dorsal/arxiv
View SchemaNear Critical Reflection of Internal Waves
| Authors | T. Dauxois, W. R. Young |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9902003 |
| URL | https://arxiv.org/abs/patt-sol/9902003 |
| DOI | 10.1017/S0022112099005108 |
| Journal | Journal of Fluid Mechanics 390, 271-295 (1999) |
Abstract
Using a matched asymptotic expansion we analyze the two-dimensional, near- critical reflection of a weakly nonlinear, internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the classical solution in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the along-slope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as $1/t$. During the course of this scale reduction, the stratification is `overturned' and the Miles-Howard condition for stratified shear flow stability is violated. However, for all slope angles, the `overturning' occurs before the Miles-Howard stability condition is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process.
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"abstract": "Using a matched asymptotic expansion we analyze the two-dimensional, near-\ncritical reflection of a weakly nonlinear, internal gravity wave from a sloping\nboundary in a uniformly stratified fluid. Taking a distinguished limit in which\nthe amplitude of the incident wave, the dissipation, and the departure from\ncriticality are all small, we obtain a reduced description of the dynamics.\nThis simplification shows how either dissipation or transience heals the\nsingularity which is presented by the classical solution in the precisely\ncritical case. In the inviscid critical case, an explicit solution of the\ninitial value problem shows that the buoyancy perturbation and the along-slope\nvelocity both grow linearly with time, while the scale of the reflected\ndisturbance is reduced as $1/t$. During the course of this scale reduction, the\nstratification is `overturned\u0027 and the Miles-Howard condition for stratified\nshear flow stability is violated. However, for all slope angles, the\n`overturning\u0027 occurs before the Miles-Howard stability condition is violated\nand so we argue that the first instability is convective. Solutions of the\nsimplified dynamics resemble certain experimental visualizations of the\nreflection process.",
"arxiv_id": "patt-sol/9902003",
"authors": [
"T. Dauxois",
"W. R. Young"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1017/S0022112099005108",
"journal_ref": "Journal of Fluid Mechanics 390, 271-295 (1999)",
"title": "Near Critical Reflection of Internal Waves",
"url": "https://arxiv.org/abs/patt-sol/9902003"
},
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