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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 2
*D* Ross by wave. We show that the method of relative dispersion is able to generate 2
^{nd} order accurate (on the scale of the discretization) hyperbolic trajectories.

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 [

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 [

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 [

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 cases [

A variation of the two point method was used to investigate transport and mixing in baroclinic vortices in atmospheric flows, particularly in the vicinity of developing eddies [

Both the two point and eight point methods have shortcomings relative to either reliability or computational implementation when applied to two dimen- sional systems. The two point method does not take into account stretching in directions orthogonal to the line segments formed by the points, while the eight point method includes duplicate information, unnecessarily increasing the number of required computations [

We use the four point method [

y n + 1 = y n + K sin ( x n + 1 ) (1)

x n + 1 = x n + y n (2)

For the Standard map, relative dispersion is evaluated on the space of [ 0,2 π ) in the x and y directions, using a mod of 2. The interval is discretized into a 628 × 628 grid and the resulting dispersion values for each gridpoint ( x , y ) are stored in the 628 × 628 matrix, R ( i , j ) . A natural conversion from ( i , j ) to ( x , y ) space is given by:

x r = i − 1 628 2 π (3)

y r = j − 1 628 2 π (4)

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,

ϕ ( R i j ) = ( 1 R i j ≥ τ 0 R i j < τ (5)

The resulting binary, filtered matrix consists of ones in the ( i , j ) locations that correspond to relative dispersion values above a given threshold and zeroes elsewhere. The threshold is determined relative to the maximum possible dispersion value. Converting the resulting grid points from ϕ ( R i j ) intensity space to ( x r , y r ) space yields the path of maximum transport relative to neighboring points for the map. Based on distance arguments, relative dispersion on the [ 0,2 ) torus is bound. For the point located at ( x , y ) , there are four neighboring points that are followed in iteration and used to calculate relative dispersion. The locations of the neighboring points are provided in

R h = ( x i + 1 , j N − x i − 1 , j N ) 2 + ( y i + 1 , j N − y i − 1 , j N ) 2 (6)

Similarly, for points ( x i , j − 1 N , y i , j − 1 N ) and ( x i , j + 1 N , y i , j + 1 N ) , the separation distance after advection is bound in the vertical direction by:

R v = ( x i , j + 1 N − x i , j − 1 N ) 2 + ( y i , j + 1 N − y i , j − 1 N ) 2 (7)

For the Standard map:

R h < 2 π 2 , R v < 2 π 2 (8)

R = R h + R v (9)

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 10 − 2 , we can consider only the dispersion after iteration. Using Equations 8 and 9 the relative dispersion response can be bound by

R < 32 π 2 ≈ 17.771 (10)

We compare the R value at each location, ( i , j ) , with the upper bound of R values to generate our threshold as a percentage:

T h r e s h = R max ( R ) (11)

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 (below the line y = K x in the interval x ∈ ( 0,3 ) ) 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 = K x are trapped outside of the chaotic islands and travel more as a packet, undergoing less dispersion relative to each other.

For initial positions near the hyperbolic trajectory as in

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:

d y d t = a 0 sin ( k x ) + b 0 sin ( ω t ) (12)

d x d t = y (13)

The equation is often used to simulate gyre motion. The system as used can be considered a simple model for a double gyre flow (

equation uses a parameter, a 0 , to control the non-linear “kick” for the system. The parameter, b 0 controls the time dependence [

ψ 0 = 1 2 y 2 − a 0 k cos ( k x ) + c (14)

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

map. There are also a large number of grid points whose relative dispersion value falls in the interval R ∈ ( 2,7 ) . While the number of grid points that result in an R > 7 decreases, the cumulative effect of relative dispersion for all values greater than R = 7 results in poor isolation of the trajectory. Other orbits are well defined at that value while values of R > 12 result in visually and

quantitatively closer approximations to the stable and unstable manifolds of the Standard map. The distribution of the 628 × 628 grid of relative dispersion values is binned in

To help evaluate how well the relative dispersion derived trajectories approximate the traditionally derived trajectories, Jacobian analyses are per- formed on each system. Using the results classifies and provides the orientations for the trajectories corresponding to the fixed point.

In the Standard map, there are fixed points at ( 2 π n , 0 ) and at ( ( 2 n − 1 ) π ,0 ) . A Jacobian analysis classifies the points as hyperbolic fixed points and centers, respectively.

