k q, p

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k (q, p)
T Tcond dT sin(p z) exp i(q x p y ) exp( s
t )
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Neutral curve
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e 0
k (q, p)
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Paramagnet
Ferromagnet
ltdTgt
Temperature
Q dT sin( p z ) exp i ( q x p y )
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Fluctuations well below the onset of convection
Structure factor square of the modulus of the
Fourier transform of the snapshot
Shadowgraph image of the pattern. The sample is
viewed from the top.In essence, the method shows
the temperature field.
p
p
Snapshot in real space
R / Rc 0.94
Movie by Jaechul Oh
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dST k2
e -0.57
-0.68
-0.78
dST k-4
k
k
Experiment J. Oh and G.A., cond-mat/0209104. Line
ar Theory J. Ortiz de Zarate and J. Sengers,
Phys. Rev. E 66, 036305 (2002).
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Squares t 200 ms Circles t 500 ms

• camera exposure
• time

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C(k, t) lt ST (k, t) ST (k, t t) gt / lt ST2 (k,
t) gt
C C0 exp( -s(k) t )
-0.14
s(k)
e -0.70
J. Oh, J. Ortiz de Zarate, J. Sengers, and G.A.,
Phys. Rev. E 69, 021106 (2004).
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Just above onset, straight rolls are
stable. Theory A. Schluter, D. Lortz, and F.
Busse, J. Fluid Mech. 23, 129 (1965). This
experiment K.M.S. Bajaj, N. Mukolobwiez, N.
Currier, and G.A., Phys. Rev. Lett. 83, 5282
(1999).
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DT
k
F. Busse and R.M. Clever, J. Fluid Mech. 91, 319
(1979) and references therein.
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Taylor
vortex flow First experiments and linear
stability analysis by G.I. Taylor in Cambridge
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The rigid top and bottom pin the phase of the
vortices. They also lead to the formation of
a sub-critical Ekman vortex. M.A.
Dominguez-Lerma, D.S. Cannell and G.A., Phys.
Rev. A 34, 4956 (1986). G. A., D.S. Cannell, M.A.
Dominguez-Lerma, and R. Heinrichs,
Physica, 23D, 202 (1986). A.M. Rucklidge and A.R.
Champneys, Physica A 191, 282 (2004).
In the interior, a vortex pair is lost or gained
when the system leaves the stable band of
states. Theory W. Eckhaus, Studies in nonlinear
stability theory, Springer, NY, 1965.
Experiment M.A. Dominguez-Lerma, D.S. Cannell
and G.A., Phys. Rev. A 34, 4956 (1986). G. A.,
D.S. Cannell, M.A. Dominguez-Lerma, and R.
Heinrichs, Physica, 23D, 202 (1986).
18
M.A. Dominguez-Lerma, D.S. Cannell and G.A.,
Phys. Rev. A 34, 4956 1986.
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At the free upper surface the pinning of the
phase is weak and a vortex pair can be gained
or lost. The Eckhaus Instability is never
reached. Experiment M. Linek and G.A.,
Phys. Rev. E 58, 3168 (1998). Theory M.C.
Cross, P.G. Daniels, P.C. Hohenberg, and E.D.
Siggia, J. Fluid Mech. 127, 155 (1983).
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Free upper surface
Rigid boundaries
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Theory H. Riecke and H.G. Paap, Phys. Rev. A 33,
547 (1986). M.C. Cross, Phys. Rev. A 29, 391
(1984). P.M. Eagles, Phys. Rev. A 31, 1955
(1985). Experiment M.A. Dominguez-Lerma, D.S.
Cannell and G.A., Phys. Rev. A 34, 4956 (1986).
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Back to Rayleigh-Benard !
Shadowgraph image of the pattern. The sample is
viewed from the top. In essence, the method shows
the temperature field.
Wavenumber Selection by Domain wall
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J.R. Royer, P. O'Neill, N. Becker, and G.A.,
Phys. Rev. E 70 , 036313 (2004).
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Experiment J. Royer, P. ONeill, N. Becker, and
G.A., Phys. Rev. E 70, 036313 (2004). Theory J.
Buell and I. Catton, Phys. Fluids 29, 1
(1986) A.C. Newell, T. Passot, and M. Souli, J.
Fluid Mech. 220, 187 (1990).
30
W 0
V. Croquette, Contemp. Phys. 30, 153 (1989). Y.
Hu, R. Ecke, and G. A., Phys. Rev. E 48, 4399
(1993)
Phys. Rev. E 51, 3263 (1995).
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W 0
32
Movie by N. Becker
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Movie by Nathan Becker
W 0
Spiral-defect chaos S.W. Morris, E. Bodenschatz,
D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026
(1993).
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Wc lt W 16
G. Kuppers and D. Lortz, J. Fluid Mech. 35, 609
(1969). R.M. Clever and F. Busse, J. Fluid Mech.
94, 609 (1979). Y.-C. Hu, R. Ecke, and G.A.,
Phys. Rev. Lett. 74 , 5040 (1995) Y. Hu, R. E.
Ecke, and G.A., Phys. Rev. E 55, 6928 (1997) Y.
Hu, W. Pesch, G.A., and R.E. Ecke, Phys. Rev. E
58, 5821 (1998).
Movies by Nathan Becker
36
Electroconvection in a nematic liquid crystal
Planar Alignment
Director
V V0 cos( wt )
Convection for V0 gt Vc
e (V0 / Vc) 2 - 1
Anisotropic !
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Director
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X.-L. Qiu G.A., Phys. Rev. Lett. 94, 087802
(2005)
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Rayleigh-Benard convection Fluctuations and
linear growth rates below onset Rotational
invariance Neutral curve Straight rolls above
onset Stability range above onset, Busse
Balloon Taylor-vortec flow Eckhaus
instability Narrower band due to reduced phase
pinning at a free surface Wavenumber selection
by a ramp in epsilon More Rayleigh-Benard Wavenu
mber selection by a domain wall Wavenumber
determined by skewed-varicose instability Onset
of spiral-defect chaos Rayleigh-Benard with
rotation Kuepers-Lortz or domain
chaos Electro-convection in a nematic Loss of
rotational invariance
Summary