Numerical Dynamo Simulations: From Basic Concepts to Realistic Models

  • Johannes Wicht
  • Stephan  Stellmach
  • Helmut Harder
Living reference work entry

Abstract

The last years have witnessed an impressive growth in the number and quality of numerical dynamo simulations. The numerical models successfully describe many aspects of the geomagnetic field and also set out to explain the various fields of other planets. The success is somewhat surprising since numerical limitations force dynamo modelers to run their models at unrealistic parameters. In particular the Ekman number, a measure for the relative importance of viscous to Coriolis forces, is many orders of magnitude too large: Earth’s Ekman number is \(\mbox{ E} = 10^{-15}\), while even today’s most advanced numerical simulations have to content themselves with \(\mbox{ E} = 10^{-6}\). After giving a brief introduction into the basics of modern dynamo simulations, we discuss the fundamental force balances and address the question how well the modern models reproduce the geomagnetic field. First-level properties like the dipole dominance, realistic Elsasser and magnetic Reynolds numbers, and an Earth-like reversal behavior are already captured by larger Ekman number simulations around \(\mbox{ E} = 10^{-3}\). However, low Ekman numbers are required for modeling torsional oscillations which are thought to be an important part of the decadal geomagnetic field variations. Moreover, only low Ekman number models seem to retain the huge dipole dominance of the geomagnetic field once the Rayleigh number has been increased to values where field reversals happen. These cases also seem to resemble the low-latitude field found at Earth’s core-mantle boundary more closely than larger Ekman number cases.

Keywords

Rayleigh Number Critical Rayleigh Number Magnetic Reynolds Number Dynamo Action Ekman Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Johannes Wicht thanks Uli Christensen for useful discussions and hints.

References

  1. Alboussière T, Deguen R, Melzani M (2010) Melting-induced stratification above the Earth’s inner core due to convective translation. Nature 466:744–747Google Scholar
  2. Amit H, Aubert J, Hulot G (2010a) Stationary, oscillating or drifting geomagnetic flux patches? J Geophys Res 115:B07108Google Scholar
  3. Amit H, Aubert J, Hulot G, Olson P (2008) A simple model for mantle-driven flow at the top of Earth’s core. Earth Planets Space 60:845–854Google Scholar
  4. Amit H, Choblet G (2009) Mantle-driven geodynamo features – effects of post-perovskite phase transition. Earth Planets Space 61:1255–1268Google Scholar
  5. Amit H, Choblet G (2012) Mantle-driven geodynamo features – effects of compositional and narrow D” anomalies. Phys Earth Planet Inter 190:34–43Google Scholar
  6. Amit H, Korte M, Aubert J, Constable C, Hulot G (2011) The time-dependence of intense archeomagnetic flux patches. J Geophys Res 116(B15):B12106Google Scholar
  7. Amit H, Leonhardt R, Wicht J (2010b) Polarity reversals from paleomagnetic observations and numerical dynamo simulations. Space Sci Rev 155:293–335Google Scholar
  8. Amit H, Olson P (2006) Time-average and time-dependent parts of core flow. Phys Earth Planet Inter 155:120–139Google Scholar
  9. Amit H, Olson P (2008) Geomagnetic dipole tilt changes induced by core flow. Phys Earth Planet Inter 166:226–238Google Scholar
  10. Aubert J (2013) Flow throughout the Earth’s core inverted from geomagnetic observations and numerical dynamo models. Geophys J Int 192:1537–556Google Scholar
