### Abstract

Sand-ripples under oscillatory water flow form periodic patterns with wave lengths primarily controlled by the amplitude d of the water motion. We present an amplitude equation for sand-ripples in one spatial dimension which captures the formation of the ripples as well as secondary bifurcations observed when the amplitude $d$ is suddenly varied. The equation has the form
h_t=- ε(h-mean(h))+((h_x)^2-1)h_(xx)- h_(xxxx)+
δ((h_x)^2)_(xx)
which, due to the first term, is neither completely local (it has long-range coupling through the average height mean(h)) nor has local sand conservation. We argue that this is reasonable and that this term (with ε = d^(-2)) stops the coarsening process at a finite wavelength proportional to $d$. We compare our numerical results with experimental observations in a narrow channel.

Original language | English |
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Publication date | 2007 |

Publication status | Published - 2007 |

Event | 60th Annual Meeting of the Division of Fluid Dynamics - Salt Lake City, UT, United States Duration: 18 Nov 2007 → 20 Nov 2007 Conference number: 60 http://meetings.aps.org/Meeting/DFD07/Content/912 |

### Conference

Conference | 60th Annual Meeting of the Division of Fluid Dynamics |
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Number | 60 |

Country | United States |

City | Salt Lake City, UT |

Period | 18/11/2007 → 20/11/2007 |

Internet address |

## Cite this

Schnipper, T. (Author), Mertens, K. (Author), Ellegaard, C. (Author), & Bohr, T. (Author). (2007). Amplitude equation for under water sand-ripples in one dimension. Sound/Visual production (digital) http://flux.aps.org/meetings/YR07/DFD07/all_DFD07.pdf