Anelastic spherical dynamos with variableconductivity
Chris Jones
Department of Applied Mathematics,
University of Leeds UK
Wieland Dietrich
MPI for Solar System Research, Göttingen, Germany

Research seminar, Kyoto, 1st December 2017

Planetary magnetic fields

Magnetic fields have been mapped for Mercury, Earth, Jupiter,
Saturn, Uranus and Neptune.
We know Ganymede has an active dynamo, and Mars had adynamo which switched off 350Myr after formation.
It has been suggested that the minor planet Vesta may have had adynamo in the past.
Likely to be dynamos in extra-solar planets, e.g. hot Jupiters.
The great diversity of planetary magnetic fields suggests that thephysical conditions in their interiors varies considerably.

(i). Observational background

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Magnetic fields of Earth and Jupiter

-1mT

1mT

Earth’s Br at CMB-1.2mT

1.2mT

Jupiter’s Br at surface

Pre-Juno Jupiter magnetic field went up to spherical harmonic
degree 4 reliably. More Earth-like at high resolution?
Both fields are broadly dipolar, with axis offset by ∼ 10◦ .
Both planets are believed to be convecting in their electricallyconducting zones. Earth may have a thin stable layer just below
CMB.
(i). Observational background

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Magnetic fields of the Ice Giants-0.12mT

0.12mT

Uranus Br at surface-0.1mT

0.1mT

Neptune Br at surface.

These magnetic fields are not mainly dipolar, they are a mixture of
non-axisymmetric quadrupolar and dipolar modes.
Fields are significantly weaker, but we don’t know much about theinternal structure of these planets.
It may be that the dynamo region is near the surface but the fields
there are really weaker than in the geodynamo, or it may be thefields are generated in the deep interior and are strong there.
(i). Observational background

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Magnetic fields of Mercury and Saturn

Field strength ∼ 1% Earth’s
Mercury Br at surface-0.06mT

0.06mT

Saturn Br at surface.

These magnetic fields are quite axisymmetric, though Mercury’s
field is equatorially asymmetric.
Dynamo generated fields must be non-axisymmetric, but they canappear axisymmetric at the surface.
It has been suggested that these planets have a stably stratified
layer above the dynamo region. If this layer is electricallyconducting and differentially rotating, the non-axisymmetriccomponents in the dynamo region are filtered out by the stably
stratified layer.
(i). Observational background

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Juno and Cassini Grand Finale

Juno is gathering gravity and magnetic field data from Jupiter.
Mission will last longer than planned: close flyby occurs every 53
days.
Gravity data may indicate the size of the core, and the internalrotation rate, hence how deep the zonal flows are.
Magnetic data looks very promising, and has already raised newquestions.
Cassini entered Saturn in September, and detected the
non-axisymmetric field component for the first time.

(i). Juno data

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Saturn’s axisymmetric magnetic field• Saturn’s field is remarkably axisymmetic, dipole tilt is only about
than 0.06◦ (Cao, 2017).
• Could be because the metallic hydrogen region is smaller thanJupiter’s, because smaller mass means 2 Mbar pressure occurs
deeper.
• Saturn has significantly less helium in its atmosphere comparedto Jupiter and the Sun. Helium might have ‘rained out’.
• Rain-out could (but not necessarily) lead to a stably stratified
zone above the dynamo region.
• Here we examine whether Saturn-like almost axisymmetric fields
can be generated solely because the metallic hydrogen layer is very
deep.
(i). Juno data

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Juno: closest view of any natural dynamo
Magnetic field outside the electrically conducting region ispotential,
V = Rs∞ m=n
X
X  Rs n+1
n=1 m=0r

Pnm (cos θ)(gnm cos mφ + hnm sin mφ)(2.1)
where B = −∇V , and r , θ and φ are spherical polar coordinates,r = Rs is the planetary surface.
Determine gnm , hnm at r = Robs , and evaluate field at top of the
dynamo region r = Rdyn .
Attenuation ∼ (Rdyn /Robs )n+1 ∼ 0.8n+1 since Juno passes only
4000 km above Jupiter’s surface.
For Earth, attenuation is 0.546n+1 , so components with n >∼ 13
are obscured by crustal magnetism.
Jupiter’s field can be seen at high resolution: may well change ourview of geomagnetic field too.
(i). Juno data

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Jupiter’s field post Juno

Jupiter’s Br at the surface

Jupiter’s Br at r = 0.85Rjup

The solid line is Juno’s fly-by path. New model constructed bysuperimposing the new data on the old Jupiter model.
Only the field close to the Juno path can be reliably estimated.
Revised field has intense flux spots, some near the equator, somenear the poles.
Models of Giant Planets

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Internal structure of the Giant Planets

Standard Model assumes Jupiter has rocky inner core of about 1%of the total mass of the planet. The French et al. 2012 model has
radius ratio 0.0923. Based on computational quantum density
functional techniques. Dynamo model extends from core to
3000km below the surface, the cut-off being for resolution reasons.
Low electricalconductivity −→Transition zone −→High electricalconductivity −→Models of Giant Planets

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Dilute core model?

