A Laboratory Planetary Core

Transport, Waves, and Magnetic Fields in Spherical Couette Flow at Very High Re.

Daniel S. Zimmerman
Santiago A. Triana
Daniel P. Lathrop

Work made possible by:
NSF/MRI EAR-0116129
NSF EAR-1114303
University of Maryland Physics/IREAP/Geology

The Experiment


  • Outer sphere maximum speed $$\Omega_o/2\pi=4\mathrm{Hz}$$
  • Inner sphere maximum speed $$\Omega_\mathrm{i}/2\pi=\pm20\mathrm{Hz}$$
  • Total rotating mass: 20 tons (18000kg)
    • 7 ton (6300kg), 3m diameter shell
    • 13 tons (12000kg) fluid (water, sodium metal)
  • Two 250kW (350HP) motors
  • Hot oil system: 120kW heating + 500kW cooling

Outer sphere at two revolutions/second

Why 13 Tons of Spinning Sodium?

  • Sodium best chance for liquid metal dynamo.
  • Fast rotation for self-organization.
  • Large and relatively slow rotating for "strong" magnetic effects.
  • Waves and other large-scale flows with low boundary friction.

Geometry & Forcing


  • Geometrically similar to Earth's core: $$\Gamma = r_\mathrm{i}/r_\mathrm{o} = 0.35$$
  • Imposed differential rotation (not convection!) to provide stirring in rotating frame.
  • Simple geometry: good for simulation
  • Common features with planetary core, not a scale model.


Dimensionless Numbers

$$Ro = \frac{\Delta\Omega}{\Omega_o},0.01<|Ro|<100 $$
$$E = \frac{\nu}{\Omega_o(r_o-r_i)^2},10^{-8}< E<10^{-6}$$
$$Re = \frac{\Delta\Omega(r_o-r_i)^2}{\nu}, 10^6 < Re < 10^8 $$
$$Rm = \frac{\Delta\Omega(r_o-r_i)^2}{\eta}, 10 < Rm < 1000$$
$$Pm = \frac{\nu}{\eta} = \frac{Rm}{Re} \sim 10^{-5}$$
$$S = \frac{B_0 (r_o-r_i)}{\eta\sqrt{\rho \mu_0}}, 0 < S < 6 $$
$$\Lambda = \frac{B_0^2}{\rho\mu_0\eta\Omega_o},0 < \Lambda < 13\:\: (2\pi/\Omega_o = 30s)$$

Hydrodynamic Outline

  • Many turbulent flow states at different Ro.
  • Some flow transitions show large changes in angular momentum transport.
  • Turbulent scaling at fixed Ro.

Torque vs. Reynolds Number, Outer Stationary

Torque vs. Reynolds Number, Outer 1.25Hz

  • $$Ro = \Delta\Omega/\Omega_o$$

State Transitions: Torque and Azimuthal Velocity

$$\small Ro = 2.33$$ Phys. Fluids 23, 065104 (2011) - http://arxiv.org/abs/1107.5082

State Transitions: Mean Flows

Phys. Fluids 23, 065104 (2011) - http://arxiv.org/abs/1107.5082

State Transitions: Waves

Velocity frequency spectra:

Angular Momentum Transport: Torque vs. Ro

$$\small G(Ro,Re) = f(Ro)G_\infty(Re)$$

MHD Outline

  • Strong generation of Bφ (Ω-effect), large Ro-dependence, peaks at Ro=+6.
  • Strong applied field: new states, reduced Ω-effect, dipole moment enhancement from "dynamo-like" feedback loop.

Internal Field and External Gauss Coefficients

$$\scriptsize B_r(r,\theta,\phi,t) = \sum_{l=0}^{l=4}\sum_{m=0}^{m=l}l(l+1)\left(\frac{r_\mathrm{o}}{r}\right)^{l+2}P_l^m(\cos{\theta})(g_l^{m,s}(t)\sin{m\phi}+g_l^{m,c}(t)\cos{m\phi})$$
$$\scriptsize B_l^m = l(l+1)g_l^m$$

Internal Magnetic Field, "Weak" Applied Field

$$S=0.39$$Note: legend typo: circles are always Bφ

Internal Magnetic Field vs. Applied Field

State Changes at Strong Field

RMS Gauss Coefficients & Torque

Dipole Bursting State

$$\small S=3.5, Ro = +6, Rm = 430, E = 1.2\times10^{-7} $$
Ro=+6.0, S=3.5, Rm=430, E = 1.2x10-7, Re = 5x107
20% actual speed

Dynamo-style feedback?

Axisymmetric flow can't induce external dipole
from axisymmetric applied field.


  • Many different turbulent flow states in high-Re spherical Couette controlled by Ro.
  • Different turbulent states have much different large scales: mean flows and waves.
  • Large Ro-dependence of Ω-effect due to hydrodynamic state changes.
  • Strong applied field: new states, reduced Ω-effect, dipole bursts with a "dynamo-like" feedback loop.
  • Dipole enhanced by large scale nonaxisymmetric waves
  • Movie code: https://github.com/danzimmerman/matlabmag

Challenges and Promises

  • Data so far can provide good quantitative tests for models: Gauss coefficients, torque, waves, azimuthal and radial field.
  • To model spherical Couette for dynamo purposes, need to capture same states. Do we? Re high enough?