Jan Abshagen

A Nonlinear Theory for El Niño Bursting

Abstract A new mechanism is proposed that explains two key features of the observed El Nino–Southern Oscillation (ENSO) phenomenon—its irregularity and decadal amplitude changes. Using a low-order ENSO model, the authors show that the nonlinearities in the tropical heat budget can lead to bursting behavior characterized by decadal occurrences of strong El Nino events. La Nina events are not affected, a feature that is also seen in ENSO observations. One key result of this analysis is that decadal variability in the Tropics can be generated without invoking extratropical processes or stochastic forcing. The El Nino bursting behavior simulated by the low-order ENSO model can be understood in terms of the concept of homoclinic and heteroclinic connections. It is shown that this new model for ENSO amplitude modulations and irregularity, although difficult to prove, might explain some features of ENSO dynamics seen in more complex climate models and the observations.

Mode competition of rotating waves in reflection-symmetric Taylor-Couette flow

We report on the results of a combined experimental and numerical study on mode interactions of rotating waves in Taylor–Couette flow. Our work shows that rotating waves which originate at a Hopf bifurcation from the steady axisymmetric Taylor vortex flow interact with this axisymmetric flow in a codimension-two fold-Hopf bifurcation. This interaction gives rise to an (unstable) low-frequency modulated wave via a subcritical Neimark–Sacker bifurcation from the rotating wave. At higher Reynolds numbers, a complicated mode interation between stable modulated waves originating at a different Neimark–Sacker bifurcation and a pair of symmetrically related rotating waves that originate at a cyclic pitchfork bifurcation is found to organize complex

Bursting dynamics due to a homoclinic cascade in Taylor-Couette flow

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Transient dynamics at the onset of Tailor vortices

-symmetry breaking of rotating waves via global bifurcations. In addition to symmetry breaking of rotating waves via a (local) cyclic pitchfork bifurcation, we found symmetry breaking of modulated waves via a saddle-node-infinite-period (SNIP) global bifurcation. Tracing these local and global bifurcation curves in Reynolds number/aspect ratio parameter space toward their apparant merging point, unexpected complexity arises in the bifurcation structure involving non-symmetric two-tori undergoing saddle-loop homoclinic bifurcations. The close agreement between the numerics and the experiment is indicative of the robustness of the observed complex dynamics.

An Organizing Center for Thermohaline Excitability

Transitions between regular oscillations and bursting oscillations that involve a bifurcational process which culminates in the creation of a relative periodic orbit of infinite period and infinite length are investigated both experimentally and numerically in a short-aspect-ratio Taylor–Couette flow. This bifurcational process is novel in that it is the accumulation point of a period-adding cascade at which the midheight reflection symmetry is broken. It is very rich and complex, involving verylow-frequency states arising via homoclinic and heteroclinic dynamics, providing the required patching between states with very different dynamics in neighbouring regions of parameter space. The use of nonlinear dynamical systems theory together with symmetry considerations has been crucial in interpreting the laboratory experimental data as well as the results from the direct numerical simulations. The phenomenon corresponds to dynamics well beyond the first few bifurcations from the basic state and so is beyond the reach of traditional hydrodynamic stability analysis, but it is not fully developed turbulence where a statistical or asymptotic approach could be employed. It is a transitional phenomenon, where the phase dynamics of the large-scale structures (jets of angular momentum emanating from the boundary layer on the rotating inner cylinder) becomes complicated. Yet the complicated phase dynamics remains accessible to an analysis of its space–time characteristics and a comprehensive mechanical characterization emerges. The excellent agreement between the experiments and the numerical simulations demonstrates the robustness of this complex bifurcation phenomenon in a physically realized system with its inherent imperfections and noise. Movies are available with the online version of the paper.

The effect of physical boundaries on oscillatory bifurcation in counterrotating Taylor–Couette flow

The effect of boundary conditions on the ‘critical dynamics’ at the onset of Taylor vortices is investigated in a combined numerical and experimental study. Numerical calculations of Navier–Stokes equations with ‘stress-free’ boundary conditions show that the Landau amplitude equation provides a good model of the transient dynamics. However, this rapidly breaks down when the ‘no-slip’ condition is approached. Apparent ‘critical’ behaviour observed in experiments is shown to have a surprising dependence on the length of the system.

Decay of turbulent bursting in enclosed flows

The bifurcation behavior of a conceptual heat‐salt oscillator model is analyzed by means of numerical continuation methods. A global (homoclinic) bifurcation acts as an organizing center for the dynamics of the simplified convective model. It originates from a codimension-2 bifurcation in an extended parameter space. Comparison with earlier work by Cessi shows that the intriguing stochastic thermohaline excitability can be understood from the bifurcation structure of the model. It is argued that global bifurcations may play a crucial role in determining long-term variability of the thermohaline circulation.

Transient turbulent bursting in enclosed flows

The results of an experimental study on the bifurcation structure of oscillatory modes in counterrotating Taylor–Couette flow with stationary end plates are presented. It is shown that the cylinder length L acts as an important geometric control parameter of the system. As a result of a supercritical Hopf bifurcation it is found that for an aspect ratio Γ=L/d>16 (d gap width) only spiral vortices appear in basic laminar flow. For Γ<10.5 spiral vortices are almost entirely replaced by two types of standing waves called SW0 and SWπ as supercritical oscillatory flow. Experimental evidence is presented that the mode exchange between standing waves SW0 and SWπ is governed by underlying Ekman induced vortices which appear as a result of stationary end plates in the flow. In this regime spiral vortices appear only from a sub- or supercritical symmetry breaking bifurcation of the standing waves. Within an “intermediate regime” between 10.5⩽Γ⩽16 spiral vortices are found to be the predominant primary oscillatory flow but small stability intervals of standing waves are also observed. Surprisingly, the experimentally determined critical Reynolds number is found to deviate not more than 2% from the numerical values for all values of aspect ratio even though they are calculated under the assumption of infinite axial length. Moreover, the critical oscillation frequency is also in agreement with the numerical values and is independent from Γ.

Symmetry Breaking Via Global Bifurcations of Modulated Rotating Waves in Hydrodynamics

In order to shed light on the transition to turbulence in shear flows, many investigations have analyzed the decay of localized turbulent structures. At low Reynolds numbers, exponential distributions of survival times have been observed in plane Couette, pipe and Taylor-Couette flows. We present a new flow state in the counter-rotating Taylor-Couette system that becomes unstable to transient turbulent bursting, which also shows an exponential distribution of lifetimes. In contrast to previous works, the turbulent state is here excited without disturbing the flow externally and appears in the centrifugally unstable regime, thus competing with several coherent states in phase space. In our combined experimental and numerical study we analyze the spatiotemporal properties and the lifetime behavior of this flow.

Taylor–Couette flow with independently rotating end plates

The origin and the transition to turbulence in wall-bounded shear flows is one of the outstanding problems of classical physics. In order to shed light on this transition, recent investigations have analyzed the decay of localized turbulent structures. At low Reynolds numbers, an exponential distribution of survival times has been observed in pipe and plane Couette flows [1, 2, 3, 4]. In phase space this is related to the decay from a chaotic saddle. However, pipe flow is an open flow, posing many experimental and numerical challenges for the study of bifurcation events. In a closed system we have found a flow state that shows transient turbulent behavior also at low Reynolds numbers. It appears as turbulent bursting in the Taylor-Couette system without an external forcing for counter-rotating cylinders above the centrifugal instability. Therefore it competes with coherent states of the system. Here we present spatiotemporal properties and lifetime behavior of this flow.