Wahl et al.
(2017)The standard French et al. model used for dynamo simulations fits
the new Juno gravity data results rather well.
However, different density functional theories give different
equations of state. Using one of these would allow a different
equilibrium structure model that still satisfied Juno constraints.
Helium rain-out might have led to a dilute core model: also might
have given rise to the stable layer suggested by Saturn’s
axisymmetric field.
Models of Giant Planets

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Jupiter electrical diffusivity profile
30

25

2

loge η (m /s)

20

15

10

+ Data points from French et al. 2012
5

Hyperbolic fit0

cut−off
−5

0

1

2

3

4

radius (metres)

5

6

7
7

x 10

Diffusivity η = 1/µσ, σ being the electrical conductivity.
+ signs are the French et al. 2012 model, the curve is a smoothed
hyperbolic fit. The French et al. conductivity drops off
super-exponentially beyond a radius of 0.85Rjup . By 0.95Rjup themagnetic Reynolds number is less than 1.

Models of Giant Planets

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Verifying the electrical conductivity profile
As more perijove passes are logged, Lowes-Mauersberger spectrum
will be determined:
Rn = R 2n+4sr(n + 1)

n h
X

2
2 i
gnm + hnm

m=0

The slope of Rn with n is expected to be flat (white noise) whenr = Rdyn ; this gives a method of checking whether this occurs
where the DFT based profiles predict.
Alternatively, compare the Juno spectrum with a dynamo model
simulation. If the slope is correct, that will confirm that the correct
conductivity profile is being used in the simulation.
As the extended mission evolves, it may be possible to detectchanges in the Gauss coefficients: the secular variation. There issome evidence Jupiter’s field does change slowly with time.
Important as it gives independent evidence of convective velocity.
Models of Giant Planets

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Anelastic equations for non-perfect gas
Mass conservation: ∇ · ρ̄u = 0. The equation of motion is


1
∂u
p̂
2
1
+ u · ∇u = −∇ − ẑ × u +(∇ × B) × B
Pm ∂t
E
E
E ρ̄
+Fν −

Pm
d T̄
RaS
r̂.
Pr
dr

The entropy equation is




DS
Pm
Pr
η̄
1
2
∇ · ρ̄T̄ ∇S + H +=(∇ × B) + Qν ,Dt
Pr ρ̄T̄
PmRaT̄ E ρ̄assuming constant kinematic entropy diffusivity and constantkinematic viscosity throughout the shell. Here PmH/Pr is thesource term from the gradual cooling of Jupiter.
Anelastic Dynamo Models

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Dimensionless parameters
Input Parameters:∆STc d 2.
νκPrandtl number Pr = ν/κ, Magnetic Prandtl number Pm = ν/ηc ,η magnetic diffusivity, ν kinematic viscosity, κ is the entropy
diffusivity, d = ro − ri .
Ekman number E = ν/Ωd 2 , Rayleigh number Ra =∆S is the imposed entropy drop across the shell. Tc and ηc arevalues at r = ri + d/2.
Inner boundary no-slip, outer boundary stress-free. Electricallyinsulating bcs. Mostly fixed entropy, some fixed flux bcs.
Outputs: Rossby number Ro = U/Ωd, Magnetic Reynolds number,
Rm = Ud/η, Elsasser number B 2 /µΩρη, magnetic and kinetic
energies, zonal flow, heat flux, dipole and quadrupole moments.
Anelastic Dynamo Models

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Choice of units and diffusivities
U ∗ ∼ 10−2 m/sec fits the secular variation data and therequirement to get the heat flux out. With French et al. diffusivity,
Rm ∼ 106 .
Simulations only possible for Rm up to 5 × 103 , so magnetic
diffusivity must be enhanced. If the typical dimensionless velocity
103 corresponds to 10−2 m/sec, the magnetic diffusion time isthen ∼ 200, 000 years.
Unit of field strength is (ρc µΩηc )1/2 ∼ 3 gauss, too low: use theenhanced magnetic diffusion? Best to interpret field strength bymatching the model to the known surface value.
Thermal and viscous diffusivities much larger than their molecularvalues. Inevitable, to cut off the turbulent spectrum before it
reaches very small scales.

