The fundamental physics and dynamics relating to two-layer convection with an infinite Prandtl number and large viscosity contrasts have not yet been quantitatively resolved by previous numerical analyses or simulations and laboratory experiments. Here, a series of high-resolution numerical simulations of Rayleigh-Bénard convection with a highly viscous outer layer (HVL) and a low-viscosity inner layer (LVL) in 2-D spherical-shell geometry were performed to investigate the dynamics of convection between the two layers with large viscosity contrasts of up to 103. To achieve a two-layer thermal convection system considering a thermally and mechanically continuous interface between the two layers without any specified boundary conditions, an “effective thermal expansion coefficient” was introduced to the buoyancy term of the momentum equation, discretized in a finite-volume-based regular grid system. In this study, the heat transport efficiency of two-layer convection was evaluated, and the coupling modes between the two layers were directly analyzed using the temperature anomaly and deviatoric stress fields near the interface. Results show that the mechanical coupling mode is dominant in two-layer convection when the absolute viscosity contrast between the two layers is sufficiently small, and it weakens, becoming closer to the thermal coupling mode, as the LVL viscosity decreases. This transition from the mechanical coupling to the thermal coupling modes is quantitatively detected even when the viscosity contrast between the two layers is 10−3, and results in the stabilizing of the convection speed and the heat transport efficiency of the HVL. Applying the mantle–outer core coupling of the present Earth with an extremely large absolute viscosity contrast, our numerical results imply that thermal convection in the mantle may control the heat transport efficiency of a layered whole solid-earth system and the convective style in the outer core.
REFERENCES
1.
G. A.
Glatzmaier
, Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation
(Princeton University Press
, Princeton, NJ, USA
, 2013
).2.
G.
Schubert
, D. L.
Turcotte
, and P.
Olson
, Mantle Convection in the Earth and Planets
(Cambridge University Press
, UK
, 2001
).3.
S.
Honda
, “A simple parameterized model of Earth’s thermal history with the transition from layered to whole mantle convection
,” Earth Planet. Sci. Lett.
131
(3-4
), 357
–369
(1995
).4.
U. R.
Christensen
, “Thermal evolution models for the Earth
,” J. Geophys. Res.
90
(B4
), 2995
–3007
, doi:10.1029/JB090iB04p02995 (1985
).5.
S.
Maruyama
, M.
Santosh
, and D.
Zhao
, “Superplume, supercontinent, and post-perovskite: Mantle dynamics and anti-plate tectonics on the core–mantle boundary
,” Gondwana Res.
11
(1-2
), 7
–37
(2007
).6.
7.
J.
Bloxham
and D.
Gubbins
, “Thermal core–mantle interactions
,” Nature
325
, 511
–513
(1987
).8.
S.
Yoshida
and Y.
Hamano
, “The westward drift of the geomagnetic field caused by length-of-day variation, and the topography of the core-mantle boundary
,” Geophys. J. Int.
114
(3
), 696
–710
(1993
).9.
K.
Zhang
and D.
Gubbins
, “On convection in the earth’s core driven by lateral temperature variations in the lower mantle
,” Geophys. J. Int.
108
(1
), 247
–255
(1992
).10.
P.
Cardin
and H.-C.
Nataf
, “Nonlinear dynamical coupling observed near the threshold of convection in a two-layer system
,” Europhys. Lett.
14
(7
), 655
–660
(1991
).11.
A.
Kageyama
and M.
Yoshida
, “Geodynamo and mantle convection simulations on the Earth Simulator using the Yin-Yang grid
,” J. Phys. Conf. Ser.
16
, 325
–338
(2005
).12.
S.
Karato
, Deformation of Earth Materials: Introduction to the Rheology of the Solid Earth
(Cambridge University Press
, UK
, 2008
).13.
M.
Kono
and P. H.
Roberts
, “Recent geodynamo simulations and observations of the geomagnetic field
,” Rev. Geophys.
40
(4
), 4-1
–4-53
, doi:10.1029/2000RG000102 (2002
).14.
G. A.
Glatzmaier
, “Geodynamo simulations—How realistic are they?
,” Annu. Rev. Earth Planet. Sci.
30
, 237
–257
(2002
).15.
E.
Dormy
, J.-P.
Valet
, and V.
Courtillot
, “Numerical models of the geodynamo and observational constraints
,” Geochem., Geophys., Geosyst.
1
(10
), 2000GC000062
, doi:10.1029/2000GC000062 (2000
).16.
M.
