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

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

Ra = (convective driving) / (viscous and thermal diffusion) Ek = (viscous diffusion) / (Coriolis effects) Pr = (viscous diffusion) / (thermal diffusion) q = (thermal diffusion) / (magnetic diffusion) Ro c = (Ra/Pr) 1/2 Ek = (convective driving) / (Coriolis effects) = N/  if stable Re = (fluid velocity) / (viscous diffusion velocity) Rm = (fluid velocity) / (magnetic diffusion velocity) Ro = (fluid velocity) / (rotational velocity) Ro m = (Alfven velocity) / (rotational velocity)

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)

Brun, Miesch, Toomre Solar dynamo model Anelastic with  bot  top  30    4 Ra = 8x10 4 Ek = Ro c = (g  T/D) 1/2 / 2  = 0.7 Spatial resolution: 128 x 512 x 1024 Re = 10 2 Rm = 4x10 2 Ro = 10 -1

Brun, Miesch, Toomre Radial velocity

Brun, Miesch, Toomre Enstrophy

Brun, Miesch, Toomre Radial magnetic field

Brun, Miesch, Toomre Toroidal magnetic field

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

Poloidal magnetic field Brun, Miesch, Toomre

Glatzmaier, Roberts Geodynamo model Anelastic with  bot  top  1.2    1   700 (with hyperdiffusion) Ra 2x10 -6 Ro c = (g  T/D) 1/2 / 2  = or    1   1 (with  too small or, ,  all too large) Spatial resolution: 65 x 32 x 64 Re < 1 Rm < 7x10 2 Ro = 2x10 -6

Geodynamo simulation Differential rotation is a thermal wind

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

Dipole moment Pole latitude kyrs

Glatzmaier Jovian dynamo model Anelastic with  bot  top  27  0.01 (  and  are constants)   1 in deep (metallic) region and up to in the upper (molecular) region Internal heating proportional to pressure Solar heating at surface Ra = 10 8 Ek = Ro c = (g  T/D) 1/2 / 2  = 0.1 Spatial resolution: 289 x 384 x 384 Re = 10 4 Rm > 10 4 Ro = 10 -2

Longitudinal velocity

Entropy In equatorial plane viewed from northern hemisphere

Kinetic energy In equatorial plane viewed from northern hemisphere

Magnetic energy In equatorial plane viewed from northern hemisphere

Longitudinal velocity In equatorial plane viewed from northern hemisphere

Anelastic Glatzmaier Jupiter dynamo simulations shallow deep Longitudinal flow

Zonal winds

Radial magnetic field

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 Model shortcomings: low resolution large diffusivities laminar flow (Boussinesq) Possible similarities in toroidal field generation: ICB / tangent cylinder tachocline hydrogen phase transition

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