Anelastic Dynamo Models

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Parameter space of Anelastic dynamos
There are severe computational constraints on compressible
dynamos. Dynamo action can only be established by integrating
for over a diffusion time. Rm must generally be of order 103 fordynamo action, so integration for many turn-over times is needed.
Low E is desirable, but less than 10−5 currently not practical. Ra
must be large enough to give vigorous convection, but not toolarge to destroy the dominance of the Coriolis force. Window ofacceptable Ra not that great at feasible E .
Pr not too severely constrained. Low Pr more realistic.
Astrophysical value of Pm should be less than unity. Larger Pm
more computationally attractive, as it allows large Rm without
large Reynolds number Re. Also gives stronger field. However, toolarge Pm leads to smaller timesteps, so a compromise is best.

Anelastic Dynamo Models

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Steady Dipoles not easy to find
Taken from Gastine et al. 2012. Red
circles dipole dominated solutions, blue
squares multipolar solutions. Nρ isnumber of scale heights in thepolytrope. No dipole dominatedsolutions at all at large Nρ , E = 10−4for constant conductivity model.
Zonal flows in the outer regions (stress-free boundaries), dynamo
becomes αω and dynamo waves break up steady dipoles.
Situation improves somewhat if variable conductivity is used, as
dynamo action doesn’t occur where zonal flow is strongest. But
zonal flow usually penetrates somewhat into dynamo region.
If dynamo has a dynamically strong field, it can suppress thedifferential rotation, and hence stay dipole dominant. Bistability is
an issue.
Anelastic Dynamo Models

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Variety of Dynamos in anelastic models• Many different types of dynamo found.
• Hemispherical dynamos, quadrupolar dynamos, steady andreversing dipolar solutions and small-scale dynamos.
• The stress-free outer boundary helps generate a wider range ofdynamos in Boussinesq models (Simitev and Busse), so this is apartial explanation.
• However, there does seem to be a greater diversity with anelastic
models. The steady dipole window in parameter space is much
reduced.

Anelastic Dynamo Models

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Pr = 0.1, E = 2.5 × 10−5 solution snapshot-1.2mT

1.2mT-0.900Jupiter surface radial Br-1.000Model radial Br snapshot
truncated at harmonic ` ≤ 5

1.000

Model radial Br (full resolution)
Anelastic Dynamo Models

0.900

Axisymmetric meridional Bφ20/42

How do we get the dipolar solutions?

With a uniform entropy source, expected if Jupiter is convecting
normally everywhere outside the small core, only found steady
dipoles at small Pr ∼ 0.1
Also need Pm ∼ 3 (or bigger) because field has to be strong
enough to suppress differential rotation.
Domain of instability might increase at lower E , but very expensive
to go there.
Jupiter’s dynamo might be reversing: expected timescale thousandsof years. But dynamo wave snapshots don’t look very Jupiter-like.

Anelastic Dynamo Models

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Gastine et al. 2014 solution
Taken from Gastine et al 2014.
Pr = 1, Pm = 0.6, E = 10−5 .
The driving is basal heating,coming out of the core. This isperhaps less realistic, but it
makes it easier to get dipole
dominated solutions, notnecessary to go to low Pr .
Despite different parameters and different driving, the Göttingenmodels look quite similar to the Leeds models.
They do have more differential rotation at the top of the dynamo
region, hence dynamo waves, but their deeper driving means that
this doesn’t disturb the deep dipole part of the dynamo.

