Flow in the earth's core and the hydromagnetic dynamo

Zusammenfassung Den Hauptgegenstand des Artikels bildet die Auswahl von Geschwindigkeiten für ein kinematisches Modell eines hydromagnetischen Dynamos. Die Ergebnisse der theoretischen Analyse betreffend die M?glichkeit der Entwicklung des magnetischen Feldes werden durch die Angaben über die Oberfl?chenstr?mung an der Grenze zwischen dem Erdkern und dem Mantel erg?rzt, die man aus der beobachteten sekul?ren ?nderung des magnetischen Feldes der Erde erwcrben hat. Das dreidimensionale Geschwindigkeitsfeld wurde in der Weise gew?hlt, damit die notwendige Bedingung der Entwicklung, d.i. das Nichtverschwinden des Braginskischen Entwicklungskoeffizienten, erfüllt werde und damit der Charakter von Str?mung auf der Kernoberfl?che den aus der sekul?ren ?nderung gewonnenen Angaben entspreche. Eine m?glichen Form des dreidimensionalen Geschwindigkeitsfeldes wird im Modell des hydromagnetischen Dynamos angewendet, das durch ein System von Integralgleichungen dargestellt wird. Die vorausgesetzte numerische L?sung ist nicht durchgeführt.

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Address: Boční II, Praha 4-Spořilov.     

A dynamo model with double diffusive convection for Mercury's core

A recent dynamo model for Mercury assumes that the upper part of the planet's fluid core is thermally stably stratified because the temperature gradient at the core–mantle boundary is subadiabatic. Vigorous convection driven by a superadiabatic temperature gradient at the boundary of a growing solid inner core and by the associated release of light constituents takes place in a deep sub-layer and powers a dynamo. These models have been successful at explaining the observed weak global magnetic field at Mercury's surface. They have been based on the concept of codensity, which combines thermal and compositional sources of buoyancy into a single variable by assuming the same diffusivity for both components. Actual diffusivities in planetary cores differ by a large factor. To overcome the limitation of the codensity model, we solve two separate transport equations with different diffusivities in a double diffusive dynamo model for Mercury. When temperature and composition contribute comparable amounts to the buoyancy force, we find significant differences to the codensity model. In the double diffusive case convection penetrates the upper layer with a net stable density stratification in the form of finger convection. Compared to the codensity model, this enhances the poloidal magnetic field in the nominally stable layer and outside the core, where it becomes too strong compared to observation. Intense azimuthal flow in the stable layer generates a strong axisymmetric toroidal field. We find in double diffusive models a surface magnetic field of the observed strength when compositional buoyancy plays an inferior role for driving the dynamo, which is the case when the sulphur concentration in Mercury's core is only a fraction of a percent.     

Density perturbations in the earth's core due to hydromagnetic oscillations

Summary The equation for density perturbation induced by plane hydromagnetic waves (responsible for geomagnetic secular variation) has been derived based on -plane formulation. It is deduced that for the usual physical properties of the earth's core such a density perturbation would be small and would not be, in the main, responsible for the recently discovered correlation between gravitation and geomagnetic potentials.     

Magnetic torque and convection in the earth's core

Summary The problem of expressing analytically the magnetic torque, acting on the electrically conducting part of the Earth's mantle, is treated as a function of the system of convection on the surface of the core. The changes of velocities in the system of convection are estimated for decadic changes of the Earth's rotation and for the perturbation of the Earth's rotation in 1897. As regards the decadic changes of the Earth's rotation a change of velocity in the system of convection at the surface of the core of the order of 10^–4 m/s corresponds, and as regards the perturbation of the Earth's rotation in 1897 (10^–3 s/year) a change of velocity of 10^–3 m/s reduced to the whole surface of the core corresponds, and 10^–2 m/s corresponds for the region of the focus of the world geomagnetic anomaly (dimension of this region is 10⁶ m).     

Hydromagnetic dynamo model with spiral convection

Summary The present paper deals with a hydromagnetic dynamo model of the generation mechanism of the Earth's magnetic field. An attempt has been made at selecting a flow-velocity field in the Earth's core which would satisfy the condition 0 for regenerating the field according to [2], and which would yield a velocity field pattern on the core surface as given in the papers by Kahle et al. [9]. These conditions are satisfied by the velocityv=V ₁+U ₂ ^c –V ₂ ^c and, geometrically, this velocity field is represented in space by a spiral convective motion. On the core surface two downflows and two upflows with the corresponding rotating cells may then be found. Only the axisymmetric harmonic component regeneration of the magnetic field has been considered. Adequate regeneration equations have been obtained by means of Braginski's method of quantity estimates in order of magnitude.     

