地磁场发电机数值模拟综述
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摘要
简要综述地磁场发电机模型,包括控制方程组、无量纲方案、初始条件边界条件、数值方法和标度律等,并以MoSST模型为例,展示地磁场发电机的数值计算结果。虽然地磁场发电机的数值模拟还存在很多问题和挑战,但它仍是研究地磁场不可或缺的方法。随着计算能力的不断提升,地磁场发电机的数值模拟将会取得更大的进展,为理解地磁场提供有力的支持。
This review simply summarizes geodynamo models, including the control equations, dimensionless scheme, initial and boundary conditions, the numerical methods, the scaling laws, and so on. MoSST model is taken as an example to show the results of geodynamo numerical model. Though in the geodynamo numerical simulation there exist many problems and challenges, it is still an indispensable method to study the geomagnetic field. With the improvement in the computing power in the near future, the geodynamo numerical simulation will achieve great development and will provide powerful supports for understanding the geomagnetic field.
This review simply summarizes geodynamo models, including the control equations, dimensionless scheme, initial and boundary conditions, the numerical methods, the scaling laws, and so on. MoSST model is taken as an example to show the results of geodynamo numerical model. Though in the geodynamo numerical simulation there exist many problems and challenges, it is still an indispensable method to study the geomagnetic field. With the improvement in the computing power in the near future, the geodynamo numerical simulation will achieve great development and will provide powerful supports for understanding the geomagnetic field.
引文
[1] Roberts P H. Theory of the Geodynamo[M].Oxford: Elsevier, 2015, 8: 57-90.
[2] Buffett B A. Earth’s Core and the Geodynamo[J]. Science, 2000, 288(5 473): 2 007-2 012.
[3] Christensen U R, Wicht J. Numerical dynamo simulations [M]. Oxford: Elsevier, 2015, 8: 245-277.
[4] Glatzmaier G A, Roberts P H. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal[J]. Nature, 1995, 377(6 546): 203-209.
[5] Glatzmaier G A, Roberts P H. A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle[J]. Physics of the Earth and Planetary Interiors, 1995, 91(1-3): 63-75.
[6] Song X R, Paul G. Seismological evidence for differential rotation of the Earth’s inner core[J]. Nature, 1996, 382(6 588): 221-224.
[7] Kageyama A, Sato T. Computer simulation of a magnetohydrodynamic dynamo. II[J]. Physics of Plasmas, 1995, 2(5): 1 421-1 431.
[8] Kuang W, Bloxham J. An Earth-like numerical dynamo model[J]. Nature, 1997, 389(6 649): 371-374.
[9] Christensen U R, Olsen P, Glatzmaier G A. Numerical modelling of the geodynamo a systematic parameter study[J]. Geophys J Int, 1999, 138(2): 393-409.
[10] Kono M, Roberts P H. Recent geodynamo simulations and observations of the geomagnetic field[J]. Reviews of Geophysics, 2002, 40(4): 1-53.
[11] Stanley S, Glatzmaier G. Dynamo models for planets other than Earth[J]. Space Science Reviews, 2010, 152: 617-650.
[12] Glatzmaier G A. Numerical simulation of stellar convective dynamos. 1. The model and methods[J]. Journal of Computational Physics, 1984, 55(3): 461-484.
[13] Omara B, Miesch M S, Featherstone N A, et al. Velocity amplitudes in global convection simulations: the role of the Prandtl number and near-surface driving[J]. Advances in Space Research, 2016, 58(8): 1 475-1 489.
[14] Jones C A. Planetary magnetic fields and fluid dynamos[J]. Annu Rev Fluid Mech, 2011, 43: 583-614.
[15] Matsui H, Heien E M, Aubert J, et al. Performance benchmarks for a next generation numerical dynamo model[J]. Geochemistry Geophysics Geosystems, 2016, 17(5): 1 586-1 607.
[16] Tilgner A. Precession-driven dynamos[J]. Physics of Fluids, 2005, 17(3): 34-104.
[17] Cowling T G. The stability of gaseous stars[J]. Monthly Notices of the Royal Astronomical Society, 1934, 94(8): 768-782.
[18] Anufriev A P, Jones C A, Soward A M. The Boussinesq and anelastic liquid approximations for convection in the Earth’s core[J]. Physics of the Earth and Planetary Interiors, 2005, 152(3): 163-190.
