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# Uncertainty in finite-time Lyapunov exponent computations

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.

Mathematics Subject Classification: Primary: 37M25, 37N10; Secondary: 60H10.

 Citation: • • 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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