Defect Formation in Convective Patterns as seen from the Regularized Cross-Newell Phase Diffusion Equation

1. Defect Formation in Convective Patterns as seen from the Regularized Cross-Newell Phase Diffusion Equation R. Indik, A.C. Newell, S. Venkataramani (Arizona) T. Passot (Nice) M. Taylor (UNC Chapel Hill) The Geometry of the Phase Diffusion Equation, J. Nonlinear Sci. 10, 223-274 (2000) Global Description of Patterns Far From Onset: A Case Study PhysicaD 184, 127-140 (2003)

2. Patterns far from threshold • In an isotropic stripe pattern-forming system, the direction of the wavevector is not chosen. Such structures can then be described by noting that the pattern locally consists of patches with a preferred wavevector k. One can then let kvary slowly in space and time • Experiment: From Y.-C. Hu, R. Ecke, & G. Ahlers, Phys. Rev. E 51, 3263 (1995). See also • http://tweedledee.ucsb.edu/~guenter/picturepage2.html

3. Convex Disclination Pr = 1.4 v.d.t.=2s courtesy of Eberhard Bodenschatz

4. Concave Disclination Aspect ratio = 30 Pr = 0.32 Courtesy of G. Ahlers, U.C. Santa Barbara http://tweedledee.ucsb.edu/~guenter/picturepage5.html

5. Twist

6. Outline I. Without Twist A. Motivation B. Swift-Hohenberg Model C. Cross-Newell Equation D. Regularization E. Results F. Self-Duality II. With Twist A. Director Fields B. Energetics C. Comparisons (Numerical and Experimental) D. Rigorous Scaling for Extended CN system

7. Swift-Hohenberg Swift-Hohenberg Eqn: ut = - (kc+D)2 u + m u – u3 SH Energy Density: - ½ ((kc+D)u)2 + ½ m u2 – ¼ u4 Family of Exact Stationary Solutions: u0(x) = a1(k) cos(q) + a2(k)cos(2q) + … k u0 = const. 2p/k

8. Modulational Ansatz

9. Cross – Newell Equation • Slowly Varying Rolls  Modulational Ansatz • u = ue (Q/e) Q = Q (X,Y,T)Q = k(X,Y,T) k = |k| • Q = e qX= e x Y =e yT = e2 t t(k2)Q  T = - ·(k B(k2)) • M.C. Cross & A.C. Newell, Convection patterns large aspect ratio systems, • Physica 10 D, 299-328 (1984)

10. Cross-Newell Energy Densityt(k2)Q  T = - ·(k B(k2)) t(k2) (kc = kB = 1) kB(k2)

11. Hodograph Solutions

12. Hodograph “Swallowtail”

13. Regularized Cross-Newell Eqn t(k2) QT = -  (k B(k2) + e2 D k) QT =k~1 -   (2 Q (1 – |Q|2) ) - e2 D D Q) RCN Energy: Fe (Q) = e(DQ)2 +1/e (1-|Q|2)2dX dY

14. Regularized CN Disclinations C. Bowman

15. Analytical Results • Fe (Q) =  e(DQ)2 +1/e (1-|Q|2)2dX dY • As e  0, minimizers Qe  Q0 in H1(W)where Q0 solves • the eikonal equation |Q0| = 1. Defects are, therefore, • supported on locally 1-dimensional sets. An example of an • eikonal solution on a domain W: Q0 (x) = dist (x , W ). • (2) There is a natural conjecture that the asymptotic value of • the minimal energy (ground state energy) = a “jump energy” • supported on the defect locus S : • lim infe 0 1/3 |[Q0]|3 ds • where [Q0] = jump in k0 across S.

16. Eikonal Viscosity Solutions N=32 q N=32 q

17. (3) The conjecture can be completely verified in the class of examples where S is a straight line segment. In this same class the asymptotic energy is supported only on the 1-dimensional eikonal defects. Lower dimensional singularities carry no energy. • Ambrosio, DeLellis, Mantegazza • DeSimone, Kohn, Müller, Otto • Ercolani, Taylor

18. Analytical Difficulties of RCN: 1) 4th order (QT = -   (2 Q (1 – |Q|2) ) - e2 D D Q) 2) Hausdorff dim of defects is > 0 (not point defects) Self-Dual Eqn:e(DQ) = (1-|Q|2) 1) Motivated by equi-partition of energy e(DQ)2 +1/e (1-|Q|2)2 2) If the Gaussian curvature of the graph of QSD vanishes, then QSD solves RCN 3) In the e-> 0 limit, the curvature of QSD concentrates in points.

19. Helmholtz Linearization • The substitution Q =  e ln Y reduces • e(DQ)  (1-|Q|2) = 0 • to the linear Helmholtz eqn: • e2DY - Y = 0

20. Knees to Chevrons

21. Self-Dual Test Functions Thm: Suppose ve solves e(D ve)+(1-| ve|2) = 0 on W with ve = q on the boundary of W where |q(x) - q(y)|  a dist(x,y) with a < 1. Then as e  0, ve converges uniformly on the closure of W to the unique viscosity solution of | v|2 - 1 = 0 on W with v = q on W.

22. Steepest Descent These conditions are equivalent to k satisfying the jump conditions for a weak solution of stationary CN.

23. Outline I. Without Twist A. Motivation B. Swift-Hohenberg Model C. Cross-Newell Equation D. Regularization E. Results F. Self-Duality II. With Twist A. Director Fields B. Energetics C. Comparisons (Numerical and Experimental) D. Rigorous Scaling for Extended CN system

24. Boussinesq Simulation courtesy of Mark Paul b/a = 1.9 e = 0.850

25. Swift-Hohenberg Equation

26. Swift-Hohenberg equation q q N=32 N=32 m= 0.5, b/a = 1.9 e = 0.850

27. Stadium

28. Double Cover

29. Comparison of knee solution to concave-convex disclination pair 8/3 sin3(a) 8/3 (1 - sin (a)) Energy cost per unit length of GB

30. Phase Topology • u() ~ cos() • k =  = (f,g) Target Space Identifications: •    + 2n • (,f,g)  (-,-f,-g) Quotient map has critical points at  = n; i.e., d = f dx + g dy behaves like a quadratic differential

31. Knee-to-Disclination Pair Transition Triangle: curvature center Diamond: critical transition

32. Comparison to Energy Density

33. Fluid Experiment b/a = 1.5 Courtesy of G. Ahlers and W. Meevasana U.C.S.B.

34. Comparison with Experiments

35. Comparison with Other Simulations Swift-Hohenberg simulation Boussinesq simulation

36. Swift-Hohenberg “Zippers”

37. Cross-Newell Zippers • cos((x,y)) is even in y and smooth in (x,y) (x,y) is an even function of y  y (x,0) = 0; or, (x,y) is an odd function of y modulo  • x  cos() as y   • Shift-periodic symmetry in x: (x+l ,y) = (x,y) +  where l = cos()

38. Boundary Conditions • (x,0) = 0 for 0  x  a • y(x,0) = 0 for a  x  l • (x+l ,y) = (x,y) +  •   xcos() + ysin() +  • as y   • Domain is S (boxed strip) •  = cos()

39. Total Energy as a Function of Disclination Separation

46. Energy of the Minimizer

47. Separation between the Concave and Convex Disclinations

48. Asymptotic Phase Shift of the Minimizer

49. Rigorous Scaling Law for Energy Minimizers • Natural small parameter here is  = cos()

50. Rigorous Scaling Law for Energy Minimizers