Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g., the α-effect, are highly anisotropic in both examples. An α²-mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.     

The geodynamo as a bistable oscillator

Abstract

Our intent is to provide a simple and quantitative understanding of the variability of the axial dipole component of the geomagnetic field on both short and long time scales. To this end we study the statistical properties of a prototype nonlinear mean field model. An azimuthal average is employed, so that (1) we address only the axisymmetric component of the field, and (2) the dynamo parameters have a random component that fluctuates on the (fast) eddy turnover time scale. Numerical solutions with a rapidly fluctuating α reproduce several features of the geomagnetic field: (1) a variable, dominantly dipolar field with additional fine structure due to excited overtones, and sudden reversals during which the field becomes almost quadrupolar, (2) aborted reversals and excursions, (3) intervals between reversals having a Poisson distribution. These properties are robust, and appear regardless of the type of nonlinearity and the model parameters. A technique is presented for analysing the statistical properties of dynamo models of this type. The Fokker-Planck equation for the amplitude a of the fundamental dipole mode shows that a behaves as the position of a heavily damped particle in a bistable potential ∝(1 ? a ²)², subject to random forcing. The dipole amplitude oscillates near the bottom of one well and makes occasional jumps to the other. These reversals are induced solely by the overtones. Theoretical expressions are derived for the statistical distribution of the dipole amplitude, the variance of the dipole amplitude between reversals, and the mean reversal rate. The model explains why the reversal rate increases with increasing secular variation, as observed. Moreover, the present reversal rate of the geodynamo, once per (2?3) × 10⁵ year, is shown to imply a secular variation of the axial dipole moment of ～ 15% (about the current value). The theoretical dipole amplitude distribution agrees well with the Sint-800 data.     

Future of geodynamo theory

Abstract

This is an attempt to predict the next fifteen years of geodynamo theory, and to assess the success potential of current directions of research. A new phenomenon, the subcriticality of a model-Z geodynamo, is described.     

Towards a realistic theory of the geodynamo

Abstract

The physics of the geodynamo is discussed. The main processes relevant for the buoyancy driven geodynamo are isolated. The successive stages of development of geodynamo theory are briefly described. The mechanism of local turbulence in the Earth's core is explained, and an estimate is presented of the turbulent transport of density inhomogeneities in the Earth's core. The significance of this turbulent transport to the geodynamo mechanism is stressed. The general scheme of the complete geodynamo theory of the future is outlined.     

The effects of different parameter regimes in geodynamo simulations

Over the past 10 years, geodynamo simulations have grown rapidly in sophistication. However, it is still necessary to make certain approximations in order to maintain numerical stability. In addition, models are forced to make assumptions about poorly known parameters for the Earth's core. Different magnetic Prandtl numbers have been used and different assumptions about the presence of radiogenic heating have been made. This study examines some of the consequences of different approximations and assumptions using the Glatzmaier–Roberts geodynamo model. Here, we show that the choice of magnetic Prandtl number has a greater influence on the character of the magnetic field produced than the addition of a plausible amount of radiogenic heating. In particular, we find that prescribing a magnetic Prandtl number of unity with Ekman number limited by current computing resources, results in magnetic fields with significantly smaller intensities and variabilities compared with the much more Earth-like results obtained from simulations with large magnetic Prandtl numbers. A magnetic Prandtl number of unity, with both the viscous and magnetic diffusivities set to the Earth's magnetic diffusivity, requires a rotation rate much smaller than that of the Earth for currently reachable Ekman numbers. This results in a reduced dominance of the Coriolis forces relative to the buoyancy forces, and therefore, a reduction in the magnetic field intensity and the variability compared to the large Prandtl number cases.     

The effect of anisotropic heat transport in the earth's core on the geodynamo

Intermediate dynamos are axisymmetric, spherical models that evade Cowling's theorem by invoking an α-effect to create the meridional magnetic field from the zonal. Usually the energy source maintaining the motions is a specified thermal wind, but here the dynamo is driven by the buoyancy created by a uniform distribution of heat sources. It has been argued by Braginsky and Meytlis (this journal, vol. 55, 1990) that, in a rapidly rotating, strongly magnetic system such as the Earth's core, heat is transported principally by a small-scale turbulence that is highly anisotropic. They conclude that the diffusion of heat parallel to the rotation axis is then significantly greater than it is in directions away from that axis. A preliminary study of the consequences of this idea is reported here. Solutions are derived numerically using both isotropic and non-isotropic thermal diffusivity tensors, and the results are compared. It is shown that even a small degree of anisotropy can materially alter the character of the dynamo.     

