Ataru Sakuraba

A numerical dynamo benchmark

We present the results of a benchmark study for a convection-driven magnetohydrodynamic dynamo problem in a rotating spherical shell. The solutions are stationary aside from azimuthal drift. One case of non-magnetic convection and two dynamos that differ in the assumptions concerning the inner core are studied. Six groups contributed numerical solutions which show good agreement. This provides an accurate reference standard with high confidence.

Effect of the inner core on the numerical solution of the magnetohydrodynamic dynamo

Abstract We report two simulation results of the magnetohydrodynamic dynamo applied to rapidly rotating spherical systems using fully nonlinear equations under Boussinesq approximation. Calculations were carried out under the same parameter conditions but for a spherical shell and a sphere. We assume that a uniform internal heat source distributed in the whole sphere drives the convection and dynamo and that the physical properties of the inner core are identical to those of the fluid outer core except for its rigidity. This treatment enables us to compare two cases under the same condition, except the existence of the inner core. Magnetic field is effectively generated by strong velocity shear and helicity of the fluid near the top (and bottom) boundaries. A stable axial dipole field develops in the case of the spherical shell because of the steady field generation at both the outer and inner boundaries, while the magnetic field in the sphere fluctuates with time from lack of the bottom boundary before it reaches the dipole dominant state at last. This result suggests that the Earths magnetic field may be stabilized as the inner core grows, even though the total energy input is the same. This study provides a first step to interpret the paleointensity data from the Archaean when there was a transition due to the growth of the inner core.

Dynamo simulation and palaeosecular variation models

In this paper we examine the simulation results of a fully nonlinear, three–dimensional dynamo and obtain inferences useful in the study of present and past geomagnetic field. This approach has importance because of the limitation in the available data of the real magnetic field: the present field is known with a high accuracy, but the time covered is only a small fraction of the time constant inherent to the geodynamo, while palaeomagnetic data provide data for a long time–span, but they are of poor quality and are distributed quite irregularly both in space and time. Thus, we compare what we see from the external field of the dynamo model with the features established or conjectured for the present or palaeomagnetic fields. We show that some of the statistical properties of the magnetic field generated by dynamo models compare well with those of the real magnetic field: dominance of the axial dipole, similar power in each degree of the harmonic at the core surface, nearly normal distribution of Gauss coefficients, and the lack of correlation among their variations. Differences were observed in other features, such as drift in the azimuthal direction and concentration of magnetic flux in small patches at the core surface. They can be attributed to either the shortness of the observational period, or to the difference in the resolution of the models, which suppresses small–scale features far below the level of the observation.

Linear Magnetoconvection in Rotating Fluid Spheres Permeated by a Uniform Axial Magnetic Field

A linear analysis of thermally driven magnetoconvection is carried out with emphasis on its application to convection in the Earths core. We consider a rotating and self-gravitating fluid sphere (or spherical shell) permeated by a uniform magnetic field parallel to the spin axis. In rapidly rotating cases, we find that five different convective modes appear as the uniform field is increased; namely, geostrophic, polar convective, magneto-geostrophic, fast magnetostrophic and slow magnetostrophic modes. The polar convective (P) and magneto-geostrophic (E) modes seem to be of geophysical interest. The P mode is characterized by such an axisymmetric meridional circulation that the fluid penetrates the equatorial plane, suggesting that generation of quadrapole from dipole fields could be explained by a linear process. The E mode is characterized by a few axially aligned columnar rolls which are almost two-dimensional due to a modified Proudman-Taylor theorem.

Effect of a uniform magnetic field on nonlinear magnetocenvection in a rotating fluid spherical shell

Abstract Dynamic interaction between magnetic field and fluid motion is studied through a numerical experiment of nonlinear three-dimensional magnetoconvection in a rapidly rotating spherical fluid shell to which a uniform magnetic field parallel to its spin axis is applied. The fluid shell is heated by internal heat sources to maintain thermal convection. The mean value of the magnetic Reynolds number in the fluid shell is 22.4 and 10 pairs of axially aligned vortex rolls are stably developed. We found that confinement of magnetic flux into anti-cyclonic vortex rolls was crucial on an abrupt change of the mode of magnetoconvection which occurred at Δ = 1 ∼ 2, where A is the Elsasser number. After the mode change, the fluid shell can store a large amount of magnetic flux in itself by changing its convection style, and the magnetostrophic balance among the Coriolis, Lorentz and pressure forces is established. Furthermore, the toroidal/poloidal ratio of the induced magnetic energy becomes less than unity, and the magnetized anti-cyclones are enlarged due to the effect of the magnetic force. Using these key ideas, we investigated the causes of the mode change of magnetoconvection. Considering relatively large magnetic Reynolds number and a rapid rotation rate of this model, we believe that these basic ideas used to interpret the present numerical experiment can be applied to the dynamics in the Earths and other planetary cores.

