Direct simulation of

1
Direct simulation of planetary and stellar
dynamos II. Future challenges (maintenance of
differential rotation)
Gary A Glatzmaier University of California, Santa
Cruz
2
(No Transcript)
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 2
case 1
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

36

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