Curvature in Chaotic and Turbulent Flows

Transcription

1 Curvature in Chaotic and Turbulent Flows Jean-Luc Thiffeault jeanluc Department of Applied Physics and Applied Mathematics Columbia University Curvature in Chaotic and Turbulent Flows p.1/16

2 Material Lines in Flows How do material lines embedded in a chaotic flow evolve? Relevance: Stretch, Twist, Fold Magnetic dynamo: evolution of magnetic field in a plasma. Chemical and biological mixing: creation of intermaterial contact area. Polymer mixing (i.e., DNA): follow material lines closely. Much is known about stretching, but less about the bending of material lines (generation of curvature and torsion). Some interesting regularities, such as a close anticorrelation between stretching and curvature. Curvature in Chaotic and Turbulent Flows p.2/16

3 Stretching and Folding Traces out the unstable foliation of the flow. Note the sharp folds that develop y.6 y.4.4 eplacements.2 PSfrag replacements.2 x y x x y x Cellular Flow Standard Map Can look surprisingly regular even in extremely chaotic cases. Curvature in Chaotic and Turbulent Flows p.3/16

4 Stretching along a Material Line Λ is the deviation from mean stretching. 2 log κ log Λ eplacements arc length Cellular Flow log κ log Λ PSfrag replacements arc length Standard Map Suppression of stretching. [Drummond & Münch (1991)] Curvature in Chaotic and Turbulent Flows p.4/16

5 Stretching and Curvature: the Dynamo A similar effect was recently observed for the magnetic dynamo. Magnetic field, B Curvature of B, κ The magnetic field and its curvature are anticorrelated [Schekochihin, Cowley, Maron & Malyshkin, Phys. Rev. E (22)] Curvature in Chaotic and Turbulent Flows p.5/16

6 Stretching vs Curvature along a Material Line Power law relation around sharp folds: The 1/3 law. 2 log Λ g replacements log Λ PSfrag replacements κ 1/3 4 κ 1/ log κ Cellular Flow log κ Standard Map The law is very robust even with varying degree of chaos and different flows (2D and 3D). Curvature in Chaotic and Turbulent Flows p.6/16

7 g replacements Enhancement to Gradients by Folding δ (a) (b) φ = φ 1 Fold 1 Assume linear gradient of φ varying from to 1; The endpoints of the line are brought to a distance δ; Enhancement in φ proportional to δ 1 ; Fluid elements in the crest of the bend do not benefit. Can explain 1/3 law with this simple model. [JLT, 22] Curvature in Chaotic and Turbulent Flows p.7/16

8 A Simple Model Very sharp bend in a material line, 1 y = f(x) = 1 2 κ x 2 + O(x 3 ) where κ = f () is the curvature at the tip. f(x) x away from the tip. Approximate the arc length τ from (, ) to (x, f(x)) by τ(x) f(x). Enhancement to gradients: Λ(x) = τ(x)/x f(x)/x. PSfrag replacements Measure of stretching (incompressible) y = f(x) δ 1 1 x Curvature in Chaotic and Turbulent Flows p.8/16

9 The curvature is κ (ˆt )ˆt, where ˆt is the unit tangent to f. To leading order this is κ(x) = κ 2 x 3 + O(x 2 ), Λ(x) = κ x + O(x 2 ). Solve for x in terms of κ, Λ κ 1/3 Problem: the 1/3 law works much better than predicted by this simple model. (Predicts breakdown near the tip, works fine in 3D... ) Curvature in Chaotic and Turbulent Flows p.9/16

10 A Foliation of Bends Some observations: Material lines are not isolated objects. Continuum of other material lines. Standard map resembles a foliation of bends. Distance between lines is not constant: Compression is not uniform. Curvature is readily computed (geometrical). How do we relate to stretching? y = f(x) PSfrag replacements δ x Curvature in Chaotic and Turbulent Flows p.1/16

11 Conservation Law for Lyapunov Exponents The tangent to the material line aligns with the unstable direction of the flow, û, the direction of maximum stretching. That direction satisfies the crucial constraint û + û log Λ, (exponentially) [JLT, 22, in press] following earlier work by [Tang & Boozer, 1996] and [JLT & Boozer, 21]. This is a conservation law on for Λ along the unstable manifold. τ log Λ + û =, τ arc length along û Convergence of û increase in Λ. Curvature in Chaotic and Turbulent Flows p.11/16

12 Assuming a foliation of bends with shape y = f(x), the divergence of û is easily computed, Derivative of Λ along û: û ˆt = ˆt x x = f f (1 + f 2 ) 3/2. τ log Λ = û log Λ = 1 (1 + f 2 ) 1/2 x log Λ, Equate and integrate to yield Λ (1 + f 2 ) 1/2. Curvature in Chaotic and Turbulent Flows p.12/16

13 To exhibit the relationship between stretching and curvature, we use κ(x) = f (x) /(1 + f 2 ) 3/2 for the magnitude of the curvature and obtain finally Λ f (x) 1/3 κ 1/3 For quadratic f, Λ (κ/κ ) 1/3, so that the power-law relation holds exactly. The shape of the bend and y-dependence of the tangent vector field will cause deviations from the 1/3 law. Curvature in Chaotic and Turbulent Flows p.13/16

14 PDF of Stretching along a Material Line log Λ replacements PSfrag replacements N Λ arc length log Λ The folding model predicts the Λ 2 tail of the probability of extremely low stretching events. Exponential ( fat ) tail: large fluctuations from the mean stretching. Curvature in Chaotic and Turbulent Flows p.14/16

15 PDF of Curvature log N PSfrag replacements 4 5 κ +1 κ 2/ log κ Stationary distribution. Tails seem independent of specific flow. Mean moves to the right in less chaotic flows. Curvature in Chaotic and Turbulent Flows p.15/16

16 Conclusions Stretching anticorrelated with curvature. Around sharp bends, observe stretching curvature 1/3. Can be explained using a conservation law for Lyapunov exponents. Ongoing work: The consequences of constraints in physical applications (for the dynamo, with A. Boozer). Evolution of torsion. Constrained, like curvature? Understand PDF of curvature. 2D special? Curvature in Chaotic and Turbulent Flows p.16/16