  11. Aubert J, Amit H, Hulot G (2007) Detecting thermal boundary control in surface flows from numerical dynamos. Phys Earth Planet Inter 160:143–156Google Scholar
  12. Aubert J, Amit H, Hulot G, Olson P (2008a) Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity. Nature 454:758–761Google Scholar
  13. Aubert J, Aurnou J, Wicht J (2008b) The magnetic structure of convection-driven numerical dynamos. Geophys J Int 172:945–956Google Scholar
  14. Aubert J, Labrosse S, Poitou C (2009) Modelling the paleo-evolution of the geodynamo. Geophys J Int 179:1414–1429Google Scholar
  15. Aubert J, Wicht J (2004) Axial versus equatorial dynamo models with implications for planetary magnetic fields. Earth Planet Sci Lett 221:409–419Google Scholar
  16. Biggin AJ, Steinberger B, Aubert J et al (2012) Possible links between long-term geomagnetic variations and wholemantle convection processes. Nat Geosci 5:674Google Scholar
  17. Bloxham J, Zatman S, Dumberry M (2002) The origin of geomagnetic jerks. Nature 420:65–68Google Scholar
  18. Braginsky S (1970) Torsional magnetohydrodynamic vibrations in the Earth’s core and variation in day length. Geomag Aeron 10:1–8Google Scholar
  19. Braginsky S, Roberts P (1995) Equations governing convection in Earths core and the geodynamo. Geophys Astrophys Fluid Dyn 79:1–97Google Scholar
  20. Breuer M, Manglik A, Wicht J et al (2010) Thermochemically driven convection in a rotating spherical shell. Geophys J Int 183:150–162Google Scholar
  21. Breuer M, Wesseling S, Schmalzl J, Hansen U (2002) Effect of inertia in Rayleigh-Bénard convection. Phys Rev E 69:026320/1–10Google Scholar
  22. Bullard EC, Gellman H (1954) Homogeneous dynamos and terrestrial magnetism. Proc R Soc Lond A A 247:213–278MathSciNetMATHGoogle Scholar
  23. Busse FH, Simitev R (2005a) Convection in rotating spherical fluid shells and its dynamo states. In: Soward AM, Jones CA, Hughes DW, Weiss NO (eds) Fluid dynamics and dynamos in astrophysics and geophysics. CRC Press, Boca Rato, pp 359–392Google Scholar
  24. Busse FH, Simitev R (2005b) Dynamos driven by convection in rotating spherical shells. Atronom Nachr 326:231–240MATHGoogle Scholar
  25. Carlut J, Courtillot V (1998) How complex is the time-averaged geomagnetic field over the past 5 Myr? Geophys J Int 134:527–544Google Scholar
  26. Chan K, Li L, Liao X (2006) Phys Modelling the core convection using finite element and finite difference methods. Earth Planet Inter 157:124–138Google Scholar
  27. Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. Clarendon Press, OxfordMATHGoogle Scholar
  28. Christensen UR (2002) Zonal flow driven by strongly supercritical convection in rotating spherical shells. J Fluid Mech 470:115–133MathSciNetMATHGoogle Scholar
  29. Christensen UR (2006) A deep dynamo generating Mercury’s magnetic field. Nature 444:1056–1058Google Scholar
  30. Christensen UR (2010) Accepted for publication at Space Sci RevGoogle Scholar
  31. Christensen U, Aubert J (2006) Scaling properties of convection-driven dynamos in rotating spherical shells and applications to planetary magnetic fields. Geophys J Int 116:97–114Google Scholar
  32. Christensen UR, Aubert J, Busse FH et al (2001) A numerical dynamo benchmark. Phys Earth Planet Inter 128:25–34Google Scholar
  33. Christensen UR, Aubert J, Hulot G (2010) Conditions for Earth-like geodynamo models. Earth Planet Sci Lett 296:487–496Google Scholar
  34. Christensen UR, Holzwarth V, Reiners A (2009) Energy flux determines magnetic field strength of planets and stars. Nature 457:167–169Google Scholar
