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December  2020, 7(2): 313-337. doi: 10.3934/jcd.2020013

Uncertainty in finite-time Lyapunov exponent computations

1. 

School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia

2. 

Department of Mechanical Engineering, Technische Universiteit Eindhoven, 5612 AZ Eindhoven, The Netherlands

Received  September 2019 Published  July 2020

Fund Project: Partial support from the Australian Research Council (grants FT130100484 and DP200101764) and the Australian Department of Education and Training (Endeavour Research Leadership Award) are gratefully acknowledged

The Finite-Time Lyapunov Exponent (FTLE) is a well-established numerical tool for assessing stretching rates of initial parcels of fluid, which are advected according to a given time-varying velocity field (which is often available only as data). When viewed as a field over initial conditions, the FTLE's spatial structure is often used to infer the nonhomogeneous transport. Given the measurement and resolution errors inevitably present in the unsteady velocity data, the computed FTLE field should in reality be treated only as an approximation. A method which, for the first time, is able for attribute spatially-varying errors to the FTLE field is developed. The formulation is, however, confined to two-dimensional flows. Knowledge of the errors prevent reaching erroneous conclusions based only on the FTLE field. Moreover, it is established that increasing the spatial resolution does not improve the accuracy of the FTLE field in the presence of velocity uncertainties, and indeed has the opposite effect. Stochastic simulations are used to validate and exemplify these results, and demonstrate the computability of the error field.

Citation: Sanjeeva Balasuriya. Uncertainty in finite-time Lyapunov exponent computations. Journal of Computational Dynamics, 2020, 7 (2) : 313-337. doi: 10.3934/jcd.2020013
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show all references

References:
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M. R. Allshouse and T. Peacock, Refining finite-time Lyapunov exponent ridges and the challenges of classifying them, Chaos, 25 (2015), 14pp. doi: 10.1063/1.4928210.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

H. Babaee, M. Farazmand, G. Haller and T. P. Sapsis, Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents, Chaos, 27 (2017), 12pp. doi: 10.1063/1.4984627.

[4]

M. BadasF. Domenichini and G. Querzoli, Quantification of the blood mixing in the left ventricle using finite time Lyapunov exponents, Meccanica, 52 (2017), 529-544.  doi: 10.1007/s11012-016-0364-8.

[5]

S. Balasuriya, Barriers and Transport in Unsteady Flows. A Melnikov Approach, Mathematical Modeling and Computation, 21, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974584.ch1.

[6]

S. Balasuriya, A numerical scheme for computing stable and unstable manifolds in nonautonomous flows, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 22pp. doi: 10.1142/S021812741630041X.

[7]

S. Balasuriya, Stochastic sensitivity: A computable Lagrangian uncertainty measure for unsteady flows, SIAM Review, in press.

[8]

S. Balasuriya and G. Gottwald, Estimating stable and unstable sets and their role as transport barriers in stochastic flows, Phys. Rev. E, 98 (2018). doi: 10.1103/PhysRevE.98.013106.

[9]

S. BalasuriyaR. Kalampattel and N. T. Ouellette, Hyperbolic neighborhoods as organizers of finite-time exponential stretching, J. Fluid Mech., 807 (2016), 509-545.  doi: 10.1017/jfm.2016.633.

[10]

S. BalasuriyaN. T. Ouellette and I. I. Rypina, Generalized Lagrangian coherent structures, Phys. D, 372 (2018), 31-51.  doi: 10.1016/j.physd.2018.01.011.

[11]

A. BergerT. S. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246 (2009), 1098-1118.  doi: 10.1016/j.jde.2008.06.036.

[12]

F. Beron-Vera, Mixing by low- and high-resolution surface geostrophic currents, J. Geophys. Res. Oceans, 115 (2010). doi: 10.1029/2009JC006006.

[13]

F. Beron-VeraM. OlascoagaM. Brown and H.. Koçak, Zonal jets as meridional transport barriers in the subtropical and polar lower stratosphere, J. Atmos. Sci., 69 (2012), 753-767.  doi: 10.1175/JAS-D-11-084.1.

[14]

A. BozorgMagham and S. Ross, Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty, Commun. Nonlin. Sci. Numer. Simul., 22 (2015), 964-979.  doi: 10.1016/j.cnsns.2014.07.011.

