The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

: The Finite size Lyapunov exponent (FSLE) has been used extensively since the late 1 1990’s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian 2 coherent structures in geophysical ﬂows and two-dimensional turbulence. Historically, the FSLE 3 was deﬁned in terms of its computational method rather than via a mathematical formulation, and 4 the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. 5 Here we propose an exact deﬁnition of the FSLE based on conditional averaging of the ﬁnite 6 amplitude growth rate (FAGR) of the particle pair separation. With this new deﬁnition, we show 7 that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before 8 the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the 9 links between the FAGR and other relevant Lagrangian metrics, such as the ﬁnite time Lyapunov 10 exponent and the second order velocity


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Lagrangian relative dispersion experiments, consisting in the simultaneous release 15 of large numbers of particle pairs and studying their separation characteristics, are a 16 powerful way to assess turbulent properties of a flow. Various metrics have been used 17 to study pair dispersion. An example is the "relative dispersion", which derives from 18 averaging squared pair separations at fixed times. Similar such time-based metrics 19 include the relative diffusivity and the separation kurtosis [1-3, e.g.].
In the limit of small perturbations, the FSLE recovers the finite time Lyapunov exponent.

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Our subsequent focus is on particles in turbulent flows, so we take δv to be the 32 distance between two particles r = |x 2 − x 1 |, where x 2 and x 1 are the individual positions 33 of the particles. Defining a series of geometrically-increasing reference separations r i 34 (i ∈ [1, N]), with α = r i+1 r i , the FSLE is: where T i is the time for the separation to grow from r i to αr i (also referred to as the "exit 36 time" [13]). Note that for finite amplitude perturbations, the FSLE depends on the chosen 37 norm [14]. In this work, we chose the Euclidian norm of the separation vector, the most 38 common norm used in two-dimensional and geophysical turbulence.

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The FSLE has been used frequently to analyze in situ data [5][6][7] This measure can be positive or negative. However, the latter authors dismissedλ(r i ) 75 as a proper proxy for FSLE because it is not strictly a separation-based metric, as the 76 relative dispersion involves averaging in time. Such averaging potentially combines 77 contributions from different dispersive regimes. The authors also noted thatλ(r i ) is potentially sensitive to the initial conditions.

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Hereafter, we propose a rigorous derivation ofλ (that we refer to as the Cencini-  2. Numerical experiment 88 We'll test the various metrics using three 2D simulations described in [22]. The code solves the 2D vorticity equation: where ψ is the streamfunction, ζ = ∇ 2 ψ the relative vorticity and J(a, b) the Jacobian 89 function. The forcing, F , is applied with random phases in an isotropic wavenumber 90 band, which is varied. Rayleigh dissipation is used, with a constant (Ekman) coefficient 91 of r = 0.1. Small scale variance is removed via an exponential cut-off filter [23]. The Hereafter, we refer to the function γ(t) as the single-realization finite amplitude growth 129 rate (FAGR). From (6), we may express γ(t) as a function of separation and time: 130 or equivalently: When ensemble-averaged over all pairs at constant separation r i , the FAGR γ(t) recovers 134 the CVEλ(t). For example, the finite time Lyapunov exponent (FTLE), λ T , is defined for an initial 141 separation r(0) = δ 0 as: [e.g. 28-30]. Integrating equation (10) from 0 to time t, we get: Dividing both sides by time, t, the FTLE is seen to be the time-average value of the FAGR 144 from the initial time to time t for infinitesimal initial separation: The second order velocity structure function, the ensemble-average squared separa-146 tion velocity at constant separation, can also be expressed in terms of γ. From equation ( 147 10), the separation velocity, u = dr dt , can be written: Ensemble averaging the squared velocity at a given separation yields: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 August 2021 doi:10.20944/preprints202108.0203.v1 From (10), we have: Using (12) and (14), we can express the diffusivity in terms of the initial separation, δ 0 , 157 the FAGR and the FTLE, λ T : If the initial separation δ 0 is continuously distributed with a probability density function, 159 p(δ 0 ), we have: Here the averaging is performed on pairs with an initial separation δ 0 at a fixed time. If 161 all pairs have the same initial separation (Dirac distribution), we have: where · t represents averaging all pairs at a given time t.

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One can furthermore use γ to express [32]'s instantaneous relative dispersion 164 coefficient, a time-independent proxy for the relative diffusivity: where · r represent averaging pairs with constant separation r. To express the FSLE λ in terms of the FAGR, γ, we use a similar procedure as for 168 the FTLE, integrating (10) from time t to a later time t + T, yielding: Introducing the time average operator · t+T t = 1 T t+T t · dτ, we have: Now assume that T i is the time required for the separation to grow from a reference 171 separation r i to another separation r i+1 . Averaging between time t and t + T i is then 172 equivalent to averaging between separations r i and r i+1 . Then for each particle pair: The reference separations are assumed to increase geometrically, i.e. r i+1 = αr i . By 174 ensemble averaging (25) over all pairs, we may define the variableλ(r i , α): Then, noting that 1 r dr dt = 1 2r 2 dr 2 dt , one finds thatλ is exactly equal to the CVEλ: where · r i is the ensemble average over all pairs at separation r i .  In the limit of small bin widths, (30) can be expressed as: .
We refer hereafter to the reciprocal of this, λ B ≡ τ −1 , as the inverse structural time (IST).

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Comparisons between the scale-averaged positive relative diffusivity K s (r) and values. While the link between K s (r) and X(r) is an interesting topic, it is beyond the 211 scope of the present paper and will remain to be investigated.
212 Figure 5 compares the FSLE computed in the asymptotic limit (equation 34) and 213 the IST. As expected from the agreement between X(r) and K s (r), the close similarity 214 of the curves confirms that FSLE in the small α limit is a proxy for the IST. While the 215 lack of an analytical link between X(r) and K s (r) in the present work prevents definitive so the IST can also be expressed in terms of S 2l (r): In turn, [31] showed that the Eulerian velocity variance spectrum was linked with the 223 longitudinal second order velocity structure function via the inverse Hankel transform: where E t is the total kinetic energy, K is the wavenumber, and J 1 is the first order Bessel 225 function. Hence, the Eulerian velocity variance spectrum and the IST λ B are linked: Thus, if confirmed in further analytical, experimental and numerical studies, the equiva-227 lence between FSLE and IST would yield intersting properties for FSLE, with a direct 228 link to the Eulerian velocity variance spectrum.

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[14] argued that the major difference between the CVE and the FSLE is that the 231 former depends on initial conditions while the latter does not. To explore this, we 232 computed the CVE and FSLE for different initial separations. In each case, all pairs have 233 the same initial separation (the initial separation PDF is a delta function). We then vary 234 the separation from the smallest to the largest reference separation threshold r i , as in Fig.   235 (2).

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The results are compared to the reference calculation (which uses all available pairs, Thus γ is a weighted average of positive and negative FAGRs: pairs with initial sepa-

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The FAGR suppresses the need for higher frequency interpolation at small separa-290 tion scales, and short-separation scales are more reliably represented. 3. The FAGR can be computed and averaged over any given separation set, and the 292 latter is not required to increase geometrically nor to be regular.