  35. Christensen U, Olson P (2003) Secular variation in numerical geodynamo models with lateral variations of boundary heat flow. Phys Earth Planet Inter 138:39–54Google Scholar
  36. Christensen U, Olson P, Glatzmaier G (1999) Numericalmodeling of the geodynamo: a systematic parameter study. Geophys J Int 138:393–409Google Scholar
  37. Christensen U, Tilgner A (2004) Power requirement of the geodynamo from Ohmic losses in numerical and laboratory dynamos. Nature 429:169–171Google Scholar
  38. Christensen U, Wicht J (2007) Numerical dynamo simulations. In: Olson P (eds) Core dynamics. Treatise on geophysics, vol 8. Elsevier, Amsterdam/Boston, pp 245–282Google Scholar
  39. Christensen UR, Wardinski I, Lesur V (2012) Time scales of geomagnetic secular acceleration in satellite field models and geodynamo models. Geophys J Int 190:243–254Google Scholar
  40. Clement B (2004) Dependency of the duration of geomagnetic polarity reversals on site latitude. Nature 428:637–640Google Scholar
  41. Clune T, Eliott J, Miesch M, Toomre J, Glatzmaier G (1999) Computational aspects of a code to study rotating turbulent convection in spherical shells. Parallel Comput 25:361–380MATHGoogle Scholar
  42. Coe R, Hongre L, Glatzmaier A (2000) An examination of simulated geomagnetic reversals from a paleomagnetic perspective. Philos Trans R Soc Lond A 358:1141–1170Google Scholar
  43. Constable C (2000) On the rate of occurence of geomagnetic reversals. Phys Earth Planet Inter 118:181–193Google Scholar
  44. Cowling T (1957) The dynamo maintainance of steady magnetic fields. Q J Mech Appl Math 10:129–136MathSciNetMATHGoogle Scholar
  45. Dormy E, Cardin P, Jault D (1998) Mhd flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field. Earth Planet Sci Lett 158:15–24Google Scholar
  46. Fearn D (1979) Thermal and magnetic instabilities in a rapidly rotating fluid sphere. Geophys Astrophys Fluid Dyn 14:103–126MATHGoogle Scholar
  47. Fournier A, Bunge H-P, Hollerbach R, Vilotte J-P (2005) A Fourier-spectral element algorithm for thermal convection in rotating axisymmetric containers. J Comput Phys 204:462–489MathSciNetMATHGoogle Scholar
  48. Gastine T, Duarte L, Wicht J (2012) Dipolar versus multipolar dynamos: the influence of the background density stratification. Astron Atrophys 546:A19Google Scholar
  49. Gastine T, Wicht J (2012) Effects of compressibility on driving zonal flow in gas giants. Icarus 219:428–442Google Scholar
  50. Gilbert AD, Frisch U, Pouquet A (1988) Helicity is unnecessary for alpha effect dynamos, but it helps. Geophys Astrophys Fluid Dyn 42(1–2):151–161MATHGoogle Scholar
  51. Gillet N, Brito D, Jault D, Nataf H (2007) Experimental and numerical studies of convection in a rapidly rotating spherical shell. J Fluid Mech 580:83MathSciNetMATHGoogle Scholar
  52. Gillet N, Jault D, Canet E, Fournier A (2010) Fast torsional waves and strong magnetic fields within the Earths core. Nature 465:74–77Google Scholar
  53. Glatzmaier G (1984) Numerical simulation of stellar convective dynamos. 1. The model and methods. J Comput Phys 55:461–484Google Scholar
  54. Glatzmaier G (2002) Geodynamo simulations how realistic are they? Annu Rev Earth Planet Sci 30:237–257Google Scholar
  55. Glatzmaier G, Coe R (2007) Magnetic polarity reversals in the core. In: Olson P (eds) Core dynamics. Treatise on geophysics, vol 8. Elsevier, Amsterdam/Boston, pp 283–297Google Scholar
  56. Glatzmaier G, Coe R, Hongre L, Roberts P (1999) The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals. Nature 401:885-890Google Scholar
