dorsal/arxiv
View SchemaQuasi-geostrophic model of the instabilities of the Stewartson layer
| Authors | Nathanael Schaeffer, Philippe Cardin |
|---|---|
| Categories | |
| ArXiv ID | physics/0312129 |
| URL | https://arxiv.org/abs/physics/0312129 |
Abstract
We study the destabilization of a shear layer, produced by differential rotation of a rotating axisymmetric container. For small forcing, this produces a shear layer, which has been studied by Stewartson and is almost invariant along the rotation axis. When the forcing increases, instabilities develop. To study the asymptotic regime (very low Ekman number $E$), we develop a quasi-geostrophic two-dimensional model, whose main original feature is to handle the mass conservation correctly, resulting in a divergent two-dimensional flow, and valid for any container provided that the top and bottom have finite slopes. We use it to derive scalings and asymptotic laws by a simple linear theory, extending the previous analyses to large slopes (as in a sphere), for which we find different scaling laws. For a flat container, the critical Rossby number for the onset of instability evolves as $E^{3/4}$ and may be understood as a Kelvin-Helmoltz shear instability. For a sloping container, the instability is a Rossby wave with a critical Rossby number proportional to $\beta E^{1/2}$, where $\beta$ is related to the slope. We also investigate the asymmetry between positive and negative differential rotation and propose corrections for finite Ekman and Rossby numbers. Implemented in a numerical code, our model allows us to study the onset over a broad range of parameters, determining the threshold but also other features such as the spatial structure. We also present a few experimental results, validating our model and showing its limits.
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"abstract": "We study the destabilization of a shear layer, produced by differential\nrotation of a rotating axisymmetric container. For small forcing, this produces\na shear layer, which has been studied by Stewartson and is almost invariant\nalong the rotation axis. When the forcing increases, instabilities develop. To\nstudy the asymptotic regime (very low Ekman number $E$), we develop a\nquasi-geostrophic two-dimensional model, whose main original feature is to\nhandle the mass conservation correctly, resulting in a divergent\ntwo-dimensional flow, and valid for any container provided that the top and\nbottom have finite slopes. We use it to derive scalings and asymptotic laws by\na simple linear theory, extending the previous analyses to large slopes (as in\na sphere), for which we find different scaling laws. For a flat container, the\ncritical Rossby number for the onset of instability evolves as $E^{3/4}$ and\nmay be understood as a Kelvin-Helmoltz shear instability. For a sloping\ncontainer, the instability is a Rossby wave with a critical Rossby number\nproportional to $\\beta E^{1/2}$, where $\\beta$ is related to the slope. We also\ninvestigate the asymmetry between positive and negative differential rotation\nand propose corrections for finite Ekman and Rossby numbers. Implemented in a\nnumerical code, our model allows us to study the onset over a broad range of\nparameters, determining the threshold but also other features such as the\nspatial structure. We also present a few experimental results, validating our\nmodel and showing its limits.",
"arxiv_id": "physics/0312129",
"authors": [
"Nathanael Schaeffer",
"Philippe Cardin"
],
"categories": [
"physics.flu-dyn",
"physics.geo-ph"
],
"title": "Quasi-geostrophic model of the instabilities of the Stewartson layer",
"url": "https://arxiv.org/abs/physics/0312129"
},
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