Yoshida
and A.
Kageyama
, “Application of the Yin-Yang grid to a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell
,” Geophys. Res. Lett.
31
(12
), L12609
, doi:10.1029/2004GL019970 (2004
).17.
L.
Vecsey
, “Chaos in thermal convection and the wavelet analysis of geophysical fields
,” Ph.D. thesis, Faculty of Mathematics and Physics, Charles University in Prague
, 2003
.18.
L.
Vecsey
, D. A.
Yuen
, E. O. D.
Sevre
, and F.
Dubuffet
, “Ultra-high Ra convection and applications of wavelet
,” in Abstracts of 8th International Workshop on Numerical Modeling of Mantle Convection and Lithospheric Dynamics, Hruba Skala, Czech Republic
(European Geosciences Union
, 2003
), Vol. 34
.19.
D. A.
Yuen
, U.
Hansen
, W.
Zhao
, A. P.
Vincent
, and A. V.
Malevsky
, “Hard turbulent thermal convection and thermal evolution of the mantle
,” J. Geophys. Res.
98
(E3
), 5355
–5373
, doi:10.1029/92JE02725 (1993
).20.
F.
Dubuffet
, D. A.
Yuen
, M. S.
Murphy
, E. O.
Sevre
, and L.
Vecsey
, “Secondary instabilities developed in upwellings of high Rayleigh number (2-D and 3-D) convection
,” Abstracts of American Geophysical Union Fall Meeting, 2001
, Abstract No. T42A-0912.21.
G.
Erlebacher
, D. A.
Yuen
, and F.
Dubuffet
, “Case study: Visualization and analysis of high Rayleigh number—3D convection in the Earth’s mantle
,” in IEEE Visualization Conference
(IEEE
, 2002
), pp. 529
–532
.22.
A. V.
Malevsky
and D. A.
Yuen
, “Plume structures in the hard-turbulent regime of three-dimensional infinite Prandtl number convection
,” Geophys. Res. Lett.
20
(5
), 383
–386
, doi:10.1029/93GL00293 (1993
).23.
M. S.
Murphy
, L.
Vecsey
, E. O.
Sevre
, and D. A.
Yuen
, “Secondary upwelling instabilities developed in high Rayleigh number convection: Possible applications to hot spots
,” Vis. Geosci.
5
(4
), 1
–10
(2000
).24.
D. A.
Yuen
and A. V.
Malevsky
, “Strongly chaotic Newtonian and non-Newtonian mantle convection
,” in Chaotic Processes in the Geological Sciences
, edited by D. A.
Yuen
(Springer-Verlag
, New York, USA
, 1992
), pp. 71
–88
.25.
M.
Breuer
and U.
Hansen
, “Turbulent convection in the zero Reynolds number limit
,” Europhys. Lett.
86
, 24004
(2009
).26.
A. P.
Vincent
and D. A.
Yuen
, “Plumes and waves in two-dimensional turbulent thermal convection
,” Phys. Rev. E
60
(3
), 2957
–2963
(1999
).27.
A. P.
Vincent
and D. A.
Yuen
, “Transition to turbulent thermal convection beyond Ra = 1010 detected in numerical simulations
,” Phys. Rev. E
61
(5
), 5241
–5246
(2000
).28.
A.
Pandey
, M. K.
Verma
, and P. K.
Mishra
, “Scaling of heat flux and energy spectrum for very large Prandtl number convection
,” Phys. Rev. E
89
(2
), 023006
(2014
).29.
A.
Pandey
, M. K.
Verma
, A. G.
Chatterjee
, and B.
Dutta
, “Similarities between 2D and 3D convection for large Prandtl number
,” Pramana
87
, 13
(2016
).30.
M. K.
Verma
, S. C.
Ambhire
, and A.
Pandey
, “Flow reversals in turbulent convection with free-slip walls
,” Phys. Fluids
27
, 047102
(2015
).31.
J.
Schmalzl
, M.
Breuer
, and U.
Hansen
, “On the validity of two-dimensional numerical approaches to time-dependent thermal convection
,” Europhys. Lett.
67
(3
), 390
–396
(2004
).32.
J.
Schmalzl
, M.
Breuer
, and U.
Hansen
, “The influence of the Prandtl number on the style of vigorous thermal convection
,” Geophys. Astrophys. Fluid Dyn.
96
(5
), 381
–403
(2002
).33.
H.-P.
Bunge
, M. A.
Richards
, and J. R.