  57. Glatzmaier G, Roberts P (1995) A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle. Phys Earth Planet Inter 91:63–75Google Scholar
  58. Glatzmaier G, Roberts P (1996) An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection. Physica D 97:81–94Google Scholar
  59. Gubbins D (2001) The Rayleigh number for convection in the Earth’s core. Phys Earth Planet Inter 128:3–12Google Scholar
  60. Gubbins D, Davies CJ (2013) The stratified layer at the core-mantle boundary caused by barodiffusion of oxygen, sulphur and silicon. Phys Earth Planet Inter 215:21–28Google Scholar
  61. Gubbins D, Kelly P (1993) Persistent patterns in the geomagnetic field over the past 2.5 ma. Nature 365:829–832Google Scholar
  62. Gubbins D, Love J (1998) Preferred vgp paths during geomagnetic polarity reversals: symmetry considerations. Geophys Res Lett 25:1079–1082Google Scholar
  63. Gubbins D, Willis AP, Sreenivasan B (2007) Correlation of Earth’s magnetic field with lower mantle thermal and seismic structure. Phys Earth Planet Inter 162:256–260Google Scholar
  64. Harder H, Hansen U (2005) A finite-volume solution method for thermal convection and dynamo problems in spherical shells. Geophys J Int 161:522–532Google Scholar
  65. Heimpel M, Aurnou J, Wicht J (2005) Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model. Nature 438:193–196Google Scholar
  66. Hejda P, Reshetnyak M (2003) Control volume method for the dynamo problem in the sphere with the free rotating inner core. Stud Geophys Geod 47:147–159Google Scholar
  67. Hejda P, Reshetnyak M (2004) Control volume method for the thermal convection problem in a rotating spherical shell: test on the benchmark solution. Stud Geophys Geod 48:741–746Google Scholar
  68. Hongre L, Hulot G, Khokholov A (1998) An analysis of the geomangetic field over the past 2000 years. Phys Earth Planet Inter 106:311–335Google Scholar
  69. Hori K, Wicht J (2013) Subcritical dynamos in the early Mars core: Implications for cessation of the past Martian dynamo. Phys Earth Planet Inter 219:21–33Google Scholar
  70. Hori K, Wicht J, Christensen UR (2010) The effect of thermal boundary conditions on dynamos driven by internal heating. Phys Earth Planet Inter 182:85–97Google Scholar
  71. Hori K, Wicht J, Christensen UR (2012) The influence of thermo-compositional boundary conditions on convection and dynamos in a rotating spherical shell. Phys Earth Planet Inter 196:32–48Google Scholar
  72. Hulot G, Bouligand C (2005) Statistical paleomagnetic field modelling and symmetry considerations. Geophys J Int 161. doi:10.1111/j.1365Google Scholar
  73. Hulot G, Finlay C, Constable C, Olsen N, Mandea M (2010) The magnetic field of planet Earth. Space Sci Rev. doi: 10.1007/s11,214–010–9644–0Google Scholar
  74. Isakov A, Descombes S, Dormy E (2004) An integro-differential formulation of magnet induction in bounded domains: boundary element-finite volume method. J Comput Phys 197:540–554MathSciNetGoogle Scholar
  75. Ivers D, James R (1984) Axisymmetric antidynamo theorems in non-uniform compressible fluids. Philos Trans R Soc Lond A 312:179–218MathSciNetMATHGoogle Scholar
  76. Jackson A (1997) Time dependence of geostrophic core-surface motions. Phys Earth Planet Inter 103:293–311Google Scholar
  77. Jackson A (2003) Intense equatorial flux spots on the surface of the Earth’s core. Nature 424:760–763Google Scholar
  78. Jackson A, Finlay C (2007) Geomagnetic secular variation and applications to the core. In: Kono M (ed) Geomagnetism. Treatise on geophysics, vol 5. Elsevier, Amsterdam, pp 147–193Google Scholar
