Simple mechanism for reversals of earth's magnetic field

Phase transitions in anisotropic turbulence

Turbulence is a widely observed state of fluid flows, characterized by complex, nonlinear interactions between motions across a broad spectrum of length and time scales. While turbulence is ubiquitous, from teacups to planetary atmospheres, oceans, and stars, its manifestations can vary considerably between different physical systems. For instance, three-dimensional turbulent flows display a forward energy cascade from large to small scales, while in two-dimensional turbulence, energy cascades from small to large scales. In a given physical system, a transition between such disparate regimes of turbulence can occur when a control parameter reaches a critical value. The behavior of flows close to such transition points, which separate qualitatively distinct phases of turbulence, has been found to be unexpectedly rich. Here, we survey recent findings on such transitions in highly anisotropic turbulent fluid flows, including turbulence in thin layers and under the influence of rapid rotation. We also review recent work on transitions induced by turbulent fluctuations, such as random reversals and transitions between large-scale vortices and jets, among others. The relevance of these results and their ramifications for future investigations are discussed.

Predictive Framework for Flow Reversals and Excursions in Turbulence

We present a dynamical framework for intermittent reversals and excursions (R&Es) of large-scale circulations in turbulence. We show that R&Es can occur when turbulent trajectories in phase space shadow invariant manifolds of certain unstable periodic orbits (UPOs). Consequently, substantial flow reorganization and extreme fluctuations in flow metrics observed during R&Es can be reconstructed by splicing the unstable manifolds of such dynamically relevant UPOs. Using this geometrical framework, we predict imminent R&Es and preemptively avert these extreme events using closed-loop control.

Wake dynamics in buoyancy-driven flows: Steady-state-Hopf-mode interaction with O(2) symmetry revisited

We present a detailed mathematical study of a truncated normal form relevant to the bifurcations observed in wake flow past axisymmetric bodies, with and without thermal stratification. We employ abstract normal form analysis to identify possible bifurcations and the corresponding bifurcation diagrams in parameter space. The bifurcations and the bifurcation diagrams are interpreted in terms of symmetry considerations. Particular emphasis is placed on the presence of attracting robust heteroclinic cycles in certain parameter regimes. The normal form coefficients are computed for several examples of wake flows behind buoyant disks and spheres, and the resulting predictions compared with the results of direct numerical flow simulations. In general, satisfactory agreement is obtained.

Training a neural network to predict dynamics it has never seen

Neural networks have proven to be remarkably successful for a wide range of complicated tasks, from image recognition and object detection to speech recognition and machine translation. One of their successes lies in their ability to predict future dynamics given a suitable training data set. Previous studies have shown how echo state networks (ESNs), a type of recurrent neural networks, can successfully predict both short-term and long-term dynamics of even chaotic systems. This study shows that, remarkably, ESNs can successfully predict dynamical behavior that is qualitatively different from any behavior contained in their training set. Evidence is provided for a fluid dynamics problem where the flow can transition between laminar (ordered) and turbulent (seemingly disordered) regimes. Despite being trained on the turbulent regime only, ESNs are found to predict the existence of laminar behavior. Moreover, the statistics of turbulent-to-laminar and laminar-to-turbulent transitions are also predicted successfully. The utility of ESNs in acting as early-warning generators for transition is discussed. These results are expected to be widely applicable to data-driven modeling of temporal behavior in a range of physical, climate, biological, ecological, and finance models characterized by the presence of tipping points and sudden transitions between several competing states.

Possible Role of Non-Stationarity of Magnetohydrodynamic Turbulence in Understanding of Geomagnetic Excursions

The natural simplifying assumptions often put forward in the theoretical investigations of the magnetohydrodynamic turbulence are that the turbulent flow is statistically isotropic, homogeneous and stationary. Of course, the natural turbulence in the planetary interiors, such as the liquid core of the Earth is neither, which has important consequences for the dynamics of the planetary magnetic fields generated via the hydromagnetic dynamo mechanism operating in the interiors of the planets. Here we concentrate on the relaxation of the assumption of statistical stationarity of the turbulent flow and study the effect of turbulent wave fields in the Earth’s core, which induces non-stationarity, on the turbulent resistivity in the non-reflectionally symmetric flow and the geodynamo effect. It is shown that the electromotive force, including the so-called α-effect and the turbulent magnetic diffusivity η¯, induced by non-stationary turbulence, evolves slowly in time. However, the turbulent α¯ coefficient, responsible for the dynamo action and η¯ evolve differently in time, thus creating periods of enhanced and suppressed turbulent diffusion and dynamo action somewhat independently. In particular, periods of enhanced α¯ may coincide with periods of suppressed diffusion, leading to a stable and strong field period. On the other hand, it is shown that when enhanced diffusion occurs simultaneously with suppression of the α-effect, this leads to a sharp drop in the intensity of the large-scale field, corresponding to a geomagnetic excursion.