Baumgardner
, “Effect of depth-dependent viscosity on the planform of mantle convection
,” Nature
379
, 436
–438
(1996
).34.
J. T.
Ratcliff
, P. J.
Tackley
, G.
Schubert
, and A.
Zebib
, “Transitions in thermal convection with strongly variable viscosity
,” Phys. Earth Planet. Int.
102
(3-4
), 201
–212
(1997
).35.
Y.
Iwase
, “Three-dimensional infinite Prandtl number convection in a spherical shell with temperature-dependent viscosity
,” J. Geomagn. Geoelectr.
48
(12
), 1499
–1514
(1996
).36.
A.
Kageyama
, T.
Sato
, K.
Watanabe
, R.
Horiuchi
, T.
Hayashi
, Y.
Todo
, T.
Watanabe
, and H.
Takamaru
, “Computer simulation of a magnetohydrodynamic dynamo. II
,” Phys. Plasmas
2
, 1421
–1431
(1995
).37.
G.
Glatzmaiers
and P. H.
Roberts
, “A three-dimensional self-consistent computer simulation of a geomagnetic field reversal
,” Nature
377
, 203
–209
(1995
).38.
G. A.
Glatzmaier
and P. H.
Roberts
, “A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle
,” Phys. Earth Planet. Int.
91
(1-3
), 63
–75
(1995
).39.
S.
Honda
, “Numerical analysis of layered convection: Marginal stability and finite amplitude analyses
,” Bull. Earthq. Res. Inst.
57
, 273
–302
(1982
).40.
K.
Ukaji
and R.
Sawada
, “Cellular convection in the two-layer fluid
,” Abstr. Meteorolog. Soc. Jpn. Meet.
17
, 96
(1970
) (in Japanese).41.
K.
Ukaji
and R.
Sawada
, “Cellular convection in the two-layer fluid (2)
,” Abstr. Meteorolog. Soc. Jpn. Meet.
18
, 37
(1970
) (in Japanese).42.
K.
Ukaji
and R.
Sawada
, “Cellular convection in the two-layer fluid (Theory 3)
,” Abstr. Meteorolog. Soc. Jpn. Meet.
20
, 2
(1971
) (in Japanese).43.
A.
Prakash
, K.
Yasuda
, F.
Otsubo
, K.
Kuwahara
, and T.
Doi
, “Flow coupling mechanisms in two-layer Rayleigh-Benard convection
,” Exp. Fluids
23
(3
), 252
–261
(1997
).44.
D.
Johnson
and R.
Narayanan
, “Geometric effects on convective coupling and interfacial structures in bilayer convection
,” Phys. Rev. E
56
(5
), 5462
–5472
(1997
).45.
J.
Boussinesq
, Théorie Analytique de la Chaleur mise en Harmonie avec la Thermodynamique et avec la Théorie Mécanique de la Lumière, Tome II
(Gauthier-Villars
, Paris
, 1903
).46.
A.
Oberbeck
, “Ueber die wärmeleitung der flüssigkeiten bei berücksichtigung der strömungen infolge von temperaturdifferenzen
,” Ann. Phys.
243
(6
), 271
–292
(1879
).47.
M.
Yoshida
, “Preliminary three-dimensional model of mantle convection with deformable, mobile continental lithosphere
,” Earth Planet. Sci. Lett.
295
(1-2
), 205
–218
(2010
).48.
C.
Adam
, M.
Yoshida
, T.
Isse
, D.
Suetsugu
, Y.
Fukao
, and G.
Barruol
, “South Pacific hotspot swells dynamically supported by mantle flows
,” Geophys. Res. Lett.
37
, L08302
, doi:10.1029/2010GL042534 (2010
).49.
L. E.
Malvern
, Introduction to the Mechanics of a Continuous Medium
(Prentice Hall
, Englewood Cliffs, NJ, USA
, 1969
).50.
K.
Lambeck
and P.
Johnston
, “The viscosity of the mantle: Evidence from the analysis of glacial rebound phenomena
,” in The Earth’s Mantle, Composition, Structure and Evolution
, edited by I.
Jackson
(Cambridge University Press
, UK
, 1998
), pp. 461
–502
.51.
U. R.
Christensen
and D. A.
Yuen
, “Layered convection induced by phase transitions
,” J. Geophys. Res.
90
(B12
), 10291
–10300
, doi:10.1029/JB090iB12p10291 (1985
).52.
F. M.
Richter
, “Finite amplitude convection through a phase boundary
,” Geophys. J. Int.
