Corrigendum to "The influence of magnetic fields in planetary dynamo models" (Earth Planet. Sci. Lett. 333-334 (2012) 9-20)

Earth and Planetary Science Letters 392 (2014) 121–123 Contents lists available at ScienceDirect Earth and Planetary Science Letters www.elsevier.com/locate/epsl Corrigendum Corrigendum to “The inﬂuence of magnetic ﬁelds in planetary dynamo models” [Earth Planet. Sci. Lett. 333–334 (2012) 9–20] K.M. Soderlund a ,∗ , E.M. King b , J.M. Aurnou c a b c Institute for Geophysics, Jackson School of Geosciences, University of Texas at Austin, Austin, TX 78758, USA Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA a r t i c l e i n f o Article history: Available online xxxx We have investigated the inﬂuence of magnetic ﬁelds on the convective properties of planetary dynamo models and the transi- tion between dynamos with dipolar and multipolar magnetic ﬁelds in Soderlund et al. (2012). We thank Uli Christensen for pointing out an error solely in our algorithm used to calculate the volu- metric average of relative axial helicity, H rel z . However, it should be noted that the main results of the paper remain intact, as dis- cussed below. Relative axial helicity is deﬁned as axial helicity normalized by its maximum possible value, H rel z = u z ω z h ( u z u z h ω z ω z h ) 1 / 2 where u z is axial velocity, ω z = ∇ × u · z ˆ is axial vorticity, and h is the volumetric average in each hemisphere excluding bound- ary layers (e.g., Olson et al., 1999; Schmitz and Tilgner, 2010). We average the helicity magnitude over each hemisphere since axial helicity tends to be anti-symmetric across the equator and assume Ekman boundary layer thicknesses of δ E / D = 3E 1 / 2 following King et al. (2012). In Soderlund et al. (2012), the | H rel z | calculations are missing a factor of sin θ in the volume integral. The erroneously calculated helicity data are shown in Fig. 1a. The effect of this missing fac- tor is to weight the contributions within the tangent cylinder too heavily. Corrected | H rel z | are shown in Fig. 1b. While instantaneous val- ues were given in Soderlund et al. (2012), here we report the average of three random snapshots in time for each case. Temporal standard deviations ( σ ) are typically near 0 . 02. The corrected val- ues differ on average from the original values by [0.02 (0.02), 0.04 (0.04), 0.07 (0.02)] for cases with E = [ 10 − 3 , 10 − 4 , 10 − 5 ] where parentheses indicate non-magnetic cases. These differences only exceed 2 σ in the lowest Ekman number dynamo cases. Tables 1 and 2 provide corrected values of | H rel z | . In addition, we have also included time averaged values of axial vorticity colum- narity C ω z , in situ Lorentz to Coriolis force ratio F L / F C , and in situ inertial to viscous force ratio F I / F V . The two main conclusions of the paper are not modiﬁed sig- niﬁcantly. They are: (i) The magnetic ﬁelds do not strongly affect relative axial helicity in cases with E 10 − 4 . This weak inﬂuence can be understood theoretically by estimating the strength of the Lorentz forces in the high magnetic Reynolds number limit with a dynamic Elsasser number, Λ d = B 2 /( 2 ρμ o Ω U B ) . Calculations of the dynamic Elsasser number correctly capture the secondary inﬂuence of Lorentz forces on convection. (ii) The breakdown of dipolar magnetic ﬁeld generation still coincides with the degra- dation of helicity in the ﬂow, even though the helicity decreases are markedly less abrupt than originally reported (see Fig. 1b). Im- portantly, the dipolarity breakdown still occurs when the inertial and viscous forces become comparable, F I / F V ∼ 1, for all Ekman numbers considered here (10 − 3 E 10 − 5 ). Because viscosity is expected to be negligible in planetary settings, the dipolarity breakdown in present day dynamo models with moderate Ekman numbers may not extrapolate to planets. Acknowledgements DOI of original article: http://dx.doi.org/10.1016/j.epsl.2012.03.038. Corresponding author. Tel.: +1 (218) 349 3006; fax: +1 (512) 471 8844. E-mail addresses: krista@ig.utexas.edu (K.M. Soderlund), eric.king@berkeley.edu (E.M. King), aurnou@ucla.edu (J.M. Aurnou). http://dx.doi.org/10.1016/j.epsl.2014.01.052 0012-821X/ © 2014 Elsevier B.V. All rights reserved. We thank Uli Christensen and Johannes Wicht for their assis- tance to verify our calculations.