Amplitude equation and long-range interactions in underwater sand ripples in one dimension

Publication: Research - peer-reviewJournal article – Annual report year: 2008

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We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form h(t)=epsilon-(h-(h) over bar) + ((h(x))(2)-1)h(xx)-h(xxxx) + delta((h(x))(2))(xx) which, due to the first term, is neither completely local (it has long-range coupling through the average height (h) over bar) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large delta (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.
Original languageEnglish
JournalPhysical Review E (Statistical, Nonlinear, and Soft Matter Physics)
Publication date2008
Volume78
Journal number4
Pages047301
ISSN1539-3755
DOIs
StatePublished

Bibliographical note

Copyright 2008 American Physical Society

CitationsWeb of Science® Times Cited: 3

Keywords

  • WAVES, PATTERN-FORMATION, DYNAMICS
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