Wolf-Gerrit Früh

Wave interactions and the transition to chaos of baroclinic waves in a thermally driven rotating annulus

A series of laboratory experiments is presented investigating regular and chaotic baroclinic waves in a high–Prandtl number fluid contained in a rotating vessel and subjected to a horizontal temperature gradient. The study focuses on nonlinear aspects of mixed–mode states at moderate values of the forcing parameters within the regular wave regime. Frequency entrainment and phase locking of resonant triads and sidebands were found to be widespread. Cases were analysed in phase space reconstructions through a singular value decomposition of multi–variate time series. Four forms of mixed–mode states were found, each in well–defined regions of parameter space: (1) a nonlinear interference vacillation associated with strong phase locking through higher harmonics; (2) a modulated amplitude vacillation showing strong phase coherence in triads involving the long wave; (3) an intermittent bursting of secondary modes; (4) an attractor switching flow, where the dominant wave number switched at irregular intervals between two possible wave numbers. Many of the mixed–mode states are suggested to arise from homoclinic bifurcations, whereas no secondary Hopf bifurcations were found. One of the postulated homoclinic bifurcations was consistent with a bifurcation through intermittency. The bifurcation sequences, however, were strongly affected by phase locking between different wave number components and frequency locking between drift and modulation frequencies. When all free frequencies were locked, the flow reduced to a limit cycle which subsequently became unstable through an incomplete period–doubling cascade. The only observed case of torus–doubling was also associated with strong phase locking. Most of the observed regimes were consistent with low–dimensional dynamics involving a limited number of domain–filling modes, which can be represented in phase space reconstructions and characterized by invariants such as attractor dimensions and the Lyapunov exponents. Some flows associated with a weak structural vacillation, however, were not consistent with low–dimensional dynamics. It appeared rather that they were the result of spatially localized instabilities consistent with high–dimensional dynamics, which can be parametrized as stochastic dynamics.

Experiments on a barotropic rotating shear layer. Part 1. Instability and steady vortices

The barotropic shear layer in a rotating fluid is studied in a laboratory experiment. Through the rotation of circular sections in the base and lid of a circular tank relative to a background rotation of the entire system, a vertical layer of strong horizontal shear develops, the Stewartson layer. Above a critical shear, the shear layer breaks up through barotropic instability, which is an inertial instability. The flow then develops a string of vortices along the shear zone. It will be shown that the transition from an axisymmetric flow to regular vortices occurs through a Hopf bifurcation. Subsequent transitions to more complex flows, such as modulated vortices, chaos and highly irregular flow, will be presented briefly, while the main points of this paper are the primary instability, steady vortices and their nonlinear dynamics. Among the issues discussed is the sensitivity of the flow to the direction of the differential shear. The experimental data will be used to test the ability of boundary layer theory and quasi-geostrophic theory to predict the onset of instability and the range of unstable wavenumbers.

Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

Three-dimensional Direct Numerical Simulation (DNS) on the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating is presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies by Maubert and Randriamampianina , the fluid used is air rather than a liquid as used in all other previous work. This study investigated a bifurcation sequence from the axisymmetric flow to a number of complex flows. The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady, fully-developed baroclinic wave to chaotic flow followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic MAV\@. In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow. Further transitions of a steady wave with a higher wave number pointed to the possibility that a barotropic instability of the side wall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.

Wave interactions and baroclinic chaos: a paradigm for long timescale variability in planetary atmospheres

Abstract Baroclinic instability is the principal mode of non-axisymmetric flow in the large-scale atmospheric circulation at mid-latitudes, and is responsible for organising the structure and behaviour of major weather systems. This instability can also be fruitfully studied in the laboratory under controlled conditions. In this paper, we review recent work carried out by the authors and collaborators on various routes to chaotic behaviour in rotating, stratified flows. Results include the discovery of new multi-mode regimes in which small ensembles of baroclinic waves interact in a nonlinear mode competition with the thermally-driven axisymmetric component of the flow, generating chaotic oscillatory variability on very long timescales. We discuss various attempts to capture this type of behaviour in simple models, and consider the significance of the phenomenon as a paradigm for understanding the nature of long timescale variability in the climates of the Earth and Mars.

