other hand,Racrit in the range ofQi/Qo <0.5is relatively smaller than those in the range of Qi/Qo ≥ 0.5. Since we computed Racrit by multiplyingRaFcrit and
∆T obtained from the simulation results, relatively smallRacrit in the range of Qi/Qo <0.5is due to the smallerRaFcrit in spite of the smaller∆T.
and 6.2 < Ra/Racrit < 7.0, andEmag < Ekin inRa/Racrit > 7.6. The overall tendency of Ekin and Emag depending on the Rayleigh number is that sustained magnetic fields are dissipated, large, and small with increasingRa. This tendency is the same as those found in the FT cases.
To assess the dipolar dominance in the sustained dynamo cases, we calculated the dipolarityfdip at the CMB and the ratio of the extrapolation from fittingfmag fit. Figs. 5.4 and 5.5 show fdip and fmag fit as a function of the Rayleigh number for different spherical shell geometries. The simulation results corresponding to Qi/Qo = 0.25,0.5,0.75,and1are indicated by stars, triangles, squares, and cir-cles, respectively. The error bars represent the standard deviation. The dipolarity gradually decreases from 0.8 to 0.3 with increasing Rayleigh number at both ra-dius ratios of ri/ro = 0.25and0.35. As we found in the FT cases, it is difficult to determine the dynamo regime based on the dipolarity only. On the contrary, fmag fit decreases from 5 to 1.2 with increasing Rayleigh number at both radius ratios. These results show that we could clearly classify sustained dynamos into dipole or non-dipole regime by referring to bothfdipandfmag fit.
Considering both fdip and fmag fit, we classified the dynamo regime in various Qi/Qo atri/ro = 0.25and0.35, as shown in Fig. 5.6, where the red circles, blue triangles, green squares, and black crosses represent the strong dipolar, weak dipo-lar, non-dipodipo-lar, and failed dynamo cases, respectively. When the magnetic energy is larger/smaller than the kinetic energy in a simulation case, we categorized this
as a strong/weak dynamo. Dotted lines in Fig. 5.6 represent the simulation results corresponding to the same heat flow Qo at the CMB. We find, in the range of Qi/Qo ≥0.5, a similar behavior of the dynamo regime against with the change of Ra/Racrit. We also find a different tendency of the dynamo regime in the range ofQi/Qo <0.5, as we discussed in the previous section.
In Qi/Qo = 0, dynamos are difficult to sustain becauseRa becomes small. We performed additional MHD simulations of Qi/Qo = 0 inRa/Racrit = 4.7and 5.2 at ri/ro = 0.35. A dynamos is failed in Ra/Racrit = 4.7, while a strong dipole dynamo is sustained in Ra/Racrit = 5.2. Hori et al. (2010) pointed out that dynamos are likely to be dipole in FF cases of zero flux at the ICB. Our simulations of zero flux at the ICB are almost failed dynamos since we set small Ra. Based on the simulation results and the above discussion, we conclude that the properties of the dynamo regime can be determined by the Rayleigh number regardless ofQi/Qo in the range ofQi/Qo ≥0.5.
Table 5.1:∆T andEkin in variousQi/Qoatri/ro = 0.15.
Qi/Qo RaF[×103] ∆T Ra[×103] Ekin
0 600 0.0781 46.8 5.95
0 650 0.0764 49.7 8.52
0 700 0.0748 52.3 11.40
0 750 0.0732 54.9 14.52
0 800 0.0716 57.3 17.87
0 850 0.0701 59.6 21.42
0.25 240 0.297 71.2 5.48
0.25 260 0.291 75.5 8.23
0.25 280 0.285 79.8 11.19
0.25 300 0.280 83.9 14.30
0.25 320 0.275 87.9 17.55
0.25 340 0.270 91.9 20.90
0.5 145 0.517 74.9 4.89
0.5 155 0.508 78.7 7.12
0.5 165 0.499 82.4 9.49
0.5 175 0.491 86.0 11.95
0.5 185 0.484 89.5 14.50
0.5 195 0.477 93.0 17.11
0.75 105 0.734 77.1 5.03
0.75 110 0.725 79.8 6.59
0.75 115 0.716 82.4 8.22
0.75 120 0.708 85.0 9.89
0.75 125 0.700 87.5 11.61
0.75 130 0.693 90.1 13.36
1 80 0.959 76.7 4.24
1 85 0.943 80.2 6.22
1 90 0.929 83.6 8.30
1 95 0.915 86.9 10.46
1 100 0.902 90.2 12.68
1 105 0.890 93.4 14.96
Table 5.2:∆T andEkin in variousQi/Qoatri/ro = 0.25.