Anelastic Dynamo Models

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Flow for Pr = 0.1 solutions: units of Rm

Axisymmetric part of uφ run A

Axisymmetric part of uφ run C

4000.02400.0800.0-800.0-2400.0-4000.0

Equatorial section of ur
Anelastic Dynamo Models

Random meridional slice of ur
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Pr = 0.1 Dynamo Movie-0.9000.900

Time-lapse movie of Br at the surface, for the Pr = 0.1 internal
heated dynamo. Fixed entropy boundaries. Ra = 1.2 × 107 ,Pm = 3.0, E = 2.5 × 10−5 . Dynamo remains dipolar dominant.
Anelastic Dynamo Models

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Flow Movie for the Pr = 0.1 Dynamo

3500.02100.0700.0-700.0-2100.0-3500.0

Time-lapse movie of ur in the equatorial plane, for the Pr = 0.1
internal heated dynamo. Fixed entropy boundaries.
Ra = 1.2 × 107 , Pm = 3.0, E = 2.5 × 10−5 . Note the difference
between the magnetically locked interior and zonal flow dominatedmolecular region.
Anelastic Dynamo Models

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Comparison with Juno field-18 Gauss

Juno field at r = Rjup

Dynamo simulation at r = Rjup

-65 Gauss

Juno field at r = 0.85Rjup
Anelastic Dynamo Models

18 Gauss

65 Gauss

Dynamo simulation at r = 0.85Rjup
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Weak polar field: signature of the core?

Weak polar patch at the N.pole
Olson and Aurnou (1999)

Jupiter model meridional
section of ur

Geodynamo models suggest this is due to the inner core:
convection rolls line up just outside the inner core equator.
Could the same thing happen in Jupiter if it has a larger core?
Rolls less columnar in simulations, but Ro overestimated.
Anelastic Dynamo Models

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Jupiter dynamo models after Juno

• Anelastic dynamo models, convecting from the core to thesurface, can produce Jupiter-like fields.
• The small scale features found by Juno were predicted by themodels.
• Models with internal heating only give dipole dominated fields in
restricted area of parameter space (low E , low Pr , moderate Pm).
• Equatorial flux spots and weak polar fields not common inexisting models.
• Models with helium separation (dilute core) worth exploring.

Conclusions

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Variable conductivity models
ro
rd

ri
hydro magnetic

rigid

mTC
TC

free slip

Schematic model

Electrical conductivity of models tested

Conductivity = 1/λ, λ magnetic diffusivity. Saturn corresponds tord = 0.65, Jupiter to rd = 0.95.
Chosen conductivity models are scaled Jupiter models (French etal. 2012). mTC is the magnetic tangent cylinder, where r = rdand Rm(r ) = 1.
Magnetic tangent cylinder

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Anelastic dynamo model parameters

The standard rotating, convection-driven, anelastic dynamo
equations are solved numerically.
Typical input Parameters: E = 5 × 10−5 , Ra = 107 , Prandtl
number Pr = 0.25, Magnetic Prandtl number Pm = 3. Normalised
at shell midpoint.
Inner boundary no-slip, outer boundary stress-free. Electricallyinsulating bcs. Outer boundary heat flux constant over surface.
Most models are internally heated (Kelvin-Helmholtz contraction),some bottom-heated for comparison.
Simulations are run till they equilibrate, usually around a diffusion
time (sometimes more). Lot of CPU consumed!

Models and results

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Solution Snapshots
Brrd = 0.95

uφBrrd = 0.75

uφ
Br
uφ

rd = 0.65
uφModels and results

Ψm helicity

Br

Bφ
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Key points from the results• Zonal flow confined to region outside mTC. Deeper and extends
to higher latitude in Saturn-like cases.
• Meridional circulation is driven by the magnetic field.
• Magnetic field reverses the sign of the helicity h = u · ω(Sreenivasan + Jones 2011 found magnetic field effects helicity inBoussinesq models: larger effect in anelastic models).
• Models between the Jupiter and Saturn cases are mostly
quadrupolar.

Models and results

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Limiting cases
Brrd = 1.0

uφrd = 0uφuφ

Ψm helicity

BrBφ

Top case has uniform electrical conductivity, bottom case isnon-magnetic. As expected, top case has a weak dipole andquadrupole component. No helicity reversal in either case. Zonal
flow strongest in non-magnetic case.
Models and results

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Types of dynamo found: dipole octupole

Radial Br as a function of latitude, evolving with time (in diffusion
times).
(a) is the Jupiter-like case, rd = 0.95, which leads to a steady
dipole (or one that reverses on a difusive time-scale).
(b) If rd is decreased to 0.85, (planet of mass between Jupiter ands
Saturn), dynamo generates a steady octupole, with positive andnegative radial field in both hemispheres.