The space-weather enterprise: past,present, and future

Space-weather impacts society in diverse ways. Societies’ responses have been correspondingly diverse. Taken together these responses constitute a space weather “enterprise”, which has developed over time and continues to develop. Technological systems that space-weather affects have grown from isolated telegraph systems in the 1840s to ocean and continent-spanning cable communications systems, from a generator electrifying a few city blocks in the 1880s to continent-spanning networks of high-tension lines, from wireless telegraphy in the 1890s to globe-spanning communication by radio and satellites. To have a name for the global totality of technological systems that are vulnerable to space weather, I suggest calling it the cyberelectrosphere. When the cyberelectrosphere was young, scientists who study space weather, engineers who design systems that space weather affects, and operators of such systems — the personnel behind the space-weather enterprise — were relatively isolated. The space-weather enterprise was correspondingly incoherent. Now that the cyberelectrosphere has become pervasive and indispensable to most segments of society, the space weather enterprise has become systematic and coherent. At present it has achieved considerable momentum, but it has barely begun to realize the level of effectiveness to which it can aspire, as evidenced by achievements of a corresponding but more mature enterprise in meteorology, a field which provides useful lessons. The space-weather enterprise will enter a new phase after it matures roughly to where the tropospheric weather enterprise is now. Then it will become indispensable for humankind's further global networking through technology and for humankind's further utilization of and expansion into space.     

Toroidal flux oscillation as possible cause of geomagnetic excursions and reversals

It is proposed that convection driven dynamos operating in planetary cores could be oscillatory even when the oscillations are not directly noticeable from the outside. Examples of dynamo simulations are pointed out that exhibit oscillations in the structure of the azimuthally averaged toroidal magnetic flux while the mean poloidal field shows only variations in its amplitude. In the case of the geomagnetic field, global excursions may be associated with these oscillations. Long period dynamo simulations indicate that the oscillations may cause reversals once in a while. No special attempt has been made to use most realistic parameter values. Nevertheless some similarities between the simulations and the paleomagnetic record can be pointed out.     

Evolutionary multiobjective optimization in water resources: The past,present, and future

This study contributes a rigorous diagnostic assessment of state-of-the-art multiobjective evolutionary algorithms (MOEAs) and highlights key advances that the water resources field can exploit to better discover the critical tradeoffs constraining our systems. This study provides the most comprehensive diagnostic assessment of MOEAs for water resources to date, exploiting more than 100,000 MOEA runs and trillions of design evaluations. The diagnostic assessment measures the effectiveness, efficiency, reliability, and controllability of ten benchmark MOEAs for a representative suite of water resources applications addressing rainfall–runoff calibration, long-term groundwater monitoring (LTM), and risk-based water supply portfolio planning. The suite of problems encompasses a range of challenging problem properties including (1) many-objective formulations with four or more objectives, (2) multi-modality (or false optima), (3) nonlinearity, (4) discreteness, (5) severe constraints, (6) stochastic objectives, and (7) non-separability (also called epistasis). The applications are representative of the dominant problem classes that have shaped the history of MOEAs in water resources and that will be dominant foci in the future. Recommendations are given for the new algorithms that should serve as the benchmarks for innovations in the water resources literature. The future of MOEAs in water resources needs to emphasize self-adaptive search, new technologies for visualizing tradeoffs, and the next generation of computing technologies.     

