We summarize new findings of this study as follow.
(i) We proposed a method of combining fdip and fmag fit to evaluate the dipolar dominance. This method is capable to assess the dipolar dominance quantitatively.
(ii) TheRarange of sustained strong dipolar dynamos became narrower with the smaller inner core. The dipolar dominance was also weakened with the smaller inner core.
(iii) Atri/ro = 0.25and0.35, theRarange of sustained strong dipolar dynamos was smaller in the heat-flow-balanced FF cases than in FT cases while the dipolar dominance was stronger in FF cases than FT cases. At ri/ro = 0.15, no strong dipole regime was represented in both FT and FF cases.
(iv) The dynamo regime was determined byRa/Racritregardless of different heat flow rates in the range ofQi/Qo ≥0.5.
fohm is the rate of Ohmic dissipation to core power. The red, blue, and green points indicate the cases ofri/ro = 0.15,0.25,and0.35, respectively. While the trend is inverse of that of Stelzer and Jackson (2013), it is consistent with their localized trend corresponding to E = 3×10−4 shown in Fig. 6.2. The physical unit of the horizontal axis of Fig. 6.1 is equivalent to the energy flux (Christensen, 2010). From Fig. 6.1, it is suggested that the magnetic field strength at different radius ratios follows a scaling law individually. This is because the tendency of the magnetic energy depending on the Rayleigh number is different at various radius ratios. In order to understand the real planetary dynamos with the use of a scal-ing law, further numerical dynamo simulations in smaller E or P mare needed.
Another advantage of revealing properties of numerical dynamos with various in-ner core size is to advance understanding the environment of planets other than the Earth. It is thought that the ancient Moon (Weiss and Tikko, 2014) and Mars (Stevenson, 2001), and the present Mercury (Ness et al., 1975; Anderson et al., 2011) have an intrinsic magnetic field. Since these rocky planets had/have the outer core of iron alloy, we can apply results of numerical geodynamo, in which magnetofluid is treated as electrically conducting Boussinesq fluid, to these plan-ets. For example, the radius ratio of the Mercury is approximately0.25from ther-mal calculation (Dumberry and Rivoldini, 2015). The radius ratio corresponds
magnetic field was observed up to degree l = 4 by MESSENGER (Anderson et al., 2012). While the information about Mercurian magnetic field is restricted so far, BepiColombo (Benkhoff et al., 2010) is expected to observe the Mercurian environment in detail near future. Combination of its observation and numeri-cal simulations is expected to reveal the process of Mercurian dynamo. In the perspective of planetary dynamos, capturing the characteristics of numerical dy-namos with different inner core size is essential.
Fig. 6.1:Magnetic field strength corrected for the relative fraction of Ohmic dissipation scaled by the same combination of parameters as Stelzer and Jackson (2013).
Fig. 6.2:Magnetic field strength corrected for the relative fraction of Ohmic dissipation scaled by the combination of four control parameters, which are equivalent to the energy flux (Stelzer and Jackson, 2013).
Bibliography
[1] Alken, P., Thebault, E., Beggan, C. D., Amit, H., Aubert, J., Baerenzung, J., Bondar, T. N., Brown, W. J., Califf, S., Chambodut, A., Chulliat, A., Cox, G. A., Finlay, C. C., Fournier, A., N., Gillet, Grayver, A., Hammer, M.
D., Holschneider, M., Huder, L., Hulot, G., Jager, T., Kloss, C., Korte, M., Kuang, W., Kuvshinov, A., Langlais, B., Leger, J.-M., Lesur, V., Livermore, P. W., Lowes, F. J., Macmillan, S., Magnes, W., Mandea, M., Marsal, S., Matzka, J., Metman, M. C., Minami, T., Morschhauser, A., Mound, J. E., Nair, M., Nakano, S., Olsen, N., Pavon-Carrasco, F. J., Petrov, V. G., Ropp, G., Rother, M., Sabaka, T. J., Sanchez, S., Saturnino, D., Schnepf, N., R., Shen, X., Stolle, C., Tangborn, A., Toffner-Clausen, L., Toh, H., Torta, J.