J = ( 0 1 k cos x 0 ) (15)

J ( 0 , 0 ) : → λ 1 = k , λ 2 = − k (16)

J ( ( 2 n − 1 ) π , 0 ) : → λ 1 = i k , λ 2 = − i k (17)

In the Rossby wave, a Jacobian analysis of the fixed points for k > 0 on the interval x ∈ ( − 2 π ,2 π ) returns 2 real eigenvalues at ( 0,0 ) and 2 purely imaginary eigenvalues at ± ( π ,0 ) . The fixed points at ( 2 π n , 0 ) are hyperbolic, while the fixed points corresponding to ( ( 2 n − 1 ) π ,0 ) result in centers.

J = ( 0 1 a 0 k cos x 0 ) (18)

J ( 0 , 0 ) : → λ 1 = a 0 k , λ 2 = − a 0 k (19)

J ( ( 2 n − 1 ) π , 0 ) : → λ 1 = i a 0 k , λ 2 = − i a 0 k (20)

With a cursory glance, one can anticipate lower dispersion values to be generated when the four points used are contained inside of either of the two constrained regions (

Though a closed form solution exists for the system, the solution for all systems may not be easily found, if it is possible at all. In the case of observed data, the equations generating the motion are often unknown. The accuracy of the method must be investigated relative to data that is typically available. As in the case of the Standard map, the assessment is conducted by identifying a comparison hyperbolic trajectory, densely discretizing along it, and iterating in time. To generate the comparison trajectory, points are initialized along the hyperbolic trajectory for x ∈ ( 0,0.1 ) , requiring the sampling rate δ x ≪ 0.1 and iterated forward in time. In the trial, δ x = 0.00001 . After iteration, the distance between neighboring points along the trajectory is calculated, δ x ′ . The distance between any two successive points after iteration is required to be less than the grid discretization that is used in calculating the relative dispersion values ( 0.01 ) .

To compare the data, a rigid transformation from ( i , j ) space to ( x , y ) space must again be applied to the data. Due to the difference in the domain size (as compared to the Standard map), a different scaling is required. The system is analyzed on the interval x ∈ ( − 2 π ,2 π ) and y ∈ ( − π , π ) . The conversion from ( i , j ) to ( x , y ) space is

x = 4 π i − 1 N x − 2 π y = 2 π j − 1 N y − π (21)

where i and j are both integers and i ∈ [ 1, N x ] and j ∈ [ 1, N y ] .

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 present the relative dispersion assessments for K = 0.971635 . Each map was iterated for N = 16 steps to normalize the comparison. The trajectory of focus can be seen starting near ( 0,0 ) , peaking near ( 4.4 , 2 ) and oscillating as it approaches ( 2,0 ) . A full discussion of the features and dynamics of the Standard map can be found in numerous articles and books, and is omitted here. Examples of the intensity images returned from relative dispersion are pre- sented in

From our assessment of R distribution (

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 R ∈ ( 14,16 ) 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 R ∈ ( 75 % max ( R ) , 90 % max ( R ) ) provide a general form for the optimal R threshold values for delineating both the trajectories of the Standard map. These values are equivalent to an interval of R ∈ ( 12.8147,15.377 ) .

In

A similar assessment is applied to the Rossby wave example. The Rossby example did not require as high a threshold as the Standard map (

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 (

The relative dispersion response for the Standard map is presented for dispersion

values greater than 15 (the maximum possible is approximately 17.77) (

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 (

Instead of using a threshold involving an arbitrarily chosen dispersion value, the threshold is defined in terms of a percentage of the maximum dispersion value. All of the dispersion values are considered and only central locations that return 80% of the maximum dispersion value over the time period are used. The response from relative dispersion is similar to that of the Standard map, though with better delineation in the unstable directions. Using a filter of 80%, there is good consistency in the localization of the points. Superimposing the responses from the relative dispersion implementation, the Runge-Kutta derived trajectory, and the flow field generated by the equations results in good alignment upon inspection (

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 deter- mining the appropriate length of time to run the model (

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 induce additional dynamics will give insight into the robustness of the method. In the larger scheme, expansion to three dimensional systems and an assessment of the method’s ability to accurately identify surfaces is a major goal.

Redd, T.C., El Moghraby, A. and Tillman, T.J. (2017) Assessing Relative Dispersion. Applied Mathematics, 8, 1572-1589. https://doi.org/10.4236/am.2017.811115