[19] Jones C A. Thermal and Compositional Convection in the Outer Core[M]. Oxford: Elsevier, 2015, 8:115-159.
[20] Christensen U R, Aubert J, Cardin P, et al. A numerical dynamo benchmark[J]. Physics of the Earth and Planetary Interiors, 2001, 128: 25-34.
[21] Gubbins D, Roberts P. Magnetohydrodynamics of the Earth’s core[M]//Jacobs J A. Geomagnetism, London: Academic Press, 1987: 1-183.
[22] Braginsky S I, Roberts P H. Equations governing convection in Earth’s core and the geodynamo[J]. Geophysical and Astrophysical Fluid Dynamics, 1995, 79: 1-97.
[23] Glatzmaier G A, Roberts P H. An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection[J]. Physica D, 1996, 97: 81-94.
[24] Kuang W, Bloxham J. Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: weak and strong field dynamo action[J]. Journal of Computational Physics, 1999, 153(1): 51-81.
[25] Gastine T, Duarte L, Wicht J. Dipolar versus multipolar dynamos: the influence of the background density stratification[J]. Astronomy and Astrophysics, 2012, 546: A19.
[26] Evonuk M, Samuel H. Simulating rotating fluid bodies: When is vorticity generation via density stratification important?[J]. Earth Planet. Science Letters, 2012, 317/318: 1-7.
[27] Glatzmaier G A, Roberts P H. Simulating the geodynamo[J]. Contemporary Physics, 1997, 38(4): 269-288.
[28] Olson P, Christensen U, Glatzmaier G A. Numerical modeling of the geodynamo: mechanisms of field generation and equilibration[J]. Journal of Geophysical Research: Solid Earth, 1999, 104(B5): 10 383-10 404.
[29] Glatzmaier G A, Roberts P H. Rotation and magnetism of Earth’s inner core[J]. Science, 1996, 274(5 294): 1 887-1 891.
[30] Makinen A M, Deuss A. Global seismic body wave observations of temporal variations in the Earth’s inner core, and implications for its differential rotation[J]. Geophysical Journal International, 2011, 187(1): 355-370.
[31] Zhang J, Song X, Li Y, et al. Inner core differential motion confirmed by earthquake waveform doublets[J]. Science, 2005, 309(5 739): 1 357-1 360.
[32] Buffett B A, Glatzmaier G A. Gravitational braking of inner-core rotation in geodynamo simulations[J]. Geophysical Research Letters, 2001, 27(19): 3 125-3 128.
[33] Hollerbach R. A spectral solution of the magneto-convection equations in spherical geometry[J]. International Journal for Numerical Methods in Fluids, 2000, 32(7): 773-797.
[34] Harder H, Hansen U. A finite-volume solution method for thermal convection and dynamo problems in spherical shells[J]. Geophysical Journal International, 2005, 161(2): 522-532.
[35] Kageyama A, Sato T. Dipole field generation by an MHD dynamo[J]. Plasma Physics and Controlled Fusion, 1997, 39: 83-92.
[36] Li J, Sato T, Kageyama A. Repeated and sudden reversals of the dipole field generated by spherical dynamo action[J]. Science, 2002, 295(5 561): 1 887-1 890.
[37] Hollerbach R, Jones C A. A geodynamo model incorporating a finitely conducting inner core[J]. Physics of the Earth and Planetary Interiors, 1993, 75(4): 317-327.
[38] Wicht J. Inner-core conductivity in numerical dynamo simulations[J]. Physics of the Earth and Planetary Interiors, 2002, 132(4): 281-302.
[39] Dharmaraj G, Stanley S. Effect of inner core conductivity on planetary dynamo models[J]. Physics of the Earth and Planetary Interiors, 2012, 212/213: 1-9.
[40] Loper D E. Some thermal consequences of the gravitationally powered dynamo[J]. Journal of Geophysical Research, 1978, 83: 5 961-5 970.
[41] Nimmo F. Energetics of the Core[M]. Oxford: Elsevier, 2015, 8: 27-55.
[42] Corgne A, Keshav S, Fei Y W, et al. How much potassium is in the Earth’s core? New insights from partitioning experiments[J]. Earth and Planetary Science Letters, 2007, 256: 567-576.
