dorsal/arxiv
View SchemaStability of Rossby waves in the beta-plane approximation
| Authors | Youngsuk Lee, Leslie M. Smith |
|---|---|
| Categories | |
| ArXiv ID | physics/0208032 |
| URL | https://arxiv.org/abs/physics/0208032 |
| DOI | 10.1016/S0167-2789(03)00010-1 |
Abstract
Floquet theory is used to describe the unstable spectrum at large scales of the beta-plane equation linearized about Rossby waves. Base flows consisting of one to three Rossby wave are considered analytically using continued fractions and the method of multiple scales, while base flow with more than three Rossby waves are studied numerically. It is demonstrated that the mechanism for instability changes from inflectional to triad resonance at an O(1) transition Rhines number Rh, independent of the Reynolds number. For a single Rossby wave base flow, the critical Reynolds number Re^c for instability is found in various limits. In the limits Rh --> infinity and k --> 0, the classical value Re^c = sqrt(2) is recovered. For Rh --> 0 and all orientations of the Rossby wave except zonal and meridional, the base flow is unstable for all Reynolds numbers; a zonal Rossby wave is stable, while a meridional Rossby wave has critical Reynolds number Re^c = sqrt(2). For more isotropic base flows consisting of many Rossby waves (up to forty), the most unstable mode is purely zonal for 2 <= Rh < infinity and is nearly zonal for Rh = 1/2, where the transition Rhines number is again O(1), independent of the Reynolds number and consistent with a change in the mechanism for instability from inflectional to triad resonance.
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"abstract": "Floquet theory is used to describe the unstable spectrum at large scales of\nthe beta-plane equation linearized about Rossby waves. Base flows consisting of\none to three Rossby wave are considered analytically using continued fractions\nand the method of multiple scales, while base flow with more than three Rossby\nwaves are studied numerically. It is demonstrated that the mechanism for\ninstability changes from inflectional to triad resonance at an O(1) transition\nRhines number Rh, independent of the Reynolds number. For a single Rossby wave\nbase flow, the critical Reynolds number Re^c for instability is found in\nvarious limits. In the limits Rh --\u003e infinity and k --\u003e 0, the classical value\nRe^c = sqrt(2) is recovered. For Rh --\u003e 0 and all orientations of the Rossby\nwave except zonal and meridional, the base flow is unstable for all Reynolds\nnumbers; a zonal Rossby wave is stable, while a meridional Rossby wave has\ncritical Reynolds number Re^c = sqrt(2). For more isotropic base flows\nconsisting of many Rossby waves (up to forty), the most unstable mode is purely\nzonal for 2 \u003c= Rh \u003c infinity and is nearly zonal for Rh = 1/2, where the\ntransition Rhines number is again O(1), independent of the Reynolds number and\nconsistent with a change in the mechanism for instability from inflectional to\ntriad resonance.",
"arxiv_id": "physics/0208032",
"authors": [
"Youngsuk Lee",
"Leslie M. Smith"
],
"categories": [
"physics.ao-ph"
],
"doi": "10.1016/S0167-2789(03)00010-1",
"title": "Stability of Rossby waves in the beta-plane approximation",
"url": "https://arxiv.org/abs/physics/0208032"
},
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