Numerical treatment of the kinematic dynamo model

Summary Since the initial equations are complicated, the treatment of the kinematic dynamo model requires the use of numerical methods. In applying them to the given problem difficulties are encountered, which are not easy to overcome. This paper deals with the analysis of the experience acquired in treating the model of a nearly symmetric dynamo. Three different methods were employed (stationary, oscillatory and general non-stationary), because a combination of several solutions will yield more comprehensive information about the model being studied. Although the results are based on the study of a single particular model, similar problems also occur in other excercises and, therefore, the conclusions have a more general validity.     

Non-steady kinematic dynamo model with spiral convection

Summary The stability of steady states, the evolution of the magnetic field and possible changes of the magnetic field under small changes of velocity are studied on a non-stationary solution of a kinematic dynamo model.     

Magnetic field at the core boundary in the nearly symmetric hydromagnetic dynamo z

Summary The behaviour of the poloidal and toroidal magnetic field at the core-mantle boundary is analysed in more detail, assuming that the conductive layer in the lowest mantle is thin. We can conclude that, in the case of the Z-model of the nearly symmetric hydromagnetic dynamo, the poloidal field may be considered potential everywhere in the mantle and that the azimuthal field depends on the geostrophic azimuthal velocity in the same manner as derived in[1] and[3].

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On the nearly axially-symmetrical model of the hydromagnetic dynamo of the earth

A short review of the present state of the nearly axially-symmetrical dynamo model is given. A simplified theory for hydromagnetic dynamos taking into account the forces acting in the Earth's core is considered. The role of weak core-mantle friction is discussed and a form of solution is suggested which is characterized by a large geostrophic velocity in the core and by a boundary layer of a new type. The consequences of such a model (called model Z) for the Earth's dynamo are discussed.     

Boundary conditions in the hydromagnetic dynamo problem

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Z-model of the nearly symmetric hydromagnetic dynamo with electromagnetic core-mantle coupling

Summary The present paper deals with theZ-model of the nearly symmetric hydromagnetic dynamo as a generation mechanism of the Earth's magnetic field. TheZ-model of Braginsky [2] was solved for viscous core-mantle coupling [3]. It is shown that a similarZ-model can also be constructed for electromagnetic core-mantle coupling, or for both effects combined. A new part of the azimuthal velocity appears in the equations, but the character of the boundary layer is not changed too much. No numerical solution is presented.

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mam auam ma aa Z-¶rt; nmu umuu¶rt;aum ¶rt;ua, ma n¶rt;mam amu au aum n. Z-¶rt; au [2] a a ¶rt; a au¶rt;mu ¶rt; ¶rt; u amu. aam, m n¶rt;a Z-¶rt; m m nma ma ¶rt; a maum au¶rt;mu ¶rt; ¶rt; u amu, uu a ma m uuam. au nm a am auma mu, aam nau m. ua u u nu¶rt;um.

     

Axially asymmetric velocities in the boundary layer of the nearly symmetric hydromagnetic dynamo

Abstract

An attempt has been made to include the axially asymmetric velocities into the calculation of Braginsky's Z-model of the nearly symmetric hydromagnetic dynamo. In this axisymmetric non-linear model dominated by Lorentz and Coriolis forces and maintained by a specified convection, the α-effect is prescribed. An example is shown of the axially asymmetric Archimedean buoyancy, which can imply an arbitrary alpha effect in the model with viscous core-mantle coupling. The formalisms of Tough and Roberts (1968) is also discussed and a modified α-effect in the Z-model is suggested.     

Numerical model of convection in sills

The numerical model of convection in magma sills is developed. The model is based on a full system of equations of fluid dynamics and includes heat transfer, buoyancy effects and diffusion of some minor component (marker). Solidification is treated as a phase transition. The results indicate that there are some qualitative differences between very thin sills with Rayleigh number Ra = 10⁵ and thin sills with Ra = 10⁶. For a basaltic magma the first case corresponds to the thickness of the sills of approximately 30 cm and the second case corresponds to the thickness of 60 cm. In the first case mixing is inefficient and conduction is the dominant form of heat transfer. In the second case mixing is efficient and convection is the dominant form of heat transfer. Some of the results can be scaled for the more viscous magmas in thicker sills.     