M., Varner, J., Vervelidou, F., Vigneron, P., Wardinski, I., Wicht, J., Woods, A., Yang, Y., Zeren, Z. and Zhou, B., 2021. International Geomagnetic Reference Field: the thirteenth generation, Earth, Planets and Space, 73:49, doi: 10.1186/s40623-020-01288-x.
[2] Al-Shamali, F. M., Heimpel, M. H. and Aurnou, J. M., 2004. Varying the spherical shell geometry in rotating thermal convection, Geophys. Astro-phys. Fluid Dyn., 98(2), 153-169, doi:10.1080/0309192041000165928.
[3] Anderson, B. J., Johnson, C. L., Korth, H., Purucker, M. E., Winslow, R.
M., Slavin, J. A., Solomon, S. C., McNutt Jr., R. L., Raines, J. M. and Zurbuchen, T. H., 2011. The Global Magnetic Field of Mercury from MESSENGER Orbital Observations, Science, 333 (6051), 1859-1862, doi:
10.1126/science.1211001.
[4] Anderson, B. J., Johnson, C. L., Korth, H., Winslow, R. M., Borovsky, J. E., Purucker, M. E., Slavin, J. A., Solomon, S. C., Zuber, M. T. and McNutt Jr., R. L., 2012. Low-degree structure in Mercury’s planetary magnetic field, J.
Geopyhs. Res., 117, E00L12, doi: 10.1029/2012JE004159.
[5] Aubert, J., Labrosse, S. and Poitou, C., 2009. Modelling the paleo-evolution of the geodynamo, Geophys. J. Int., 179, 1414-1428, doi: 10.1111/j.1365-246X.2009.04361.x.
[6] Benkhoff, J., Casteren, J. v., Hayakawa, H., Fujimoto, M., Laakso, H., Novara, M., Ferri, P., Middleton, H., R. and Ziethe, R., 2010. Bepi-Colombo - Comprehensive exploration of Mercury: Mission overview and science goals, Planetary and Space Science, 58 (1-2), 2-20, doi:
10.1016/j.pss.2009.09.020.
[7] Biggin, A. J., Piispa, E. J., Holme, R., Paterson, G. A., Veikkolao-nen, T. and Tauxe L., 2015. Paleomagnetic field intensity variations suggest Mesoproterozoic inner-core nucleation, Nature, 526, 245-248, doi:10.1038/nature15523.
[8] Blackett, P. M. S., 1952. A negative experiment relating to Magnetism and the Earth’s Rotation, Phil. Trans. Roy. Soc. London A, 245, 309-370, doi:
https://doi.org/10.1098/rsta.1952.0024.
[9] Bullard, E. C., 1948. The magnetic field within the earth, Proc. ROy. Soc.
London A, 197, 433-453.
[10] Bullard, E. C. and Gellman, H., 1954. Homogeneous Dynamos and Ter-restrial Magnetism, Phil. Trans. Roy. Soc. London A, Math. and Phys. Sci. , 247 (928), 213-278.
[11] Bullard, E. C., 1955. The stability of a homopolar dynamo, Proc. of the Cambridge Philos. Soc., 51 (4), 744-760, doi:
10.1017/S0305004100030814.
[12] Bullen, K. E., 1936. The variation of density and the ellipticities of strata of equal density within the earth, Geophysical Supplements to the Monthly Notices of the Roy. Astron. Soc., 3 (9), 395-401, doi: 10.1111/j.1365-246X.1936.tb01747.x.
[13] Busse, F. H., 1970. Thermal instabilities in rapidly rotating systems, Journal of Fluid Mechanics, 44 (3), 441-460, doi: 10.1017/S0022112070001921.
