximating the desired orbits. While values closer to ( ) max R would seem ideal, the accuracy relative to location that is gained from a higher threshold  shows the total number of grid points returning a given dispersion value. The 31,752 points are grouped based on the dispersion value that is generated for their initial locations.
value sacrifices density of data points used to delineate the trajectory. The loss of additional data points introduces room for interpolation errors. Therefore we have found that there is an inherent balancing act that must be performed by the investigator based on the desired outcome and intended use of the data. Values of ( ) 14,16 R ∈ were found to provide the best balance between accuracy of approximation and density of data.
In an effort to generalize R thresholds for other maps and flows, the threshold has also been investigated as a percentage relative to the maximum dispersion value.
We found values of   Figure 10. The Relative Dispersion generated intensity field, the locations that return a value above 80% of the maximum value, and the locations returning 80% of the maximum value superimposed on the orbits generated by a 4th order Runge-Kutta method.
In Figure 9, the dark blue regions correspond to small dispersion values.
Lighter blue and white correspond to larger dispersion values. The grid points that satisfy a given threshold value are highlighted in green. In each image, the corresponding trajectory (stable or unstable) is superimposed in red. In many of the images, the green is difficult to observe due to its (desired) coincidence with the superimposed, comparison trajectory, in red.
A similar assessment is applied to the Rossby wave example. The Rossby example did not require as high a threshold as the Standard map ( Figure 8). A threshold that isolates particles that have traveled in the top 50% of total T. C. Redd et al.  distances eliminates many of the excess paths, while increasing to 80% refines the results. Increasing to 90% does not improve the results in the unstable directions, while losing some information in the stable directions ( Figure 10).

Discussion
The relative dispersion response for the Standard map is presented for dispersion T. C. Redd et al.  relative dispersion than point quartets that are wholly located either inside or outside of the chaotic islands of the Standard map. The mean error is second order accurate. The dispersion error is calculated as the system evolves and the relative dispersion response is calculated, which contributes to the error convergence ( Figure 12). The results may allow one to determine how long in time or iteration is necessary to allow a system to run to return sufficient representation of its behavior.
When only the final time step is considered, and the per point error is plotted, additional information can be isolated. Initial locations near fixed points in the stable direction immediately undergo dispersion, and it grows with each time step. Values that begin near the unstable trajectory travel as a packet in the short term and undergo increasing amounts of dispersion as they near the next fixed point.
In the Rossby wave example, the oscillations in the stable direction that occur in the Standard map are not present ( Figure 13). Upon inspection, the relative dispersion generated approximation appears to be better. The mean error is improved, but more importantly is also on the scale of the discretization. The To gauge the degree of accuracy of the approximation, an error calculation is performed. For each time step, the total registration error between the points generated by relative dispersion and Runge-Kutta is computed. The error decreases with time. There is very quick convergence of the relative dispersion response to the Runge-Kutta response, which can aid investigators in determining the appropriate length of time to run the model ( Figure 14). Depending on the needs of the project, the model may be able to be run for as few as 10 time steps rather than 45 depending on desired accuracy.
In the future, an application to other time independent systems would provide further insight into the viability of the method for identifying stable and unstable trajectories and other orbits. While relative dispersion can and has been applied to time dependent systems, additional methods are needed for evaluating the results and may be informative. Modifying the parameters of the systems to T. C. Redd et al.