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

t

x

y

Wu

Ws

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 surfacesHaller and Yuan, 2000

Finite-time instability requires exponential separation on arbitrarily short time intervalsM is an unstable material surface (in x*y*t) on the time interval I

u if there is a

positive exponent lusuch that for any close enough initial condition

p(t) = (x(t),t) and for any small time step h>0 we have, fort and t +h from Iu

dist p h ,M dist p ,M exp uh

or N x h ,h DF h x0

N x0, exp u h

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 (x0,t

0), the

instability time Tu(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 aresuch that where t

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

distt I

M,M' C e

u

u

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 = g 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 hyperbolicityAssume that on a closed time interval I

1 u02 det u0

Then u has real eigenvalues t 0 t .

Definemin

mintI

t .

The eigenvectors e1

t and e2

t corresponds to t and t ,andT e1

t , e2

t .

DefinemintI

det T and maxt I

T .

Theorem (Haller, 2000; Haller & Yuan 2000): Suppose that for a fluid trajectory x(t), we have

a det u x t , t 0 and b min

2 2

Then x(t) is contained in an unstable material line on the time interval I. Furthermore the instability exponent can be estimated as

u

min O

This criterion leads to a new definition of the instability time Tu for

the trajectory passing in (x0, t

0)as the maximum time in the interval

[t0, 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.

det u0

Tu1

t1 t0

det u0dt

T'udet u0

t dt

3D generalization in Haller, 2001b

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

Transform the dispersion equation from the fixed frame to the frame of the strain axis using

the rotation matrix R defined by the orthonormal basis of the eigenvectors of the strain matrix

u T u .

If the line element satisfies u in the fixed frame, the rotated element ' R1 satisfies

' R1 uRR1 R ' ustrain

' ,

where the matrix u strain is a function of the strain rate and the effective rotation.

Defining r effective rotation

strain

vorticity - strain axis rotation

strain, the relative orientation of

the line element with the strain axis satisfies r cos

while the length of the line element satisfies

1

2

D

Dt 2 sin

Lapeyre, Klein & Hua, 1999; Klein,Lapeyre & Hua, 2000

Assuming that r is slowing varyingr 1 defines the hyperbolic regionH where the line element is growing exponentiallyr 1 defines the elliptic regionE where the line element does not grow or grows weakly

In the general time dependent case, the conditiondet u0 can be replaced bydet u

strain0 or equivalently r 1 to get an improved hyperbolicity condition.

Lapeyre, Hua &Legras,2001.

(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,

I E

0 x t ,t dt2

where0 x t ,t min + x t ,t ,- x t ,t , 1

2

,M

S,

+ and - are the zero modes of the strain matrix S u T u i.e , S 0 , andM is

the strain acceleration tensorM SS u .

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 ITheorem 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

I E

0 x t ,t dt2

.

Tu

S 1

t1 t

0

r 1dt

TuS ' r 1

t 1r2 dt

Detection of hyperbolic material lines

From the eigenvalues ofMaxima of

u From the eigenvalues ofMaxima of

ustrain

Tu1

t1 t0

det u0dt

T'udet u0

t dt

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.

Tu(vorticity < strain) T

uS (effective vort. < strain) T

uS ' eff. vort. < strain

S dt

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 IIIStirring 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 displacement Lyapunov exponent FSLE

Example IV:2-D experimental periodic flow (Voth, Haller & Gollub, 2001)

Experiment in a stratified electrolytic flow forced periodically. ParametersRe = UL/n, 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 forthe 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

PVgradient

Densities ds and du of thestable and unstable lines versus PV gradient and effective diffusivity.

The stable or unstablematerial lines are NOT the boundary of the polar vortex.

ds

du

Koh and Legras, 2001

Duration over which the necessary condition for hyperbolicity is satisfiedfor forward and backward integrations of 9 days from 11 October 1996.

forward backward

In

t 1r 2 dt where Inis the set of intervals in t 0 ,t 1 , and t1

,t 0 respectively,

where the necessary hyperbolicity condition of Haller (2001b)is satisfied for forward and backward integration respectively

forward

In

t 1r 2 dt where Inis the set of intervals in t 0 ,t 1 , and t1

,t 0 respectively,

where the necessary hyperbolicity condition of Haller (2001b)is satisfied for forward and backward integration respectively

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].)

Tu

S 1

t1 t1

r 1dt T

uS ' r 1

t 1r 2 dt

Sufficient condition for ellipticity over 9 days inforward 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?