- Email: [email protected]
Corrigendum to “Iron snow dynamo models for Ganymede” [Icarus 247 (2015) 248–259]
Icarus 256 (2015) 63–65
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Corrigendum
Corrigendum to ‘‘Iron snow dynamo models for Ganymede’’ [Icarus 247 (2015) 248–259] Ulrich R. Christensen Max Planck Institute for Solar System Research, Göttingen, Germany
a r t i c l e
i n f o
Article history: Received 9 March 2015 Revised 30 March 2015 Accepted 11 April 2015 Available online 2 May 2015 Keywords: Ganymede Magnetic fields Interiors
a b s t r a c t An error was detected in the numerical code that has been used for the simulations of dynamo models with a ‘snow layer’, in Ganymede’s core which can explain the low observed quadrupole-to-dipole ratio in Ganymede’s magnetic field. After correcting the error it is found that the stability of such strongly dipolar dynamos is even larger than found before. In contrast to a previous conclusion, they exist in a wide range of thicknesses for the snow layer. Ó 2015 Elsevier Inc. All rights reserved.
In a recent paper (Christensen, 2015) I presented numerical dynamo models for a possible ‘iron-snow’ regime in the core of Ganymede (Hauck et al., 2006). In this scenario iron crystallizes in an outer layer of Ganymede’s core, sinks, and remelts at the bottom of the snow layer. This leads to sulfur enrichment in the snowforming layer. A compositional gradient is set up that very strongly stabilizes the layer against radial overturn. In the compositionally homogeneous central region of the core the influx of melting iron from above drives a dynamo by compositional convection. The dynamo models aim at explaining the properties of Ganymede’s magnetic field determined by magnetometer measurements during flybys of the Galileo mission. This is in particular the very low quadrupole moment in relation to the dipole moment, which is unusual in comparison to the ratio at other planetary dynamos (Kivelson et al., 2002; Saur et al., 2015). Technically, the dynamo model consists of two separate shells: a convecting inner shell where the dynamo operates and an outer shell representing the snow layer in which only horizontal fluid motion is possible. The magnetic field and the flow in the two shells are coupled by continuity conditions at the interface. The model results show that in a parameter regime where the dynamo-generated magnetic field is dominantly dipolar, the presence of the outer shell reduces the ratio of quadrupole power R2 to dipole power R1 significantly. With the outer shell, the values of R2 =R1 (calculated at the surface of Ganymede’s core) are well below 0.04, which is the upper limit suggested by the Galileo measurements. In reference models DOI of original article: http://dx.doi.org/10.1016/j.icarus.2014.10.024 E-mail address: [email protected] http://dx.doi.org/10.1016/j.icarus.2015.04.022 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
without the stable layer the ratio is larger than, or at best marginally compatible with, the upper limit. The model results also suggested that for a thick snow layer, whose width exceeds approximately 25% of Ganymede’s core radius, the dipolar dynamo mode is replaced by a hemispherical mode (e.g. Landeau and Aubert, 2011) that generates a field with dipolar and quadrupolar components of about equal strength. Such field would clearly be incompatible with observation and I concluded that the snow layer can only have a moderate depth extent. After publication I detected an error in the numerical dynamo code. At the interface between the two shells the induction term r ðu BÞ was corrupted (u is velocity and B magnetic field). In the spectral transform method employed in the dynamo code (Christensen and Wicht, 2007), the non-linear term ðu BÞ is evaluated on a grid in physical space. Its calculation has been skipped for the grid point on the outer radius of the