# Hyperbolic Covariant Coherent Structures in Two Dimensional Flows

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Covariant Lyapunov Vectors

#### Hyperbolic Covariant Coherent Structures

**Definition**

**1**(Hyperbolic Covariant Coherent Structures (HCCSs)).

## 3. Numerical Examples

#### 3.1. A Simple Autonomous Hamiltonian System

#### 3.2. Double Gyre

#### 3.3. Bickley Jet

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Description of the Algorithm

- Initialization (T1): This preliminary step is used to find the initial backward Lyapunov vector bases $\{{\mathbf{l}}_{1}^{-}\left(t\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{l}}_{2}^{-}\left(t\right)\}$ for a whole set of initial conditions [73]. A set of initial conditions $\left\{{\mathbf{x}}_{\mathbf{0}}\right\}\in D$ and two sets of initial orthonormal random bases are defined in the tangent spaces at every point at time ${t}_{0}$. The second set is necessary to check the convergence to the backward Lyapunov vector bases. The initial conditions and the random bases are evolved respectively with Equations (1) and (6) until the convergence of the two bases is reached with the desired accuracy. The convergence toward the Lyapunov vectors is typically exponential in time [51,74]. At every time step, for every initial condition, the evolved vectors are stored as a column of a matrix that is decomposed with a QRdecomposition. The last passage is implemented in order to find the new orthogonal basis at every time step and the upper triangular matrix containing the coefficients that allow one to express the old basis in terms of the new one.
- Forward transition (T2): The backward Lyapunov bases are evolved from time t to ${t}^{\prime}$. The evolution is done with the help of Equations (1) and (6) and the $QR$ decomposition for every evolution step. We indicate with $\mathbf{X}\left({t}_{k}\right)$ the matrix for which columns contain the new bases at time ${t}_{k}$ and with $\mathbf{R}({t}_{k-1},\phantom{\rule{0.166667em}{0ex}}{t}_{k})$ the correspondent upper triangular matrix. During this step, both the local Lyapunov bases and upper triangular matrices are stored. The diagonal elements ${\left(\mathbf{R}({t}_{k-1},\phantom{\rule{0.166667em}{0ex}}{t}_{k})\right)}_{ii}$ of the upper triangular matrices give information about the local growth rates of the basis vectors at a given time ${t}_{k}$, and they are used to compute the FTLEs as a time average:$${\lambda}_{i}=\frac{1}{{t}^{\prime}-t}\sum _{k=0}^{N-1}\mathrm{log}{\left(\mathbf{R}({t}_{k-1},\phantom{\rule{0.166667em}{0ex}}{t}_{k})\right)}_{ii},$$$$\mathbf{G}(t,{t}^{\prime})=\mathbf{F}{(t,{t}^{\prime})}^{\top}\mathbf{F}(t,{t}^{\prime}).$$This operator is also known as the deformation tensor, whose eigenvalues ${\mu}_{i}({t}_{0},t)$ and eigenvectors ${\mathit{\xi}}_{i}({t}_{0},t)$ satisfy:$$\begin{array}{c}\mathbf{G}(t,{t}^{\prime}){\xi}_{i}(t,{t}^{\prime})={\mu}_{i}(t,{t}^{\prime}){\mathit{\xi}}_{i}(t,{t}^{\prime}),\hfill \end{array}$$$$\begin{array}{c}{\mu}_{1}(t,{t}^{\prime})>{\mu}_{2}(t,{t}^{\prime})>0,\hfill \end{array}$$$$\begin{array}{c}{\mathit{\xi}}_{1}(t,{t}^{\prime})\phantom{\rule{0.166667em}{0ex}}\perp \phantom{\rule{0.166667em}{0ex}}{\mathit{\xi}}_{2}(t,{t}^{\prime}).\hfill \end{array}$$From a geometric point of view, a set of initial conditions corresponding to the unit sphere is mapped by the dynamics into an ellipsoid, with the principal axis aligned in the direction of the eigenvectors of the CGT and with length determined by the correspondent eigenvalues. The eigenvalues of the CGT determine the FTLEs as:$${\lambda}_{i}(t,{t}^{\prime})=\frac{1}{2({t}^{\prime}-t)}\mathrm{log}\left({\mu}_{i}(t,{t}^{\prime})\right),$$
- Forward dynamics (T3): In this step, the trajectories and the bases are further evolved from time ${t}^{\prime}$ to time ${t}^{\u2033}$ using Equations (1) and (6). This time interval should grant convergence, during the backward dynamic, to the CLVs. During this step, only the upper triangular matrices are stored and are used to continue the computation of the FTLEs using Equation (A4).
- Backward transition (T4): In this step, random upper triangular matrices are generated for every point of the grid, $\mathbf{C}$. These matrices contain the expansion coefficients of a set of two generic vectors (expressed as the column of a matrix) in terms of the ${L}^{-}$ bases. Using the stored matrices $\mathbf{R}$ of Step 3, these matrices are evolved backward in time, until time ${t}^{\prime}$, using the following relation:$$\mathbf{C}\left({t}_{n}\right)={\mathbf{R}}^{-1}({t}_{n},{t}_{n+1})\mathbf{C}\left({t}_{n+1}\right)\mathbf{D}({t}_{n},{t}_{n+1}),$$
- Backward dynamics (T5): In this final part of the algorithm, Equation (A5) is used with the $\mathbf{R}$ matrices of Step 2 to evolve backward the upper triangular matrices $\mathbf{C}$, from time ${t}^{\prime}$ to time t. In this phase, the backward Lyapunov bases stored can be used to write the CLVs. The matrix containing in each column the different CLVs at a given point in space and time, $\mathbf{W}$, can thus be written as:$$\mathbf{W}\left({t}_{n}\right)=\mathbf{C}\left({t}_{n}\right)\mathbf{X}\left({t}_{n}\right).$$For a two-dimensional system, the algorithm could be optimized making use of (18). One can follow the first step (T1) of the previous algorithm to find the convergence toward the backward Lyapunov bases in the time interval $[{t}_{0},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}t]$. After this first step, it is possible to carry out the evolution of the vectors, as in the second step (T2) during the time interval $[t,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{t}^{\prime}]$, without saving the triangular matrices. In the same way, it is possible to repeat the step (T1), but for a backward evolution during the time interval $[{t}^{\u2033},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{t}^{\prime}]$ to find the forward Lyapunov bases and then continue the backward evolution as in the step (T2) during the time interval $[{t}^{\prime},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}t]$. At this stage, it is possible to consider directly the first backward Lyapunov vectors and the second forward Lyapunov vectors in the time interval $[t,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{t}^{\prime}]$ as the CLVs. In this algorithm, there are just four steps and not five as in the one presented above, but for two times, one has to consider the convergence step (T1), which is more time consuming with respect to Step (T4) or (T5). It should be noted that in this algorithm, the forward and the backward evolutions could be done in parallel. The comparison between this algorithm and the one used for this study is left for future studies.