  79. Jackson A, Jonkers A, Walker M (2000) Four centuries of geomagnetic secular variation from historical records. Philos Trans R Soc Lond A358:957–990Google Scholar
  80. Jault D (2003) Electromagnetic and topographic coupling, and LOD variations. In: Jones CA, Soward AM, Zhang K (eds) Earth’s core and lower mantle. Taylor & Francis, London/New York, pp 56–76Google Scholar
  81. Jault D, Gire C, LeMouël J-L (1988) Westward drift, core motion and exchanges of angular momentum between core and mantle. Nature 333:353–356Google Scholar
  82. Johnson C, Constable C (1995) Time averaged geomagnetic field as recorded by lava flows over the past 5 Myr. Geophys J Int 122:489–519Google Scholar
  83. Johnson C, Constable C, Tauxe L (2003) Mapping long-term changed in Earth’s magnetic field. Science 300:2044–2045Google Scholar
  84. Johnson CL, McFadden P (2007) Time-averaged field and paleosecular variation. In: Kono M (ed) Geomagnetism. Treatise on geophysics, vol 5. Elsevier, Amsterdam, pp 217–254Google Scholar
  85. Jones C (2007) Thermal and compositional convection in the outer core. In: Olson P (eds) Core dynamics. Treatise on geophysics, vol 8. Elsevier, Amsterdam/Boston, pp 131–186Google Scholar
  86. Jones CA, Boronski P, Brun AS et al (2011) Anelastic convection-driven dynamo benchmarks. Icarus 216:120–135Google Scholar
  87. Jonkers A (2003) Long-range dependence in the cenozoic reversal record. Phys Earth Planet Inter 135:253–266Google Scholar
  88. Julien K, Knobloch E (1998) Strongly nonlinear convection cells in a rapidly rotating fluid layer: the tilted f-plane. J Fluid Mech 360:141–178MathSciNetMATHGoogle Scholar
  89. Julien K, Knobloch E, Werne J (1998) A new class of equations for rotationally constrained flows. Theor Comput Fluid Dyn 11(3–4):251–261MATHGoogle Scholar
  90. Julien K, Rubio A, Grooms I, Knobloch E (2012) Statistical and physical balances in low Rossby number Rayleigh–Bénard convection. Geophys Astrophys Fluid Dyn 106(4–5):392–428MathSciNetGoogle Scholar
  91. Kageyama A, Miyagoshi T, Sato T (2008) Formation of current coils in geodynamo simulations. Nature 454:1106–1109Google Scholar
  92. Kageyama A, Sato T (1995) Computer simulation of a magnetohydrodynamic dynamo. II. Phys Plasmas 2:1421–1431Google Scholar
  93. Kageyama A, Sato T (1997) Generation mechanism of a dipole field by a magnetohydrodynamic dynamo. Phys Rev E 55:4617–4626MathSciNetGoogle Scholar
  94. Kageyama A, Watanabe K, Sato T (1993) Simulation study of a magnetohydrodynamic dynamo: convection in a rotating shell. Phys Fluids B 24(8):2793–2806Google Scholar
  95. Kageyama A, Yoshida M (2005) Geodynamo and mantle convection simulations on the Earth simulator using the yin-yang grid. J Phys Conf Ser 16:325–338Google Scholar
  96. Kaiser R, Schmitt P, Busse F (1994) On the invisible dynamo. Geophys Astrophys Fluid Dyn 77:93–109Google Scholar
  97. Kelly P, Gubbins D (1997) The geomagnetic field over the past 5 million years. Geophys J Int 128:315–330Google Scholar
  98. Kono M, Roberts P (2002) Recent geodynamo simulations and observations of the geomagnetic field. Rev Geophys 40:1013. doi:10.1029/2000RG000102Google Scholar
  99. Korte M, Constable C (2005) Continuous geomagnetic field models for the past 7 millennia: 2. cals7k. Geochem Geophys Geosys 6:Q02H16Google Scholar
  100. Korte M, Constable C, Donadini F, Holme R (2011) Reconstructing the Holocene geomagnetic field. Earth Planet Sci Lett 312:497–505Google Scholar
  101. Korte M, Genevey A, Constable C, Frank U, Schnepp E (2005) Continuous geomagnetic field models for the past 7 millennia: 1. A new global data compilation. Geochem Geophys Geosyst 6:Q02H15Google Scholar
