Zonal jets

This dissertation is a theoretical study of the formation and evolution of zonal jets and of their effect on passive tracer dispersion.; I present a two-dimensional barotropic model of zonal jet formation. The model is forced at a small scale by a sinusoidal shear flow, the Kolmogorov flow. The effect of differential rotation is implemented via the β-plane approximation. I show that the β-effect significantly modifies the linear stability of the Kolmogorov flow. The Reynolds' number, an adimensional parameter relating the strength of the forcing to the viscous term, controls the instability. With no rotation the critical Reynolds' number is $2$. With differential rotation the critical Reynolds' number is 0 for most orientations of the Kolmogorov flow with respect to the meridional direction. Stability curves and regions of instability in the wavenumber space are calculated using analytic and numerical methods.; For a purely meridional Kolmogorov flow I derive an amplitude equation which describes the non-linear evolution of disturbances of the Kolmogorov flow. These disturbances organize into zonal jets that are similar to those observed in geophysical flows. Eastward jets are faster and narrower than westward jets. Their time evolution is guaranteed to reach an asymptotic state by the existence of a Lyapunov functional. With no bottom drag, the jets evolve through a series of merger events, decreasing their number until only one eastward and one westward jet remain. Bottom drag arrests this one-dimensional inverse cascade when the width of the jets reaches a length scale which depends on the strength of the drag coefficient.; In the last chapter I present an analytic and numerical study of a one-dimensional stochastic model, the generalized telegraph model, which mimics the effect of jets and vortices on the dispersion of passive tracers. A characterization in terms of the long time behaviour of the moments of particle displacement is introduced. The model can present normal or anomalous diffusion, and strong or weak self-similar diffusion, depending on the parameters used. A numerical exploration of the “kicked Harper map” shows a similar behaviour for a range of parameter values.