Self-consistent dynamo models driven by hydromagnetic instabilities

The dynamics of the Earth's core are dominated by a balance between Lorentz and Coriolis forces. Previous studies of possible (magnetostrophic) hydromagnetic instabilities in this regime have been confined to geophysically unrealistic flows and fields. In recent papers we have treated rather general fields and flows in a spherical geometry and in a computationally simple plane-layer model. These studies have highlighted the importance of differential rotation in determining the spatial structure of the instability. Here we have proceeded to use these results to construct a self-consistent dynamo model of the geomagnetic field. An iterative procedure is employed in which an α-effect is calculated from the form of the instability and is then used in a mean field dynamo model. The mean zonal field calculated there is then input back into the hydromagnetic stability problem and a new α-effect calculated. The whole procedure is repeated until the input and output zonal fields are the same to some tolerance.     

A nonlinear dynamo driven by rapidly rotating convection

In Kim et al. (Kim, E., Hughes, D.W. and Soward, A.M., “An investigation into high conductivity dynamo action driven by rotating convection”, Geophys. Astrophys. Fluid Dynam. 91, 303–332 ().) we investigated kinematic dynamo action driven by rapidly rotating convection in a cylindrical annulus. Here we extend this work to consider self-consistent nonlinear dynamo action in which the back-reaction of the Lorentz force on the flow is taken into account. In particular, we investigate, as a function of magnetic Prandtl number, the evolution of an initially weak magnetic field in two different types of convective flow – one chaotic and the other integrable. On saturation, the latter shows a systematic dependence on the magnetic Prandtl number whereas the former appears not to. In addition, we show how, in keeping with the findings of Cattaneo et al. (Cattaneo, F., Hughes, D.W. and Kim, E., “Suppression of chaos in a simplified nonlinear dynamo model”, Phys. Rev. Lett. 76, 2057–2060 ().), saturation of the growth of the magnetic field is brought about, for the originally chaotic flow, by a strong suppression of chaos.     

On a dynamo driven topographically by longitudinal libration

Variation in the angular velocity Ω of a planetary body is called libration or longitudinal libration when the Ω-axis is fixed in direction. This motion of the body's solid mantle drives motions in its fluid core, either by viscous coupling across the core-mantle interface S, or topographically when S is asymmetric with respect to the Ω-axis, the only case considered in this article. A significant topographically-driven flow is identified having uniform vorticity within S and no component parallel to the Ω-axis. Its dynamic stability depends on the amplitude, Ω ₁, of the sinusoidally varying part of Ω and on the ratio, b/a, of the lengths of the principal axes of S, assumed spheroidal. In (Ω ₁/Ω ₀, b/a) parameter space where Ω ₀ is the average Ω, islands are shown to exist where the constant vorticity states are dynamically unstable. These are surrounded by a sea in which they are stable. When the fluid is slightly viscous, a state in the stable sea retains its uniform vorticity structure except in a viscous boundary layer on S in which the flow acquires a component parallel to the Ω-axis. For (Ω ₁/Ω ₀, b/a) on an island where the uniform vorticity state is unstable, an “alternative flow” exists, which is three-dimensional and is examined here. Assuming that the core is electrically conducting, kinematic dynamos are sought. Uniform vorticity flow appears to be non-regenerative but, when it is stable and viscosity acts to create a sufficiently strong boundary layer flow, dynamo action may occur. It is shown that the alternative flow that exists on an instability island in (Ω ₁/Ω ₀,?b/a) space can be vigorously regenerative.     

Geoneutrinos and the energy budget of the Earth

The total energy loss of the Earth is well constrained by heat flux measurements on land, the plate cooling model for the oceans, and the buoyancy flux of hotspots. It amounts to 46 ± 2 TW. The main sources that balance the total energy loss are the radioactivity of the Earth's crust and mantle, the secular cooling of the Earth's mantle, and the energy loss from the core. Only the crustal radioactivity is well constrained. The uncertainty on each of the other components is larger than the uncertainty of the total heat loss. The mantle energy budget cannot be balanced by adding the best estimates of mantle radioactivity, secular cooling of the mantle, and heat flux from the core. Neutrino observatories in deep underground mines can detect antineutrinos emitted by the radioactivity of U and Th. Provided that the crustal contribution to the geoneutrino flux can be very precisely calculated, it will be possible to put robust constraints on mantle radioactivity and its contribution to the Earth's energy budget. Equally strong constraints could be obtained from a deep ocean observatory without the need of crustal correction. In the future, it may become possible to obtain directional information on the geoneutrino flux and to resolve radial variations in concentration of heat producing elements in the mantle.     

Fast dynamo action in the Ponomarenko dynamo

Abstract

An analysis of small-scale magnetic fields shows that the Ponomarenko dynamo is a fast dynamo; the maximum growth rate remains of order unity in the limit of large magnetic Reynolds number. Magnetic fields are regenerated by a “stretch-diffuse” mechanism. General smooth axisymmetric velocity fields are also analysed; these give slow dynamo action by the same mechanism.     