main convecting shell; here ðu BÞ was set to zero. This is appropriate in the simple case without an outer shell where u ¼ 0 on the boundary. All reported results without an outer shell are therefore correct. In the models with an outer shell the horizontal velocity does not vanish at the interface. Setting ðu BÞ to zero at this radius leads, in conjunction with the radial derivative in the curl operator, to a large spurious spike in the source term for the toroidal magnetic field (Eqs. (36) and (37) in Christensen and Wicht, 2007). The spike is most pronounced at the radial level next to the interface boundary in the dynamo shell. Several representative model cases have been rerun with the corrected code, covering the range of control parameters (Ekman number E, Rayleigh number Ra, magnetic Prandtl number Pm and relative outer shell thickness do ) that has been studied in
64
U.R. Christensen / Icarus 256 (2015) 63–65
Table 1 Model parameters and results. Case 1.4
E
Pm 4
3 10
5
Ra 1:5 10
do 6
Type
Rm
Nu
Brms
Bdip
Rz =R
Tilt
R2 =R1
0.4
D
136
1.84
3.16
0.183
.991
1.4
(133) 235
(1.91) 3.78
(3.03) 4.25
(0.192) 0.459
(.993) 0.92
(1.7 ) 3.5
0.0030 (0.0047) 0.0167
3.2
3 104
4
6:0 106
0.2
(D) D
3.4
3 104
4
6:0 106
0.4
(D) D
(232) 235
(3.90) 3.85
(4.10) 4.40
(0.407) 0.285
(0.93) .985
(3.2 ) 1.8
(0.0141) 0.0044
A.8
3 104
4
6:0 106
0.8
(H) D
(277) 236
(3.36) 3.86
(2.81) 4.35
(0.071) 0.149
(0.94) .999
(5.6 ) 0.8
(1.165) 0.0011
4.2
3 104
4
9:0 106
0.2
D
290
4.61
4.45
0.459
0.94
2.7
0.0145
(350) 149
(4.19) 4.65
(2.96) 3.14
(0.127) 0.137
(0.85) .999
(8.7 ) 0.8
(0.814) 0.0006
B.8
3 104
2
9:0 106
0.8
(H) D
C.8
3 104
4
9:0 106
0.8
H
374
4.23
4.25
0.051
.998
1.0
1.933
D.2
3 104
2
1:5 107
0.2
D
199
5.79
2.97
0.370
.949
2.6
0.0158
E.2
3 104
4
1:5 107
0.2
DH
406
5.95
4.72
0.347
.958
3.5
0.084
E.4
3 104
4
1:5 107
0.4
D
438
5.91
4.40
0.180
.993
1.7
0.012
E.8
3 104
4
1:5 107
0.8
QH
524
5.80
5.28
0.015
1.00
2.1
8.97
F.4
3 104
2
2:5 106
0.8
DH
282
7.54
3.50
0.184
.995
1.4
0.054
7.2
104
3
1:0 107
0.2
D
141
2.61
3.40
0.373
0.92
3.0
0.0140
(139) 82
(2.78) 4.19
(3.20) 2.16
(0.379) 0.348
(0.93) 0.96
(3.1 ) 1.6
(0.0119) 0.0084
8.2
104
1
2:5 107
0.2
(D) D
9.2
104
3
2:5 107
0.2
(D) D
(83) 231
(4.31) 4.76
(2.05) 3.26
(0.336) 0.446
(0.94) 0.94
(2.4 ) 3.1
(0.0082) 0.0136
9.4
104
3
2:5 107
0.4
(D) D
(231) 231
(4.60) 4.48
(3.95) 4.24
(0.435) 0.305
(0.93) .989
(3.2 ) 1.5
(0.0117) 0.0030
9.8
104
3
2:5 107
0.8
(D) D
(232) 237
(4.64) 4.50
(3.96) 4.26
(0.295) 0.155
(.985) .998
(2.1 ) 0.4
(0.0033) 0.0003
11.2
104
3
9:0 107
0.2
(H) D
(279) 472
(3.77) 8.03
(4.06) 4.76
(0.037) 0.472
(.993) 0.96
(1.9 ) 3.1
(1.415) 0.0104
G.4
104
3
9:0 107
0.4
(H) D
(598) 467
(7.44) 8.06
(2.85) 4.92
(0.090) 0.324
(0.90) 0.991
(11 ) 1.6
(2.15) 0.0048
G.8
104
3
9:0 107
0.8
D
498
8.14
4.72
0.139
0.999
0.6
0.0008
H.2
4
10
1
1:5 10
8
0.2
D
217
10.0
2.55
0.397
0.971
2.0
0.0068
H.4
104
1
1:5 108
0.4
D
219
10.1
2.53
0.266
0.994
1.5
0.0067
J.2
104
3
1:5 108
0.2
DH
643
10.4
5.20
0.397
0.975
3.3
0.0606
K.2
104
1
2:5 108
0.2
H
408
12.4
1.81
0.036
0.921
6.8
0.537
14.4
3 105
3
2:5 108
0.4
D
459
7.55
5.26
0.330
0.994
1.1
0.0016
(D)
(476)
(7.85)
(4.90)
(0.294)
(0.983)
(2.7 )
(0.0036)
Cases designated by pure numbers are reruns of models reported before. The result obtained with the defective code is given in parentheses. New cases have a leading capital letter. do is the thickness of the outer shell normalized with that of the dynamo shell, Rm is magnetic Reynolds number, Nu is the Nusselt number, Brms is the mean magnetic field strength in the dynamo shell, Bdip is the rms dipole field strength on the outer boundary, Rz =R is the ratio of power in the zonal field to that in the total field on the outer boundary, Tilt is the time-average dipole tilt relative to the rotation axis. Solution types: D = dipolar, Q = quadrupolar, H = hemispherical; combinations indicate hybrid types. For definitions and scaling see Christensen (2015).