[15]

A. E. BozorgMaghamS. D. Ross and D. G. Schmale, Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: An application and error analysis, Phys. D, 258 (2013), 47-60.  doi: 10.1016/j.physd.2013.05.003.

[16]

S. L. Brunton and C. W. Rowley, Fast computation of finite-time Lyapunov exponent fields for unsteady flows, Chaos, 20 (2010), 12pp. doi: 10.1063/1.3270044.

[17]

T. F. Dauche, C. Ates, T. Rapp, M. C. Keller and G. Chaussonnet, et al., Analyzing the interaction of vortex and gas-liquid interface dynamics in fuel spray nozzles by means of Lagrangian-coherent structures (2D), Energies, 12 (2019). doi: 10.3390/en12132552.

[18]

F. d'Ovidio, V. Fernández, E. Hernández-Garcia and C. López, Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents, Geophys. Res. Lett., 31 (2004). doi: 10.1029/2004GL020328.

[19]

J. Finn and S. V. Apte, Integrated computation of finite-time Lyapunov exponent fields during direct numerical simulation of unsteady flows, Chaos, 23 (2013), 14pp. doi: 10.1063/1.4795749.

[20]

A. FortinT. Briffard and A. Garon, A more efficient anisotropic mesh adaptation for the computation of Lagrangian coherent structures, J. Comp. Phys., 285 (2015), 100-110.  doi: 10.1016/j.jcp.2015.01.010.

[21]

D. Garaboa-PazJ. Eiras-Barca and V. Perez-Munuzuri, Climatology of Lyapunov exponents: The link between atmospheric rivers and large-scale mixing variability, Earth System Dyn., 8 (2017), 865-873.  doi: 10.5194/esd-8-865-2017.

[22]

H. GuoW. HeT. PeterkaH.-W. ShenS. Collis and J. Helmus, Finite-time Lyapunov exponents and Lagrangian coherent structures in uncertain unsteady flows, IEEE Trans. Visual. Comp. Graphics, 22 (2016), 1672-1682.  doi: 10.1109/TVCG.2016.2534560.

[23]

A. Hadjighasem, M. Farazmand, D. Blazevski, G. Froyland and G. Haller, A critical comparison of Lagrangian methods for coherent structure detection, Chaos, 27 (2017), 25pp. doi: 10.1063/1.4982720.

[24]

G. Haller, A variational theory of hyperbolic Lagrangian coherent structures, Phys. D, 240 (2011), 574-598.  doi: 10.1016/j.physd.2010.11.010.

[25]

G. Haller, Lagrangian coherent structures, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., 47, Annual Reviews, Palo Alto, CA, 2015,137–162. doi: 10.1146/annurev-fluid-010313-141322.

[26]

G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent, Chaos, 21 (2011), 7pp. doi: 10.1063/1.3579597.

[27]

G.-S. HeC. PanL.-H. FengQ. Gao and J.-J. Wang, Evolution of Lagrangian coherent structures in a cylinder-wake disturbed flat plate boundary layer, J. Fluid Mech., 792 (2016), 274-306.  doi: 10.1017/jfm.2016.81.

[28]

I. Hernández-CarrascoC. LópezE. Hernández-Garcia and A. Turiel, How reliable are finite-size Lyapunov exponents for the assessment of ocean dynamics?, Ocean Model., 36 (2011), 208-218.  doi: 10.1016/j.ocemod.2010.12.006.

[29]

P. Holmes and D. Whitley, On the attracting set for Duffing's equation, Phys. D, 7 (1983), 111-123.  doi: 10.1016/0167-2789(83)90121-5.

[30]

Z. Y. Hu and M. Zhou, Lagrangian analysis of surface transport patterns in the northern south China sea, Deep Sea Res. II: Topical Stud. Oceanography, 167 (2019), 4-13.  doi: 10.1016/j.dsr2.2019.06.020.

[31]

F. HuhnA. von KamekeS. Allen-PerkinsP. MonteroA. Venancio and V. Pérez-Muñuzuri, Horizontal Lagrangian transport in a tidal-driven estuary–Transport barriers attached to prominent coastal boundaries, Continental Shelf Res., 39-40 (2012), 1-13.  doi: 10.1016/j.csr.2012.03.005.