[43] Rama Murthy V, van Westrenen W, Fei Y. Experimental evidence that potassium is a substantial radioactive heat source in planetary cores[J]. Nature, 2003, 423(6 936): 163-165.
[44] Driscoll P, Olson P. Polarity reversals in geodynamo models with core evolution[J]. Earth and Planetary Science Letters, 2009, 282: 24-32.
[45] Sakuraba A, Roberts P H. Generation of a strong magnetic field using uniform heat flux at the surface of the core[J]. Nature Geoscience, 2009, 2(11): 802-805.
[46] Hori K, Wicht J, Christensen U R. The effect of thermal boundary conditions on dynamos driven by internal heating[J]. Physics of the Earth and Planetary Interiors, 2010, 182: 85-97.
[47] Hori K, Wicht J, Christensen U R. The influence of thermo-compositional boundary conditions on convection and dynamos in a rotating spherical shell[J]. Physics of the Earth and Planetary Interiors, 2012, 196/197: 32-48.
[48] Alboussière T, Deguen R, Melzani M. Melting-induced stratification above the Earth’s inner core due to the convective translation[J]. Nature, 2010, 466: 744-747.
[49] Bullard E C, Gellman H. Homogeneous dynamos and terrestrial magnetism[J]. Proceedings of the Royal Society of London A, 1954, 247(928): 213-278.
[50] Zhang K, Busse F H. Convection driven magnetohydrodynamic dynamos in rotating spherical shells[J]. Geophysical and Astrophysical Fluid Dynamics, 1989, 49: 97-116.
[51] Zhang K, Busse F H. Generation of magnetic fields by convection in a rotating spherical fluid shell of infinite Prandtl number[J]. Physics of the Earth and Planetary Interiors, 1990, 59(3): 208-222.
[52] Sakuraba A, Kono M. Effect of the inner core on the numerical simulation of the magnetohydrodynamic dynamo[J]. Physics of the Earth and Planetary Interiors, 1999, 111: 105-121
[53] Stanley S, Bloxham J. Convective-region geometry as the cause of Uranus’ and Neptune’s unusual magnetic fields[J]. Nature, 2004, 428(6 979): 151-153.
[54] Kageyama A, Yoshida M. Geodynamo and mantle convection simulations on the Earth Simulator using the Yin-Yang grid[J]. Journal of Physics: Conference Series, 2005, 16(1): 325-338.
[55] Wicht J, Stellmach S, Harder H. Numerical models of the geodynamo: from fundamental Cartesian models to 3D simulations of field reversals[M]. Berlin: Springer, 2009, 8: 107-158.
[56] Christensen U R, Aubert J. Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields[J]. Geophysical Journal International, 2006, 166(1): 97-114.
[57] Christensen U R. A sheet-metal geodynamo[J]. Nature, 2008, 454(7 208): 1 058-1 059.
[58] Davies C J, Gubbins D, Jimack P K, et al. Scalability of pseudospectral methods for geodynamo simulations[J].Concurrency and Computation: Practice and Experience, 2011,23(1):38-56.
[59] Christensen U R. Dynamo scaling laws and applications to the planets[J]. Space Science Reviews, 2010, 152(1-4): 565-590.
[60] Christensen U R, Tilgner A. Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos[J]. Nature, 2004, 429(6 988): 169-171.
[61] Aubert J. Steady zonal flows in spherical shell dynamos[J]. Journal of Fluid Mechanics, 2005, 542(1): 53-67.
[62] Kuang W, Jiang W, Wang T. Sudden termination of Martian dynamo?: Implications from subcritical dynamo simulations[J]. Geophysical Research Letters, 2008, 35(14):L14204.
[63] 王天媛, Wei Z G, Jiang W Y,等.数值模拟火星古发电机湮灭前磁场特征[J]. 中国科学: 地球科学, 2013, 43(1): 155-162.
[64] Sreenivasan B, Sahoo S, Dhama G. The role of buoyancy in polarity reversals of the geodynamo[J]. Geophysical Journal International, 2014, 199(3): 1 698-1 708.
[65] Sarson G R, Jones C A, Longbottom A W. Convection driven geodynamo models of varying Ekman number[J]. Geophysical & Astrophysical Fluid Dynamics, 1998, 88(1-4): 225-259.
