lecture 2: nonlinear equations from symmetry and conservation: application to sand ripples

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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

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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)

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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?