Global Dynamo Models: Planetary and Stellar Dynamos

1. Direct simulation of planetary and stellar dynamos I. Methods and results Gary A Glatzmaier University of California, Santa Cruz

2. Solar interior (anelastic models) Gilman, Miller Glatzmaier Miesch, Clune, Brun, Toomre, … Earth’s fluid outer core (mostly Boussinesq models) Zhang, Busse, … Kageyama, … Glatzmaier, Roberts Jones, … Kuang, Bloxham Kida, … Tilgner, Busse, … Christensen, Olson, Glatzmaier Sakuraba, Kono Aubert, Cardin, Dormy, … Katayama, … Hollerbach Livermore Hejda, Reshetnyak Giant planet interiors (mostly Boussinesq models) Glatzmaier Sun, Schubert Busse, … Christensen, … Aurnou, Olson, … Stanley, Bloxham

5. Ra = (convective driving) / (viscous and thermal diffusion) Ek = (viscous diffusion) / (Coriolis effects) Pr = (viscous diffusion) / (thermal diffusion) q = (thermal diffusion) / (magnetic diffusion) Roc = (Ra/Pr)1/2 Ek = (convective driving) / (Coriolis effects) = N/W if stable Re = (fluid velocity) / (viscous diffusion velocity) Rm = (fluid velocity) / (magnetic diffusion velocity) Ro = (fluid velocity) / (rotational velocity) Rom = (Alfven velocity) / (rotational velocity)

6. Numerical method • poloidal / toroidal decomposition of momentum density • and magnetic field • spherical harmonics and Chebyshev polynomials • - spectral transform method, Chebyshev collocation and a semi-implicit time integration • parallel (MPI)

7. Solar dynamo model Anelastic withrbot / rtop = 30 n/k = 0.125 n/h =4 Ra = 8x104 Ek = 10-3 Roc = (gaDT/D)1/2 / 2W = 0.7 Spatial resolution:128 x 512 x 1024 Re = 102 Rm = 4x102 Ro = 10-1 Brun, Miesch, Toomre

8. Radial velocity Brun, Miesch, Toomre

9. Brun, Miesch, Toomre Enstrophy

10. Radial magnetic field Brun, Miesch, Toomre

11. Brun, Miesch, Toomre Toroidal magnetic field

12. Solar dynamo simulation Differential rotation and meridional circulation Anelastic Brun, Miesch, Toomre

13. Poloidal magnetic field Brun, Miesch, Toomre

14. Geodynamo model Anelastic withrbot / rtop = 1.2 k/h =1n/k = 700 (with hyperdiffusion) Ra < 109 Ek > 2x10-6 Roc = (gaDT/D)1/2 / 2W = 0.002 or k/h =1n/k = 1 (with W too small or n, k, h all too large) Spatial resolution:65 x 32 x 64 Re < 1 Rm < 7x102 Ro = 2x10-6 Glatzmaier, Roberts

17. Geodynamo simulation Differential rotation is a thermal wind

18. Inner core super-rotation with gravitational coupling between inner core and mantle

21. Dipole moment Pole latitude kyrs

24. Jovian dynamo model Anelastic withrbot / rtop = 27 n/k = 0.01 (n and k are constants) n/h =1 in deep (metallic) region and up to 0.001 in the upper (molecular) region Internal heating proportional to pressure Solar heating at surface Ra = 108 Ek = 10-6 Roc = (gaDT/D)1/2 / 2W = 0.1 Spatial resolution:289 x 384 x 384 Re = 104 Rm > 104 Ro = 10-2 Glatzmaier

25. Longitudinal velocity

26. Entropy In equatorial plane viewed from northern hemisphere

27. Kinetic energy In equatorial plane viewed from northern hemisphere

28. Magnetic energy In equatorial plane viewed from northern hemisphere

29. Longitudinal velocity In equatorial plane viewed from northern hemisphere

30. Jupiter dynamo simulations Longitudinal flow Anelastic shallow deep Glatzmaier

31. Zonal winds

32. Radial magnetic field

34. Current 3D global MHD dynamo models for the Earth, Jupiter and the sun Many differences: dimension, mass, rotation rate, equation of state, heat flux, force balance, energy balance, differential rotation, magnetic reversals Possible similarities in toroidal field generation: Earth @ ICB / tangent cylinder Sun @ tachocline Jupiter @ hydrogen phase transition Model shortcomings: low resolution large diffusivities laminar flow (Boussinesq)

35. Challenges for the next generation of global dynamo models high spatial resolution in 3D small diffusivities turbulent flow density stratification gravity waves in stable regions phase transitions massively parallel computing improved numerical methods anelastic equations sub-grid scale models