Hydromagnetic energy balance equations for the solar atmosphere

Summary Following the pattern established in meteorology, a system of energy balance equations for the solar atmosphere is presented. Since both hydromagnetic and thermodynamic processes are considered, the system includes kinetic, potential, thermal and magnetic forms of energy. Ionization energy is indirectly included in the treatment. The spatial distribution of the energy forms and the processes transforming them are separated into zonal means and departures from such means in order to depict turbulent eddy effects. Where available data concerning solar processes are used in order to appraise certain of the terms which arise.     

A new model for the fracture energy budget of phreatomagmatic explosions

The fragmentation of magma and of the hosting country rocks is a major process in explosive eruptions. It is important to quantify the mechanical energy needed for fragmentation in order to assess the physical processes of this volcanic phenomenon. This paper presents a method to calculate the fragmentation energy of country rock using granulometry data of a typical phreatomagmatic Eifel maar volcano explosion. The total fracture area of country rock fragments in one tephra layer was quantified and related to the critical fragmentation energy of these country rocks. The rock parameters critical shear stress and critical fragmentation energy were determined experimentally, whereas the pre-volcanic crack inventory was measured in the field. The paper concludes with the calculation of the energy balance (i.e. partitioning of thermal energy into kinetical energy and mechanical energy of the fragmentation) of one Eifel maar volcanic explosion.     

Necessary conditions for the magnetohydrodynamic dynamo

Abstract

A magnetohydrodynamic, dynamo driven by convection in a rotating spherical shell is supposed to have averages that are independent of time. Two cases are considered, one driven by a fixed temperature difference R and the other by a given internal heating rate Q. It is found that when q, the ratio of thermal conductivity to magnetic diffusivity, is small, R must be of order q ^?4/3 and Q of order q ^?2 for dynamo action to be possible; q is small in the Earth's core, so it is hoped that the criteria will prove useful in practical as well as theoretical studies of dynamic dynamos. The criteria can be further strengthened when the ohmic dissipation of the field is significant in the energy balance. The development includes the derivation of two necessary conditions for dynamo action, both based on the viscous dissipation rate of the velocity field that drives the dynamo.     

Understanding the solar dynamo

Paul Bushby and Joanne Mason take a look at the workings of the Sun's dynamo, from the development of physical theory through current ideas and methods, to the possibilities for future understanding of this enigmatic magnetic engine.     

On pressure-work,viscous dissipation and the energy balance relation for geothermal reservoirs

This article examines the conditions under which the pressure-work and viscous dissipation terms should be retained in the energy balance relation for single (liquid water or vapor) and two-phase (liquid water and vapor) fluid flow through porous media. It is shown that if one wishes to retain the pressure-work term, then one must also keep the viscous dissipation term in the energy balance. Consideration of steady non-isothermal radial flow demonstrates that both pressure-work and viscous dissipation are liable to have negligibly small effects in single phase liquid water and in two-phase liquid-vapor systems. This conclusion is, however, not generally valid for pure vapor systems; in this case, pressure-work and viscous dissipation can produce significant variations in the computed reservoir response.     

Budgeting for the budget

R esearch N otes
The prospect of advanced astrometric satellites raises the tantalizing possibility that we will be able to make astrometric measurements at such a precision as to place constraints on dark energy parameters. That is the premise outlined by Fiona Ding and Rupert Croft of Carnegie Mellon University, Pittsburgh, USA, in Monthly Notices of the Royal Astronomical Society ,     

Variations in snowmelt energy and energy balance characteristics with larch forest density on Mt Iwate,Japan: observations and energy balance analyses

The variation in snowmelt energy and energy components were evaluated with respect to forest density. Surface snowmelt rates, surface evaporation from snow cover and meteorological elements were measured in the open and under sparse (411 trees/ha) and dense (1433 trees/ha) larch canopies. The surface snowmelt rate decreased as the forest density increased. Based on the observations and energy balance analyses, we concluded the following. (1) Albedo decreased while the bulk coefficient for latent heat increased with forest density. (2) The duration of snowmelt increased with forest density because the energy for nocturnal cooling of the snow cover decreased. (3) When comparing the open and forested sites, the changes in snowmelt energy with forest density were caused by sensible heat flux. However, the contribution of net radiation was highest in the forested sites. Therefore, the effects of forest cover on the snowmelt energy were different when comparing both the open and forested sites and the sparse and densely forested sites. (4) The ratio of net radiation to snowmelt energy increased with forest density; although both snowmelt energy and net radiation decreased with increased forest density, the snowmelt energy decreased more rapidly. Sensible heat also decreased as forest density increased. Both albedo and downward long‐wave radiation influenced net radiation. Copyright © 1999 John Wiley & Sons, Ltd.     

Analytical study of the energy rate balance equation for the magnetospheric storm-ring current

We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, E_y, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of E_y, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter compatible with the observed Dst peak values. To facilitate the analytical integration we assume a constant value for through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed Dst the most appropriate values of are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, _m, are found to be very sensitive to the adopted efficiency factor, , decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of and the resulting values for the decay time parameter, _r, do not vary greatly as varies. As a consequence, the relative values of between the main and the recovery phase, _m/_r, can be greater or smaller than one as varies from 0.1 to 1.     