Time-depence of the models

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Dipolar waves, steady quadrupolar dynamo

Radial Br as a function of latitude, evolving with time (in diffusion
times).
(c) has Saturn-like rd , rd = 0.65. Fairly periodic dynamo waves,dipolar field that reverses on a difusive time-scale.
(e) Here rd = 0.75 and the Rayleigh number is a little higher thanin the octupolar (b). For intermediate rd higher Rayleigh numbers
lead to quadupolar dynamos, but very high Ra gives a small scale
dynamo. Increasing Ra slightly makes the central belt oscillate
sinuously (small dipole wave added).
Time-depence of the models

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Multipolar dynamo, hemispherical dynamo

Radial Br as a function of latitude, evolving with time (in diffusion
times).
(d) is an irregular multipolar dynamo. This type of solution isuniversal if the Rayleigh number is high enough to make the
Rossby number O(1). To keep in the rapidly rotating regime wehave to restrict Ra at fixed E .
(g) is a hemispherical dynamo wave. Here rd = 0.65, Ra = 5 × 106
and Pr = 0.15.
Time-depence of the models

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Dipole component against quadrupole component

Dipole Gauss coefficient is g10 (t), quadrupole coefficient is g20 (t).
Plot is the time-dependent phase diagram for the different
dynamos.
I - Steady dipole: III - Dipolar wave: IV - Multipolar dynamo:
V - Steady quadrupole: VI - Wobbling quadrupole:
VII - Hemispherical dynamo.
Time-depence of the models

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Hemispherical dynamo dynamics (i)Time-averaged meridional sections of the hemispherical dynamo
run.
Zonal flow ūφ , entropy S̄ and the latitudinal gradient of theentropy are all rather symmetric about the equator.
Meridional circulation is not symmetric. This suggests that theasymmetric magnetic field is generating an asymmetric meridional
circulation which maintains the asymmetric field.
Hemispherical dynamos

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Hemispherical dynamo dynamics (ii)Time-averaged meridional sections of the angular momentum
balance for the hemispherical dynamo run.
Reynolds stress and viscous terms are symmetric about theequator, but Maxwell stress term is very asymmetric, as expected.
For this hemispherical dynamo, the asymmetric Maxwell stressmaintains the asymmderic meridional circulation. A small
asymmetric perturbation in the magnetic field will grow.
Hemispherical dynamos

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Trigram plot
Ra =

5e6

1.2e7

6e6

1.5e7

7e6

2e7

8e6

2.5e7

9e6

3e7

D
★ Saturn★ Earth★ Jupiter

rd =

0.65

★ Mercury

0.15
0.2

0.75

0.25

0.85
0.95

∞

ax

mo

s

1e7

internal heatedbottom heated

sy
nti
eq

s
mo

.a

na

Q

D=

dy

mm
e

ic

etr

tric

dy

mm

na

isy

Jupiter-model
by Jones, 2014

Pr =

0.7

D=

★ Neptune

NA

5

NA

Q=NA

Uranus★

eq. symmetric dynamosQ

D/Q refer to the magnetic energy in the dipole and quadrupole
contributions, NA to the energy in the non-axisymmetriccomponents. An almost purely dipolar dynamo, like Saturn, sitsnear point D. The green square dynamo is mostly quadrupolar,
with a small non-axisymmetric component. Uranus and Neptune
are primarily non-axisymmetric.
Summary of models found

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Saturn model dynamo?
With a Saturn-like conductivity profile,the dipolar solutions are oscillatory. The
period corrresponds to many thousandsof years, so this is consistent withobservations.
Meridional sections ofzonally averaged radial
and azimuthal field,
and surface radial fieldat four times in thecycle.
There are times when the field is somewhat Saturn-like, but the
non-axisymmetric components are always larger than Saturn’s.
Summary of models found

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Conclusions• Remarkable diversity of magnetic field morphology. Large bandof rd values between Jupiter and Saturn, which are quadrupolar
rather than dipolar. Both Jupiter and Saturn lie just outside thequadrupolar belt.
• Saturn-like fields possible, but tilt angle of field typically 1◦ ∼ 2◦ ,whereas Saturn’s tilt is certainly less than 0.06◦ . A stably stratified
layer is necessary to model Saturn’s very axisymmetric field.
• Broader equatorial jet found in Saturn in agreement withobservation. Our jets are slightly broader than those in the giant
planets: better modelling of small scale turbulence needed?
• rd ∼ 0.65 dipolar dynamos are oscillatory reversing dynamos withlong periods. Shear gets inside the mTC creating dynamo waves.
• Helicity reversal occurs in the metallic regions of the anelastic
models. May be connected with Maxwell stresses and theconsequent meridional circulation.
Conclusions

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