  102. Korte M, Holme R (2010) On the persistence of geomagnetic flux lobes in global Holocene field models. Phys Earth Planet Inter 182:179–186Google Scholar
  103. Kuang W, Bloxham J (1997) An Earth-like numerical dynamo model. Nature 389:371–374Google Scholar
  104. Kuang W, Bloxham J (1999) Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: weak and strong field dynamo action. J Comput Phys 153:51–81MathSciNetMATHGoogle Scholar
  105. Kuang W, Jiang W, Wang T (2008) Sudden termination of martian dynamo? Implications from subcritical dynamo simulations. Geophys Res Lett 35(14):14,202Google Scholar
  106. Kutzner C, Christensen U (2000) Effects of driving mechanisms in geodynamo models. Geophys Res Lett 27:29–32Google Scholar
  107. Kutzner C, Christensen U (2002) From stable dipolar to reversing numerical dynamos. Phys Earth Planet Inter 131:29–45Google Scholar
  108. Kutzner C, Christensen U (2004) Simulated geomagnetic reversals and preferred virtual geomagnetic pole paths. Geophys J Int 157:1105–1118Google Scholar
  109. Lhuillier F, Fournier A, Hulot G, Aubert J (2011) The geomagnetic secular variation timescale in observations and numerical dynamo models. Geophys Res Lett 38:L09306Google Scholar
  110. Lillis R, Frey H, Manga M (2008) Rapid decrease in martian crustal magnetization in the noachian era: implications for the dynamo and climate of early mars. Geophys Res Lett 35(14):14,203Google Scholar
  111. Manglik A, Wicht J, Christensen UR (2010) A dynamo model with double diffusive convection for Mercurys core. Earth Planet Sci Lett 289:619–628Google Scholar
  112. Matsui H, Buffett B (2005) Sub-grid scale model for convection-driven dynamos in a rotating plane layer. Phys Earth Planet Inter 153:74–82Google Scholar
  113. Miyagoshi T, Kageyama A, Sato T (2010) Zonal flow formation in the Earth’s core. Nature 463(7282):793–796Google Scholar
  114. Miyagoshi T, Kageyama A, Sato T (2011) Formation of sheet plumes, current coils, and helical magnetic fields in a spherical magnetohydrodynamic dynamo. Phys Plasmas 18:072901Google Scholar
  115. Monnereau M, Calvet M, Margerin L, Souriau A (2010) Lopsided growth of Earth’s inner core. Science 328:1014Google Scholar
  116. Morin V, Dormy E (2009) The dynamo bifurcation in rotating spherical shells. Int J Mod Phys B 23(28n29):5467–5482Google Scholar
  117. Olsen N, Haagmans R, Sabaka TJ et al (2006) The Swarm End-to-End mission simulator study: a demonstration of separating the various contributions to Earth’s magnetic field using synthetic data. Earth Planets Space 58:359–370Google Scholar
  118. Olson P, Christensen U (2002) The time-averaged magnetic field in numerical dynamos with nonuniform boundary heat flow. Geophys J Int 151:809–823Google Scholar
  119. Olson P, Christensen U (2006) Dipole moment scaling for convection-driven planetary dynamos. Earth Planet Sci Lett 250:561–571Google Scholar
  120. Olson P, Christensen UR, Driscoll PE (2012) From superchrons to secular variation: a broadband dynamo frequency spectrum for the geomagnetic dipole. Earth Planet Sci Lett 319–320:75–82Google Scholar
  121. Olson P, Christensen U, Glatzmaier G (1999) Numerical modeling of the geodynamo: mechanism of field generation and equilibration. J Geophys Res 104:10383–10404Google Scholar
  122. Pozzo M, Davies C, Gubbins D, Alfè D (2012) Thermal and electrical conductivity of iron at Earth’s core conditions. Nature 485:355–358Google Scholar
