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

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.

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

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  1. Lecture 2: nonlinearequations fromsymmetry and conservation: application to sandripples ChaouqiMISBAH LIPhy (Laboartoire Interdisciplinaire de Physique) Univ. J. Fourier Grenoble and CNRS France

  2. Geometrical formulation normal velocity Remark: in 3D add Gauss curvature and use surface operator

  3. Conservation constraints Csahok, C.M., Valance Physica D 128 (1999) 87–100 1) No conservation 1) Mass conservation If anisotropy:

  4. Dense pattern Snowflacke Star fish

  5. « Weakly » nonlinearequations 1) No conservation z x h(x) Kuramoto-Sivashinsky

  6. Spatiotemporal chaos

  7. KS equation and this one canbe made free of parameter

  8. 2) Mass conservation Case C=0 or small Similar to situation encountered in crystalgrowth; O. Pierre-Louis, Phys. Rev. Lett. 1998 Recentanalysis by Guedda and Benlahsen

  9. Indefiniteincrease of the amplitude

  10. 3) No conservation withanisotropy Benneyequation (KS+KDV)

  11. Benneyeq. derived for stepbunching by C.M. and O. Pierre-Louis (PRE, 1998); seealso C.M. et al. Review of Modern Physics 2010. And for sandripplesundererosionusing a modified model of Bouchaud et el. 1994. Valanace and C.M., (PRE 2003)

  12. 4) Mass conservation withanisotropy (case of sandripples, dunes) Modified BCRE model (Csahok, C.M., Rioual, Valance, EPJE 2000)

  13. Spatio-temporal portait

  14. No consevation C=0 anisotropy consevation anisotropy

  15. Conclusion • Classes of equationsderivedfromsymmetries and conservations • Eqscanbeweakly or highlynonlinear; identification by scaling • This provides a powerfull basis to guide the analysis • Eqs. are consistent withthosederivedfrom « microscopic » models • Application to dunes wouldbeinteresting • Next lecture: wheniscoarseningexpected?

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