Christensen (2015). In addition, several simulations at parameter combinations that had not been covered previously have been performed in order to delineate the boundary between the different dynamo regimes. In Table 1, the corrected model results are listed (numbering of cases as in Table 1 of Christensen (2015)). The previous (incorrect) values are given in parentheses. New simulations are designated by capital letters. In all cases that had been classified before as dipolar dynamos, the differences between the faulty results and the corrected ones are relatively small. In particular, the ratios of quadrupole power to dipole power at the outer boundary of the model R2 =R1 do not change by much and remain well below 0.04 when a stably stratified outer shell is present. Also other characteristic values of the solution, such as the mean dipole field strength or the mean dipole tilt angle, remain rather similar in most cases. Furthermore, the structure of the zonal flow and of the axisymmetric magnetic field is qualitatively unchanged. This includes the thin anomalous region of prograde flow at the layer interface in equatorial latitudes, shown in Fig. 5c of the original paper, which is presumably caused by the Lorentz forces of electrical current sheets. These localized currents are therefore not an artefact created by the error in the original code.
However, the error in the code had an impact on the selection of the dynamo mode. After correction, in all tested models for which a hemispherical dynamo had been reported and which had a stable outer layer (cases 3.4, 4.2, 9.8 and 11.2), now a dipolar magnetic field is found. The quadrupole-to-dipole ratio R2 =R1 is well below 0.04 also in these models. Only when, for a given value of the Ekman number, the Rayleigh number is increased to higher, previously untested values, solutions with a significant quadrupole contribution arise. In Christensen (2015) the parameter combination RaE9=4 was found useful to separate the different dynamo regimes. In Fig. 1 (the equivalent to Fig. 12 in Christensen (2015)) all corrected and additional results are plotted as function of this parameter and the outer shell thickness do (which is normalized with the thickness of the dynamo region). Open (unfilled) symbols indicate cases with R2 =R1 < 0:04. The stability range of these strongly dipole-dominated dynamos is expanded towards higher values of the Rayleigh number compared to the previously determined stability limit, which is shown as a broken line in Fig. 1. With the previous incorrect simulations it seemed that a large thickness of the stable layer (do > 0:2) favors hemispherical solutions. With the corrected code dipole-dominated dynamos are stabilized by
U.R. Christensen / Icarus 256 (2015) 63–65
1 0.9 0.8 0.7
do
0.6 0.5 0.4 0.3 0.2 0.1 0 −2
−1
10
Ra E
10 9/4
0
10
Fig. 1. Regime diagram for dynamo solutions. Open symbols strongly dipolar with R2 =R1 < 0:04, symbols with light grey fill dominantly dipolar but with significant quadrupole contribution 0:04 < R2 =R1 < 0:4, with dark grey fill hemispherical or other non-dipolar dynamo (R2 =R1 > 0:4). Triangles are for E ¼ 3 104 , squares for E ¼ 104 and circles for E ¼ 3 105 . Cases with Pm 6 2 are slightly offset in vertical direction from the lines indicating the outer shell thickness do whereas cases with Pm P 3 are plotted on the line. The broken line is the stability limit for dipolar solutions reported in Christensen (2015), the full thick line is a tentative stability boundary for the correct results.