  123. Proctor M (1994) Convection and magnetoconvection in a rapidly rotating sphere. In: Proctor MRE, Gilbert AD (eds) Lectures on solar and planetary dynamos, vol 1. Cambridge University Press, Cambridge/New York, p 97Google Scholar
  124. Roberts P (1972) Kinematic dynamo models. Philos Trans R Soc Lond A 271:663–697Google Scholar
  125. Roberts P (2007) Theory of the geodynamo. In: Olson P (eds) Core dynamics. Treatise on geophysics, vol 8. Elsevier, Amsterdam/Boston, pp 245–282Google Scholar
  126. Ryan DA, Sarson GR (2007) Are geomagnetic field reversals controlled by turbulence within the Earth’s core? Geophys Res Lett 34:2307Google Scholar
  127. Sakuraba A (2002) Linear magnetoconvection in rotating fluid spheres permeated by a uniform axial magnetic field. Geophys Astrophys Fluid Dyn 96:291–318MathSciNetMATHGoogle Scholar
  128. Sakuraba A, Kono M (2000) Effect of a uniform magnetic field on nonlinear magnetocenvection in a rotating fluid spherical shell. Geophys Astrophys Fluid Dyn 92:255–287MathSciNetGoogle Scholar
  129. Sakuraba A, Roberts P (2009) Generation of a strong magnetic field using uniform heat flux at the surface of the core. Nat Geosci 2:802–805Google Scholar
  130. Schmalzl J, Breuer M, Hansen U (2002) The influence of the Prandtl number on the style of vigorous thermal convection. Geophys Astrophys Fluid Dyn 96:381–403MATHGoogle Scholar
  131. Simitev R, Busse F (2005) Prandtl-number dependence of convection-driven dynamos in rotating spherical fluid shells. J Fluid Mech 532:365–388MathSciNetMATHGoogle Scholar
  132. Simitev RD, Busse FH (2009) Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells. Europhys Lett 85:19001Google Scholar
  133. Soderlund KM, King E, Aurnou JM (2012) The influence of magnetic fields in planetary dynamo models. Earth Planet Sci Lett 333–334:9–20Google Scholar
  134. Sprague M, Julien K, Knobloch E, Werne J (2006) Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J Fluid Mech 551:141–174MathSciNetMATHGoogle Scholar
  135. Sreenivasan B (2009) On dynamo action produced by boundary thermal coupling. Phys Earth Planet Inter 177:130–138Google Scholar
  136. Sreenivasan B, Jones CA (2006) The role of inertia in the evolution of spherical dynamos. Geophys J Int 164:467–476Google Scholar
  137. Sreenivasan B, Jones CA (2011) Helicity generation and subcritical behaviour in rapidly rotating dynamos. J Fluid Mech 688:5–30MathSciNetMATHGoogle Scholar
  138. St Pierre M (1993) The strong field branch of the Childress-Soward dynamo. In: Proctor MRE et al (eds) Solar and planetary dynamos, Cambridge University Press, Cambridge, pp 329–337Google Scholar
  139. Stanley S, Bloxham J (2004) Convective-region geometry as the cause of Uranus’ and Neptune’s unusual magnetic fields. Nature 428:151–153Google Scholar
  140. Stanley S, Bloxham J, Hutchison W, Zuber M (2005) Thin shell dynamo models consistent with mercurys weak observed magnetic field. Earth Planet Sci Lett 234:341–353Google Scholar
  141. Stanley S, Glatzmaier G (2010) Dynamo models for planets other than Earth. Space Sci Rev 152:617–649Google Scholar
  142. Stellmach S, Hansen U (2004) Cartesian convection-driven dynamos at low ekman number. Phys Rev E 70:056312Google Scholar
  143. Stelzer Z, Jackson A (2013, in press) Extracting scaling laws from numerical dynamo models. Geophys J IntGoogle Scholar
  144. Stieglitz R, Müller U (2001) Experimental demonstration of the homogeneous two-scale dynamo. Phys Fluids 1:561–564Google Scholar
  145. Takahashi F, Matsushima M (2006) Dipolar and non-dipolar dynamos in a thin shell geometry with implications for the magnetic field of Mercury. Geophys Res Lett 33:L10202Google Scholar