Qi/Qo RaF[×103] ∆T Ra[×103] Ekin
0 275 0.136 37.5 5.34
0 300 0.134 40.1 7.65
0 325 0.131 42.7 10.22
0 350 0.129 45.1 13.04
0 375 0.127 47.5 16.07
0 400 0.124 49.8 19.29
0.25 120 0.346 41.5 3.26
0.25 130 0.341 44.3 5.08
0.25 140 0.336 47.0 7.08
0.25 150 0.331 49.6 9.25
0.25 160 0.326 52.2 11.57
0.25 170 0.321 54.6 14.02
0.5 80 0.551 44.1 3.74
0.5 90 0.538 48.4 6.78
0.5 100 0.526 52.6 10.20
0.5 110 0.514 56.5 13.94
0.5 120 0.503 60.4 17.96
0.5 130 0.493 64.1 22.22
0.75 60 0.756 45.3 4.04
0.75 65 0.743 48.3 6.11
0.75 70 0.731 51.2 8.35
0.75 75 0.720 54.0 10.76
0.75 80 0.709 56.7 13.31
0.75 85 0.698 59.3 15.99
1 50 0.952 47.6 5.28
1 55 0.932 51.3 8.07
1 60 0.914 54.8 11.12
1 65 0.896 58.2 14.40
1 70 0.879 61.6 17.87
1 75 0.864 64.8 21.52
Table 5.3:∆T andEkin in variousQi/Qoatri/ro = 0.35.
Qi/Qo RaF[×103] ∆T Ra[×103] Ekin
0 175 0.190 33.3 7.11
0 190 0.188 35.6 9.38
0 200 0.186 37.1 11.02
0 210 0.184 38.6 12.75
0 225 0.181 40.8 15.59
0 235 0.179 42.1 17.46
0.25 90 0.382 34.4 5.65
0.25 100 0.376 37.6 8.32
0.25 110 0.369 40.6 11.29
0.25 120 0.363 43.6 14.55
0.25 130 0.357 46.4 18.07
0.25 140 0.351 49.1 21.85
0.5 60 0.575 34.5 5.07
0.5 65 0.567 36.9 6.96
0.5 70 0.560 39.2 9.02
0.5 75 0.553 41.5 11.24
0.5 80 0.546 43.7 13.60
0.5 85 0.539 45.8 16.12
0.75 47 0.762 35.8 5.77
0.75 50 0.733 36.7 7.31
0.75 52 0.748 38.9 8.40
0.75 55 0.741 40.7 10.12
0.75 58 0.733 42.5 11.93
0.75 60 0.728 3.7 13.20
1 37 0.955 35.3 5.24
1 40 0.942 37.7 7.14
1 42 0.934 39.2 8.49
1 45 0.922 41.5 10.66
1 48 0.911 43.7 12.97
1 50 0.903 45.2 14.59
Table 5.4:The critical Rayleigh numbers (RaFcritandRacrit) in variousQi/Qowith dif-ferent geometry.
ri/ro Qi/Qo RaFcrit[×103] Racrit[×103]
0.15 0 511 42.5
0.15 0.25 206 64.4
0.15 0.50 126 68.0
0.15 0.75 90.2 69.5
0.15 1 70.4 70.4
0.25 0 231 33.3
0.25 0.25 106 38.0
0.25 0.50 71.3 40.9
0.25 0.75 52.1 41.0
0.25 1 41.9 42.6
0.35 0 135 27.5
0.35 0.25 74.0 29.9
0.35 0.50 49.1 29.6
0.35 0.75 37.2 29.0
0.35 1 30.0 30.1
Table5.5:ResultsofMHDsimulationsatri/ro=0.25. ri/roQi/QoRaF[×103]Ra[×103]Ra/RacritEkinEmagfdipfmagfitM[ZAm2] 0.25023533.11.01.422.75×10−6−−− 0.25038251.21.520.75.86×10−6−−− 0.25042657.71.732.08.75×10−6−−− 0.25052970.22.163.27.20×10−6−−− 0.25073593.02.8146.62.05×10−4 −−− 0.250.2523569.61.832.27.35×10−6−−− 0.250.253821062.893.30.963−−− 0.250.254261183.185.4736.50.7501.210218.6 0.250.255291534.0190.04.63×10−3−−− 0.250.257352135.6278.8278.00.4901.79551.29 0.250.523599.22.474.34.94×10−6 −−− 0.250.53821543.8173.30.268−−− 0.250.54261764.3217.50.0557−−− 0.250.55292235.52205460.5752.48388.88 0.250.57353087.5483.3119.40.3821.24625.39 0.250.752351283.159.6527.50.8372.585154.3 0.250.753822035.0142.6661.00.6582.574123.6 0.250.754262235.4222.5482.50.6182.13484.38 0.250.755292776.7285.3715.50.5501.379112.9 0.250.757353859.4657.6129.40.3771.35728.67 0.2512351333.187.9810.20.8574.003136.4 0.2513822365.5191.0787.30.6551.761136.3 0.2514262626.2338.5118.50.5352.16533.26 0.2515293387.9494.0104.00.4541.64332.42 0.25173546711.0766.6259.10.3791.21746.23
Table5.6:ResultsofMHDsimulationsatri/ro=0.35. ri/roQi/QoRaF[×103]Ra[×103]Ra/RacritEkinEmagfdipfmagfitM[ZAm2] 0.35015730.51.14.629.09×10−5−−− 0.35026746.41.723.91.55×10−4−−− 0.35040362.12.360.21.81×10−4−−− 0.35073690.13.3177.71.69×10−2−−− 0.35017381414.7536.75.43×10−3 −−− 0.35026601575.272110850.3112.442109.5 0.350.2515753.71.828.61.32×10−6−−− 0.350.2526779.62.782.61.86×10−3−−− 0.350.254031083.6159.60.261−−− 0.350.257361685.6339.232.00.4582.94913.78 0.350.515772.92.563.34.86×10−4 −−− 0.350.52671123.876.1879.90.8452.434222.7 0.350.54031464.9140.9713.40.6021.550139.7 0.350.57362247.6440.5144.90.3881.37332.94 0.350.7515790.63.1101.60.208−−− 0.350.752671324.5158.7356.20.6102.05091.77 0.350.754031796.2191.6814.10.5932.087149.2 0.350.757362759.5560.7146.80.3541.58834.14 0.3511571063.5128.534.170.7055.64218.43 0.3512671645.4141.3882.40.6972.470154.9 0.3514032127.0272.4576.30.5382.367107.6 0.35173631710.5712.0104.40.2991.26920.66
Fig. 5.1:The kinetic energy density in spherical shells as a function of the flux Rayleigh number for variousQi/Qo and different geometries.