the presence of an outer stable layer nearly irrespectively of the layer thickness in the range 0:2 6 do 6 0:8. For the results obtained with the defective code there was a dichotomy between dipolar solutions with small R2 =R1 and the hemispherical solutions for which R2 =R1 was of order one or larger. With the correct code the transition is more gradual. Some cases in light grey near the rectified stability boundary (full thick line in Fig. 1) have a magnetic field that is still dominated by the axial dipole but also shows some traits of a hemispherical dynamo. Their time-average magnetic field contains a persistent axial quadrupole contribution of moderate amplitude (compared to that of the axial dipole) and a ratio R2 =R1 > 0:04. The magnetic Prandtl number Pm (the ratio of viscosity to magnetic diffusivity) also has an influence on the dynamo mode. In three model pairs where one case differs from the other only in the value of Pm (cases B.8 and C.8, D.2 and E.2, H.2 and J.2), a purely dipolar field with R2 =R1 < 0:04 is found at the smaller value of Pm 6 2, whereas at higher Pm P 3 the field is hemispherical or
65
of mixed dipolar-hemispherical type. Models with Pm 6 2 are distinguished in Fig. 1 by plotting the symbol with a slight offset in vertical directions above the lines for do ¼ 0; 0:2; 0:4; 0:8. Finally, in Christensen (2015) bistability was reported to exist for at least one case (case 9.2), which stayed on the dipolar branch when started with a strong dipole field but settled on the hemispherical branch when started with a weak irregular seed field. With the corrected code the magnetic field turned dipolar also with the latter initial condition. Also in other cases no indication of bistability has been found: cases 3.4, 4.2, 9.8 and 11.2 have been started with the hemispherical solution obtained with the defective code as initial condition, but all turned dipolar. In summary, the stability of strongly dipolar dynamos with a low quadrupole-to-dipole ratio R2 =R1 < 0:04 in models with a stably stratified layer is significantly higher than what had been found with the faulty code. Otherwise the properties of dipolar solutions reported in Christensen (2015) have not been much affected by the error in the code. The main conclusion in that paper, that an iron snow dynamo in Ganymede offers a viable scenario for explaining the known and inferred properties of Ganymede’s magnetic field, therefore still holds and is even strengthened by the expansion of the stability range of strongly dipolar dynamos towards higher values of RaE9=4 . The newly found trend towards more dipolar solutions at lower values of the magnetic Prandtl number further strengthens the case, since in planetary dynamos Pm 1. Also most other conclusions in Christensen (2015) still hold, e.g. an iron snow model is compatible with the observed magnitude of Ganymede’s dipole moment and secular variation time scales are larger compared to those of a dynamo without a stable outer layer. Since the previously claimed preference for hemispherical dynamos in case of a thick stable layer does not exist, the only conclusion that turns out to be unfounded is that Ganymede’s low quadrupole moment would indicate that the snow layer cannot be much thicker than 200 km.
References Christensen, U.R., 2015. Iron snow dynamo models for Ganymede. Icarus 247, 248–259. Christensen, U.R., Wicht, J., 2007. Numerical dynamo simulations. In: Schubert, G. (Ed.), Treatise on Geophysics, Core Dynamics, vol. 8. Elsevier, Amsterdam, pp. 245–282. Hauck, S.A., Aurnou, J.M., Dombard, A.J., 2006. Sulfur’s impact on core evolution and magnetic field generation on Ganymede. J. Geophys. Res. 111, E09008. Kivelson, M., Khurana, K., Volwerk, M., 2002. The permanent and induced magnetic moments of Ganymede. Icarus 157, 507–522. Landeau, M., Aubert, J., 2011. Equatorially asymmetric convection inducing a hemispherical magnetic field in rotating spheres and implications for the past martian dynamo. Phys. Earth Planet. Inter. 185, 61–73. Saur, J., Duling, S., Roth, L., Jia, X., Strobel, D., Feldman, P., Christensen, U., Retherford, K., McGrath, M., Musacchio, F., Wennmacher, A., Neubauer, F., Simon, S., Hartkorn, O., 2015. The search for a subsurface ocean in Ganymede with Hubble space telescope observations of its auroral oval. J. Geophys. Res. 120. http:// dx.doi.org/10.1002/2014JA020778.