  146. Takahashi F, Matsushima M, Honkura Y (2008a) Scale variability in convection-driven MHD dynamos at low Ekman number. Phys Earth Planet Inter 167:168–178Google Scholar
  147. Takahashi F, Tsunakawa H, Matsushima M, Mochizuki N, Honkura Y (2008b) Effects of thermally heterogeneous structure in the lowermost mantle on the geomagnetic field strength. Earth Planet Sci Lett 272:738–746Google Scholar
  148. Taylor J (1963) The magneto-hydrodynamics of a rotating fluid and the Earth’s dynamo problem. Proc R Soc Lond A 274:274–283MATHGoogle Scholar
  149. Tilgner A (1996) High-Rayleigh-number convection in spherical shells. Phys Rev E 53:4847–4851Google Scholar
  150. Vallis GK (2006) Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation. Cambridge University Press, CambridgeGoogle Scholar
  151. Wicht J (2002) Inner-core conductivity in numerical dynamo simulations. Phys Earth Planet Inter 132:281–302Google Scholar
  152. Wicht J (2005) Palaeomagnetic interpretation of dynamo simulations. Geophys J Int 162:371–380Google Scholar
  153. Wicht J, Aubert J (2005) Dynamos in action. GWDG-Bericht 68:49–66Google Scholar
  154. Wicht J, Christensen UR (2010) Torsional oscillations in dynamo simulations. Geophys J Int 181:1367–1380Google Scholar
  155. Wicht J, Mandea M, Takahashi F et al (2007) The origin of Mercurys internal magnetic field. Space Sci Rev 132:261–290Google Scholar
  156. Wicht J, Olson P (2004) A detailed study of the polarity reversalmechanism in a numerical dynamo model. Geochem Geophys Geosyst 5. doi:10.1029/2003GC000602Google Scholar
  157. Wicht J, Stellmach S, Harder H (2009) Numerical models of the geodynamo: from fundamental Cartesian models to 3d simulations of field reversals. In: Glassmeier K, Soffel H, Negendank J (eds) Geomagnetic field variations – space-time structure, processes, and effects on system Earth. Springer monograph. Springer, Berlin/Heidelberg/NewYork, pp 107–158Google Scholar
  158. Wicht J, Tilgner A (2010) Theory and modeling of planetary dynamos. Space Sci Rev 152:501–542Google Scholar
  159. Willis AP, Sreenivasan B, Gubbins D (2007) Thermal core mantle interaction: exploring regimes for locked dynamo action. Phys Earth Planet Inter 165:83–92Google Scholar
  160. Yadav RK, Gastine T, Christensen UR (2013) Scaling laws in spherical shell dynamos with freeslip boundaries. Icarus 225:185–193Google Scholar
  161. Zatman S, Bloxham J (1997) Torsional oscillations and the magnetic field within the Earth’s core. Nature 388:760–761Google Scholar
  162. Zhang K-K, Busse F (1988) Finite amplitude convection and magnetic field generation in in a rotating spherical shell. Geophys Astrophys Fluid Dyn 44:33–53MATHGoogle Scholar
  163. Zhang K, Gubbins D (2000a) Is the geodynamo process intrinsically unstable? Geophys J Int 140:F1–F4Google Scholar
  164. Zhang K, Gubbins D (2000b) Scale disparities and magnetohydrodynamics in the Earth’s core. Philos Trans R Soc Lond A 358:899–920MathSciNetMATHGoogle Scholar
  165. Zhang K, Schubert G (2000) Magnetohydrodynamics in rapidly rotating spherical systems. Annu Rev Fluid Mech 32:409–443MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Johannes Wicht
    • 1
  • Stephan  Stellmach
    • 2
  • Helmut Harder
    • 3
  1. 1.Max-Planck Intitut für SonnensystemforschungKaltenburg-LindauGermany
  2. 2.Institut für Geophysik, Westfählische Wilhelms-UniversitätMünsterGermany
  3. 3.Institut für Geophysik, Westfählische Wilhelms-UniversitätMünsterGermany

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