Planetary Fluid Mechanics in the Lab

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

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


Stats:

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

Outer sphere at two revolutions/second


Dynamos

Westinghouse


Homogeneous


Glatzmaier and Roberts Reversing Dynamo Simulation 1995

Why 13 Tons of Spinning Sodium?



  • Sodium best chance for liquid metal dynamo.
  • Fast rotation is very important in planetary dynamo.

Geometry & Forcing


Design:

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

Instrumentation

Dimensionless Numbers


$$Ro = \frac{\Delta\Omega}{\Omega}, |Ro|<100 $$
$$Re = \frac{\Delta\Omega(r_o-r_i)^2}{\nu}, Re\sim10^8$$
$$Rm = \frac{\Delta\Omega(r_o-r_i)^2}{\eta}, Rm\sim10^3$$
$$Pm = \frac{\nu}{\eta} = \frac{Rm}{Re} \sim 10^{-5}$$
$$S = \frac{B_0 L}{\eta\sqrt{\rho \mu_0}}, S\sim6$$
$$\Lambda = \frac{B_0^2}{\rho\mu_0\eta\Omega},\Lambda\sim15$$
$$Ha = \frac{B_0 L}{\sqrt{\rho \mu_0 \eta \nu}},Ha\sim2\times10^3$$

Hydrodynamic Preview

  • Torque, G: common turbulent scaling with Re
  • State changes: dozens of states depending on Ro
  • Turbulent rotating shear flow torque:

    $$\small G(Re,Ro)=f(Ro)g(Re)$$
    $$\small g(Re)=C_f Re^2, Re\rightarrow\infty$$

Torque vs. Reynolds Number, Outer Stationary

Torque vs. Reynolds Number, Outer 1.25Hz

  • $$Ro = \Delta\Omega/\Omega$$

Torque and Azimuthal Velocity - State Changes

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

Probability Distribution of Torque

$$\small Ro = 2.13$$ Phys. Fluids 23, 065104 (2011) - http://arxiv.org/abs/1107.5082

Different Large-Scale Flow States

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

Turbulent Torque Rossby Dependence

Magnetohydrodynamic Preview

  • Differential rotation generates strong internal azimuthal field (strong ω-effect)- important for dynamo.
  • Large Ro-dependence from hydrodynamic state changes: ω-effect peaks at Ro=+6.
  • Strong applied field: new states, reduced ω-effect, field bursting with a "dynamo-like" feedback loop.

Internal Field and External Gauss Coefficients

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

Internal Magnetic Field, "Weak" Applied Field

$$\small S=0.39$$

Internal Magnetic Field vs. Applied Field

State Changes at Strong Field

Field Bursts

$$\small S=3.5, Ro = +6, Rm = 477$$
Ro=+6.0, S=3.5, Rm=477

Summary

  • Torque, G: common turbulent scaling with Re
  • Dozens of turbulent flow states depending on Ro
  • Turbulent rotating shear flow torque:

    $$\small G(Re,Ro)=f(Ro)g(Re)$$
    $$\small g(Re)=C_f Re^2, Re\rightarrow\infty$$
  • Differential rotation gives strong ω-effect- important for dynamo.
  • Large Ro-dependence of ω-effect due to hydrodynamic state changes.
  • Strong applied field: new states, reduced ω-effect, field bursting (Ro=+6) with a "dynamo-like" feedback loop.
  • Opportunities for good quantitative tests for spherical turbulent codes (fluid Rossby relatively small!)