[14] Chandrasekhar, S., 1961. Hydrodynamic and hydromagnetic stability, Ox-ford University Press.
[15] Christensen, U. R., Aubert J., Cardin, P., Dormy, E., Gibbon, S., Glatz-maier, G. A., Grote, E., Honkura, Y., Jones, C., Kono, M., Matsushima, M., Sakuraba, A., Takahashi, F., Tilgner, A., Wicht, J. and Zhnag K., 2001.
A numerical dynamo benchmark, Phys. Earth planet. Int., 128, 25-34,
[16] Christensen, U. R. and Aubert, J., 2006. Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields, Geophys. J. Int., 166 (1), 97-144, doi: 10.1111/j.1365-246X.2006.03009.x.
[17] Christensen, U. R., 2010. Dynamo Scaling Laws and Applications to the Planets, Space Sci. Rev., 152, 565-590
[18] Cowling, T. G., 1933. The Magnetic Field of Sunspots, Monthly No-tices Roy. Astron. Soc., 94 (1), 39-48, doi: 10.1007/s11214-009-9553-210.1093/mnras/94.1.39.
[19] Driscoll, P. E., 2016. Simulating 2 Ga of geodynamo history, Geophys. Res.
Lett., 43, 5680-5687, doi: 10.1002/2016GL068858.
[20] Dumberry, M. and Rivoldini, A., 2015. Mercury’s inner core size and core-crystallization regime, Icarus, 248, 254-268, doi:
10.1016/j.icarus.2014.10.038.
[21] Dziewonski, A. M. and Anderson, D. L., 1981. Preliminary reference Earth model, Physics of the Earth and Planetary Interiors, 25 (4), 297-356, doi:
10.1016/0031-9201(81)90046-7.
[22] Elsasser, W. M., 1946. Induction Effects in Terrestrial Magnetism, Phys.
Rev., 69, 106-116.
[23] Glatzmaier, G. and Roberts, P. H., 1995. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal, Nature, 377, 203-209.
[24] Heimpel, M. H., Aurnou, J. M., Al-Shamali, F. M. and Perez N. G., 2005. A numerical study of dynamo action as a function of spherical shell geometry, Earth planet. Sci. Lett., 236, 542-557, doi:10.1016/j.epsl.2005.04.032.
[25] Hori, K., Wicht, J. and Christensen U. R., 2010. The effect of thermal boundary conditions on dynamo driven by internal heating, Phys. Earth planet. Int., 182, 85-97, doi:10.1016/j.pepi.2010.06.011.
[26] IAGA Commission 2 Working Group No. 4, 1969. The international Geo-magnetic Reference Field 1965.0, Journal of Geomagnetism. and Geoelec-tricity, 21 (2), 569-571.
[27] Jacob, J. A., 1987. Geomagnetism Volume 2, Academic Press.
[28] Kageyama, A, Sato, T. and the Complexity Simulation Group, 1995, Com-puter simulation of a magnetohydrodynamic dynamo. 2, Physics of Plasmas 2, 1421-1431, doi: 10.1063/1.871485.
[29] Kageyama, A. and Sato, T., 1997. Generation mechanism of a dipole field by a magnetohydrodynamic dynamo, Phys. Rev. E, 55, 4617-4626, doi:
10.1103/PhysRevE.55.4617.
[30] Kono, M., 1987. Rikitake two-disk dynamo and paleomagnetism, Geophys.
Res. Lett., 14 (1), 21-24, doi: 10.1029/GL014i001p00021.
[31] Kono, M. and Roberts, P. H., 2002. Recent geodynamo simulations and observations of the geomagnetic field, Rev. Geophys., 40(4), 1-53, doi:
10.1029/2000RG000102.
[32] Kono, M. and Stevenson, D. J., 2003. Dynamo Process and the Mag-netic Field of the Earth and Planets, Zisin 2, 56 (3), 311-315, doi:
10.4294/zisin1948.56.3 311, in Japanese.