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Icarus journal homepage: www.elsevier.com/locate/icarus
Corrigendum
Corrigendum to ‘‘Iron snow dynamo models for Ganymede’’ [Icarus 247 (2015) 248–259] Ulrich R. Christensen Max Planck Institute for Solar System Research, Göttingen, Germany
a r t i c l e
i n f o
Article history: Received 9 March 2015 Revised 30 March 2015 Accepted 11 April 2015 Available online 2 May 2015 Keywords: Ganymede Magnetic fields Interiors
a b s t r a c t An error was detected in the numerical code that has been used for the simulations of dynamo models with a ‘snow layer’, in Ganymede’s core which can explain the low observed quadrupole-to-dipole ratio in Ganymede’s magnetic field. After correcting the error it is found that the stability of such strongly dipolar dynamos is even larger than found before. In contrast to a previous conclusion, they exist in a wide range of thicknesses for the snow layer. Ó 2015 Elsevier Inc. All rights reserved.
In a recent paper (Christensen, 2015) I presented numerical dynamo models for a possible ‘iron-snow’ regime in the core of Ganymede (Hauck et al., 2006). In this scenario iron crystallizes in an outer layer of Ganymede’s core, sinks, and remelts at the bottom of the snow layer. This leads to sulfur enrichment in the snowforming layer. A compositional gradient is set up that very strongly stabilizes the layer against radial overturn. In the compositionally homogeneous central region of the core the influx of melting iron from above drives a dynamo by compositional convection. The dynamo models aim at explaining the properties of Ganymede’s magnetic field determined by magnetometer measurements during flybys of the Galileo mission. This is in particular the very low quadrupole moment in relation to the dipole moment, which is unusual in comparison to the ratio at other planetary dynamos (Kivelson et al., 2002; Saur et al., 2015). Technically, the dynamo model consists of two separate shells: a convecting inner shell where the dynamo operates and an outer shell representing the snow layer in which only horizontal fluid motion is possible. The magnetic field and the flow in the two shells are coupled by continuity conditions at the interface. The model results show that in a parameter regime where the dynamo-generated magnetic field is dominantly dipolar, the presence of the outer shell reduces the ratio of quadrupole power R2 to dipole power R1 significantly. With the outer shell, the values of R2 =R1 (calculated at the surface of Ganymede’s core) are well below 0.04, which is the upper limit suggested by the Galileo measurements. In reference models DOI of original article: http://dx.doi.org/10.1016/j.icarus.2014.10.024 E-mail address: [email protected] http://dx.doi.org/10.1016/j.icarus.2015.04.022 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
without the stable layer the ratio is larger than, or at best marginally compatible with, the upper limit. The model results also suggested that for a thick snow layer, whose width exceeds approximately 25% of Ganymede’s core radius, the dipolar dynamo mode is replaced by a hemispherical mode (e.g. Landeau and Aubert, 2011) that generates a field with dipolar and quadrupolar components of about equal strength. Such field would clearly be incompatible with observation and I concluded that the snow layer can only have a moderate depth extent. After publication I detected an error in the numerical dynamo code. At the interface between the two shells the induction term r ðu BÞ was corrupted (u is velocity and B magnetic field). In the spectral transform method employed in the dynamo code (Christensen and Wicht, 2007), the non-linear term ðu BÞ is evaluated on a grid in physical space. Its calculation has been skipped for the grid point on the outer radius of the main convecting shell; here ðu BÞ was