Secular variation in numerical geodynamo models with lateral variations of boundary heat flow

We study magnetic field variations in numerical models of the geodynamo, with convection driven by nonuniform heat flow imposed at the outer boundary. We concentrate on cases with a boundary heat flow pattern derived from seismic anomalies in the lower mantle. At a Rayleigh number of about 100 times critical with respect to the onset of convection, the magnetic field is dominated by the axial dipole component and has a similar spectral distribution as Earth’s historical magnetic field on the core-mantle boundary (CMB). The time scales of variation of the low-order Gauss coefficients in the model agree within a factor of two with observed values. We have determined the averaging time interval needed to delineate deviations from the axial dipole field caused by the boundary heterogeneity. An average over 2000 years (the archeomagnetic time scale) is barely sufficient to reveal the long-term nondipole field. The model shows reduced scatter in virtual geomagnetic pole positions (VGPs) in the central Pacific, consistent with the weak secular variation observed in the historical field. Longitudinal drift of magnetic field structures is episodic and differs between regions. Westward magnetic drift is most pronounced beneath the Atlantic in our model. Although frozen flux advection by the large-scale flow is generally insufficient to explain the magnetic drift rates, there are some exceptions. In particular, equatorial flux spot pairs produced by expulsion of toroidal magnetic field are rapidly advected westward in localized equatorial jets which we interpret as thermal winds.     

Mantle regulation of core cooling: A geodynamo without core radioactivity?

The case for radioactivity in the core based on the power requirements of the geodynamo is re-evaluated. Previous calculations of mantle regulation of core thermal evolution have used an inappropriate formula. New calculations with a more appropriate formula yield lower core heat loss in the past, thus mitigating the implication of unreasonably high past core and mantle temperatures. Multiple thermal evolutions leading to present heat flows are also demonstrated, depending on the efficiency of mantle removal of core heat, some with moderately high past core heat loss and some with low and steady core heat loss. The latter would permit a low- or moderate-power dynamo without core radioactivity. Key uncertainties are the efficiency of core cooling by the mantle, the thermal conductivity of the core and the energy or entropy flow required to maintain the dynamo. The present rate of heat loss from the core is argued to be still rather uncertain, and a commonly used estimate of the thermal conductivity of the core is shown plausibly to be too high and in any case to be uncertain by perhaps a factor of 2. The geochemical difficulties associated with postulating radioactive heat sources in the core are stressed.     

Evaluation of past,present and future tools for radar-based flash-flood prediction in the USA

Abstract

The societal impacts of flash floods are more significant than any other weather-related hazard. They are often manifested in the form of damage to infrastructure, flooding of roadways and bridges, creating deadly hazards to motorists and inundation of crops and pasture. Some of these hazards can be anticipated and thus mitigated given effective warning systems. This study describes the tools proposed over recent decades in the USA to predict flash flooding and evaluates them using a common observational data set. Design recommendations for flash-flood forecasting systems are provided, taking into account today's availability of high-resolution rainfall data at scales commensurate with flash flooding, their archives, spatial data sets to describe physiographic properties, and ever-increasing computational resources.

Editor D. Koutsoyiannis; Guest editor R.J. Moore

Citation Gourley, J.J., Flamig, Z.L., Hong, Y., and Howard, K.W., 2014. Evaluation of past, present and future tools for radar-based flash-flood prediction in the USA. Hydrological Sciences Journal, 59 (7), 1377–1389. http://dx.doi.org/10.1080/02626667.2014.919391     

Broken ergodicity,magnetic helicity,and the MHD dynamo

We consider an unforced, incompressible, turbulent magnetofluid constrained by concentric inner and outer spherical surfaces. We define a model system in which normal components of the velocity, magnetic field, vorticity, and electric current are zero on the boundaries. This choice allows us to find a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity and current. The model dynamical system represents magnetohydrodynamic (MHD) turbulence in a spherical domain and is analyzed by the methods similar to those applied to homogeneous MHD turbulence. We find a statistical theory of ideal (i.e. no dissipation) MHD turbulence analogous to that found in the homogeneous case, including the prediction of coherent structure in the form of a large-scale quasistationary magnetic field. This MHD dynamo depends on broken ergodicity, an effect that is enhanced when total magnetic helicity is increased relative to total energy. When dissipation is added and large scales are only weakly damped, quasiequilibrium may occur for long periods of time, so that the ideal theory is still pertinent on a global scale. Over longer periods of time, the selective decay of energy over magnetic helicity further enhances the effects of broken ergodicity. Thus, broken ergodicity is an essential mechanism and relative magnetic helicity is a critical parameter in this model MHD dynamo theory.     