35
, 265
–276
(1973
).53.
M.
Yoshida
, Y.
Iwase
, and S.
Honda
, “Generation of plumes under a localized high viscosity lid in 3-D spherical shell convection
,” Geophys. Res. Lett.
26
(7
), 947
–950
, doi:10.1029/1999GL900147 (1999
).54.
M.
Liu
, D. A.
Yuen
, W.
Zhao
, and S.
Honda
, “Development of diapiric structures in the upper mantle due to phase transitions
,” Science
252
(5014
), 1836
–1839
(1991
).55.
S.
Honda
, D. A.
Yuen
, S.
Balachandar
, and D.
Reutelcr
, “Three-dimensional instabilities of mantle convection with multiple phase transitions
,” Science
259
(5099
), 1308
–1311
(1993
).56.
S.
Honda
and D. A.
Yuen
, “Model for convective cooling of mantle with phase changes: Effects of aspect ratios and initial conditions
,” J. Phys. Earth
42
(2
), 165
–186
(1994
).57.
M.
Wolstencroft
and J. H.
Davies
, “Influence of the Ringwoodite-Perovskite transition on mantle convection in spherical geometry as a function of Clapeyron slope and Rayleigh number
,” Solid Earth
2
, 315
–326
(2011
).58.
J. X.
Mitrovica
, “Haskell [1935] revisited
,” J. Geophys. Res.
101
(B1
), 555
–569
, doi:10.1029/95JB03208 (1996
).59.
D. L.
Turcotte
and E. R.
Oxburgh
, “Finite amplitude convective cells and continental drift
,” J. Fluid Mech.
18
(1
), 29
–42
(1967
).60.
A.
Davaille
, “Two-layer thermal convection in miscible viscous fluids
,” J. Fluid Mech.
379
, 223
–253
(1999
).61.
M.
Le Bars
and A.
Davaille
, “Large interface deformation in two-layer thermal convection of miscible viscous fluids
,” J. Fluid Mech.
499
, 75
–110
(2004
).62.
M.
Le Bars
and A.
Davaille
, “Stability of thermal convection in two superimposed miscible viscous fluids
,” J. Fluid Mech.
471
, 339
–363
(2002
).63.
A.
Davaille
, M.
Le Bars
, and C.
Carbonne
, “Thermal convection in a heterogeneous mantle
,” C. R. Geosci.
335
(1
), 141
–156
(2003
).64.
P.
Machetel
and P.
Weber
, “Intermittent layered convection in a model mantle with an endothermic phase change at 670 km
,” Nature
350
, 55
–57
(1991
).65.
P. J.
Tackley
, D. J.
Stevenson
, G. A.
Glatzmaier
, and G.
Schubert
, “Effects of an endothermic phase transition at 670 km depth in a spherical model of convection in the Earth’s mantle
,” Nature
361
(6414
), 699
–704
(1993
).66.
D. L.
Turcotte
and G.
Schubert
, Geodynamics
, 3rd ed. (Cambridge University Press
, UK
, 2014
).67.
V. S.
Solomatov
, “Scaling of temperature- and stress-dependent viscosity convection
,” Phys. Fluids
7
, 266
–274
(1995
).68.
J. M.
Hewitt
, D. P.
Mckenzie
, and N. O.
Weiss
, “Dissipative heating in convective flows
,” J. Fluid Mech.
68
(4
), 721
–738
(1975
).69.
P. H.
Roberts
and J. M.
Aurnou
, “On the theory of core-mantle coupling
,” Geophys. Astrophys. Fluid Dyn.
106
(2
), 157
–230
(2012
).70.
P.
Olson
, R.
Deguen
, M. L.
Rudolph
, and S.
Zhong
, “Core evolution driven by mantle global circulation
,” Phys. Earth Planet. Int.
243
, 44
–55
(2015
).71.
M.
Petry
and F. H.
Busse
, “Theoretical study of flow coupling mechanisms in two-layer Rayleigh-Bénard convection
,” Phys. Rev. E
68
(1
), 016305
(2003
).72.
S. V.
Diwakar
, S.
Tiwari
, S. K.
Das
, and T.
Sundararajan
, “Stability and resonant wave interactions of confined two-layer Rayleigh–Bénard systems
,” J. Fluid Mech.
754
, 415
–455
(2014
).73.
Y.-C.
Xie
and K.-Q.
Xia
, “Dynamics and flow coupling in two-layer turbulent thermal convection
,” J. Fluid Mech.
728
, R1
(2013
).74.
