Defect Formation in Convective Patterns as seen from the Regularized CrossNewell Phase Diffusion Equ

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 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)

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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 k vary
  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.h
  tml

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Convex Disclination
Pr 1.4 v.d.t.2s
courtesy of Eberhard Bodenschatz
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Concave Disclination
Aspect ratio 30 Pr 0.32
Courtesy of G. Ahlers, U.C. Santa
Barbara http//tweedledee.ucsb.edu/guenter/pictur
epage5.html
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Twist
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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
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Swift-Hohenberg
Swift-Hohenberg Eqn ut - (kcD)2 u m u
u3 SH Energy Density - ½ ((kcD) 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
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Modulational Ansatz

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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 q X e x Y
  e y T 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)

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Cross-Newell Energy Densityt(k2) ?Q? ? T -
?(k B(k2))
t(k2)
(kc kB 1)
kB(k2)
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Hodograph Solutions
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Hodograph Swallowtail
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Regularized Cross-Newell Eqn

t(k2) QT - ?? (k B(k2) e2 D k)
QT k1 - ? ? (2 ?Q (1 ?Q2) ) - e2
D D Q) RCN Energy Fe (Q) e(DQ)2
1/e (1-?Q2)2 dX dY
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Regularized CN Disclinations
C. Bowman
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Analytical Results

• Fe (Q) ? e(DQ)2 1/e
  (1-?Q2)2 dX 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
  ?Q03 ds
• where ?Q0 jump in k0 across S.

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Eikonal Viscosity Solutions
q
q
N32
N32
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(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
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Analytical Difficulties of RCN 1) 4th order
(QT - ? ? (2 ?Q (1 ?Q2) ) - e2 D D Q) 2)
Hausdorff dim of defects is gt 0 (not point
defects) Self-Dual Eqn e(DQ) ?(1-?Q2) 1)
Motivated by equi-partition of energy
? e(DQ)2 1/e (1-?Q2)2 2) If the Gaussian
curvature of the graph of QSD vanishes,
then QSD solves RCN 3) In the e-gt 0 limit, the
curvature of QSD concentrates in points.

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Helmholtz Linearization

• The substitution Q ? e ln Y reduces
• e(DQ) ? (1-?Q2) 0
• to the linear Helmholtz eqn
• e2DY - Y 0

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Knees to Chevrons
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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 lt 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.

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Steepest Descent
These conditions are equivalent to k satisfying
the jump conditions for a weak solution of
stationary CN.
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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
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Boussinesq Simulation
courtesy of Mark Paul
b/a 1.9 e 0.850
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Swift-Hohenberg Equation
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Swift-Hohenberg equation
q
q
N32
N32
m 0.5, b/a 1.9 e 0.850
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Stadium
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Double Cover
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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
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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

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Knee-to-Disclination Pair Transition
Triangle curvature center Diamond
critical transition
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Comparison to Energy Density
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Fluid Experiment
b/a 1.5
Courtesy of G. Ahlers and W. Meevasana U.C.S.B.
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Comparison with Experiments
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Comparison with Other Simulations
Swift-Hohenberg simulation
Boussinesq simulation
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Swift-Hohenberg Zippers
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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
• ?(xl ,y) ?(x,y) ? where l
  ??cos(?)

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Boundary Conditions

• ?(x,0) 0 for 0 ? x ? a
• ?y(x,0) 0 for a ? x ? l
• ?(xl ,y) ?(x,y) ?
• ? ? xcos(?) ysin(?) ?
• as y ? ?
• Domain is S? (boxed strip)
• ? cos(?)

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Total Energy as a Function of Disclination
Separation
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Energy of the Minimizer
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Separation between the Concave and Convex
Disclinations
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Asymptotic Phase Shift of the Minimizer
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Rigorous Scaling Law for Energy Minimizers

• Natural small parameter here is ? cos(?)

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Rigorous Scaling Law for Energy Minimizers