On Thermal Driving of the Geodynamo

It is widely believed that the main geomagnetic field is created by the dynamo action of motions in the Earth’s fluid core that are driven by thermal and compositional buoyancy. Early numerical simulations of the geodynamo that succeeded in generating strong, Earth-like dipole magnetic fields had to assume, for computational reasons, an unrealistically high viscosity for the core fluid. Some recent high-resolution models have used more realistic, smaller viscosities, but have unexpectedly produced only non-dipolar or dipolar but comparatively weak magnetic fields, which are less Earth-like. We recently advanced a possible explanation for this paradoxical behavior: we argued that these models had used the geophysically unrealistic outer boundary condition of uniform temperature on the core-mantle interface. In support of this opinion, we integrated two otherwise identical models, in one of which we applied the uniform temperature condition and in the other the more realistic condition of horizontally uniform heat flux. In the latter model, we obtained large-scale convective flows and a comparatively strong dipole-type magnetic field; for the former, we found solutions resembling those obtained by other models that had assumed uniform temperature on the core-mantle boundary. Further explanations for the very different character of the solutions are given here.

Free oscillations of a fluid sphere in an infinite elastic medium and long-period volcanic earthquakes

A source model of long-period volcanic earthquakes is presented. We consider that a fluid-filled spherical cavity surrounded by an infinite elastic medium is excited into resonance like the Earth’s free oscillations. The eigenequation of this system is derived in a general manner, making use of the spherical harmonic and spherical Bessel expansions. The solution is given as a complex number; its real part is the eigenfrequency and the imaginary part represents the attenuation coefficient of the oscillation. The eigenmodes are classified into five groups: (1) the compressional modes in a fluid sphere, (2) the compressional modes in a solid medium, (3) the shear modes in a solid medium, (4) the Stoneley modes, and (5) the torsional modes. We apply them to the long-period volcanic earthquake observed at Asama volcano, Japan. Estimating the characteristic frequencies and attenuation coefficients of the observed vibrations and assuming that the primary component (f = 1.73 Hz) corresponds to the fundamental translation mode of a fluid sphere as one of the compressional modes in fluid, we conclude that the resonator which is a spherical cavity of diameter 220 m filled with steam of temperature 500°C and pressure 170 atm is favorable.

A jet-like structure revealed by a numerical simulation of rotating spherical-shell magnetoconvection

Numerical results on thermally driven nonlinear magnetoconvection in a rapidly rotating fluid spherical shell are reported. A uniform magnetic field that is parallel to the rotation axis is imposed externally. The Ekman number is 2 x 10 -6 , representing a state of negligible viscosity, as in the Earths core. The convection pattern is characterized by a few large-scale vortex columns superimposed on a fast westward (retrograde) zonal flow. In the equatorial region, an anticyclonic vortex is intensified, in which an induced axial magnetic field is stored. Interaction between the magnetized vortex and the zonal flow leads to a thin jet at the western side of the vortex. The jet is also characterized by a thin electric current sheet caused by a steep gradient of the axial magnetic field. Because of this structure, the jet region can be designated as a magnetic front by analogy with fronts in mid-latitude atmospheric cyclones. It can be estimated from an order-of-magnitude analysis that the jet width decreases in inverse proportion to the zonal flow speed, and that the jet speed and the sheet-like electric current are proportional to the square of the zonal flow speed.

Linear stability of plane Poiseuille flow in an infinite elastic medium and volcanic tremors

The linear stability of a plane compressible laminar (Poiseuille) flow sandwiched between two semi-infinite elastic media was investigated with the aim of explaining the excitation of volcanic tremors. Our results show that there are several regimes of instability, and the nature of stability significantly depends on the symmetry of oscillatory fluid and solid motion. It has been shown that long-wave symmetric instability occurs at a very small value of the Reynolds number, but it is unlikely that this is the cause of volcanic tremors. We show that antisymmetric (flexural) instability also occurs, involving two parallel Rayleigh waves traveling against the Poiseuille flow, but the critical flow speed is faster than that of symmetric instability. However, if the basic flow profile is nonparabolic because of a nonuniform driving force or nonuniform viscosity, the critical flow speed of antisymmetric instability can be considerably slower than that of symmetric instability. Based on numerical calculations and analytical consideration, we conclude that this anomalous antisymmetric instability is possibly produced by a basaltic magma flow of a few meters per second through a dike with thickness of 1 m and extending for several kilometers; this origin can explain some of the characteristics of volcanic tremors.