Amplitude equation for under water sand-ripples in one dimension

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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 languageEnglish
Publication date2007
Publication statusPublished - 2007
Event60th Annual Meeting of the Division of Fluid Dynamics - Salt Lake City, UT, United States
Duration: 18 Nov 200720 Nov 2007
Conference number: 60
http://meetings.aps.org/Meeting/DFD07/Content/912

Conference

Conference60th Annual Meeting of the Division of Fluid Dynamics
Number60
CountryUnited States
CitySalt Lake City, UT
Period18/11/200720/11/2007
Internet address

ID: 2974507