SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION Supplementary Figure 1 Nondimensional magnetic and kinetic energy densities averaged in the fluid core plotted as functions of time. Black and red lines show results of the UHFM and USTM, respectively. Thick (thin) lines represent the magnetic (kinetic) energy. One dipole diffusion time is c 2 /π 2 η.1c 2 /η = 19,4 years, if η =2m 2 /s and c = 348 km. Supplementary Figure 2 Contribution of Coriolis, Lorentz and inertial terms in the vorticity equation. We calculated the Coriolis term (F C = r (u e z ); black line), the Lorentz term (F L = r (j b); red line), and the inertial term (F I = r [ Pm 1 Eu ( u) ] ; blue line) along the latitude circle (z =.1c, r =.675c), for (a) the UHFM and (b) the USTM. The buoyancy term is exactly zero in this expression. The data are the same as illustrated in Fig. 1 in the main text. The magnetic and kinetic energy densities are also shown by red and blue lines, respectively. In the UHFM, the magnetic energy is much larger than the kinetic energy and the Lorentz term tends to be balanced by the Coriolis term. Their relation to the inertial term is not obvious, with the exception of a few thin zones where jet-like flows exist (e.g., at φ = 1 ). In the USTM, the magnetic and kinetic energy densities are of the same order (equipartition), and the Lorentz term behaves similarly to the inertial term, with the exception of a few places where a relatively large-scale magnetic field is generated (e.g., at φ = 9 ). In general, the Lorentz and inertial forces are respectively written as µ 1 B 2 /L and ρu 2 /L according to an order-of- 1 nature geoscience 1

2 magnitude estimation, where B is the field intensity, U is the velocity and L is the length scale common to both fields. Then, equipartition, which yields 1 2 µ 1 B ρu 2, means that the Lorentz and the inertial forces are of the same order. Supplementary Figure 3 Balance between azimuthal forces averaged in time and longitude on the equatorial plane. The Lorentz, Coriolis and inertial forces in our UHFM are shown as functions of the nondimensional distance from the z-axis by black, blue and light-blue lines, respectively. The red line represents j s b z, which well approximates the total Lorentz force j z b s j s b z, where denotes the average over time and longitude. This approximation would be valid if either b s or j z were nearly zero. In our case, both b s and j z are nearly zero on the equatorial plane because the solution has near mirror symmetry across the equatorial plane. In more turbulent cases like the Earth s core, this approximation would become worse, but a similar argument could be still made as long as a symmetry of dipole-type existed in the solution. Supplementary Movie1 Radial magnetic field at the CMB in Mollweide projection for the UHFM. The time span is.48 t.52 in our nondimensional unit, corresponding to about 7,7 years for the Earth s core (see Supplementary Fig. 1). 2 nature geoscience

3 Supplementary Movie 2 Radial magnetic field at the CMB in Mollweide projection for the USTM. The time span is.195 t.225 in our nondimensional unit, corresponding to about 5,8 years for the Earth s core (see Supplementary Fig. 1). nature geoscience 3

4 Supplementary Figure 1 5 magnetic and kinetic energies time 4 nature geoscience

5 Supplementary Figure 2 (a) nature geoscience 5

6 Supplementary Figure 2 (b) nature geoscience

7 Supplementary Figure 3 8 azimuthal force nondimensional radius nature geoscience 7