[66] Simitev R, Busse F H. Prandtl-number dependence of convection-driven dynamos in rotating spherical fluid shells[J]. Journal of Fluid Mechanics, 2005, 532: 365-388.
[67] Calkins M A, Aurnou J M, Eldredge J D, et al. The influence of fluid properties on the morphology of core turbulence and the geomagnetic field[J]. Earth Planet Sci Lett, 2012, 359/360: 55-60.
[68] Starchenko S V, Jones C A. Typical velocities and magnetic field strengths in planetary interiors[J]. Icarus, 2002, 157(2): 426-435.
[69] 董超,焦立果,张怀,等. 外核黏性对地磁场发电机数值模型的影响[J].地球物理学报, 2018,61(4): 1 366-1 377.
[2] Buffett B A. Earth’s Core and the Geodynamo[J]. Science, 2000, 288(5 473): 2 007-2 012.
[3] Christensen U R, Wicht J. Numerical dynamo simulations [M]. Oxford: Elsevier, 2015, 8: 245-277.
[4] Glatzmaier G A, Roberts P H. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal[J]. Nature, 1995, 377(6 546): 203-209.
[5] Glatzmaier G A, Roberts P H. A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle[J]. Physics of the Earth and Planetary Interiors, 1995, 91(1-3): 63-75.
[6] Song X R, Paul G. Seismological evidence for differential rotation of the Earth’s inner core[J]. Nature, 1996, 382(6 588): 221-224.
[7] Kageyama A, Sato T. Computer simulation of a magnetohydrodynamic dynamo. II[J]. Physics of Plasmas, 1995, 2(5): 1 421-1 431.
[8] Kuang W, Bloxham J. An Earth-like numerical dynamo model[J]. Nature, 1997, 389(6 649): 371-374.
[9] Christensen U R, Olsen P, Glatzmaier G A. Numerical modelling of the geodynamo a systematic parameter study[J]. Geophys J Int, 1999, 138(2): 393-409.
[10] Kono M, Roberts P H. Recent geodynamo simulations and observations of the geomagnetic field[J]. Reviews of Geophysics, 2002, 40(4): 1-53.
[11] Stanley S, Glatzmaier G. Dynamo models for planets other than Earth[J]. Space Science Reviews, 2010, 152: 617-650.
[12] Glatzmaier G A. Numerical simulation of stellar convective dynamos. 1. The model and methods[J]. Journal of Computational Physics, 1984, 55(3): 461-484.
[13] Omara B, Miesch M S, Featherstone N A, et al. Velocity amplitudes in global convection simulations: the role of the Prandtl number and near-surface driving[J]. Advances in Space Research, 2016, 58(8): 1 475-1 489.
[14] Jones C A. Planetary magnetic fields and fluid dynamos[J]. Annu Rev Fluid Mech, 2011, 43: 583-614.
[15] Matsui H, Heien E M, Aubert J, et al. Performance benchmarks for a next generation numerical dynamo model[J]. Geochemistry Geophysics Geosystems, 2016, 17(5): 1 586-1 607.
[16] Tilgner A. Precession-driven dynamos[J]. Physics of Fluids, 2005, 17(3): 34-104.
[17] Cowling T G. The stability of gaseous stars[J]. Monthly Notices of the Royal Astronomical Society, 1934, 94(8): 768-782.
[18] Anufriev A P, Jones C A, Soward A M. The Boussinesq and anelastic liquid approximations for convection in the Earth’s core[J]. Physics of the Earth and Planetary Interiors, 2005, 152(3): 163-190.
[19] Jones C A. Thermal and Compositional Convection in the Outer Core[M]. Oxford: Elsevier, 2015, 8:115-159.
[20] Christensen U R, Aubert J, Cardin P, et al. A numerical dynamo benchmark[J]. Physics of the Earth and Planetary Interiors, 2001, 128: 25-34.
[21] Gubbins D, Roberts P. Magnetohydrodynamics of the Earth’s core[M]//Jacobs J A. Geomagnetism, London: Academic Press, 1987: 1-183.
[22] Braginsky S I, Roberts P H. Equations governing convection in Earth’s core and the geodynamo[J]. Geophysical and Astrophysical Fluid Dynamics, 1995, 79: 1-97.
