| 1. |
中島涼輔, 吉田 茂生, Shallow water MHD waves trapped near the poles in a stably stratified outermost Earth's core, JpGU-AGU Joint Meeting 2020, 2020.07, Magnetohydrodynamic (MHD) shallow water linear waves are investigated over a rotating sphere with an imposed equatorially antisymmetric toroidal magnetic field: B0Φ=B0sinθcosθ, where B0 is a constant, θ is the colatitude and Φ is the azimuth. This system can imitatively represent the dynamics of a liquid metal within a stably stratified layer at the top of the Earth's core, which was detected through seismological surveys (e.g. Helffrich & Kaneshima, 2010[1]) and also has been deduced from geophysical and geochemical knowledge (e.g. Buffett & Seagle, 2010[2]; Pozzo et al., 2012[3]; Gubbins & Davies, 2013[4]; Brodholt & Badro, 2017[5]). Because slowly propagating waves in the liquid core can result in geomagnetic secular variations, comparison between exhaustive studies of MHD waves in a rotating stratified fluid and observations of geomagnetic fluctuations should provide constraints on the obscure stratified layer in the outermost core (e.g. Braginsky, 1993[6]; Buffett, 2014[7]).
The adopted configuration of the background field complicates solving the eigenvalue problem of linear waves due to the emergence of an Alfvén continuum and critical latitudes unless dissipation effects are taken into account. These result from non-dissipative Alfvén resonance, which occurs only when B0Φ/sinθ depends on θ, that is, regular singular points appear in the differential equation of linear problems. The solutions of the continuum are required to express the transient evolution of an arbitrary initial disturbance (e.g. Case, 1960[8]; Goedbloed & Poedts, 2004[9]). We can confirm numerically and analytically that introducing magnetic diffusion eliminates these Alfvén continuous modes and their singular structures around critical latitudes (Nakashima, Ph.D. thesis, 2020[10]).
For the Earth's core-like parameter (B0≃0.5—5mT and magnetic diffusivity η≃1m2/s), westward polar trapped modes are obtained as eigenmodes, which have a period of around from several to 1000 years. We may be able to observe these modes as geomagnetic secular variations in high latitude regions, if the strength of stratification in the stratified layer is close to the estimate of Buffett (2014)[7]. The analyses of recent geomagnetic models and paleomagnetic data in terms of such waves could confirm the robustness of previous estimates of the properties of the layer.
[ Reference ]
[1] Helffrich, G., Kaneshima, S. (2010) Nature, 468, 807. doi: 10.1038/nature09636
[2] Buffett, B. A., Seagle, C. T. (2010) J. Geophys. Res., 115, B04407. doi: 10.1029/2009JB006751
[3] Pozzo, M., Davies, C., Gubbins, D., Alfè, D. (2012) Nature, 485, 355. doi: 10.1038/nature11031
[4] Gubbins, D., Davies, C. J. (2013) Phys. Earth Planet. Inter., 215, 21. doi: 10.1016/j.pepi.2012.11.001
[5] Brodholt, J., Badro, J. (2017) Geophys. Res. Lett., 44, 8303. doi: 10.1002/2017GL074261
[6] Braginsky, S. I. (1993) J. Geomag. Geoelectr., 45, 1517. doi: 10.5636/jgg.45.1517
[7] Buffett, B. (2014) Nature, 507, 484. doi: 10.1038/nature13122
[8] Case, K. M. (1960) Phys. Fluids, 3, 143. doi: 10.1063/1.1706010
[9] Goedbloed, J. P., Poedts, S. (2004) Principles of magnetohydrodynamics: with applications to laboratory and astrophysical plasmas, Cambridge Univ. Press, Cambridge.
[10] Nakashima, R. (2020) Ph.D. thesis, Kyushu University. http://dyna.geo.kyushu-u.ac.jp/HomePage/nakashima/pdf/doctoral_thesis.pdf. |
| 2. |
Shallow water MHD waves trapped in the polar regions on a rotating sphere with an imposed azimuthal magnetic field. |
| 3. |
中島涼輔, 吉田 茂生, The effect of an azimuthal background magnetic field on waves in a stably stratified layer at the top of the Earth's outer core, 日本地球惑星科学連合2018年大会, 2018.05, We investigated waves in a stably stratified thin layer in a rotating sphere with an imposed magnetic field. This represents the stably stratified outermost Earth's core or the tachocline of the Sun. Recently, many geophysicists focus on the stratification of the outermost outer core evidenced through seismological studies (e.g. Helffrich and Kaneshima, 2010) and an interpretation of the 60-year geomagnetic secular
variations with Magnetic-Archimedes-Coriolis (MAC) waves (Buffett, 2014).