## Appendix B. Technical Details of the Numerical Examples

#### Appendix B.1. Hamiltonian System

**Figure A1.**Convergence of the averaged scalar product for the two random bases chosen for the initialization of the numerical algorithm for the Hamiltonian system Equation (27).

#### Appendix B.2. Double Gyre

**Table A1.**The five temporal windows of the numerical algorithm used for the two experiments DV1 and DV2.

T1 | T2 | T3 | T4 | T5 | |
---|---|---|---|---|---|

DV1 | $0\to 5$ | $5\to 10$ | $10\to 15$ | $10\leftarrow 15$ | $5\leftarrow 10$ |

DV2 | $0\to 10$ | $10\to 20$ | $20\to 30$ | $20\leftarrow 30$ | $10\leftarrow 20$ |

#### Appendix B.3. Bickley Jet

**Table A2.**The five temporal windows of the algorithm used for the two experiments BJ1 and BJ2. Times have been rescaled with ${L}_{x}/U$.

T1 | T2 | T3 | T4 | T5 | |
---|---|---|---|---|---|

BJ1 | $0\to 1.89$ | $1.89\to 3.79$ | $3.79\to 5.68$ | $3.79\leftarrow 5.68$ | $1.89\leftarrow 3.79$ |

BJ2 | $0\to 3.79$ | $3.79\to 7.57$ | $7.57\to 11.36$ | $7.57\leftarrow 11.36$ | $3.79\leftarrow 7.57$ |

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**Figure 1.**Orientation of Covariant Lyapunov Vectors (CLVs) on a most attracting material line (upper line) and on a neighboring material line (lower line). The distance $\parallel {P}_{2}-{P}_{1}\parallel $ decreases more quickly than $\parallel {P}_{4}-{P}_{3}\parallel $ since in the second curve, the normal vector connecting the points ${P}_{3}$ and ${P}_{4}$ is expressed as a linear combination of both CLVs.