L.
Cserepes
, M.
Rabinowicz
, and C.
Rosemberg-Borot
, “Three-dimensional infinite Prandtl number convection in one and two layers with implications for the Earth’s gravity field
,” J. Geophys. Res.
93
(B10
), 12009
–12025
, doi:10.1029/JB093iB10p12009 (1988
).75.
K.
Ellsworth
and G.
Schubert
, “Numerical models of thermally and mechanically coupled two-layer convection of highly viscous fluids
,” Geophys. J. Int.
93
(2
), 347
–363
(1988
).76.
G. A.
Glatzmaier
and G.
Schubert
, “Three-dimensional spherical models of layered and whole mantle convection
,” J. Geophys. Res.
98
(B12
), 21969
–21976
, doi:10.1029/93JB02111 (1993
).77.
P. H.
Roberts
and G. A.
Glatzmaier
, “Geodynamo theory and simulations
,” Rev. Mod. Phys.
72
(4
), 1081
–1123
(2000
).78.
N.
Coltice
, T.
Rolf
, P. J.
Tackley
, and S.
Labrosse
, “Dynamic causes of the relation between area and age of the ocean floor
,” Science
336
(6079
), 335
–338
(2012
).79.
B. R.
Phillips
, H.-P.
Bunge
, and K.
Schaber
, “True polar wander in mantle convection models with multiple, mobile continents
,” Gondwana Res.
15
(3–4
), 288
–296
(2009
).80.
T.
Lay
, J.
Hernlund
, and B. A.
Buffett
, “Core–mantle boundary heat flow
,” Nat. Geosci.
1
, 25
–32
(2008
).81.
G. A.
Glatzmaier
, R. S.
Coe
, L.
Hongre
, and P. H.
Roberts
, “The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals
,” Nature
401
, 885
–890
(1999
).82.
A.
Sakuraba
and P. H.
Roberts
, “Generation of a strong magnetic field using uniform heat flux at the surface of the core
,” Nat. Geosci.
2
, 802
–805
(2009
).83.
Z. X.
Li
, S. V.
Bogdanova
, A. S.
Collins
, A.
Davidson
, B. D.
Waele
, R. E.
Ernst
, I. C. W.
Fitzsimons
, R. A.
Fuck
, D. P.
Gladkochub
, J.
Jacobs
, K. E.
Karlstrom
, S.
Lu
, L. M.
Natapov
, V.
Pease
, S. A.
Pisarevsky
, K.
Thrane
, and V.
Vernikovsky
, “Assembly, configuration, and break-up history of Rodinia: A synthesis
,” Precambrian Res.
160
(1-2
), 179
–210
(2008
).84.
W. J.
Collins
, “Slab pull, mantle convection, and Pangaean assembly and dispersal
,” Earth Planet. Sci. Lett.
205
(3-4
), 225
–237
(2003
).85.
F.
Nimmo
, “Energetics of the core
,” in Treatise on Geophysics, Core Dynamics
, 2nd ed., edited by P.
Olson
(Elsevier
, Amsterdam, The Netherlands
, 2015
), Vol. 8
, pp. 27
–55
.86.
C. J.
Davies
, “Cooling history of Earth’s core with high thermal conductivity
,” Phys. Earth Planet. Int.
247
, 65
–79
(2015
).87.
A. J.
Biggin
, E. J.
Piispa
, L. J.
Pesonen
, R.
Holme
, G. A.
Paterson
, T.
Veikkolainen
, and L.
Tauxe
, “Palaeomagnetic field intensity variations suggest mesoproterozoic inner-core nucleation
,” Nature
526
, 245
–248
(2015
).88.
T.
Lay
, J.
Hernlund
, E. J.
Garnero
, and M. S.
Thorne
, “A post-perovskite lens and D″ heat flux beneath the central pacific
,” Science
314
(5803
), 1272
–1276
(2006
).89.
J. W.
Hernlund
, C.
Thomas
, and P. J.
Tackley
, “A doubling of the post-perovskite phase boundary and structure of the Earth’s lowermost mantle
,” Nature
434
, 882
–886
(2005
).90.
R. D.
van der Hilst
, M. V.
de Hoop
, P.
Wang
, S.-H.
Shim
, P.
Ma
, and L.
Tenorio
, “Seismostratigraphy and thermal structure of Earth’s core-mantle boundary region
,” Science
315
(5820
), 1813
–1817
(2007
).91.
K.
Ohta
, Y.
Kuwayama
, K.
Hirose
, K.
