150 likes | 222 Views
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.
E N D
Lecture 2: nonlinearequations fromsymmetry and conservation: application to sandripples ChaouqiMISBAH LIPhy (Laboartoire Interdisciplinaire de Physique) Univ. J. Fourier Grenoble and CNRS France
Geometrical formulation normal velocity Remark: in 3D add Gauss curvature and use surface operator
Conservation constraints Csahok, C.M., Valance Physica D 128 (1999) 87–100 1) No conservation 1) Mass conservation If anisotropy:
Dense pattern Snowflacke Star fish
« Weakly » nonlinearequations 1) No conservation z x h(x) Kuramoto-Sivashinsky
KS equation and this one canbe made free of parameter
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
3) No conservation withanisotropy Benneyequation (KS+KDV)
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)
4) Mass conservation withanisotropy (case of sandripples, dunes) Modified BCRE model (Csahok, C.M., Rioual, Valance, EPJE 2000)
No consevation C=0 anisotropy consevation anisotropy
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?