**Figure 2.**(

**a**) Trajectories of the Hamiltonian system Equation (28) (solid line). The trajectory $y=0$ represents a repulsive barrier for the system. Notice that the arrows indicate the flow velocity field, which is expressed on the right-hand side of Equation (27), and not the CLVs, which are shown separately. (

**b**,

**c**) Streamlines for the double gyre and the Bickley jet, respectively.

**Figure 3.**Finite Time Lyapunov Exponents (FTLEs) and CLVs fields for the Hamiltonian system Equation (27). (

**a**) Maximum FTLE field computed from Equation (A1); (

**b**) $\theta $ field; (

**c**) zoom of the domain to show the CLV fields. Arrows are not used due to the arbitrariness of the orientation of the CLVs. The red lines are associated with ${\mathbf{w}}_{1}$, the expansion direction, while the blue lines are associated with ${\mathbf{w}}_{2}$, the contraction direction.

**Figure 4.**FTLEs, CLVs and joint probability plot for the experiments DV1 (left panels) and DV2 (right panels) of the double gyre system. (

**a**,

**b**) show the maximum FTLE fields computed from (A1); (

**c**,

**d**) show the angle between the CLVs for the two experiments; (

**e**,

**f**) show the joint probability between FTLEs and the angles shown in the previous panels.

**Figure 5.**Comparison between the Hyperbolic Covariant Coherent Structures (HCCSs) at time $t=5$, green lines, computed for the experiment DV2 and the FTLE field shown in Figure 4b. In the right panel there is a zoom of a portion of the domain in which are also shown the CLVs characterizing that region. The blue direction is contracting, while red is expanding. Since the contour in this region is aligned with the second CLVs, the blue one, the character of the HCCSs here is repulsive.

**Figure 6.**Comparison of the angle between CLVs, $\theta $, computed for the two experiments DV1 and DV2 at the same time $t=10$. (

**a**) shows the evolution of the angle obtained in DV1 at $t=10$; (

**b**) shows the difference between the angle computed at the beginning of the experiment DV2 and the one computed at the end of the interval of the CLVs computation for the experiment DV1.

**Figure 7.**PDF of $\theta $ for the DV2 experiment computed at $t=10$ (dashed line) and at $t=20$ (full line).

**Figure 8.**Evolution of the first four moments of the PDF for $\theta $ for the DV2 experiments in the interval $T2=\left[10\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}20\right]$.

**Figure 9.**FTLEs, CLVs and joint probability plot for the experiments BJ1 (left panels) and BJ2 (right panels) of the Bickley Jet. (

**a**,

**b**) show the maximum FTLE fields computed from (A1); (

**c**,

**d**) show the angle between the CLVs for the two experiments; (

**e**,

**f**) show the joint probability between FTLEs and the angles shown in the previous panels.

**Figure 10.**Comparison between the Hyperbolic Covariant Coherent Structures (HCCSs) at time $t=3.79$, green lines, computed for the experiment BJ2 and the FTLE field shown in Figure 9b. In the right panel there is a zoom of a portion of the domain in which are also shown the CLVs characterizing that region. The blue direction is contracting, while red is expanding. Since the contour in this region is aligned with the second CLVs, the blue one, the character of the HCCSs here is repulsive.

**Figure 11.**Comparison of the angle between CLVs, $\theta $, computed for the two experiments BJ1 and BJ2 at the same time $t=3.79$. (

**a**) shows the evolution of the angle obtained in BJ1 at $t=3.79$; (

**b**) shows the difference between the angle computed at the beginning of the experiment BJ2 and the one computed at the end of the interval of the CLVs computation for the experiment BJ1.

**Figure 12.**PDF of $\theta $ for the DV2 experiment computed at $t=3.79$ (dashed line) and at $t=20$ (full line).

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Conti, Giovanni, and Gualtiero Badin. 2017. "Hyperbolic Covariant Coherent Structures in Two Dimensional Flows" *Fluids* 2, no. 4: 50.
https://doi.org/10.3390/fluids2040050