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

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) Ro c = (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) Ro m = (Alfven velocity) / (rotational velocity)

Geodynamo simulation Differential rotation is a thermal wind

Jupiter Saturn Surface zonal winds

Boussinesq Christensen Ro c = 0.04 Ro c = 0.21 equatiorial plane meridian plane T zz zz vv

Glatzmaier Jovian dynamo model Anelastic with  bot  top  27  0.01 everywhere   1 in lower part and at top Internal heating proportional to pressure Solar heating at surface Ra = 10 8 Ek = Ro c = (g  T/D) 1/2 / 2  = Spatial resolution: 289 x 384 x 384

Anelastic Glatzmaier Jupiter dynamo simulations shallow deep Longitudinal flow

Solar differential rotation

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

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

Convection turbulent vs laminar compressible vs incompressible

2D anelastic rotating magneto-convection 2001 x 4001 P r =  =  = 1.0, 0.1 E k = / 2  D 2  = , Ra = g  TD 3 /  = 10 6, Re = v D / = 10 3, 10 6 Ro = v / 2  D = 10 -1, 10 -3

large diffusivities small density stratification Laminar convection

Turbulent convection small diffusivities large density stratification

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

height mean entropy Ra = 3x10 Ek = large diffusivities small density stratification Laminar convection

height mean entropy Ra = 3x10 Ek = small diffusivities small density stratification Turbulent convection

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

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

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

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

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

case 1 Turbulent Boussinesq convection in a 2D disk

Rotating anelastic convection in a 2D disk  bot /  top = 7 (h H = 0) 961 x 2160 Ra = 2 x (10 times critical) Ek = Pr = 0.5 Ro c = 0.02 Re = 10 5 (10 revolutions by zonal flow so far) Ro = 10 -2

density hh hh radius Reference state profiles for rotating convection in a 2D disk case 1case 2

Convergence of prograde angular momentum flux near the inner boundary, where the h  effect is greatest case 1

Convergence of prograde angular momentum flux near the outer boundary, where the h  effect is greatest case 2

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

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

h  is comparable to h H 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 h H effect is relevant for laboratory experiments and is seen in many 3D simulations of rotating laminar convection. However, if the Ek 1/3 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 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.

The h  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 h  experienced by a fluid parcel as it moves will depend on the latitude of its trajectory, phase transitions, magnetic field, … Therefore, the h H 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.

Sub-grid scale corrections to advection terms

Similarity subgrid-scale method R a = 10 8 E k = density ratio = 27

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