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Mixing and coherence are fundamental issues at the heart of understanding fluid dynamics and other non-autonomous dynamical systems. Recently the notion of coherence has come to a more rigorous footing, in particular, within the studies of finite-time nonautonomous dynamical systems. Here we recall “shape coherent sets” which is proven to correspond to slowly evolving curvature, for which tangency of finite time stable foliations (related to a “forward time” perspective) and finite time unstable foliations (related to a “backwards time” perspective) serve a central role. We compare and contrast this perspective to both the variational method of geodesics [17], as well as the coherent pairs perspective [12] from transfer operators.

Understanding and describing mixing and transport in two-dimensional fluid flows have been a classic problem in dynamical systems for decades. Here we focus on three theories related to coherence in finite-time nonautonomous dynamical systems: 1) Shape coherent sets [

In this paper, we contrast three perspectives of coherence by calculating several objective measures which are a) the evolution of arc length; b) the relative coherence pairing; c) foliation angles; d) change of curvature; e) registration of shapes; and f) the shape coherence factor for sets developed from each of these methods, two of which are respectively shown in Tables 1-3. We show that each of these three perspectives of coherence may keep its advantages on its corresponding measure. According to a geodesic perspective, arc length should vary

Theory | Measures | Design | Related Keywords |
---|---|---|---|

Shape coherent sets | Shape coherence factor | Preserving shape. Regularity link between curvature and shape coherence | Finite-time stable/unstable foliation, nonhyperbolic splitting angles, slow evolving curvature, FTC. |

Geodesic transport barrier | Arc length evolution | Stationary values of the averaged strain and the averaged shear | Hyperbolic, elliptic and parabolic barriers Lagrangian coherent structures. [ |

Coherent pairs | Coherent pair number | Minimizing density loss, Frobenius-Perron operator by Ulam-Galerkin’s matrix, SVD | Coherent pairs, hierarchical partitions, Galerkin-Ulam matrices. [ |

Theory | Arc Length Change (%) | Specification | ||
---|---|---|---|---|

Shape coherent sets | 0.9637 | 1.14% | 1 | The grid size is 200 by 200. See |

Geodesic transport barrier | 0.9362 | 0.93% | 1 | Calculated by LCStool [ |

Coherent pairs | 0.9210 | 1.9% | 0.995 | 50,000 by 50,000 matrix with 20 million sample points. See |

Theory | Arc Length Change (%) | Specification | ||
---|---|---|---|---|

1^{st} Shape coherent sets | 0.9184 | 3.69 % | 1 | The grid size is 2000 by 200. See |

2^{nd} Shape coherent sets | 0.9525 | 2.44 % | 1 | The grid size is 2000 by 200. See |

Geodesic transport barrier | 0.9156 | 1.2159 % | 1 | Calculated by LCStool [ |

Coherent pairs | 0.9193 | 1.2174 % | 0.992 | 80,000 by 80,000 matrix with 20 million sample points. See |

slowly. On the other hand, according to the perspective of shape coherent sets, shape should be roughly preserved and the nonlinear flow restricted to that set should appear as a rigid body motion on that scale of time and space. This suggests that arc length may vary, however generally slowly. For coherent pairs, the definition [

It may seem striking that each method performs well on the measures of the other methods. There are of course differences between the methods as well, since each tends to identify sets that the other two do not, with significant difference in the non-corresponding measure. Note that we have used LCSTool [

For completeness, in brief detail we review the three methods to be compared, with references to the originating literature for greater detail. Let (

We use ^{1}.

Here we review shape coherence [^{2}, motivated by an intuitive idea of sets that “hold together” through finite-time. We connected shape coherent sets to slowly evolving boundary curvature by studying the tangency of finite-time stable and unstable foliations, which relate to forward and backward time perspectives respectively. The zero-angle curves are developed from the nonhyperbolic splitting of stable and unstable foliations by continuation methods that relate to the implicit function theorem. These closed zero-angle curves preserve their shapes in the time- dependent flow by a relatively small change of curvatures. See [

Definition 1 [

where

Stated simply, the stable foliation at a point describes the dominant direction of local contraction in forward time, and the unstable foliation describes the dominant direction of contraction in “backward” time. See

To compute the major axis of ellipsoids corresponding to how discs evolve under the action of matrices, we

may refer to the singular value decomposition [

where

matrix

We summarize, the stable foliation and unstable foliation at

where

Definition 2 [

We give a comprehensive discussion of how the non-hyperbolic splitting angle of foliations preserves curvature of a curve in [

To calculate the shape coherence number is a form of “image registration”, which is the process of transforming different image data into one coordinate system. See [

Next, we review the geodesic theory of Lagrangian coherent structures. Consider an evolving material line

where

where

vector fields, which are defined as,

barrier if

In this section, we briefly review the coherent pairs. Relatively coherent pairs [

Definition 3 [

where the pair

Then we build a relative measure on

Definition 4 [

To find coherent pairs in time-dependent dynamical systems, we use the Frobenius-Perron operator. If

for all

and let

where the sequence

We next apply the three methods to the Rossby wave system and double gyre system, both of which have become classic examples for studying and contrasting coherence and transport, [

Consider the nonautonomous double gyre system,

where

Consider the example sets shown. Note that the LCS has the smallest arc length change, but the shape coherent sets have the highest shape coherence factor. While the coherent pair number of both the shape coherence derived set and the geodesic transport derived set are shown as 1, since if the image in definition Equation (9) is its image, as the pairing, then 1 will always be the result, but the theory is properly interpreted the entire definition of coherent pairs requires a more careful pairing of sets as discussed above. What is striking here is that three different methods can find comparable sets, as reflected that the measures are similar.On the other hand, not shown here, is that each can find different sets, but the measures between them would ne- cessarily be dramatically different. See

The second benchmark problem is a quasiperiodic system which represents an idealized zonal stratospheric flow [

Let

In this paper, we have reviewed three complementary theories of coherence that come with similar goals but from different perspectives. These are the geodesic theory of LCS which emphasizes stationarity of averaged strain and shear, coherent pairs which describes “very small leakage” of sets and shape coherent sets which emphasizes those sets that preserve their shape by a slowly evolving curvature.

We have then presented two benchmark examples, the double gyre and the Rossby wave, to compare the different methods. In the examples described here, it has been illustrated that all three methods have reasonable and similar numerical results which agree with their own theories, and comparable results between given that similar sets were identified. For arc length, LCS always has the least change; and with respect to shape coherence, the zero-splitting curve has the best shape coherence number; coherent pairs has results very close to them. Notice that here we only compared similar elliptic shapes for all three methods, to allow each the best possibility of doing well relative to each other. On the other hand, each of the three methods can and does find clearly different sets and therefore with significant differences of performance measure. See

We have not found an a priori reason to expect that each of the three definitions of coherence will usually, or even often, find the same sets. A theorem directly connecting them is lacking. Indeed sometimes we find that each finds sets that the others do not, but in such a case, a numerical comparison of the measures included here only points out dramatic differences. Stated in terms of choice, a practitioner may ask which method to use for their own applied problem, but there is no magic bullet or best method to compute coherence. By offering the contrasting perspective of three different concepts of the general idea of coherence, in the same light, we hope that this discussion offers the practitioner a richer mathematical perspective of what is the outcome of what they are computing, no matter what method they choose to use.

Tian Ma,Erik Bollt, (2015) Contrast of Perspectives of Coherency. Journal of Applied Mathematics and Physics,03,781-791. doi: 10.4236/jamp.2015.37095