A multi-layered kinematic dynamo model: implications of a stratified upper layer in the Earth's core

It has been suggested that there exists a stably stratified electrically conducting layer at the top of the Earth's outer fluid core and that lateral temperature gradients in the lower mantle is capable of a driving thermal-wind-type flow near the core–mantle boundary. We investigate how such a flow in a stable layer could influence the geomagnetic field and the geodynamo using a very simple two-dimensional kinematic dynamo model in Cartesian geometry. The dynamo has four layers representing the inner core, convecting lower outer core, stable upper core, and insulating mantle. An α² dynamo operates in the convecting outer core and a horizontal shear flow is imposed in the stable layer. Exact dynamo solutions are obtained for a range of parameters, including different conductivities for the stable layer and inner core. This allows us to connect our solutions with known, simpler solutions of a single-layer α² dynamo, and thereby assess the effects of the extra layers. We confirm earlier results that a stable, static layer can enhance dynamo action. We find that shear flows produce dynamo wave solutions with a different spatial structure from the steady α² dynamos solutions. The stable layer controls the behavior of the dynamo system through the interface conditions, providing a new means whereby lateral variations on the boundary can influence the geomagnetic field.     

Forced magnetohydrodynamic oscillations of the earth's core

¶rt;m uu maua anu, ¶rt;mu aau ¶rt;a u amuu a aumu¶rt;¶rt;uau ¶rt;uu ¶rt; u. a auum anu m u ¶rt;am au u mu u u¶rt;ua aum n. a nuu ¶rt; ¶rt;u naa, m u anmam u¶rt; . m uu u u ¶rt;uunauu a anmau mu . uu a u¶rt;, m m¶rt; a mu u m aum mm aumaua maum n ¶rt; u.     

Magnetohydrodynamic disturbances in the earth's core,II

Summary In paper I, (Mohandis [2]²)), the author contributes to the discussion of the origin of the secular variation of the earth's magnetic field. A mathematical solution of magnetohydrodynamic disturbances and fluid motion due to the sudden introduction of an oscillating dipole in the earth's core has been obtained. Only the symmetrical case of the problem, where the axis of the dipole is placed perpendicular to the mantle and parallel to a poloidal field, has been discussed.In this paper, the source of disturbance is still considered to be the oscillating dipole, but the exciting field is taken as a toroidal field always parallel to the mantle. Two unsymmetric different cases of the problem are considered but the disturbed field is sonsidered only in the mantle. It is worth to note here that simpler results can be obtained by applying more conditions than those used in the present work. The new method will be illustrated in a forthcoming paper of this series, Magnetolydrodynamic disturbances in the earth's core, IV where another case of the problem will be discussed.     

Magnetohydrodynamic disturbances in the earth's core,IV

Summary In Paper III (Mohandis [1]²) we considered the sudden introduction of amagnetic dipole in the earth's core to act as a source of disturbance to the exitation field taken as a poloidal one. A symmetrical case was considered where the dipole axis is placed parallel to the original field and perpendicular to the earth's mantle. In the present work, we consider an unsymmetric case where the axis of themagnetic dipole is placed perpendicular to both the mantle and the exitation field which is taken as a toroidal one. A mathematical study is made for the resulting fluid motion in the core as well as for the generated hydromagnetic perturbations in both the mantle and the earth's fluid core. A more powerful method has been adopted than those used in previous cases.     

Magnetohydrodynamic disturbances in the earth's core,III

Summary It is in the hope of establishing a theorem contributing to the origin of the Secular Variation of the geomagnetic field that this third paper under the same heading, is written, —In this paper it is supposed that amagnetic dipole is suddenly introduced in the earth's core to act as a source of disturbance to the exciting field taken as a poloidal one. The dipole axis is placed parallel to the original field and perpendicular to the mantle. Mathematical solutions have been obtained for the resulting fluid motion in the core as well as for the generated magnetohydrodynamic perturbations in the earth's core and in the mantle. It can be seen from the mathematical results obtained that although the disturbances in the core are so complicated, yet they are much less complicated in the mantle and specially at the plane boundary separating core and mantle.     

A geophysical aspect of a disturbed fluid motion in the earth's core

Summary A magnetic dipole is supposed to be suddenly introduced in the earth's core (taken as an inviscid, incompressible fluid of finite electrical conductivity) to act as a source of disturbance. It is shown that although in the symmetric case of the problem the disturbed fluid is stagnant in any direction at the interface separating core from the solid insulating mantle; yet it should slip in any tangential direction at the interface, in each of the three unsymmetric cases considered here.     

A quantitative kinematic theory of convection currents in the earth's mantle

Summary Based on a system of structurally simple postulations the kinematics of mantle convection is derived. With regard to the strain-stress relations valid in the mantle and the energy source of the convection the theory is without any presumptions. In compliance with recent hydrodynamic investigations the flow is introduced rather as roll currents than as a hexagonal cell pattern. From the feasible types of current a theoretical topography is derived which is in quantitative agreement with the observed one. Also the distribution of the seismic discontinuities substantiates the validity of the expression. Finally, some suggestions are given for a hydrodynamic theory of mantle convection. This paper contains that part of the theory which is necessary for testing the calculations, while the relationship of the theory to other geological and geophysical problems are dealt with by the author [26]²).