Shimizu
, and Y.
Ohishi
, “Experimental determination of the electrical resistivity of iron at Earth’s core conditions
,” Naure
534
, 95
–98
(2016
).92.
Z.
Konôpková
, R. S.
McWilliams
, N.
Gómez-Pérez
, and A. F.
Goncharov
, “Direct measurement of thermal conductivity in solid iron at planetary core conditions
,” Nature
534
, 99
–101
(2016
).93.
J.
Aubert
, H.
Amit
, G.
Hulot
, and P.
Olson
, “Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity
,” Nature
454
, 758
–761
(2008
).94.
M.
Yoshida
, “Formation of a future supercontinent through plate motion–driven flow coupled with mantle downwelling flow
,” Geology
44
(9
), 755
–758
(2016
).95.
M.
Yoshida
and Y.
Hamano
, “Pangea breakup and northward drift of the Indian subcontinent reproduced by a numerical model of mantle convection
,” Sci. Rep.
5
, 8407
(2015
).96.
M.
Yoshida
and M.
Santosh
, “Supercontinents, mantle dynamics and plate tectonics: A perspective based on conceptual vs numerical models
,” Earth-Sci. Rev.
105
(1-2
), 1
–24
(2011
).97.
D. F.
Argus
, R. G.
Gordon
, and C.
Demets
, “Geologically current motion of 56 plates relative to the no-net-rotation reference frame
,” Geochem., Geophys., Geosyst.
12
(11
), Q11001
, doi:10.1029/2011GC003751 (2011
).98.
F. H.
Busse
, “Thermal instabilities in rapidly rotating systems
,” J. Fluid Mech.
44
(3
), 441
–460
(1970
).99.
H.
Gomi
, K.
Ohta
, K.
Hirose
, S.
Labrosse
, R.
Caracas
, M. J.
Verstraete
, and J. W.
Hernlund
, “The high conductivity of iron and thermal evolution of the Earth’s core
,” Phys. Earth Planet. Int.
224
, 88
–103
(2013
).100.
R. A.
Secco
, “Viscosity of the outer core
,” in Mineral Physics and Crystallography: A Handbook of Physical Constants
, AGU Reference Shelf 2
, edited by T. J.
Ahrens
(American Geophysical Union
, Washington, D.C., USA
, 1995
).101.
M.
Kono
and P. H.
Roberts
, “Definition of the Rayleigh number for geodynamo simulation
,” Phys. Earth Planet. Int.
128
(1-4
), 13
–24
(2001
).102.
R.
Krishnamurti
, “On the transition to turbulent convection. Part 2. The transition to time-dependent flow
,” J. Fluid Mech.
42
(2
), 309
–320
(1970
).103.
R.
Krishnamurti
and L. N.
Howard
, “Large-scale flow generation in turbulent convection
,” Proc. Natl. Acad. Sci. U. S. A.
78
(4
), 1981
–1985
(1981
).104.
M.
Manga
and D.
Weeraratne
, “Experimental study of non-Boussinesq Rayleigh–Bénard convection at high Rayleigh and Prandtl numbers
,” Phys. Fluids
11
(10
), 2969
–2976
(1999
).105.
F. D.
Stacey
and P. M.
Davis
, Physics of the Earth
, 4th ed. (Cambridge University Press
, UK
, 2008
).106.
P. H.
Roberts
and G. A.
Glatzmaier
, “The geodynamo, past, present and future
,” Geophys. Astrophys. Fluid Dyn.
94
(1-2
), 47
–84
(2001
).107.
G. F.
Davies
, Dynamic Earth: Plates, Plumes and Mantle Convection
(Cambridge University Press
, Cambridge, UK
, 1999
).108.
S.
Honda
and Y.
Iwase
, “Comparison of the dynamic and parameterized models of mantle convection including core cooling
,” Earth Planet. Sci. Lett.
139
(1-2
), 133
–145
(1996
).109.
J. T.
Ratcliff
, G.
Schubert
, and A.
Zebib
, “Steady tetrahedral and cubic patterns of spherical shell convection with temperature-dependent viscosity
,” J. Geophys. Res.
101
(B11
), 25473
–25484
, doi:10.1029/96JB02097 (1996
).110.
P.
Wessel
, W. H. F.
Smith
, R.
Scharroo
, J. F.
Luis
, and F.
Wobbe
, “Generic mapping tools: Improved version released
,” Trans., Am. Geophys. Union
94
(45
), 409
–410
(2013
).© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