Márquez-Artavia et al.(2017) studied the effect of a toroidal magnetic field on shallow water waves over a rotating sphere as the model of this stratified layer. On the other hand, MAC waves are strongly affected by a radial field (e.g. Knezek and Buffett, 2018). We added a non-zero radial magnetic perturbation and magnetic diffusion to Márquez-Artavia et al.(2017)'s equations. Unlike their paper's formulation, we applied velocity potential and stream function for both fluid motion and magnetic perturbation, which is similar to the first method of Longuet-Higgins(1968). In the non-diffusive case, the dispersion relation obtained with the azimuthal equatorially symmetric field (Bφ(θ) ∝sinθ, where θis colatitude) is almost the same as Márquez-Artavia et al.(2017)'s result, which includes magneto-inertia gravity (MIG) waves, fast magnetic Rossby waves, slow MC Rossby waves and an unexpected instability. In particular, we replicate the transition of the propagation direcition of zonal wavenumber m=1 slow MC Rossby waves from eastward to westward with increasing Lamb parameter (ε=4Ω^2 a^2/gh, where Ω, a, g and h is the rotation rate, the sphere radius, the acceleration of gravity and an equivalent depth, respectively) and Lehnert number (α=v_A/2Ωa, where v_A is Alfvén wave speed). As a consequence, fast magnetic Rossby waves and slow MC Rossby waves interact, and the non-diffusive instability occur.
Next, we are examining the case with an equatorially antisymmetric background field, which is more realistic in the Earth's core. In this case, if the magnetic diffusion is ignored, the continuous spectrums appear owing to Alfvén waves resonance (similar to the continuous spectrums in inviscid shear flow, e.g. Balmforth and Morrison, 1995). To solve this difficulty, our numerical model includes the magnetic diffusion term.. |
| 4. |
A possible stably stratified layer at the top of the outer core --the compositional evolution of the outer core. |
| 5. |
Equatorial waves modified by the presence of a toroidal magnetic field within the stably stratified layer at the top of the Earth’s outer core. |
| 6. |
Empirical mode decomposition of the decadal variation of the Gauss coefficients. |
| 7. |
The boundary mode of axially symmetric MAC waves can exist in the stratified layer at the top of the Earth’s outer core. |
| 8. |
Spontaneous rotation of a block ice melting on metal surface. |
| 9. |
Spontaneous rotation of a melting ice block. |
| 10. |
Spontaneous rotation of a melting ice block. |
| 11. |
Spontaneous rotation of a block of ice on a flat surface of a warm metal column. |
| 12. |
Spontaneous rotation of a block of ice on a flat surface of a warm metal column. |
| 13. |
Hirotaka Hohokabe, Shigeo Yoshida, Linear stability analysis of two-layer thermal convection and the generation of surface flow of gas giants, The 3rd International Congress on Natural Sciences (ICNS 2013), 2013.10, Surface flows observed on gas giants are dominated by zonal wind, which is particularly strong in equatorial region and peaks at approximately 400 m/s on Neptune and 300 m/s on Saturn. The directions of the equatorial jets on Jupiter and Saturn are prograde, but are retrograde on Uranus and Neptune. What makes the difference in the direction is not well understood. The dynamics of jets depend on their depth. Existing models of global flow are based on the assumption of either shallow two-dimensional turbulence or deep columnar thermal convection. Observational evidence supporting the latter for Neptune is obtained by Karkoschka (2010), who found deep-seated features on the surface. However, it has been established that single-layer deep columnar convection generates prograde equatorial flow by the Reynolds stress, because the topographic β effect increasing outward, which originates from the spherical form of the planet surface, tilt the convection cells so as to transport the angular momentum outward.
We envisage that the interaction of the convection in the outer gas envelope and that in the inner conducting layer is important in determining the direction of the angular momentum transfer. The effective topographic β effect should become negative near the boundary between the inner and outer shells, and could lead to inward angular momentum transfer. The interaction should be strong if the depth of the convection in the inner shell is small, as suggested by Stanley and Bloxham (2004, 2006) for Uranus and Neptune. The depth of the convection in the dynamo region might be the key for explaining the difference between ice giants (Uranus and Neptune) and gas giants (Jupiter and Saturn).
To confirm this hypothesis, we perform linear stability analyses of the two-layer system. We use a two-layer annuli model with constant β for each layer to focus on the effect of the interaction of the two layers. We find that thermally coupled two-layer columnar convection system can produce both prograde and retrograde equatorial flow, depending on the ratio of the radii of the inner and outer annuli.
We are extending the results for spherical shells, which hopefully we can show at the meeting.