set to zero. This is appropriate in the simple case without an outer shell where u ¼ 0 on the boundary. All reported results without an outer shell are therefore correct. In the models with an outer shell the horizontal velocity does not vanish at the interface. Setting ðu BÞ to zero at this radius leads, in conjunction with the radial derivative in the curl operator, to a large spurious spike in the source term for the toroidal magnetic field (Eqs. (36) and (37) in Christensen and Wicht, 2007). The spike is most pronounced at the radial level next to the interface boundary in the dynamo shell. Several representative model cases have been rerun with the corrected code, covering the range of control parameters (Ekman number E, Rayleigh number Ra, magnetic Prandtl number Pm and relative outer shell thickness do ) that has been studied in
64
U.R. Christensen / Icarus 256 (2015) 63–65
Table 1 Model parameters and results. Case 1.4
E
Pm 4
3 10
5
Ra 1:5 10
do 6
Type
Rm
Nu
Brms
Bdip
Rz =R
Tilt
R2 =R1
0.4
D
136
1.84
3.16
0.183
.991
1.4
(133) 235
(1.91) 3.78
(3.03) 4.25
(0.192) 0.459
(.993) 0.92
(1.7 ) 3.5
0.0030 (0.0047) 0.0167
3.2
3 104
4
6:0 106
0.2
(D) D
3.4
3 104
4
6:0 106
0.4
(D) D
(232) 235
(3.90) 3.85
(4.10) 4.40
(0.407) 0.285
(0.93) .985
(3.2 ) 1.8
(0.0141) 0.0044
A.8
3 104
4
6:0 106
0.8
(H) D
(277) 236
(3.36) 3.86
(2.81) 4.35
(0.071) 0.149
(0.94) .999
(5.6 ) 0.8
(1.165) 0.0011
4.2
3 104
4
9:0 106
0.2
D
290
4.61
4.45
0.459
0.94
2.7
0.0145
(350) 149
(4.19) 4.65
(2.96) 3.14
(0.127) 0.137
(0.85) .999
(8.7 ) 0.8
(0.814) 0.0006
B.8
3 104
2
9:0 106
0.8
(H) D
C.8
3 104
4
9:0 106
0.8
H
374
4.23
4.25
0.051
.998
1.0
1.933
D.2
3 104
2
1:5 107
0.2
D
199
5.79
2.97
0.370
.949
2.6
0.0158
E.2
3 104
4
1:5 107
0.2
DH
406
5.95
4.72
0.347
.958
3.5
0.084
E.4
3 104
4
1:5 107
0.4
D
438
5.91
4.40
0.180
.993
1.7
0.012
E.8
3 104
4
1:5 107
0.8
QH
524
5.80
5.28
0.015
1.00
2.1
8.97
F.4
3 104
2
2:5 106
0.8
DH
282
7.54
3.50
0.184
.995
1.4
0.054
7.2
104
3
1:0 107
0.2
D
141
2.61
3.40
0.373
0.92
3.0
0.0140
(139) 82
(2.78) 4.19
(3.20) 2.16
(0.379) 0.348
(0.93) 0.96
(3.1 ) 1.6
(0.0119) 0.0084
8.2
104
1
2:5 107
0.2
(D) D
9.2
104
3
2:5 107
0.2
(D) D
(83) 231
(4.31) 4.76
(2.05) 3.26
(0.336) 0.446
(0.94) 0.94
(2.4 ) 3.1
(0.0082) 0.0136
9.4
104
3
2:5 107
0.4
(D) D
(231) 231
(4.60) 4.48
(3.95) 4.24
(0.435) 0.305
(0.93) .989
(3.2 ) 1.5
(0.0117) 0.0030
9.8
104
3
2:5 107
0.8
(D) D
(232) 237
(4.64) 4.50
(3.96) 4.26
(0.295) 0.155
(.985) .998
(2.1 ) 0.4
(0.0033) 0.0003
11.2
104
3
9:0 107
0.2
(H) D
(279) 472
(3.77) 8.03
(4.06) 4.76
(0.037) 0.472
(.993) 0.96
(1.9 ) 3.1
(1.415) 0.0104
G.4
104
3
9:0 107
0.4
(H) D
(598) 467
(7.44) 8.06
(2.85) 4.92
(0.090) 0.324
(0.90) 0.991
(11 ) 1.6
(2.15) 0.0048
G.8
104
3
9:0 107
0.8
D
498
8.14
4.72
0.139
0.999
0.6
0.0008
H.2
4
10
1
1:5 10
8
0.2
D
217
10.0
2.55
0.397
0.971
2.0
0.0068
H.4
104
1
1:5 108
0.4
D
219
10.1
2.53
0.266
0.994
1.5
0.0067
J.2
104
3
1:5 108
0.2
DH
643
10.4
5.20
0.397
0.975
3.3
0.0606
K.2
104
1
2:5 108
0.2
H
408
12.4
1.81
0.036
0.921
6.8
0.537
14.4
3 105
3
2:5 108
0.4
D
459
7.55
5.26
0.330
0.994
1.1
0.0016
(D)
(476)
(7.85)
(4.90)
(0.294)
(0.983)
(2.7 )
(0.0036)
Cases designated by pure numbers are reruns of models reported before. The result obtained with the defective code is given in parentheses. New cases have a leading capital letter. do is the thickness of the outer shell normalized with that of the dynamo shell, Rm is magnetic Reynolds number, Nu is the Nusselt number, Brms is the mean magnetic field strength in the dynamo shell, Bdip is the rms dipole field strength on the outer boundary, Rz =R is the ratio of power in the zonal field to that in the total field on the outer boundary, Tilt is the time-average dipole tilt relative to the rotation axis. Solution types: D = dipolar, Q = quadrupolar, H = hemispherical; combinations indicate hybrid types. For definitions and scaling see Christensen (2015).
