Defect Formation in Convective Patterns
as seen from the Regularized CrossNewell 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)
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.html
Convex Disclination
Pr = 1.4
v.d.t.=2s
courtesy of Eberhard Bodenschatz
Concave Disclination
Aspect ratio = 30
Pr = 0.32
Courtesy of G. Ahlers, U.C. Santa Barbara
http://tweedledee.ucsb.edu/~guenter/picturepage5.html
Twist
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
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
Modulational Ansatz
Cross – Newell Equation
Slowly Varying Rolls
u = ue (Q/e)

Q = Q (X,Y,T)
Q=eq
X=ex
Modulational Ansatz
 Q = k(X,Y,T)
Y=ey
k = |k|
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)
Cross-Newell Energy Density
t(k2) Q  T = - ·(k B(k2))
t(k2)
(kc = kB = 1)
kB(k2)
Hodograph Solutions
Hodograph “Swallowtail”
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)2 dX dY

Regularized CN Disclinations
C. Bowman
Analytical Results
Fe (Q) =  e(DQ)2 +1/e (1-|Q|2)2 dX dY
(1) As e  0, minimizers Qe  Q0 in H1() 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 : Q0 (x) = dist (x ,   ).
(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
where
[Q0]

S
= jump in
|[Q0]|3 ds
k0 across
S.
Eikonal Viscosity Solutions
N=32
q
N=32
q
(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
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.
Helmholtz Linearization
The substitution Q =  e ln Y reduces
e(DQ)  (1-|Q|2) = 0
to the linear Helmholtz eqn:
e2DY - Y = 0
Knees to Chevrons
Self-Dual Test Functions
Thm: Suppose ve solves e (D ve)+(1-| ve |2) = 0 on  with ve = q
on the boundary of  where |q(x) - q(y)|  a dist(x,y) with a < 1.
Then as e  0, ve converges uniformly on the closure of  to the
unique viscosity solution of | v |2 - 1 = 0 on  with
v = q on .
Steepest Descent




X (s ) - X
k = v X = 

X (s ) - X
( )




wher e





X (s ) - X


 - v X (s )   X (s ) = 0
X (s ) - X

(
)
These conditions are equivalent to k satisfying the jump
conditions for a weak solution of stationary CN.
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
Boussinesq Simulation
courtesy of Mark Paul
b/a = 1.9 e = 0.850
Swift-Hohenberg Equation
Swift-Hohenberg equation
N=32
q
N=32
m = 0.5, b/a = 1.9 e = 0.850
q
Stadium
Double Cover
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
Phase Topology
• u(q) ~ cos(q)
• k = q = (f,g)
Target Space Identifications:
• q  q + 2np
• (q,f,g)  (-q,-f,-g)
Quotient map has critical points at q = np;
i.e., dq = f dx + g dy behaves like a quadratic differential
Knee-to-Disclination Pair
Transition
Triangle: curvature center
Diamond: critical transition
Comparison to Energy Density
Fluid Experiment
Courtesy of G. Ahlers and W. Meevasana U.C.S.B.
b/a = 1.5
Comparison with Experiments
Comparison with Other
Simulations
Swift-Hohenberg simulation
Boussinesq simulation
Swift-Hohenberg “Zippers”
Cross-Newell Zippers
• cos(q(x,y)) is even in y and smooth in (x,y)
q(x,y) is an even function of y  qy (x,0) = 0;
or, q(x,y) is an odd function of y modulo p
• qx  cos(a) as y  
• Shift-periodic symmetry in x:
q(x+l ,y) = q(x,y) + p
where l = p/cos(a)
Boundary Conditions
•q(x,0) = 0 for 0  x  a
•qy(x,0) = 0 for a  x  l
•q(x+l ,y) = q(x,y) + p
•q  xcos(a) + ysin(a) + 
as y  
Domain is Se (boxed strip)
e = cos(a)
Total Energy as a Function of
Disclination Separation
Energy of the Minimizer
Separation between the Concave and
Convex Disclinations
Asymptotic Phase Shift of the
Minimizer
Rigorous Scaling Law for Energy Minimizers
• Natural small parameter here is e = cos(a)

(
Minimize F (q , a ) =  Dq + 1 - q
e
2
2
)  dxdy
Se
over all a  [0,1) and q  W 2, 2 (S e ) subject to " zipper" b.c.' s
4p (1 - e 2 )
e
F (q sd , a ) =
3e
3/ 2
Rigorous Scaling Law for Energy Minimizers
Upper Boun d : There is a constant E0 such that
F e (q e , a e )  E0 for all e  (0,1].
Lower Bound : There is a constant E1 > 0 such that
F e (q e , a e )  E1 for sufficient ly small e .
Moreover, there are constants 0  a1  a 2 such that
1 - a 2e  a e  1 - a1e for sufficient ly small e