[23] Glatzmaier G A, Roberts P H. An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection[J]. Physica D, 1996, 97: 81-94.
[24] Kuang W, Bloxham J. Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: weak and strong field dynamo action[J]. Journal of Computational Physics, 1999, 153(1): 51-81.
[25] Gastine T, Duarte L, Wicht J. Dipolar versus multipolar dynamos: the influence of the background density stratification[J]. Astronomy and Astrophysics, 2012, 546: A19.
[26] Evonuk M, Samuel H. Simulating rotating fluid bodies: When is vorticity generation via density stratification important?[J]. Earth Planet. Science Letters, 2012, 317/318: 1-7.
[27] Glatzmaier G A, Roberts P H. Simulating the geodynamo[J]. Contemporary Physics, 1997, 38(4): 269-288.
[28] Olson P, Christensen U, Glatzmaier G A. Numerical modeling of the geodynamo: mechanisms of field generation and equilibration[J]. Journal of Geophysical Research: Solid Earth, 1999, 104(B5): 10 383-10 404.
[29] Glatzmaier G A, Roberts P H. Rotation and magnetism of Earth’s inner core[J]. Science, 1996, 274(5 294): 1 887-1 891.
[30] Makinen A M, Deuss A. Global seismic body wave observations of temporal variations in the Earth’s inner core, and implications for its differential rotation[J]. Geophysical Journal International, 2011, 187(1): 355-370.
[31] Zhang J, Song X, Li Y, et al. Inner core differential motion confirmed by earthquake waveform doublets[J]. Science, 2005, 309(5 739): 1 357-1 360.
[32] Buffett B A, Glatzmaier G A. Gravitational braking of inner-core rotation in geodynamo simulations[J]. Geophysical Research Letters, 2001, 27(19): 3 125-3 128.
[33] Hollerbach R. A spectral solution of the magneto-convection equations in spherical geometry[J]. International Journal for Numerical Methods in Fluids, 2000, 32(7): 773-797.
[34] Harder H, Hansen U. A finite-volume solution method for thermal convection and dynamo problems in spherical shells[J]. Geophysical Journal International, 2005, 161(2): 522-532.
[35] Kageyama A, Sato T. Dipole field generation by an MHD dynamo[J]. Plasma Physics and Controlled Fusion, 1997, 39: 83-92.
[36] Li J, Sato T, Kageyama A. Repeated and sudden reversals of the dipole field generated by spherical dynamo action[J]. Science, 2002, 295(5 561): 1 887-1 890.
[37] Hollerbach R, Jones C A. A geodynamo model incorporating a finitely conducting inner core[J]. Physics of the Earth and Planetary Interiors, 1993, 75(4): 317-327.
[38] Wicht J. Inner-core conductivity in numerical dynamo simulations[J]. Physics of the Earth and Planetary Interiors, 2002, 132(4): 281-302.
[39] Dharmaraj G, Stanley S. Effect of inner core conductivity on planetary dynamo models[J]. Physics of the Earth and Planetary Interiors, 2012, 212/213: 1-9.
[40] Loper D E. Some thermal consequences of the gravitationally powered dynamo[J]. Journal of Geophysical Research, 1978, 83: 5 961-5 970.
[41] Nimmo F. Energetics of the Core[M]. Oxford: Elsevier, 2015, 8: 27-55.
[42] Corgne A, Keshav S, Fei Y W, et al. How much potassium is in the Earth’s core? New insights from partitioning experiments[J]. Earth and Planetary Science Letters, 2007, 256: 567-576.
[43] Rama Murthy V, van Westrenen W, Fei Y. Experimental evidence that potassium is a substantial radioactive heat source in planetary cores[J]. Nature, 2003, 423(6 936): 163-165.
[44] Driscoll P, Olson P. Polarity reversals in geodynamo models with core evolution[J]. Earth and Planetary Science Letters, 2009, 282: 24-32.
[45] Sakuraba A, Roberts P H. Generation of a strong magnetic field using uniform heat flux at the surface of the core[J]. Nature Geoscience, 2009, 2(11): 802-805.
[46] Hori K, Wicht J, Christensen U R. The effect of thermal boundary conditions on dynamos driven by internal heating[J]. Physics of the Earth and Planetary Interiors, 2010, 182: 85-97.