Christensen (2015). In addition, several simulations at parameter combinations that had not been covered previously have been performed in order to delineate the boundary between the different dynamo regimes. In Table 1, the corrected model results are listed (numbering of cases as in Table 1 of Christensen (2015)). The previous (incorrect) values are given in parentheses. New simulations are designated by capital letters. In all cases that had been classified before as dipolar dynamos, the differences between the faulty results and the corrected ones are relatively small. In particular, the ratios of quadrupole power to dipole power at the outer boundary of the model R2 =R1 do not change by much and remain well below 0.04 when a stably stratified outer shell is present. Also other characteristic values of the solution, such as the mean dipole field strength or the mean dipole tilt angle, remain rather similar in most cases. Furthermore, the structure of the zonal flow and of the axisymmetric magnetic field is qualitatively unchanged. This includes the thin anomalous region of prograde flow at the layer interface in equatorial latitudes, shown in Fig. 5c of the original paper, which is presumably caused by the Lorentz forces of electrical current sheets. These localized currents are therefore not an artefact created by the error in the original code.
However, the error in the code had an impact on the selection of the dynamo mode. After correction, in all tested models for which a hemispherical dynamo had been reported and which had a stable outer layer (cases 3.4, 4.2, 9.8 and 11.2), now a dipolar magnetic field is found. The quadrupole-to-dipole ratio R2 =R1 is well below 0.04 also in these models. Only when, for a given value of the Ekman number, the Rayleigh number is increased to higher, previously untested values, solutions with a significant quadrupole contribution arise. In Christensen (2015) the parameter combination RaE9=4 was found useful to separate the different dynamo regimes. In Fig. 1 (the equivalent to Fig. 12 in Christensen (2015)) all corrected and additional results are plotted as function of this parameter and the outer shell thickness do (which is normalized with the thickness of the dynamo region). Open (unfilled) symbols indicate cases with R2 =R1 < 0:04. The stability range of these strongly dipole-dominated dynamos is expanded towards higher values of the Rayleigh number compared to the previously determined stability limit, which is shown as a broken line in Fig. 1. With the previous incorrect simulations it seemed that a large thickness of the stable layer (do > 0:2) favors hemispherical solutions. With the corrected code dipole-dominated dynamos are stabilized by
U.R. Christensen / Icarus 256 (2015) 63–65
1 0.9 0.8 0.7
do
0.6 0.5 0.4 0.3 0.2 0.1 0 −2
−1
10
Ra E
10 9/4
0
10
Fig. 1. Regime diagram for dynamo solutions. Open symbols strongly dipolar with R2 =R1 < 0:04, symbols with light grey fill dominantly dipolar but with significant quadrupole contribution 0:04 < R2 =R1 < 0:4, with dark grey fill hemispherical or other non-dipolar dynamo (R2 =R1 > 0:4). Triangles are for E ¼ 3 104 , squares for E ¼ 104 and circles for E ¼ 3 105 . Cases with Pm 6 2 are slightly offset in vertical direction from the lines indicating the outer shell thickness do whereas cases with Pm P 3 are plotted on the line. The broken line is the stability limit for dipolar solutions reported in Christensen (2015), the full thick line is a tentative stability boundary for the correct results.