Fig. 5.2:The kinetic energy density in spherical shells as a function of the Rayleigh num-ber for variousQi/Qo and different geometries.
Fig. 5.3:The kinetic and magnetic energy density in spherical shells as a function of the Rayleigh number with different geometries. Red and blue symbols denote Ekin andEmag, respectively. The inverted triangle, star, triangle, square, and circle mean the ratio of heat flow at the ICB to that at the CMB, Qi/Qo = 0,0.25,0.5,0.75,and1, respectively.
Fig. 5.4:Dipolarityfdipas a function of the Rayleigh number for different spherical shell geometries. The star, triangle, square, and circle mean the ratio of heat flow at the ICB to that at the CMB,Qi/Qo = 0.25,0.5,0.75,and1, respectively. The bar represents the standard deviation.
Fig. 5.5:The ratio of the dipolar magnetic energy density at the CMB to the extrapolation ofl= 1,fmag fit, as a function of the Rayleigh number for different geometries.
The star, triangle, square, and circle mean the ratio of heat flow at the ICB to that at the CMB,Qi/Qo = 0.25,0.5,0.75,and1, respectively. The bar represents the standard deviation.
Fig. 5.6:Dynamo regime in various Qi/Qo at ri/ro = 0.25 and 0.35. Red circles, blue triangles, green squares, and black crosses represent strong dipolar, weak dipolar, non-dipolar, and failed dynamo cases, respectively. Dotted lines mean the same heat flow at the CMB Qo; Qo = 8.2,8.9,9.5,10.4, and 11.8 in ri/ro = 0.25, andQo = 14.9,16.3,18.3,and22.5inri/ro = 0.35.
Chapter 6 Conclusion
6.1 Concluding remarks
As the inner core has been growing for approximately one billion years, sustained dynamo conditions with different inner core sizes are required to understand the past Earth environment. To understand the geometry effect, we performed nu-merical dynamo simulations with three different radius ratios:ri/ro = 0.15,0.25, and0.35. To evaluate the morphology of the magnetic field, especially the dipole component dominancy, we combined two indices: (i) dipolarity fdip, which is widely used for assessing the relative dipole strength in numerical dynamos, and (ii) the ratio of a dipolar extrapolation from the fitting curve of higher degrees fmag fit; this method is often used to explain Earth’s observed dipolar-dominated field. We verified that the ratio of extrapolation from fitting was valid for deter-mining the dynamo regime. Around the transition between the dipolar and non-dipolar regimes (fdip ≈ 0.35), we assessed the dipolar dominance using the ratio
of extrapolation from fitting. Based on both fdip andfmag fit, we limited theRa range of the sustained dynamo for all radius ratios. We found that theRarange for the strong dipole became narrower with a smaller inner core size. While the dependence of fdip on the Rayleigh number is similar at ri/ro = 0.25and 0.35 the dipolar dominance becomes weaker with the smaller inner core. There were no strong dipolar dynamos but non-dipolar dynamos at10.1< Ra/Racrit <15.6, only at ri/ro = 0.15 in FT cases. Our results indicate that changes in the ra-dius ratio largely influence the dynamo regime in numerical dynamos with a fixed temperature boundary condition.
To understand the boundary effect, we also performed numerical dynamo simu-lations with five different heat flow rates: Qi/Qo = 0,0.25,0.5,0.75,and1. The simulation results showed that the kinetic energy becomes large basically with in-creasingRa/Racrit. The simulation results also revealed that the magnetic energy dissipated for the smaller Ra/Racrit range, and increased significantly with in-creasingRa/Racrit, followed by the decrease of the magnetic energy in the larger Ra/Racritrange. While the dynamo regime cannot be determined only based on fdip, which decreases gradually with increasingRa, the use offmag fit enables us to determine the dynamo regime clearly and quantitatively. These tendencies are the same as we found in the FT cases at ri/ro = 0.25 and 0.35. Based on the
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.