Simple stochastic model for geomagnetic excursions and reversals reproduces the temporal asymmetry of the axial dipole moment

We present a simple model for the axial dipole moment (ADM) of the geomagnetic field based on a stochastic differential equation for two coupled particles in a biquadratic potential, subjected to Gaussian random perturbations. This model generates aperiodic reversals and excursions separated by stable polarity periods. The model reproduces the temporal asymmetry of geomagnetic reversals, with slower decaying rates before the reversal and faster growing rates after it. This temporal asymmetry is possible because our model is out of equilibrium. The existence of a thermal imbalance between the two particles sets a preferential sense for the energy flux and renders the process irreversible.

A model for the geomagnetic field reversal rate and constraints on the heat flux variations at the core-mantle boundary

A striking feature of many natural magnetic fields generated by dynamo action is the occurrence of polarity reversals. Paleomagnetic measurements revealed that the Earth's magnetic field has been characterised by few hundred stochastic polarity switches during its history. The rate of reversals changes in time, maybe obeying some underlying regular pattern. While chaotic dynamical systems can describe the short-term behaviour of the switches of the Earth's magnetic polarity, modelling the long-term variations of the reversal rate is somewhat problematic, as they occur on timescales of tens to hundreds of millions of years, of the order of mantle convection timescales. By investigating data of geomagnetic reversal rates, we find the presence of cycles with variable frequency and show that the transition towards periods where reversals do not occur for tens of million years (superchrons) can be described by a second-order phase transition that we interpret to be driven by variations of the heat flux at the core-mantle boundary (CMB). The model allows us to extract from the reversal sequence quantitative information on the susceptibility of the reversal rate caused by changes in the CMB heat flux amplitude, thus providing direct information on the deep inner layers of the Earth.

Evidence for crisis-induced intermittency during geomagnetic superchron transitions

The geomagnetic field's dipole undergoes polarity reversals in irregular time intervals. Particularly long periods without reversals (of the order of 10^{7} yr), called superchrons, have occurred at least three times in the Phanerozoic (since 541 million years ago). We provide observational evidence for high non-Gaussianity in the vicinity of a transition to and from a geomagnetic superchron, consisting of a sharp increase in high-order moments (skewness and kurtosis) of the dipole's distribution. Such an increase in the moments is a universal feature of crisis-induced intermittency in low-dimensional dynamical systems undergoing global bifurcations. This implies a temporal variation of the underlying parameters of the physical system. Through a low-dimensional system that models the geomagnetic reversals, we show that the increase in the high-order moments during transitions to geomagnetic superchrons is caused by the progressive destruction of global periodic orbits exhibiting both polarities as the system approaches a merging bifurcation. We argue that the non-Gaussianity in this system is caused by the redistribution of the attractor around local cycles as global ones are destroyed.

Transition to chaos in wave memory dynamics in a harmonic well: Deterministic and noise-driven behavior

A walker is the association of a sub-millimetric bouncing drop moving along with a co-evolving Faraday wave. When confined in a harmonic potential, its stable trajectories are periodic and quantised both in extension and mean angular momentum. In this article, we present the rest of the story, specifically the chaotic paths. They are chaotic and show intermittent behaviors between an unstable quantised set of attractors. First, we present the two possible situations we find experimentally. Then, we emphasise theoretically two mechanisms that lead to unstable situations. It corresponds either to noise-driven chaos or low-dimensional deterministic chaos. Finally, we characterise experimentally each of these distinct situations. This article aims at presenting a comprehensive investigation of the unstable paths in order to complete the picture of walkers in a two dimensional harmonic potential.

