In the vicinity of the point of minimum critical shear of a quasi-geostrophic two-level model on the $ beta$-plane, the weakly nonlinear dynamics of developing baroclinic vortices can be described in terms of a nonlinear critical layer problem which, in the inviscid case, can be solved analytically. When the supercritical shear $ delta$ is such that 0 $<$ $ delta$ $ ll$ 1 and the initial conditions are sufficiently small, finite amplitude equilibration occurs even though the potential vorticity field in the bottom layer Q(X,Y,t) remains transient, the potential enstrophy being transferred to smaller and smaller scales. It is shown that the inviscid equilibrium amplitude of the unstable wave is larger by a factor of $ surd$2 than the one found by Pedlosky (1982-b) in the limit of small dissipation. This indicates that the limits t $ to$ $ infty$ and r $ to$ 0 are not interchangeable. Inviscid equilibration occurs when the mixing in the lowest layer results in the streamwise homogenization of the coarse-grained average (Q) of the potential vorticity which means that (Q) $ to$ f($ psi$), $ psi$ being the streamfunction. When $ delta$ and the initial conditions are equally important, depending on the nature of the latter, periodic solutions and finite equilibration are both possible. An example is given of a periodic case when $ delta$ = 0. The potential vorticity field then reversibly wraps and un-wraps around the streamlines and mixing does not occur. Finally, these exact solutions are used to judge the reliability of numerical results obtained from truncated spectral models. For cases where finite equilibration occurs, the resolution of a truncated model is only adequate for a finite period of time while for periodic cases, a model with sufficient resolution can represent correctly the exact solution for any length of time.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.75835 |
| Date | January 1988 |
| Creators | Gauthier, Pierre. |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | French |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Doctor of Philosophy (Department of Meteorology.) |
| Rights | All items in eScholarship@McGill are protected by copyright with all rights reserved unless otherwise indicated. |
| Relation | alephsysno: 000730564, proquestno: AAINL48508, Theses scanned by UMI/ProQuest. |
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