Descent Rate Models of the Synchronization of the Quasi-Biennial Oscillation by the Annual Cycle in Tropical Upwelling

The response of the quasi-biennial oscillation (QBO) to an imposed mean upwelling with a periodic modulation is studied, by modeling the dynamics of the zero wind line at the equator using a class of equations known as descent rate models. These are simple mathematical models that capture the essence of QBO synchronization by focusing on the dynamics of the height of the zero wind line. A heuristic descent rate model for the zero wind line is described and is shown to capture many of the synchronization features seen in previous studies of the QBO. It is then demonstrated using a simple transformation that the standard Holton–Lindzen model of the QBO can itself be put into the form of a descent rate model if a quadratic velocity profile is assumed below the zero wind line. The resulting nonautonomous ordinary differential equation captures much of the synchronization behavior observed in the full Holton–Lindzen partial differential equation. The new class of models provides a novel framework within which to understand synchronization of the QBO, and we demonstrate a close relationship between these models and the circle map well known in the mathematics literature. Finally, we analyze reanalysis datasets to validate some of the predictions of our descent rate models and find statistically significant evidence for synchronization of the QBO that is consistent with model behavior.

BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART II

This paper is Part II of an earlier paper dealing with the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, and an external periodical excitation of period τ = 2π/ω (amplitude E). In the absence of this excitation, this equation of Duffing type does not give rise to self-oscillations. Part I was essentially devoted to analyze the harmonics behavior of period τ solutions, more precisely the behavior of rank-p harmonics according to the points of the parameter plane (ω,E). The present Part II deals with period kτ solutions related to a cascade of closed fold bifurcation curves related to fractional harmonics p/k, k = 3, p = 3,4,…. With respect to the organization of bifurcation curves associated with rank-p harmonics of the basic period τ, this study shows that the situation is a lot more complex for the sequence of bifurcation curves related to rank-p/3 harmonics.

BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART I

This paper is devoted to the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, submitted to an external periodical excitation of period τ = 2π/ω , having an amplitude E. In absence of this excitation the equation of Duffing type does not give rise to self oscillations. It may be a model of a series R-L-C electrical circuit with validity conditions in the parameter space. The purpose is essentially an analysis of the harmonics behavior of the period τ solutions according to points of the parameter plane (ω, E). Let r be the place occupied by a rank-m harmonic from an ordering based on line amplitudes of a frequency spectrum in descending order. Domains of the (ω, E) plane, for which the amplitude of the rank-m harmonic has the place r in the ordering mentioned above, are defined from properties of their boundaries. When r = 2 they are regions of predominance for the rank-m harmonic. When r = 1 they are regions of full predominance for the harmonic m, m = 2,3,4,…, which contain a set of points leading to a resonance for this harmonic. These regions fulfil the following important property: each of them is directly related to a fold bifurcation structure, called rank-mlip (two folds curves joining at two cusps), associated with a well-defined rank-m harmonic. The set of such structures, with m = 2,3,4,…, constitutes an isoordinal lips cascade. As an opening to Part II (to be published) period kτ solutions related to fractional harmonics behavior are also considered for k = 2,3.