Chaos as an intermittently forced linear system

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear.

The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.

Optimized setup for two-dimensional convection experiments in thin liquid films

We present a novel experimental setup to investigate two-dimensional thermal convection in a freestanding thin liquid film. Such films can be produced in a controlled way on the scale of 5-1000 nm. Our primary goal is to investigate convection patterns and the statistics of reversals in Rayleigh-Bénard convection with varying aspect ratio. Additionally, questions regarding the physics of liquid films under controlled conditions can be investigated, like surface forces, or stability under varying thermodynamical parameters. The film is suspended in a frame which can be adjusted in height and width to span an aspect ratio range of Γ = 0.16-10. The top and bottom frame elements can be set to specific temperature within T = 15 °C to 55 °C. A thickness to area ratio of approximately 10(8) enables only two-dimensional fluid motion in the time scales relevant for turbulent motion. The chemical composition of the film is well-defined and optimized for film stability and reproducibility and in combination with carefully controlled ambient parameters allows the comparison to existing experimental and numerical data.

Deciphering records of geomagnetic reversals

Polarity reversals of the geomagnetic field are a major feature of the Earth's dynamo. Questions remain regarding the dynamical processes that give rise to reversals and the properties of the geomagnetic field during a polarity transition. A large number of paleomagnetic reversal records have been acquired during the past 50 years in order to better constrain the structure and geometry of the transitional field. In addition, over the past two decades, numerical dynamo simulations have also provided insights into the reversal mechanism. Yet despite the large paleomagnetic database, controversial interpretations of records of the transitional field persist; they result from two characteristics inherent to all reversals, both of which are detrimental to an ambiguous analysis. On the one hand, the reversal process is rapid and requires adequate temporal resolution. On the other hand, weak field intensities during a reversal can affect the fidelity of magnetic recording in sedimentary records. This paper is aimed at reviewing critically the main reversal features derived from paleomagnetic records and at analyzing some of these features in light of numerical simulations. We discuss in detail the fidelity of the signal extracted from paleomagnetic records and pay special attention to their resolution with respect to the timing and mechanisms involved in the magnetization process. Records from marine sediments dominate the database. They give rise to transitional field models that often lead to overinterpret the data. Consequently, we attempt to separate robust results (and their subsequent interpretations) from those that do not stand on a strong observational footing. Finally, we discuss new avenues that should favor progress to better characterize and understand transitional field behavior.

Chaos driven by interfering memory

The transmission of information can couple two entities of very different nature, one of them serving as a memory for the other. Here we study the situation in which information is stored in a wave field and serves as a memory that pilots the dynamics of a particle. Such a system can be implemented by a bouncing drop generating surface waves sustained by a parametric forcing. The motion of the resulting "walker" when confined in a harmonic potential well is generally disordered. Here we show that these trajectories correspond to chaotic regimes characterized by intermittent transitions between a discrete set of states. At any given time, the system is in one of these states characterized by a double quantization of size and angular momentum. A low dimensional intermittency determines their respective probabilities. They thus form an eigenstate basis of decomposition for what would be observed as a superposition of states if all measurements were intrusive.

Decay rates of magnetic modes below the threshold of a turbulent dynamo

We measure the decay rates of magnetic field modes in a turbulent flow of liquid sodium below the dynamo threshold. We observe that turbulent fluctuations induce energy transfers between modes with different symmetries (dipolar and quadrupolar). Using symmetry properties, we show how to measure the decay rate of each mode without being restricted to the one with the smallest damping rate. We observe that the respective values of the decay rates of these modes depend on the shape of the propellers driving the flow. Dynamical regimes, including field reversals, are observed only when the modes are both nearly marginal. This is in line with a recently proposed model.

Evidence for forcing-dependent steady states in a turbulent swirling flow

We study the influence on steady turbulent states of the forcing in a von Karman flow, at constant impeller speed, or at constant torque. We find that the different forcing conditions change the nature of the stability of the steady states and reveal dynamical regimes that bear similarities to low-dimensional systems. We suggest that this forcing dependence may be applicable to other turbulent systems.

