HYPERBOLIC MATERIAL LINES AND STIRRING OVER FINITE TIME INTERVALS Motivations B. Legras, GEOMIX, Carg ₩ se Finding generalisations of hyperbolic trajectories, and stable & unstable manifolds for aperiodic flows and finite time intervals Relate the hyperbolic structures to stirring & flow partition Generalise rate of strain partition

References Miller et al., 1997, Quantifying transport in numerically generated velocity fields, Physica D 110, Malhotra, Mezic & Wiggins, 1998, Patchiness a new diagnostic for Lagrangian trajectory analysis in time-dependent fluid flows, Int. J. Bifurcation Chaos Appl. Sci. Eng. 8, Haller and Poje, 1998, Finite time transport in aperiodic flows, Physica D 119, Poje, Haller and Mezic, 1999, The geometry and statistics of mixing in aperiodic flows, Phys. Fluids 11, Bowman, 2000, Manifolds geometry and mixing in observed atmospheric flows, preprint Klein, Hua & Lapeyre, 2000, Alignment of tracer gradient vectors in two-dimensional turbulence, Physiac D146, 246 Koh and Plumb, 2000, Lobe dynamics applied to barotropic Rossby-wave breaking, Phys. Fluids 12, Haller and Yuan, 2000, Lagrangian coherent structures and mixing in 2D turbulence, Physica D 147, Haller, 2000, Finding finite-time invariant manifolds in 2D velocity fields, Chaos 10, Haller, 2001a, Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D 149, Haller, 2001b, Lagrangian structures and the rate of strain in a partition of 2D turbulence, Phys. Fluids, in press. Lapeyre, Hua & Legras, 2001, Comments on 'Finding finite-time invariant manifolds in 2D velocity fields', Chaos, in press. Joseph & Legras, 2001, On the relation between kinematic boundaries, stirring and barriers for the Antarctic polar vortex, J. Atmos. Sci., in press. Voth, Haller & Gollub, 2001, Precision measurements of stretching and compression in fluid mixing, preprint

t x y WuWu WsWs Hyperbolic trajectories and invariant manifolds Hyperbolic trajectories are non stationary and frame independent extensions of stagnation points. Trajectories contained in the stable manifold W s converge to the hyperbolic trajectory as time -> ' and trajectories contained in the unstable manifold W u converge to the hyperbolic trajectory as time -> -'.

Definition of hyperbolic lines and surfaces Haller and Yuan, 2000 Finite-time instability requires exponential separation on arbitrarily short time intervals M is an unstable material surface (in x*y*t) on the time interval I u if there is a positive exponent u  such that for any close enough initial condition p(  ) = (x  ),  ) and for any small time step h>0 we have, for  and  +h from I u

A stable material surface is a smooth material surface that is unstable backward in time. An unstable (stable) material line is the t=constant section of an unstable (stable) material surface. Both referred as hyperbolic material surfaces (lines) If the flow is incompressible, trajectories must converge to each other on M.

Definition of coherent structure boundaries For a given initial initial condition (x 0,t 0 ), the instability time T u (x 0,t 0 ) is the maximum time within [t 0, t 1 ] over which the instability condition holds. Similarly, the stability time T s (x 0,t 0 )is the maximum time within [t -1, t 0 ] over which the instability condition holds for backward integration in time. It is proposed (Haller and Yuan, 2000)that the coherent structure boundaries are given by the stable and unstable material lines along which T u or T s attains local maximum.

The definition of finite-time hyperbolic material lines implies non uniqueness. Two nearby unstable surfaces are such that where  u is the length of I u. (Haller and Poje, 1998) Material lines that are hyperbolic for long enough time intervals will appear to be locally unique up to exponentially small errors. Unstable material lines stable manifolds of hyperbolic trajectories Stable material lines unstable manifolds of hyperbolic trajectories

Detection of hyperbolic lines by finite-time statistics Local extrema of patchiness (average displacement in one direction) [Malhotra et al., 1998; Poje et al., 1999] Local extrema of Lyapunov exponents Local extrema of relative dispersion or finite-size Lyapunov exponents [Bowman, 2000; Joseph & Legras, 2001]

Finite size Lyapounov exponents (FSLE)

Pair separation or FSLE do not distinguish any line for purely linear flows (e.g. u =  x, v = -g y) for which all trajectories have the same stability properties. Lines of maximum separation only occurs for non linear flows where recurrence is combined with hyperbolicity (typically: strong separation followed by weak shear like transport). Possible artefacts due to large shear regions.