[33] Kulakov, E. V., Sprain, C. J., Doubrovine, P. V., Smirnov, A. V. Paterson, G.
A., Hawkins, L., Fairchild, L., Piispa, E. J. and Biggin A. J., 2019. Anal-ysis of an Updated Paleointensity Database (QPI-PINT) for 65-200 Ma:
Implications for the Long-Term History of Dipole Moment Through the Mesozoic, J. Geophys. Res. Solid Earth, Magnetism in the Geosciences -Advances and Perspectives, 9999-10022, doi: 10.1029/2018JB017287.
[34] Kutzner, C. and Christensen, U. R., 2002. From stable dipolar towards reversing numerical dynamos, Phys. Earth planet. Int., 131(1), 29-45, doi:
10.1016/S0031-9201(02)00016-X.
[35] Labrosse, S., Poirier, J.-P. and Mouel, J.-L. L., 2001. The age of the inner core, Earth planet. Sci. Lett., 190 (3)-(4), 111-123, doi: 10.1016/S0012-821X(01)00387-9.
[36] Langel, R. A. and Estes, R. H., 1982. A geomagnetic field spectrum, Geo-phys. Res. Lett., 9(4), 250-253.
[37] Larmor, J., 1919. How could a rotating body such as the Sun become a magnet, Rep. Brit. Adv. Sci., 159-160, 1919.
[38] Lhuillier, F., Hulot, G., Gallet, Y. and Schwaiger T., 2019. Impact of
inner-[39] Lowes, F. J., 1974. Spatial power spectrum of the main geomagnetic field, and extrapolation to the core, Geophys. J. Int., 36(3), 717-730, doi:
10.1111/j.1365-246X.1974.tb00622.x.
[40] Matsushima, M., 2005, Fluid Motion in the Core Estimation from the Earth’s Magnetic Field, Journal of Geography, 114 (2), 132-141, doi:
10.5026/jgeography.114.2 132, in Japanese.
[41] Matsui, H., 2000. Studies on the Basic Process of Magnetic Field Gener-ation Based on MHD SimulGener-ation in the Rotating Spherical Shell, Tohoku University, Ph. D. thesis.
[42] Matsui, H., King, E. and Buffett B., 2014. Multiscale convec-tion in a geodynamo simulaconvec-tion with uniform heat flux along the outer boundary, Geochem. Geophys. Geosys., 15, 3212-3225, doi:10.1002/2014GC005432.
[43] Matsui H., Calypso User Manual Version 1.1.
[44] Merrill, R. T., McElhinny, M. W. and McFadden, P. L., 1996. The Mag-netic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle, Academic Press.
[45] Ness, N. F., Behannon, K. W. and Lepping R. P., 1975. The Mag-netic Field of Mercury, 1, J. Geophys. Res., 80 (19), 2708-2716, doi:
10.1029/JA080i019p02708.
[46] Olson, P. L., Glatzmaier, G. A. and Coe R. S., 2011. Complex polarity reversals in a geodynamo model, Earth planet. Sci. Lett., 304 (1)-(2), 168-179, doi: 10.1016/j.epsl.2011.01.031.
[47] Parker, E. N., 1955. Hydromagnetic dynamo models, Astrophys. Journal, 122, 293-314, doi: 10.1086/146087.
[48] Rikitake, T., 1958. Oscillations of a system of disk dynamos, Math-ematical Proc. of the Cambridge Philos. Soc., 54 (1), 89-105, doi:
10.1017/S0305004100033223.
[49] Roberts, P. H., 1968. On the thermal instability of a rotating-fluid sphere containing heat sources, Philoso. Trans. for the Roy. Soc. of London. Series A, Math. and Phys. Sci., 263 (1136), 93-117, doi:
10.1098/rsta.1968.0007.
[50] Russell, C. T., 1978. Re-evaluating Bode’s law of planetary magnetism, Nature, 272, 147-148.