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A physical conjecture for the dipolar–multipolar dynamo transition

  • B. R. McDermott (a1) and P. A. Davidson (a1)

Abstract

In numerical simulations of planetary dynamos there is an abrupt transition in the dynamics of both the velocity and magnetic fields at a ‘local’ Rossby number of 0.1. For smaller Rossby numbers there are helical columnar structures aligned with the rotation axis, which efficiently maintain a dipolar field. However, when the thermal forcing is increased, these columns break down and the field becomes multi-polar. Similarly, in rotating turbulence experiments and simulations there is a sharp transition at a Rossby number of ${\sim}0.4$ . Again, helical axial columnar structures are found for lower Rossby numbers, and there is strong evidence that these columns are created by inertial waves, at least on short time scales. We perform direct numerical simulations of the flow induced by a layer of buoyant anomalies subject to strong rotation, inspired by the equatorially biased heat flux in convective planetary dynamos. We assess the role of inertial waves in generating columnar structures. At high rotation rates (or weak forcing) we find columnar flow structures that segregate helicity either side of the buoyant layer, whose axial length scale increases linearly, as predicted by the theory of low-frequency inertial waves. As the rotation rate is weakened and the magnitude of the buoyant perturbations is increased, we identify a portion of the flow which is more strongly three-dimensional. We show that the flow in this region is turbulent, and has a Rossby number above a critical value $Ro^{crit}\sim 0.4$ , consistent with previous findings in rotating turbulence. We suggest that the discrepancy between the transition value found here (and in rotating turbulence experiments), and that seen in the numerical dynamos ( $Ro^{crit}\sim 0.1$ ), is a result of a different choice of the length scale used to define the local $Ro$ . We show that when a proxy for the flow length scale perpendicular to the rotation axis is used in this definition, the numerical dynamo transition lies at $Ro^{crit}\sim 0.5$ . Based on this we hypothesise that inertial waves, continually launched by buoyant anomalies, sustain the columnar structures in dynamo simulations, and that the transition documented in these simulations is due to the inability of inertial waves to propagate for $Ro>Ro^{crit}$ .