Analytic criterion for finite-time hyperbolicity Assume that on a closed time interval I Theorem (Haller, 2000; Haller & Yuan 2000): Suppose that for a fluid trajectory x(t), we have Then x(t) is contained in an unstable material line on the time interval I. Furthermore the instability exponent can be estimated as

This criterion leads to a new definition of the instability time T u for the trajectory passing in (x 0, t 0 )as the maximum time in the interval [t 0, t 1 ] for which, that is the Okubo-Weiss condition is satisfied (cf L. Hua's lecture) along the material trajectory, plus condition (b). Another derived criterion is to maximize the instability time weighted by the strain. 3D generalization in Haller, 2001b

Generalised criterion in the strain basis (Lapeyre, Hua & Legras, 2001; Haller, 2001b)

(Haller 2001b) Theorem 1 (sufficient condition for Lagrangian hyperbolicity) Suppose that a trajectory x(t) does not leave the set H during a time interval I, then x(t) is contained in a hyperbolic material surface over I. Theorem 2 (necessary condition for Lagrangian hyperbolicity) Suppose that a trajectory x(t) is contained in a hyperbolic material line over the time interval I. Then if I E denotes the time interval that the trajectory spends in E, Defining an elliptic material line over I as non-hyperbolic material lines that either stay in E or stay in regions of zero strain over I Theorem 3 (sufficient condition for Lagrangian ellipticity). Suppose that a trajectory x(t) is contained in the set E and Then x(t) is contained in an elliptic material line over I

Detection of hyperbolic material lines From the eigenvalues of Maxima of From the eigenvalues of Maxima of Patchiness Maxima of the finite-time Lyapunov exponent Maxima of separation or finite-size Lyapunov exponents

Example I: The Kida ellipse: Elliptic uniform vorticity patch submitted to external pure strain. The ellipse rotates with constant angular velocity. In the rotating frame, the flow is stationary with two stagnation points. The criteria are applied in the fixed frame where the flow is not stationary.

Example II Unstable manifold of the forced Duffing equation

Detection of the hyperbolic lines for Duffing system (Haller, 2000) ( a) separation (b) hyperbolic persistence T u

Unstable manifold Finite-size Lyapunov exponents (Joseph and Legras, 2001)

Relation with flow partition and stirring - are the hyperbolic lines partitioning the flow or - are they spanning a region of strong stirring, the partition occurring on the boundary of this region ('stochastic layer' and 'invariant tori')

vertical displacement Lyapounov exponent Example III Stirring in mantle plumes (Farnetani, Legras & Tackley, 2001) Convection induced by the instability of the D” thermal layer above the core mantle boundary. Case with viscosity jump at 660km

Case with a chemically dense layer vertical displacementLyapunov exponent FSLE

Example IV:2-D experimental periodic flow (Voth, Haller & Gollub, 2001) Experiment in a stratified electrolytic flow forced periodically. Parameters Re = UL/, p = UT/L (mean path length) Map of one component of velocity at two different times (Re = 45, p=1) Poincar₫ map

A. Lines of maximum finite-time Lyapunov exponents for the backward map (compression or 'unstable manifold') B. Concentration after 30 periods in the same phase as A C. Superposition of the images A and B

Intersection of stable and unstable material lines indicating the hyperbolic trajectories of the flow; Superposition of the compression lines and the concentration, Re=115, p=5

Example V: The Antarctic polar vortex (Joseph & Legras, 2001) Stable material lines (unstable manifold) and unstable material lines (stable manifold) shown as points with largest FSLE after backward and forward time integration over 9 days. 25 October 1996, 450K.

The intersection of stable and unstable lines generate lobes.

forward integration backward integration Turnstile process

Distribution of the points on the unstable and sable material surfaces as a function of equivalent latitude effective diffusivity PV gradient Densities ds and du of the stable and unstable lines versus PV gradient and effective diffusivity. The stable or unstable material lines are NOT the boundary of the polar vortex. ds du

Koh and Legras, 2001 Duration over which the necessary condition for hyperbolicity is satisfied for forward and backward integrations of 9 days from 11 October forwardbackward

forward backward

Same as previously but blanking out the particles which are known NOT to be hyperbolic after 9 days. forward backward

Sufficient condition for hyperbolicity (both forward and backward in time, i.e. over the whole interval [t 1,t -1 ].)

Sufficient condition for ellipticity over 9 days in forward time.

Unstable material lines stretching stable manifolds Stable material lines compression, folding unstable manifolds Detected by finite-time Lyapunov exponents and FSLE, but no rigorous basis. Rigorous criteria still too weak. Theory needs progress. (using higher order terms in the temporal evolution?) In some cases, the tracer exhibit large gradients across the stable material lines which behaves like barriers separating several regions of the flow (typical of short time evolution for blob dispersion). In some other cases, the largest gradients are located at the periphery of a region where hyperbolic lines intersect many times (typical for established regime). The hyperbolic lines are the physical support for large deviations in the Lyapunov exponent and stretching. Relation with statistical theory?