We analyze the double Hopf bifurcations which occur in two geophysical fluid dynamics
models: (1) a two-layer quasigeostrophic potential vorticity model with forcing and (2) a
mathematical model of the differentially heated rotating annulus experiment. The bifurcations
occur at the transition between axisymmetric steady solutions and non-axisymmetric
travelling waves. For both models, the results indicate that, close to the transition, there
are regions in parameter space where there are multiple stable waves. Hysteresis of these
waves is predicted. For each model, center manifold reduction and normal form theory are
used to deduce the local behaviour of the full system of partial differential equations from
a low-dimensional system of ordinary differential equations.
In each case, it is not possible to compute the relevant eigenvalues and eigenfunctions
analytically. Therefore, the linear part of the equations is discretized and the eigenvalues
and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However,
the projection onto the center manifold and reduction to normal form can be done
analytically. Thus, a combination of analytical and numerical methods are used to obtain
numerical approximations of the normal form coefficients, from which the dynamics are
deduced.
The first model differs from those previously studied with bifurcation analysis since
it supports a steady solution which varies nonlinearly with latitude. The results indicate
that the forcing does not qualitatively change the behaviour. However, the form of the
bifurcating solution is affected.
The second model uses the Navier-Stokes equations in the Boussinesq approximation, in
cylindrical geometry. In addition to the double Hopf bifurcation analysis, a detailed axisymmetric
to non-axisymmetric transition curve is produced from the computed eigenvalues. A
quantitative comparison with experimental data finds that the computed transition curve,
critical wave numbers and drift rates of the bifurcating waves are reasonably accurate. This
indicates that the analysis, as well as the approximations which are made, are valid. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/10801 |
| Date | 05 1900 |
| Creators | Lewis, Gregory M. |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Format | 6865987 bytes, application/pdf |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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