Direct simulation of planetary and stellar dynamos

II. Future challenges

(maintenance of differential rotation)

Gary A Glatzmaier

University of California, Santa Cruz

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Differential rotation depends on many factors

geometry: depth of convection zone, size of inner coreboundary conditions: thermal, velocity and magneticstratification: thermal (stable and unstable regions) density (number of scale heights and profile) composition and phase changesdiffusion coefficients: amplitudes and radial profilesmagnetic field: Lorentz forces oppose differential rotationparameters: 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)

Geodynamo simulation

Differential rotation is a thermal wind

Jupiter Saturn

Surface zonal winds

Boussinesq

Christensen

Roc = 0.04

Roc = 0.21

equatiorial plane meridian plane

T z z v

Glatzmaier

Jovian dynamo model

Anelastic with bottop27

0.01 everywhere

1 in lower part and 0.001 at top

Internal heating proportional to pressureSolar heating at surface

Ra = 108 Ek = 10-6 Roc = (gT/D)1/2 / 2 = 10-1

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 bottop30

0.125

4

Ra = 8x104 Ek = 10-3 Roc = (gT/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 anelasticrotating magneto-convection

2001 x 4001

Pr = = = 1.0, 0.1

Ek = / 2D2= 10-4, 10-9

Ra = gTD3 / = 106, 1012

Re = v D / = 103, 106

Ro = v / 2D = 10-1, 10-3

large diffusivities

small density stratification

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Laminar convection

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Turbulent convection

small diffusivities

large density stratification

small diffusivities

small density stratification

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Turbulent convectionwith rotation and magnetic field

heig

ht

mean entropy

Ra = 3x10 Ek = 106 -4

large diffusivities

small density stratification

Laminar convection

heig

ht

mean entropy

Ra = 3x10 Ek = 1012 -9

small diffusivities

small density stratification

Turbulent convection

Anelastic vorticity equation (curl of the momentum equation)

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∇• v = 0 ∇ • v = −hρ vr

ω =∇ x v

hρ =1

ρ

dρ

dr

hH =1

H

dH

dr

∂ω

∂t=∇ x(vx(2Ω + ω)) + ...

= −(v • ∇)ω + (2Ω + ω) hρ +hH( ) vr +g

r

∂C

∂φ+ ν ∇ 2ω

H = height of Taylor column aboveequatorial plane for 2D parameterization

(2D disk)

vorticity

i.e.,

inverse density scale height

Anelastic Taylor-Proudman Theorem

Assume geostrophic balance for the momentum equationand take its curl:

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0 = −∇(p /ρ + U) − 2Ω xv

⇒∂v

∂z= −hρ vr ˆ z

Anelastic potential vorticity theorem

Assume a balance among the inertial, pressure gradientand Coriolis terms in the 2D momentum equation:

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∂∂t

+ (v • ∇)ω = (2Ω + ω)hρ vr

dω

dt= (2Ω + ω)

d lnρ

dt

d

dt

2Ω + ω

ρ

⎛

⎝ ⎜

⎞

⎠ ⎟= 0

The spiral pattern at the boundary having the greatest h effect

eventually spreads throughout the convection zone.

Rising parcelexpands and gainsnegative vorticity

Sinking parcelcontracts and gainspositive vorticity

Density stratified flow in equatorial plane

Incompressible columnar convection

The shape of the boundary determines the tiltof the columns, which determines theconvergence of angular momentum flux,which maintains the differential rotation.

Zonal flow is prograde in outer partand retrograde in inner part.

Zonal flow is retrograde in outer partand prograde in inner part.

Busse

case 1

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TurbulentBoussinesqconvectionin a 2D disk

Rotating anelastic convection in a 2D disk

bot / top = 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

density density

hh

radius radius

radiusradius

Reference state profiles for rotating convection in a 2D disk

case 1 case 2

Convergence of progradeangular momentum fluxnear the inner boundary,where the h effect is greatest

case 1

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case 1

Convergence of progradeangular momentum fluxnear the outer boundary,where the h effect is greatest

case 2

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case 2

Differential rotation

radius radius

radiusradius

Maintenace of differential rotation byconvergence of angular momentum flux

case 1 case 2

density

radius

density

radius

case 2case 1

Transport of angular momentum by rotating turbulent convection

h 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 convectivevelocity 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 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.

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Similarity Model

∂v

∂t+ (v • ∇)v + (v • ∇)v − ( v • ∇) v[ ] = −∇P + ν (∇ 2v + ...) + forces

Similar sgs corrections for the nonlinear terms in the other prognostic equations.

The thermodynamic perturbations and the poloidal and toroidal parts of the

momentum density and magnetic field are filtered in spectral space.

if f (r,θ,φ, t) = flnm∑ (t) Y

lm (θ,φ) T

n(r)

then f (r,θ,φ, t) = flnm (t)∑ Y

lm (θ,φ) T

n(r)

where flnm = fln

m e−(n2Δ

RAD

2 /6 + l (l + 1)ΔANG

2 /24)

and ΔRAD = 2Δr /D ΔANG = 2Δφ

Sub-grid scale corrections to advection terms

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LANS Alpha Model

If Boussinesq : v = (1−α 2∇ 2) v α ≈ 2 rmid Δφmid

∇ • v = 0 and ∇ • v = 0 and ∇ x v = (1−α 2∇ 2)∇ x v

and same boundary conditions on v and v.

∂v

∂t− v x(∇ x v) +∇ v • v − 1

2v

2− α 2

2∇v

2 ⎡

⎣ ⎢

⎤

⎦ ⎥= −∇p + ν ∇ 2v + forces

Update v, then solve Helmholtz equation for v .

Similarity subgrid-scale method

Ra = 108

Ek = 10-5

density ratio = 27

Challenges for the next generation of global dynamo models

high spatial resolution in 3Dsmall diffusivitiesturbulent flow

density stratification gravity waves in stable regions phase transitions

massively parallel computingimproved numerical methods

anelastic equationssub-grid scale models