On the genesis of the Earth's magnetism

Few areas of geophysics are today progressing as rapidly as basic geomagnetism, which seeks to understand the origin of the Earth's magnetism. Data about the present geomagnetic field pours in from orbiting satellites, and supplements the ever growing body of information about the field in the remote past, derived from the magnetism of rocks. The first of the three parts of this review summarizes the available geomagnetic data and makes significant inferences about the large scale structure of the geomagnetic field at the surface of the Earth's electrically conducting fluid core, within which the field originates. In it, we recognize the first major obstacle to progress: because of the Earth's mantle, only the broad, slowly varying features of the magnetic field within the core can be directly observed. The second (and main) part of the review commences with the geodynamo hypothesis: the geomagnetic field is induced by core flow as a self-excited dynamo. Its electrodynamics define 'kinematic dynamo theory'. Key processes involving the motion of magnetic field lines, their diffusion through the conducting fluid, and their reconnection are described in detail. Four kinematic models are presented that are basic to a later section on successful dynamo experiments. The fluid dynamics of the core is considered next, the fluid being driven into motion by buoyancy created by the cooling of the Earth from its primordial state. The resulting flow is strongly affected by the rotation of the Earth and by the Lorentz force, which alters fluid motion by the interaction of the electric current and magnetic field. A section on 'magnetohydrodynamic (MHD) dynamo theory' is devoted to this rotating magnetoconvection. Theoretical treatment of the MHD responsible for geomagnetism culminates with numerical solutions of its governing equations. These simulations help overcome the first major obstacle to progress, but quickly meet the second: the dynamics of Earth's core are too complex, and operate across time and length scales too broad to be captured by any single laboratory experiment, or resolved on present-day computers. The geophysical relevance of the experiments and simulations is therefore called into question. Speculation about what may happen when computational power is eventually able to resolve core dynamics is given considerable attention. The final part of the review is a postscript to the earlier sections. It reflects on the problems that geodynamo theory will have to solve in the future, particularly those that core turbulence presents.

On the dynamic nature of azimuthal thermoacoustic modes in annular gas turbine combustion chambers

This paper deals with the dynamics of standing and rotating azimuthal thermoacoustic modes in annular combustion chambers. Simultaneous acoustic measurements have been made at multiple circumferential positions in an annular gas turbine combustion chamber. A detailed statistical analysis of the spatial Fourier amplitudes extracted from these data reveals that the acoustic modes are continuously switching between standing, clockwise and counter-clockwise travelling waves. A theoretical framework from which the modal dynamics can be explained is proposed and supported by real gas turbine data. The stochastic differential equations that govern these systems have been derived and used as a basis for system identification of the measured engine data. The model describes the probabilities of the two azimuthal wave components as a function of the random source intensity, the asymmetry in the system and the strength of the thermoacoustic interaction. The solution of the simplified system is in good agreement with experimental observations on a gas turbine combustion chamber.

Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counterrotating Taylor-Couette flow

The nonlinear dynamics of Taylor-Couette flow in a small-aspect-ratio wide-gap annulus in the counterrotating regime is investigated by solving the full three-dimensional Navier-Stokes equations. The system is invariant under arbitrary rotations about the axis, reflection about the annulus midplane, and time translations. A systematic investigation is presented both in terms of the flow physics elucidated from the numerical simulations and from a dynamical system perspective provided by equivariant normal form theory. The dynamics are primarily associated with the behavior of the jet of angular momentum that emerges from the inner cylinder boundary layer at about the midplane. The sequence of bifurcations as the differential rotation is increased consists of an axisymmetric Hopf bifurcation breaking the reflection symmetry of the basic state leading to an axisymmetric limit cycle with a half-period-flip spatiotemporal symmetry. This undergoes a Hopf bifurcation breaking axisymmetry, leading to quasiperiodic solutions evolving on a 2-torus that is setwise symmetric. These undergo a further Hopf bifurcation, introducing a third incommensurate frequency leading to a 3-torus that is also setwise symmetric. On the 3-torus, as the differential rotation is further increased, a saddle-node-invariant-circle bifurcation takes place, destroying the 3-torus and leaving a pair of symmetrically related 2-tori states on which all symmetries of the system have been broken.

