Direct simulation of planetary and stellar dynamos II. Future challenges

1. Direct simulation of planetary and stellar dynamos II. Future challenges (maintenance of differential rotation) Gary A Glatzmaier University of California, Santa Cruz

3. Differential rotation depends on many factors geometry: depth of convection zone, size of inner core boundary conditions: thermal, velocity and magnetic stratification: thermal (stable and unstable regions) density (number of scale heights and profile) composition and phase changes diffusion coefficients: amplitudes and radial profiles magnetic field: Lorentz forces oppose differential rotation parameters: 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) Re = (fluid velocity) / (viscous diffusion velocity) Rm = (fluid velocity) / (magnetic diffusion velocity) Ro = (fluid velocity) / (rotational velocity) Rom = (Alfven velocity) / (rotational velocity)

4. Geodynamo simulation Differential rotation is a thermal wind

5. Surface zonal winds Jupiter Saturn

6. T wz wz vf Boussinesq Roc = 0.04 equatiorial plane meridian plane Roc = 0.21 Christensen

7. Jovian dynamo model Anelastic with rbot / rtop = 27 n/k = 0.01 everywhere n/h =1 in lower part and 0.001 at top Internal heating proportional to pressure Solar heating at surface Ra = 108 Ek = 10-6 Roc = (gaDT/D)1/2 / 2W = 10-1 Spatial resolution: 289 x 384 x 384 Glatzmaier

8. Jupiter dynamo simulations Longitudinal flow Anelastic shallow deep Glatzmaier

9. Solar differential rotation

10. Solar dynamo model Anelastic with rbot / 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 Brun, Miesch, Toomre

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

12. Convection turbulent vs laminar compressible vs incompressible

13. 2D anelastic rotating magneto-convection 2001 x 4001 Pr = n / k = n / h = 1.0, 0.1 Ek = n/ 2WD2= 10-4, 10-9 Ra = gaDTD3/ nk = 106, 1012 Re = v D / n = 103, 106 Ro = v / 2WD = 10-1, 10-3

14. Laminar convection large diffusivities small density stratification

15. Turbulent convection small diffusivities large density stratification

16. Turbulent convection with rotation and magnetic field small diffusivities small density stratification

17. Laminar convection 6 -4 Ra = 3x10 Ek = 10 height mean entropy large diffusivities small density stratification

18. Turbulent convection 12 -9 Ra = 3x10 Ek = 10 height mean entropy small diffusivities small density stratification

19. Anelastic vorticity equation (curl of the momentum equation) i.e., vorticity inverse density scale height H = height of Taylor column above equatorial plane for 2D parameterization (2D disk)

20. Anelastic Taylor-Proudman Theorem Assume geostrophic balance for the momentum equation and take its curl:

21. Anelastic potential vorticity theorem Assume a balance among the inertial, pressure gradient and Coriolis terms in the 2D momentum equation:

22. Density stratified flow in equatorial plane Sinking parcel contracts and gains positive vorticity Rising parcel expands and gains negative vorticity The spiral pattern at the boundary having the greatest hr effect eventually spreads throughout the convection zone.

23. Incompressible columnar convection The shape of the boundary determines the tilt of the columns, which determines the convergence of angular momentum flux, which maintains the differential rotation. Zonal flow is prograde in outer part and retrograde in inner part. Zonal flow is retrograde in outer part and prograde in inner part. Busse

24. Turbulent Boussinesq convection in a 2D disk case 1

25. Rotating anelastic convection in a 2D disk • rbot / rtop = 7 (hH= 0) 961 x 2160 • Ra = 2 x 1010 (10 times critical) • Ek = 10-7 • Pr = 0.5 • Roc = 0.02 • Re = 105(10 revolutions by zonal flow so far) • Ro = 10-2

26. Reference state profiles for rotating convection in a 2D disk density density case 1 case 2 radius radius hr hr radius radius

27. case 1 Convergence of prograde angular momentum flux near the inner boundary, where the hr effect is greatest

28. case 1

29. case 2 Convergence of prograde angular momentum flux near the outer boundary, where the hr effect is greatest

30. case 2

31. Differential rotation case 1 case 2 radius radius Maintenace of differential rotation by convergence of angular momentum flux radius radius

32. Transport of angular momentum by rotating turbulent convection case 1 case 2 density density radius radius

33. hr is comparable to hH when there are about two density scale heights across the convection zone, assuming laminar flow and long narrow Taylor columns spanning the convection zone without buckling. The hH effect is relevant for laboratory experiments and is seen in many 3D simulations of rotating laminar convection. However, if the Ek1/3 scaling is assumed for columns in Jupiter, they would be a million times longer than wide; or if some eddy viscosity were invoked they may be only a thousand times longer. If instead a Rhines scaling is assumed (balance Coriolis and inertia), they would be 100 to 10000 times longer than wide. The smaller the convective velocity the greater the rotational constraint and the thinner the columns. The larger the convective velocity the greater the turbulent Reynolds stresses. These thin columns are forced to contract and expand by the spherical surfaces, which are not impermeable. The density is smallest and the turbulence is the greatest near the surface.

34. Therefore, the hH effect may not be relevant for the density-stratified strongly-turbulent fluid interiors of stars and giant planets, where flows are likely characterized by small-scale vortices and plumes detached from the boundaries, not long thin Taylor columns that span the globe. The hr effect, however, does not require intact Taylor columns or laminar flow. It exists for all buoyant blobs and vortices, including strong turbulence uninfluenced by distant boundaries. The hr experienced by a fluid parcel as it moves will depend on the latitude of its trajectory, phase transitions, magnetic field, …

35. Sub-grid scale corrections to advection terms

37. Similarity subgrid-scale method Ra = 108 Ek = 10-5 density ratio = 27

38. 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