Unbiased station grid for study of the kinetic energy balance of the atmosphere

Summary For the past several years observational studies of the northern hemisphere kinetic energy balance and related subjects have been performed by the Planetary Circulation Project of the Massachusetts Institute of Technology. Numerous integrals required were recently evaluated directly from a five-year period of observations using a network of nearly 800 stations. The stations, however, are concentrated primarily over temperate latitude continents, and the data from maritime and tropical areas were comparatively sparse. The question then arises whether the results are representative. The problem discussed in this paper is to select a subset of more uniformly spaced stations and to recompute the zonal kinetic energy balance. This is accomplished and the results are presented herewith.The research reported in this paper was sponsored by the U.S. National Science Foundation under Grant No. GA-1310X.     

Linking urban water balance and energy balance models to analyse urban design options

Using a water balance modelling framework, this paper analyses the effects of urban design on the water balance, with a focus on evapotranspiration and storm water. First, two quite different urban water balance models are compared: Aquacycle which has been calibrated for a suburban catchment in Canberra, Australia, and the single‐source urban evapotranspiration‐interception scheme (SUES), an energy‐based approach with a biophysically advanced representation of interception and evapotranspiration. A fair agreement between the two modelled estimates of evapotranspiration was significantly improved by allowing the vegetation cover (leaf area index, LAI) to vary seasonally, demonstrating the potential of SUES to quantify the links between water sensitive urban design and microclimates and the advantage of comparing the two modelling approaches. The comparison also revealed where improvements to SUES are needed, chiefly through improved estimates of vegetation cover dynamics as input to SUES, and more rigorous parameterization of the surface resistance equations using local‐scale suburban flux measurements. Second, Aquacycle is used to identify the impact of an array of water sensitive urban design features on the water balance terms. This analysis confirms the potential to passively control urban microclimate by suburban design features that maximize evapotranspiration, such as vegetated roofs. The subsequent effects on daily maximum air temperatures are estimated using an atmospheric boundary layer budget. Potential energy savings of about 2% in summer cooling are estimated from this analysis. This is a clear ‘return on investment’ of using water to maintain urban greenspace, whether as parks distributed throughout an urban area or individual gardens or vegetated roofs. Copyright © 2007 John Wiley & Sons, Ltd.     

Energy requirement for a working dynamo

There has for many years been interest in finding necessary conditions for dynamo action. These are usually expressed in terms of bounds on integrated properties of the flow. The bounds can clearly be improved when the flow structure can be taken into account. Recent research presents techniques for finding optimised dynamos (that is with the lowest dynamo threshold) subject to constraints, (e.g. with fixed mean square vorticity). It is natural to ask if such an optimum solution can exist when the mean square velocity is fixed. The aim of this note is to show that this is not the case and in fact that a steady or periodic dynamo can exist in a bounded conductor with an arbitrarily small value of the kinetic energy.     

Turbulent kinetic energy budget in a gravel-bed channel flow

The present experimental investigation focuses on the characteristics of near bed turbulence in a fully rough, uniform open-channel flow over a gravel-type bed. Due to bed topography small scale heterogeneity, the flow is not uniform locally in the near bed region and a double averaging methodology is applied over a length scale much larger than the gravel size. The double-averaged Turbulent Kinetic Energy (TKE) budget derived in the context of the present flow over a gravel bed differs from the TKE budget written for flow over a vegetation canopy. The non-constant shape of the roughness function measured in our gravel bed leads to an additional bed-induced production term which is null for vertical roughness elements, such as simplified vegetation elements. The experimental estimation of the terms of the TKE budget reveals that the maximum turbulent activity takes place away from the reference plane, near the roughness crests. However, within the interface sublayer the work of the bed induced velocity fluctuations against the Reynolds stress is of the same magnitude as the main turbulence production term. Consequently, the characteristics of the TKE budget have similarities with uniform flows over canopies and strongly differ from uniform flows over smooth and transitionally rough flows over sedimentlike beds.     

A method for the study of the zonal kinetic energy balance in the atmosphere

Summary The atmospheric balance of the kinetic energy of the zonally averaged zonal motion is investigated from five years of daily data at 800 stations for the northern hemisphere. The basic equation for such energy is used, together with the simplifying assumption that the frictional destruction is due in the main to stresses acting across horizontal surfaces, being thus presumably related to the vertical shear of the mean zonal wind, although no further details are needed in the analysis. The five-year averages of various terms as well as their seasonal means appear to give reasonable results.The research reported in this paper was sponsored by the U.S. National Science Foundation under Grant No. GA-1310X.     

On formation compaction and energy balance law for liquid-saturated porous media reservoirs

In most treatments of liquid flow through porous media, it is assumed that the pore fluid and formation motions are decoupled except that porosity φ is taken to be a function of fluid pressure. This article examines the conditions under which such a decoupling is valid. More specifically, it is shown that the usual statements of the energy balance law ignore the work performed by the overburden as the formation compaction progresses. This work term is shown to be negligibly small in practical cases.