the presence of an outer stable layer nearly irrespectively of the layer thickness in the range 0:2 6 do 6 0:8. For the results obtained with the defective code there was a dichotomy between dipolar solutions with small R2 =R1 and the hemispherical solutions for which R2 =R1 was of order one or larger. With the correct code the transition is more gradual. Some cases in light grey near the rectified stability boundary (full thick line in Fig. 1) have a magnetic field that is still dominated by the axial dipole but also shows some traits of a hemispherical dynamo. Their time-average magnetic field contains a persistent axial quadrupole contribution of moderate amplitude (compared to that of the axial dipole) and a ratio R2 =R1 > 0:04. The magnetic Prandtl number Pm (the ratio of viscosity to magnetic diffusivity) also has an influence on the dynamo mode. In three model pairs where one case differs from the other only in the value of Pm (cases B.8 and C.8, D.2 and E.2, H.2 and J.2), a purely dipolar field with R2 =R1 < 0:04 is found at the smaller value of Pm 6 2, whereas at higher Pm P 3 the field is hemispherical or
65
of mixed dipolar-hemispherical type. Models with Pm 6 2 are distinguished in Fig. 1 by plotting the symbol with a slight offset in vertical directions above the lines for do ¼ 0; 0:2; 0:4; 0:8. Finally, in Christensen (2015) bistability was reported to exist for at least one case (case 9.2), which stayed on the dipolar branch when started with a strong dipole field but settled on the hemispherical branch when started with a weak irregular seed field. With the corrected code the magnetic field turned dipolar also with the latter initial condition. Also in other cases no indication of bistability has been found: cases 3.4, 4.2, 9.8 and 11.2 have been started with the hemispherical solution obtained with the defective code as initial condition, but all turned dipolar. In summary, the stability of strongly dipolar dynamos with a low quadrupole-to-dipole ratio R2 =R1 < 0:04 in models with a stably stratified layer is significantly higher than what had been found with the faulty code. Otherwise the properties of dipolar solutions reported in Christensen (2015) have not been much affected by the error in the code. The main conclusion in that paper, that an iron snow dynamo in Ganymede offers a viable scenario for explaining the known and inferred properties of Ganymede’s magnetic field, therefore still holds and is even strengthened by the expansion of the stability range of strongly dipolar dynamos towards higher values of RaE9=4 . The newly found trend towards more dipolar solutions at lower values of the magnetic Prandtl number further strengthens the case, since in planetary dynamos Pm 1. Also most other conclusions in Christensen (2015) still hold, e.g. an iron snow model is compatible with the observed magnitude of Ganymede’s dipole moment and secular variation time scales are larger compared to those of a dynamo without a stable outer layer. Since the previously claimed preference for hemispherical dynamos in case of a thick stable layer does not exist, the only conclusion that turns out to be unfounded is that Ganymede’s low quadrupole moment would indicate that the snow layer cannot be much thicker than 200 km.
References Christensen, U.R., 2015. Iron snow dynamo models for Ganymede. Icarus 247, 248–259. Christensen, U.R., Wicht, J., 2007. Numerical dynamo simulations. In: Schubert, G. (Ed.), Treatise on Geophysics, Core Dynamics, vol. 8. Elsevier, Amsterdam, pp. 245–282. Hauck, S.A., Aurnou, J.M., Dombard, A.J., 2006. Sulfur’s impact on core evolution and magnetic field generation on Ganymede. J. Geophys. Res. 111, E09008. Kivelson, M., Khurana, K., Volwerk, M., 2002. The permanent and induced magnetic moments of Ganymede. Icarus 157, 507–522. Landeau, M., Aubert, J., 2011. Equatorially asymmetric convection inducing a hemispherical magnetic field in rotating spheres and implications for the past martian dynamo. Phys. Earth Planet. Inter. 185, 61–73. Saur, J., Duling, S., Roth, L., Jia, X., Strobel, D., Feldman, P., Christensen, U., Retherford, K., McGrath, M., Musacchio, F., Wennmacher, A., Neubauer, F., Simon, S., Hartkorn, O., 2015. The search for a subsurface ocean in Ganymede with Hubble space telescope observations of its auroral oval. J. Geophys. Res. 120. http:// dx.doi.org/10.1002/2014JA020778.