lecture 2: nonlinear equations from symmetry and conservation: application to sand ripples
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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 France. Geometrical formulation. normal velocity. - PowerPoint PPT PresentationTRANSCRIPT
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Lecture 2: nonlinear equationsfrom symmetry and conservation: application to sand ripplesChaouqi MISBAH
LIPhy (Laboartoire Interdisciplinaire de Physique) Univ. J. Fourier Grenoble and CNRS France
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nu normal velocity
,...),( ssnu arclength curvature, s
Remark: in 3D add Gauss curvature and use surface operator :)( nnIs
Geometrical formulation
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Conservation constraintsCsahok, C.M., Valance Physica D 128 (1999) 87–100
1) No conservation
ssn baaaCu 3221
1) Mass conservation
][ 13
32
21 ssssn baaaCu
If anisotropy: )( );( );( ii baC
0 dsun
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small )];6cos(1[)( 32
ssnu
ss
n Cu
3
2
][ 2ssss
n Cu
Snowflacke Dense pattern Star fish
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« Weakly » nonlinear equations
1) No conservation
ssn baaaCu 3221
)(1 Oa
x
z
h(x)21/ xtn hhu
2/32 ]1/[ xxx hh
211 2 xxxxxxxt hChbhaCh Kuramoto-Sivashinsky
)(1 Oa htx ;/1 ;/1 2
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211 2 xxxxxxxt hChbhaCh
Spatiotemporal chaos
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211 2 xxxxxxxt hChbhaCh
?0C
2211 xxxxxxxxt hahbhah
1 ;/1 ;/1 2 htx )(1 Oa
KS equation and this one can be made free of parameter
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2) Mass conservation
][ 13
32
21 ssssn baaaCu
Case C=0 or small)(1 Oa /1 ;/1 ;/1 2 htx
)])1(
(1
)(tan[2/322
111
x
xxxx
x
xxt hh
h
bhah
Similar to situation encountered in crystal growth; O. Pierre-Louis, Phys. Rev. Lett. 1998
Recent analysis by Guedda and Benlahsen
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Indefinite increase of the amplitude
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3) No conservation with anisotropy
ssn baaaCu 3221
)( );( );( baC i
2xxxxxxt hhhh
Benney equation (KS+KDV)tion transformaGallilean a viaabsorbed term-0v
n)KDV(solito toleads large v 2xxxxt hvhh
chaos) temporal-(spatio KS toleads small v
20 xxxxxxxxxxxt hvhhhhvh
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2xxxxxxxxxt hvhhhh
enough large venough small v
Benney eq. derived for step bunching by C.M. and O. Pierre-Louis (PRE, 1998); see alsoC.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)
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4) Mass conservation with anisotropy (case of sand ripples, dunes)
][ 13
32
21 ssssn baaaCu
)( );( );( baC i
xxxxxxxxxxxxxxxt hhhvhhhh )()()( 322
Modified BCRE model (Csahok, C.M., Rioual, Valance, EPJE 2000)
),( hRVRR xt ),( hRht ejdep
ejwind
ejimp
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Spatio-temporal portait
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2
2 xxxxxxxt hChhh
22 xxxxxxxxt hahhh
)])1(
(1
)(tan[2/322
111
x
xxxx
x
xxt hh
h
bhah
2xxxxxxxxxxt hvhhhh
xxxxxxxxxxxxxxxt hhhvhhhh )()()( 322
No consevation
C=0
consevationanisotropy
anisotropy
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Conclusion
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?