Direct numerical simulation of transitions towards structural vacillation in an air-filled, rotating, baroclinic annulus

The route to chaos of baroclinic waves in a rotating, stratified fluid subjected to lateral heating can occur via several possible routes, involving either low-dimensional, quasiperiodic states or via a series of secondary small-scale instabilities. In a recent paper, we have discussed direct numerical simulations (DNS) of the low-dimensional route to chaos in a baroclinic annulus filled with air as the working fluid and compared results to those obtained in the laboratory for high Prandtl number liquids. In the present paper, we consider further DNS in the air-filled annulus at higher rotation rates. A transition in the flow structure is observed, where the centrifugal acceleration exceeds gravity and the dominant physical process changes from baroclinic instability to convection due to radial buoyancy. The transition of this convection to chaotic behavior is fundamentally different from that observed in the transition to the chaotic flow observed at lower rotation rates. Rather than via a sequence of low-dimensional, quasiperiodic states, the large-scale convection developed small-scale instabilities, which has been previously suggested as the origin of structural vacillation on the transition to geostrophic turbulence.

Lattice Boltzmann model for the simulation of interfacial phenomena in magnetic fluids

A very simple lattice Boltzmann model was developed to describe the competition between surface tension and dipolar interaction in magnetic fluids. The model was used to simulate the deformation of magnetic fluid drops in a nonmagnetic fluid, as well as the deformation of a gas bubble in a magnetic fluid under the action of an external magnetic field.

The force on an object passing through a magnetic fluid seal

Forces on solid objects passed through a magnetic liquid plug in a tube are measured. A simple one-dimensional model is developed based on hydrostatic and magnetic pressures. The results demonstrate its potential to be used to separate two fluids while allowing solids to pass from one fluid to the other.

A comparison of empirical orthogonal decomposition methods in baroclinic flows

Abstract The relative merits of three contrasting empirical orthogonal decomposition methods in common use (namely, Proper Orthogonal Decomposition, Biorthogonal Decomposition and Multivariate Singular Systems Analysis) are considered as applied to baroclinic flow data. The regimes analysed are a steady, drifting wave, a modulated amplitude vacillating wave flow and a neighbouring multi-mode state which exhibits intermittency. The results are used to make a qualitative comparison of the methods in terms of convergence properties, variance capture and eigenfunction structure. The feasibility of using the resulting empirical orthogonal functions to transform partial differential equations to ordinary differential equations by Galerkin projection is mentioned.

Management and environmental risk study of the physicochemical parameters of ballast water.

Shipping is a vital industry for the global economy. Stability of ships, provided by ballast water, is a crucial factor for cargo loading and unloading processes. Ballast water treatment has practical significance in terms of environmental issues, ecosystem, and human health, because ships discharge this water into the environment before loading their cargos. This study reviews the common methods for ballast water management - exchange, heating, filtration, ultrasonic treatment, ultraviolet irradiation, chemicals, and gas supersaturation - to select the best one. This study compares water temperature, salinity, dissolved oxygen, polycyclic aromatic hydrocarbons (PAHs), and heavy metals (Co, Cr, Ni, Pb) for ballast tanks of selected ships with the recipient port environment in the Persian Gulf as a case study. The exchange of ballast water in the ocean and/or its treatment on board to prevent inadvertent effects on the environments physicochemical conditions is related to vessel characteristics, legislation, and the environmental condition. Ecological risk study showed that the salt content in ballast water is close to that of seawater, but the values of Cr (2.1mg/l) and Ni (0.029mg/l) in ballast water are higher than those in seawater (1 and 0.004mg/l, respectively).

Game-theoretic modeling of curtailment rules and network investments with distributed generation

Renewable energy has achieved high penetration rates in many areas, leading to curtailment, especially if existing network infrastructure is insufficient and energy generated cannot be exported. In this context, Distribution Network Operators (DNOs) face a significant knowledge gap about how to implement curtailment rules that achieve desired operational objectives, but at the same time minimise disruption and economic losses for renewable generators. In this work, we study the properties of several curtailment rules widely used in UK renewable energy projects, and their effect on the viability of renewable generation investment. Moreover, we propose a new curtailment rule which guarantees fair allocation of curtailment amongst all generators with minimal disruption. Another key knowledge gap faced by DNOs is how to incentivise private network upgrades, especially in settings where several generators can use the same line against the payment of a transmission fee. In this work, we provide a solution to this problem by using tools from algorithmic game theory. Specifically, this setting can be modelled as a Stackelberg game between the private transmission line investor and local renewable generators, who are required to pay a transmission fee to access the line. We provide a method for computing the equilibrium of this game, using a model that captures the stochastic nature of renewable energy generation and demand. Finally, we use the practical setting of a grid reinforcement project from the UK and a large dataset of wind speed measurements and demand to validate our model. We show that charging a transmission fee as a proportion of the feed-in tariff price between 15% and 75% would allow both investors to implement their projects and achieve desirable distribution of the profit. Overall, our results show how using game-theoretic tools can help network operators to bridge the knowledge gap about setting the optimal curtailment rule and determining transmission charges for private network infrastructure.