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Corresponding author

Email address for correspondence: pad3@eng.cam.ac.uk

References

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Baqui, Y. B. & Davidson, P. A. 2015 A phenomenological theory of rotating turbulence. Phys. Fluids 27 (2), 025107.
Bardsley, O. P. & Davidson, P. A. 2016 Inertial–Alfvén waves as columnar helices in planetary cores. J. Fluid Mech. 805, R2.
Bardsley, O. P. & Davidson, P. A. 2017 The dispersion of magnetic-Coriolis waves in planetary cores. Geophys. J. Intl 210 (1), 1826.
Bouffard, M., Labrosse, S., Choblet, G., Fournier, A., Aubert, J. & Tackley, P. J. 2017 A particle-in-cell method for studying double-diffusive convection in the liquid layers of planetary interiors. Comput. Phys. 346, 552571.
Bracewell, R. N. 1986 The Fourier Transform and its Applications, 2nd edn. McGraw-Hill.
Busse, F. H. 1975 A model of the geodynamo. Geophys. J. Intl 42 (2), 437459.
Christensen, U. R. & Aubert, J. 2006 Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields. Geophys. J. Intl 166 (1), 97114.
Dallas, V. & Tobias, S. M. 2016 Forcing-dependent dynamics and emergence of helicity in rotating turbulence. J. Fluid Mech. 798, 682695.
Davidson, P. A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.
Davidson, P. A. 2014 The dynamics and scaling laws of planetary dynamos driven by inertial waves. Geophys. J. Intl 198 (3), 18321847.
Davidson, P. A. 2016 Dynamos driven by helical waves: scaling laws for numerical dynamos and for the planets. Geophys. J. Intl 207 (2), 680690.
Davidson, P. A. & Ranjan, A. 2015 Planetary dynamos driven by helical waves – II. Geophys. J. Intl 202 (3), 16461662.
Davidson, P. A. & Ranjan, A. 2018 Are planetary dynamos driven by helical waves? J. Plasma Phys. 84 (3), 735840304.
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135144.
Dormy, E., Oruba, L. & Petitdemange, L. 2018 Three branches of dynamo action. Fluid Dyn. Res. 50 (1), 011415.
Driscoll, P. & Olson, P. 2009 Effects of buoyancy and rotation on the polarity reversal frequency of gravitationally driven numerical dynamos. Geophys. J. Intl 178 (3), 13371350.
Garcia, F., Oruba, L. & Dormy, E. 2017 Equatorial symmetry breaking and the loss of dipolarity in rapidly rotating dynamos. Geophys. Astrophys. Fluid Dyn. 111 (5), 380393.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Gubbins, D. 2001 The Rayleigh number for convection in the Earth’s core. Phys. Earth Planet. Inter. 128 (1-4), 312.
Hide, R., Ibbetson, A. & Lighthill, M. J. 1968 On slow transverse flow past obstacles in a rapidly rotating fluid. J. Fluid Mech. 32 (2), 251272.
Jackson, A., Jonkers, A. R. T. & Walker, M. R. 2000 Four centuries of geomagnetic secular variation from historical records. Phil. Trans. R. Soc. Lond. A 358 (1768), 957990.
Kageyama, A., Miyagoshi, T. & Sato, T. 2008 Formation of current coils in geodynamo simulations. Nature 454 (7208), 11061109.
Kutzner, C. & Christensen, U. R. 2002 From stable dipolar towards reversing numerical dynamos. Phys. Earth Planet. Inter. 131 (1), 2945.
Lighthill, M. J. 1970 The theory of trailing Taylor columns. Proc. Camb. Phil. Soc. 68 (2), 485491.
Mininni, P. D., Alexakis, A. & Pouquet, A. 2009 Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers. Phys. Fluids 21 (1), 015108.
Mininni, P. D. & Pouquet, A. 2009a Finite dissipation and intermittency in magnetohydrodynamics. Phys. Rev. E 80 (2), 025401.
Mininni, P. D. & Pouquet, A. 2009b Helicity cascades in rotating turbulence. Phys. Rev. E 79 (2), 026304.
Olson, P. & Christensen, U. R. 2006 Dipole moment scaling for convection-driven planetary dynamos. Earth Planet. Sci. Lett. 250 (3-4), 561571.
Olson, P., Christensen, U. R. & Glatzmaier, G. A. 1999 Numerical modeling of the geodynamo: mechanisms of field generation and equilibration. J. Geophys. Res. 104 (B5), 1038310404.
Oruba, L. & Dormy, E. 2014 Transition between viscous dipolar and inertial multipolar dynamos. Geophys. Res. Lett. 41 (20), 71157120.
Ranjan, A. & Davidson, P. A. 2014 Evolution of a turbulent cloud under rotation. J. Fluid Mech. 756, 488509.
Ranjan, A., Davidson, P. A., Christensen, U. R. & Wicht, J. 2018 Internally driven inertial waves in geodynamo simulations. Geophys. J. Intl 213 (2), 12811295.
Roberts, P. H. & King, E. M. 2013 On the genesis of the Earth’s magnetism. Rep. Prog. Phys. 76 (9), 096801.
Sahoo, G., Perlekar, P. & Pandit, R. 2011 Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence. New J. Phys. 13 (1), 013036.
Sakuraba, A. & Roberts, P. H. 2009 Generation of a strong magnetic field using uniform heat flux at the surface of the core. Nat. Geosci. 2 (11), 802805.
Sano, M., Wu, X. Z. & Libchaber, A. 1989 Turbulence in helium-gas free convection. Phys. Rev. A 40 (11), 64216430.
Schaeffer, N., Jault, D., Nataf, H.-C. & Fournier, A. 2017 Turbulent geodynamo simulations: a leap towards Earth’s core. Geophys. J. Intl 211 (1), 129.
Sheyko, A., Finlay, C. C. & Jackson, A. 2016 Magnetic reversals from planetary dynamo waves. Nature 539 (7630), 551554.
Soderlund, K. M., King, E. M. & Aurnou, J. M. 2012 The influence of magnetic fields in planetary dynamo models. Earth Planet. Sci. Lett. 333, 920.
Soderlund, K. M., King, E. M. & Aurnou, J. M. 2014 Corrigendum to ‘the influence of magnetic fields in planetary dynamo models’ [Earth Planet. Sci. Lett. 333–334 (2012) 9–20]. Earth Planet. Sci. Lett. 392, 121123.
Sreenivasan, B. & Davidson, P. A. 2008 On the formation of cyclones and anticyclones in a rotating fluid. Phys. Fluids 20 (8), 085104.
Sreenivasan, B. & Jones, C. A. 2011 Helicity generation and subcritical behaviour in rapidly rotating dynamos. J. Fluid Mech. 688, 530.
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81105.
Sumita, I. & Olson, P. 2000 Laboratory experiments on high Rayleigh number thermal convection in a rapidly rotating hemispherical shell. Phys. Earth Planet. Inter. 117 (1-4), 153170.
Yarom, E. & Sharon, E. 2014 Experimental observation of steady inertial wave turbulence in deep rotating flows. Nat. Phys. 10 (7), 510514.
Yeung, P. K. & Zhou, Y. 1998 Numerical study of rotating turbulence with external forcing. Phys. Fluids 10 (11), 28952909.
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A physical conjecture for the dipolar–multipolar dynamo transition

  • B. R. McDermott (a1) and P. A. Davidson (a1)

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