. |
| 14. |
Shigeo Yoshida, Introduction to the geodynamo theory and the alpha effect, The 3rd International Congress on Natural Sciences (ICNS 2013), 2013.10, I review the geodynamo theory for non-specialists, with a paticular emphasis on the alpha effect, which is considered to be a key concept for magnetic-field generation.
\begin{wrapfigure}{r}{5cm}
\includegraphics[width=5cm]{alpha-effect.eps}
%\includegraphics[width=8cm]{example.eps}
\caption{Schematic illustration of the alpha effect}
\end{wrapfigure}
The time-evolution of the magnetic field is governed by the induction equation
\begin{equation}
\left( \frac{\partial}{\partial t} - \lambda \nabla^2 \right) \bm{b} = \nabla \times ( \bm{u} \times \bm{b} ),
\end{equation}
where $\bm{b}$ is the magnetic field and $\bm{u}$ is the velocity. Although this equation looks simple, its actual behavior is complicated, because small scale eddies represented by $\bm{u}$ can generate a large scale magnetic field. It is thus natural to apply some averaging to the induction equation. The averaging leads to the effective electric field, called the electromotive force, usually represented as
\begin{equation}
\bm{E}_\textrm{emf} = \alpha \bm{B}.
\label{eqn:simple-alpha}
\end{equation}
This is called the alpha effect, through which an effective electric field is generated in the direction of the large scale magnetic field $\bm{B}$. The alpha effect is induced by helical flows, as schematically illustrated in Fig.1. The magnetic field is twisted by a helical flow, and the resulting loop of the field line can be interpreted as an effective electric field. If this effect is present, the magnetic field can be generated through
\begin{equation}
\left( \frac{\partial}{\partial t} - \lambda \nabla^2 \right) \bm{B} = \bm{E}_\textrm{emf},
\end{equation}
which enables us to interpret the generation mechanism of the magnetic field relatively easily.
However, the exact alpha effect is not as simple as represented by equation (\ref{eqn:simple-alpha}). It is non-local and non-instantaneous, and represented as
\begin{equation}
\bm{E}_\textrm{emf} = \int^t_{-\infty} \int^{\infty}_{\infty} \phi (t-\tau, x-\xi) \bm{B} (\tau, \xi) d\tau d\xi
- \int^t_{-\infty} \int^{\infty}_{\infty} \psi (t-\tau, x-\xi) \nabla \times \bm{B} (\tau, \xi) d\tau d\xi.
\end{equation}
We have explicitly calculated the response function $\phi$ and $\psi$ for a simple helical flow, and have revealed its non-local behaviour, which enhances magnetic-field generation to some degree [Hori and Yoshida, 2008].
\vspace{5mm}
Hori, K. and Yoshida, S. (2008) Non-local memory effects of the electromotive force by fluid motion with helicity
and two-dimensional periodicity, \textit{Geophysical and Astrophysical Fluid Dynamics}, \textbf{102}, 601-632.
doi:10.1080/03091920802260466. |
| 15. |
A model of modern science and its working: Dual Feedback-Loop Operator. |
| 16. |
The evolution of behavioral modernity and the evolution of science. |
| 17. |
How to write the history of geoscience -right and wrong of Whig interpretation of history-. |
| 18. |
The history of the study on the Earth’s inner core with the aid of a scientometric method. |
| 19. |
From philosophy of science to science of science - A casestudy on earth science. |
| 20. |
Where and how did science come from? A cognitive approach.. |
| 21. |
A natural view of the World in philosophy of science provided by interpretation of the Earth’s evolution history. |
| 22. |
How to launch the Science of Science. |
| 23. |
Model, where earth science and the philosophy of science meets. |
| 24. |
Methodology of science studies and the science of future. |
| 25. |
The reason we began to study the philosophy of science -- as an extension of the project "Decoding the Earth's Evolution". |
| 26. |
The alpha effect in the dynamo theory. |
| 27. |
Models and Simulations in Geosciences. |
| 28. |
Historical reconstruction in science. |
| 29. |
Evaluating Akiho Miyashiro's philosophy of science. |
| 30. |
Usage of the term "model" in science. |
| 31. |
Takako Sakurai-Amano, Yuko Sato, Shigeki Kobayashi, Mikio Takagi, Shuhei Okubo, Shigeo Yoshida, Scene feature matching analysis of JERS-1 SAR images, Proceedings of the 1998 IEEE International Geoscience and Remote Sensing Symposium, IGARSS. Part 1 (of 5), 1998.07, A method to examine the shifts in individual surface features in multi-temporal SAR images has been proposed. First, line-like and boundary-like features were extracted from SAR imagery using the pixel swapping method. The extracted feature images were then tied to observe an overall shift in individual features. We applied this method to two JERS-1 SAR images of the same path-row observed on different dates. We observed that the shift in flat areas was simple, but that in mountainous regions, there was a clear difference in the direction of the shift in ridges and valleys caused by the slightly different orbital characteristics of the satellite between the two images. In addition, care in choosing appropriate tie points was found to be important. The method used in this investigation is expected to be useful in planning an appropriate strategy for the coregistration of images, such as in SAR images used for change detection in environmental monitoring, and InSAR.. |