Bistability between equatorial and axial dipoles during magnetic field reversals

Numerical simulations of the geodynamo in the presence of heterogeneous heating are presented. We study the dynamics and the structure of the magnetic field when the equatorial symmetry of the flow is broken. If the symmetry breaking is sufficiently strong, the m=0 axial dipolar field is replaced by a hemispherical magnetic field, dominated by an oscillating m=1 magnetic field. Moreover, for moderate symmetry breaking, a bistability between the axial and the equatorial dipole is observed. In this bistable regime, the axial magnetic field exhibits chaotic switches of its polarity, involving the equatorial dipole during the transition period. This new scenario for magnetic field reversals is discussed within the framework of Earth's dynamo.

Experimental observation of spatially localized dynamo magnetic fields

We report the first experimental observation of a spatially localized dynamo magnetic field, a common feature of astrophysical dynamos and convective dynamo simulations. When the two propellers of the von Kármán sodium experiment are driven at frequencies that differ by 15%, the mean magnetic field's energy measured close to the slower disk is nearly 10 times larger than the one close to the faster one. This strong localization of the magnetic field when a symmetry of the forcing is broken is in good agreement with a prediction based on the interaction between a dipolar and a quadrupolar magnetic mode.

Dynamics and symmetries of flow reversals in turbulent convection

Based on direct numerical simulations and symmetry arguments, we show that the large-scale Fourier modes are useful tools to describe the flow structures and dynamics of flow reversals in Rayleigh-Bénard convection (RBC). We observe that during the reversals, the amplitude of one of the large-scale modes vanishes, while another mode rises sharply, very similar to the "cessation-led" reversals observed earlier in experiments and numerical simulations. We find anomalous fluctuations in the Nusselt number during the reversals. Using the structures of the RBC equations in the Fourier space, we deduce two symmetry transformations that leave the equations invariant. These symmetry transformations help us in identifying the reversing and nonreversing Fourier modes.

Dipole-quadrupole dynamics during magnetic field reversals

The shape and the dynamics of reversals of the magnetic field in a turbulent dynamo experiment are investigated. We report the evolution of the dipolar and the quadrupolar parts of the magnetic field in the VKS experiment, and show that the experimental results are in good agreement with the predictions of a recent model of reversals: when the dipole reverses, part of the magnetic energy is transferred to the quadrupole, reversals begin with a slow decay of the dipole and are followed by a fast recovery, together with an overshoot of the dipole. Random reversals are observed at the borderline between stationary and oscillatory dynamos.

Mechanisms for magnetic field reversals

We present a review of the different models that have been proposed to explain reversals of the magnetic field generated by a turbulent flow of an electrically conducting fluid (fluid dynamos). We then describe a simple mechanism that explains several features observed in palaeomagnetic records of the Earth's magnetic field, in numerical simulations and in a recent dynamo experiment. A similar model can also be used to understand reversals of large-scale flows that often develop on a turbulent background.

Influence of counter-rotating von Kármán flow on cylindrical Rayleigh-Bénard convection

The axisymmetric flow in an aspect-ratio-one cylinder whose upper and lower bounding disks are maintained at different temperatures and rotate at equal and opposite velocities is investigated. In this combined Rayleigh-Bénard/von Kármán problem, the imposed temperature gradient is measured by the Rayleigh number Ra and the angular velocity by the Reynolds number Re. Although fluid motion is present as soon as Re not equal 0 , a symmetry-breaking transition analogous to the onset of convection takes place at a finite Rayleigh number higher than that for Re=0 . For Re<95 , the transition is a pitchfork bifurcation to a pair of steady states, while for Re>95 , it is a Hopf bifurcation to a limit cycle. The steady states and limit cycle are connected via a pair of saddle-node infinite-period bifurcations except very near the Takens-Bogdanov codimension-two point, where the scenario includes global bifurcations. Detailed phase portraits and bifurcation diagrams are presented, as well as the evolution of the leading part of the spectrum, over the parameter ranges 0<or=Re<or=120 and 0<or=Ra<or=30 000 .

From reversing to hemispherical dynamos

We show that hemispherical dynamos can result from weak equatorial symmetry breaking of the flow in the interior of planets and stars. Using a model of spherical dynamo, we observe that the interaction between a dipolar and a quadrupolar mode can localize the magnetic field in only one hemisphere when the equatorial symmetry is broken. This process is shown to be related to the one that is responsible for reversals of the magnetic field. These seemingly very different behaviors are thus understood in a unified framework.