Eutrophication in the Polish coastal zone: the past, present status and future scenarios

In the Baltic Sea eutrophication processes have accelerated in the past 50 years of the 20th century and presently there exists a major ecological problem for this sea. The Polish coastal zone of the southern Baltic Sea is the recipient of riverine inputs from two major sources, namely the Odra and Vistula, as well as a number of smaller rivers along the central coast. Hence, the entire coastal zone remains under severe anthropogenic pressure. The variability of nutrient concentrations, especially the winter nutrient pool in the euphotic zone, summer level of total nitrogen and total phosphorus, together with such eutrophication indicators as water oversaturation with oxygen and the summer oxygen minimum, were analysed in the data time series 1959-2001. The temporal trends were investigated using linear regression and the non-parametric Whirsch test. The future characteristics of the Baltic Sea are discussed taking into account the development of driving forces.     

Evolution of aligned states within nonlinear dynamos

ABSTRACT

The Archontis dynamo is a rare example of an MHD dynamo within which forcing drives a dynamo where the flow and magnetic fields are almost perfectly aligned and the energies are approximately equal. In this paper, I expand upon our knowledge of the dynamo by showing that the intermediate steady states of the kinetic and magnetic energies observed by Cameron and Galloway are not a necessary feature of aligned dynamos. Furthermore, I show that the steady state into which the flow and magnetic fields eventually evolve is remarkably robust to the addition of time dependence and asymmetry to the forcing.     

Small Scale Kinematic Dynamos: Beyond the α-Effect

In geodynamo simulations which simulate the generation of an axial dipolar magnetic field, the generation mechanism appears to be adequately described as an α²-dynamo with an anisotropic α-effect. The anisotropy in the α-effect favors an equatorial dipole field, however, which calls into question the interpretation in terms of an α²-dynamo. It is shown in this article with kinematic dynamo calculations and exemplary velocity fields with an anisotropic α-effect that both types of dipolar magnetic field can be generated. Two examples of working dynamos in a sphere with flows with zero α-effect are also provided.     

The DRESDYN project: liquid metal experiments on dynamo action and magnetorotational instability

ABSTRACT

Magnetic fields of planets, stars and galaxies are generated by self-excitation in moving electrically conducting fluids. Once produced, magnetic fields can play an active role in cosmic structure formation by destabilising rotational flows that would be otherwise hydrodynamically stable. For a long time, both hydromagnetic dynamo action as well as magnetically triggered flow instabilities had been the subject of purely theoretical research. Meanwhile, however, the dynamo effect has been observed in large-scale liquid sodium experiments in Riga, Karlsruhe and Cadarache. In this paper, we summarise the results of liquid metal experiments devoted to the dynamo effect and various magnetic instabilities such as the helical and the azimuthal magnetorotational instability and the Tayler instability. We discuss in detail our plans for a precession-driven dynamo experiment and a large-scale Tayler–Couette experiment using liquid sodium, and on the prospects to observe magnetically triggered instabilities of flows with positive shear.     

Detecting thermal boundary control in surface flows from numerical dynamos

The geomagnetic field and secular variation exhibit asymmetrical spatial features which are possibly originating from an heterogeneous thermal control of the Earth's lower mantle on the core. The identification of this control in magnetic data is subject to several difficulties, some of which can be alleviated by the use of core surface flow models. Using numerical dynamos driven by heterogeneous boundary heat flux, we confirm that within the parameter space accessible to simulations, time average surface flows obey a simple thermal wind equilibrium between the Coriolis and buoyancy forces, the Lorentz, inertial and viscous forces playing only a secondary role, even for Elsasser numbers significantly larger than 1. Furthermore, we average the models over the duration of three vortex turnovers, and correlate them with a longer time average which fully reveals the signature of boundary heterogeneity. This allows us to quantify the possibility of observing mantle control in core surface flows averaged over a short time period. A scaling analysis is performed in order to apply the results to the Earth's core. We find that three vortex turnovers could represent between 100 and 360 years of Earth time, and that the heat flux heterogeneity at the core-mantle boundary could be large enough to yield an observable signature of thermal mantle control in a time average core surface flow within reach of the available geomagnetic data.