[32]

F. Huhn, A. von Kameke, V. Pérez-Muñuzuri, M. J. Olascoaga and F. J. Beron-Vera, The impact of advective transport by the South Indian Ocean Countercurrent on the Madagascar plankton bloom, Geophys. Res. Lett., 39 (2012). doi: 10.1029/2012GL051246.

[33]

H. KafiabadP. Chan and G. Haller, Lagrangian detection of wind shear for landing aircraft, J. Atmos. Ocean. Tech., 30 (2013), 2808-2819.  doi: 10.1175/JTECH-D-12-00186.1.

[34]

E. KaiV. RossiJ. SudreH. WeirmerskirchC. LopezE. Hernandez-GarciaF. Marsac and V. Garcon, Top marine predators track Lagrangian coherent structures, Proc. Nat. Acad. Sci., 106 (2009), 8245-8250.  doi: 10.1073/pnas.0811034106.

[35]

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Figure 1.  Double-gyre computations with $ L_r = 0.005 $ and $ v_r = 2 \times 10^{-5} $: (a) FTLE field $ \Lambda $, and (b) the error field $ {\mathcal E} $
Figure 2.  As in Fig. 1, but with $ L_r = 0.05 $ and $ v_r = 0.002 $
Figure 3.  (a, b, c) Three (of the $ 100,000 $) sample FTLE fields, and (d) the mean FTLE field, computed when using stochastic simulation of (6) for the double-gyre system with $ L_r = 0.05 $ and $ v_r = 0.002 $
Figure 4.  (a) The standard deviation of the FTLE fields computed from the $ 100,000 $ stochastic simulations, and (b) the theoretical error field (the same as Fig. 2(b), but scaled to elucidate variation at values of FTLE comparable to (a))
Figure 5.  The probability density of the FTLE value from the stochastic simulations of the double-gyre at two selected points indicated in Figs. 4(a) and 3(d): (a) maximum standard-deviation point (at red 'x'), and (b) minimum standard deviation point (at red circle)
Figure 6.  The probability density of the FTLE value from the stochastic simulations of the double-gyre with $ L_r = 0.05 $ and $ v_r = 0.002 $; the red star is the value from the standard FTLE computation: (a) at $ (1.05,0.1) $ and (b) at $ (1.15,0.1) $
Figure 7.  Wobbly Duffing from from $ t_0 = 4 $ with time-of-flow $ T = 4 $, with $ L_r = 0.2 $ and $ v_r = 0.002 $: (a) FTLE field, and (b) FTLE error field
Figure 8.  Scatter plots of the theoretical error with (left) the standard deviation, and (right) the range, obtained from $ 1000 $ stochastic simulations. Each point (red circle) in the plot corresponds to a point in the spatial domain at time $ t_0 $, and the blue line is a linear fit. The values chosen are (top) $ L_r = 0.2 $ and $ v_r = 0.002 $, and (bottom) $ L_r = 0.05 $ and $ v_r = 0.00002 $
Figure 9.  The $ v_r $ dependence analyzed for the theoretical and stochastically-determined errors in the FTLE field, for the wobbly Duffing system with $ L_r = 0.2 $: (a) theoretical (red-solid), standard deviation (green-dashed) and range (blue-dot-dashed) norms, and (b-d) investigations on the dependence of $ v_r $ on the FTLE spreading measures $ \| {\mathrm{theory}} \| $, $ \| {\mathrm{std}} \| $ and $ \| {\mathrm{range}} \| $ respectively
Figure 10.  The $ L_r $ dependence analyzed for the theoretical and stochastically-determined errors in the FTLE field, for the wobbly Duffing system with $ v_r = 0.00002 $: (a) theoretical (red-solid), standard deviation (green-dashed) and range (blue-dot-dashed) norms, and (b-d) investigations on the dependence of $ v_r $ on the FTLE spreading measures $ \| {\mathrm{theory}} \| $, $ \| {\mathrm{std}} \| $ and $ \| {\mathrm{range}} \| $ respectively
Figure 11.  FTLE fields (left) and FTLE error fields (right) for several situations based on the Copernicus data: (top) from 30 January 2016 to 29 February 2016 (middle) 25 February 2011 to 26 April 2011, and (bottom) 25 February 2011 to 13 September 2011
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