Lecture 2 nonlinear equations from symmetry and conservation application to sand ripples Chaouqi MISBAH LIPhy Laboartoire Interdisciplinaire de Physique Univ J Fourier Grenoble and CNRS ID: 564886
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Slide1
Errachidia 2011
Lecture 2: nonlinear equationsfrom symmetry and conservation: application to sand ripples
Chaouqi
MISBAH
LIPhy
(
Laboartoire
Interdisciplinaire de Physique)
Univ
. J. Fourier Grenoble and CNRS
FranceSlide2
Errachidia 2011
normal
velocity
Remark
: in 3D
add
Gauss
curvature
and use surface
operator
Geometrical
formulationSlide3
Errachidia 2011Conservation constraints
Csahok, C.M., Valance Physica D 128 (1999) 87–1001) No conservation
1) Mass conservation
If
anisotropy
: Slide4
Errachidia 2011
Snowflacke
Dense pattern
Star
fishSlide5
Errachidia 2011« Weakly
» nonlinear equations1) No conservation
x
z
h(x)
Kuramoto
-
SivashinskySlide6
Errachidia 2011
Spatiotemporal chaosSlide7
Errachidia 2011
KS
equation
and
this one can be made
free of parameterSlide8
Errachidia 2011
2) Mass conservationCase C=0 or small
Similar
to situation
encountered
in
crystal growth; O. Pierre-Louis, Phys.
Rev. Lett. 1998
Recent analysis by
Guedda and BenlahsenSlide9
Errachidia 2011
Indefinite increase of the amplitude Slide10
Errachidia 2011
3) No conservation with anisotropy
Benney
equation
(KS+KDV)Slide11
Errachidia 2011
Benney
eq
. derived
for step bunching
by C.M. and O. Pierre-Louis (PRE, 1998); see also
C.M. et al. Review of Modern
Physics 2010.And for
sand ripples
under erosion using a modified model of
Bouchaud et el. 1994.Valanace
and C.M., (PRE 2003)Slide12
Errachidia 2011
4) Mass conservation with anisotropy (case of sand ripples, dunes)
Modified
BCRE model (
Csahok
, C.M.,
Rioual, Valance, EPJE 2000) Slide13
Errachidia 2011
Spatio-temporal portaitSlide14
Errachidia 2011
No
consevation
C=0
consevation
anisotropy
anisotropySlide15
Errachidia 2011Conclusion
Classes of equations derived from symmetries and conservations Eqs can be weakly or highly nonlinear; identification by
scaling
This
provides
a powerfull basis to guide the analysis
Eqs. are consistent with those
derived from « microscopic
» models
Application to dunes would be interesting
Next lecture: when is coarsening expected
?