[47] Hori K, Wicht J, Christensen U R. The influence of thermo-compositional boundary conditions on convection and dynamos in a rotating spherical shell[J]. Physics of the Earth and Planetary Interiors, 2012, 196/197: 32-48.
[48] Alboussière T, Deguen R, Melzani M. Melting-induced stratification above the Earth’s inner core due to the convective translation[J]. Nature, 2010, 466: 744-747.
[49] Bullard E C, Gellman H. Homogeneous dynamos and terrestrial magnetism[J]. Proceedings of the Royal Society of London A, 1954, 247(928): 213-278.
[50] Zhang K, Busse F H. Convection driven magnetohydrodynamic dynamos in rotating spherical shells[J]. Geophysical and Astrophysical Fluid Dynamics, 1989, 49: 97-116.
[51] Zhang K, Busse F H. Generation of magnetic fields by convection in a rotating spherical fluid shell of infinite Prandtl number[J]. Physics of the Earth and Planetary Interiors, 1990, 59(3): 208-222.
[52] Sakuraba A, Kono M. Effect of the inner core on the numerical simulation of the magnetohydrodynamic dynamo[J]. Physics of the Earth and Planetary Interiors, 1999, 111: 105-121
[53] Stanley S, Bloxham J. Convective-region geometry as the cause of Uranus’ and Neptune’s unusual magnetic fields[J]. Nature, 2004, 428(6 979): 151-153.
[54] Kageyama A, Yoshida M. Geodynamo and mantle convection simulations on the Earth Simulator using the Yin-Yang grid[J]. Journal of Physics: Conference Series, 2005, 16(1): 325-338.
[55] Wicht J, Stellmach S, Harder H. Numerical models of the geodynamo: from fundamental Cartesian models to 3D simulations of field reversals[M]. Berlin: Springer, 2009, 8: 107-158.
[56] Christensen U R, Aubert J. Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields[J]. Geophysical Journal International, 2006, 166(1): 97-114.
[57] Christensen U R. A sheet-metal geodynamo[J]. Nature, 2008, 454(7 208): 1 058-1 059.
[58] Davies C J, Gubbins D, Jimack P K, et al. Scalability of pseudospectral methods for geodynamo simulations[J].Concurrency and Computation: Practice and Experience, 2011,23(1):38-56.
[59] Christensen U R. Dynamo scaling laws and applications to the planets[J]. Space Science Reviews, 2010, 152(1-4): 565-590.
[60] Christensen U R, Tilgner A. Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos[J]. Nature, 2004, 429(6 988): 169-171.
[61] Aubert J. Steady zonal flows in spherical shell dynamos[J]. Journal of Fluid Mechanics, 2005, 542(1): 53-67.
[62] Kuang W, Jiang W, Wang T. Sudden termination of Martian dynamo?: Implications from subcritical dynamo simulations[J]. Geophysical Research Letters, 2008, 35(14):L14204.
[63] 王天媛, Wei Z G, Jiang W Y,等.数值模拟火星古发电机湮灭前磁场特征[J]. 中国科学: 地球科学, 2013, 43(1): 155-162.
[64] Sreenivasan B, Sahoo S, Dhama G. The role of buoyancy in polarity reversals of the geodynamo[J]. Geophysical Journal International, 2014, 199(3): 1 698-1 708.
[65] Sarson G R, Jones C A, Longbottom A W. Convection driven geodynamo models of varying Ekman number[J]. Geophysical & Astrophysical Fluid Dynamics, 1998, 88(1-4): 225-259.
[66] Simitev R, Busse F H. Prandtl-number dependence of convection-driven dynamos in rotating spherical fluid shells[J]. Journal of Fluid Mechanics, 2005, 532: 365-388.
[67] Calkins M A, Aurnou J M, Eldredge J D, et al. The influence of fluid properties on the morphology of core turbulence and the geomagnetic field[J]. Earth Planet Sci Lett, 2012, 359/360: 55-60.
[68] Starchenko S V, Jones C A. Typical velocities and magnetic field strengths in planetary interiors[J]. Icarus, 2002, 157(2): 426-435.
[69] 董超,焦立果,张怀,等. 外核黏性对地磁场发电机数值模型的影响[J].地球物理学报, 2018,61(4): 1 366-1 377.
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