Scaling laws for planetary
dynamos driven by helical
waves
P. A. Davidson
A. Ranjan
Cambridge
What keeps planetary magnetic fields alive? (Earth, Mercury, Gas giants)
• Two ingredients of the early theories: Ω-effect, α-effect
• Ω-effect by itself does not produce a self-sustaining dynamo
α-Effect, Gene Parker (1955), Keith Moffatt
Self-sustaining α2 dynamo needs sign of helicity opposite in north and south of core
Why should the flow in the core be like that ?
• Right-handed helical flow induces
current anti-parallel to B
• Left-handed helical flow induces
current parallel to B
B
Current
Observed helicity in numerical solutions
azimuthal average of h
In computer simulations the helicity is observed to be (mostly)
–ve in the north, +ve in the south outside tangent cylinder
The α2 dynamo
This is a zero-order model of most numerical dynamos
5
But its not that simple……..
Helicity: azimuthal average Helicity: vertical slice
Numerical simulations can reproduce planet-like magnetic fields
But a long way from the correct regime:
- too viscous by a factor of 109
- underpowered by a factor of 103
Typical results of dynamo simulations
Weakly forced
10 times critical
Moderately forced
50 times critical
Note the Earth is a 105 times critical !
Numerical simulations are getting something correct but much that is wrong !
Alternating cyclones &
anti-cyclones is seat of dynamo
action. Helicity is observed to be –ve
in the north, +ve in the south
flow B
How do we get an asymmetric distribution of helicity, and hence dynamo?
A popular cartoon for geo-dynamo based on weakly-forced, highly-viscous
simulations
This explanation consistent with observation that h<0 in the north and h>0 in south
What is the source of helicity in real planets ?
4 problems for the viscous mechanism • Viscous stress is tiny, Ek ~ 10-15
• Mercury, Earth, Jupiter, Saturn have similar B-fields, both
in structure (dipolar, aligned with Ω) and magnitude
Suggests similar dynamo mechanisms despite
different interior structures…?
• As forcing gets stronger, lose the ‘Swiss-watch’
assembly of convection rolls
• Slip B.C. on mantle still gives dynamo
More realistic model of helicity generation should be: • Independent of viscosity
• Internally driven (independent of interior structures)
• Physically robust but dynamically random
Planet/
star
Mercury Earth Jupiter Saturn V374
Pegasi
5.5 x 10-6 13 x 10-6
5.2 x 10-6
2.2 x 10-6 17 x 10-6
Results from numerical simulations (Sakuruba & Roberts, 2009)
Note the strong equatorial jet
Can the equatorial plumes ‘fuel’ the required asymmetric helicity distribution ?
A clue ? Axial vorticity, equatorial slice
Temperature
An old idea recycled …(G I Taylor, 1921)
Rotation-dominated flow: pressure gradient balances Coriolis force
Ωu 2p Geostrophic balance
0
z
u 2D flow requires
How does the fluid know to move with the object ?
Incompressible rotating fluids can sustain
internal wave motion (Coriolis force
provides restoring force) called
Inertial waves.
Towed object acts like radio antenna
Reason: angular momentum conservation.
Eddy grows and propagates at the group velocity
of zero-frequency inertial wave packets
Davidson et. al (JFM, 2006)
Spontaneous formation of
columnar eddy from a
localised disturbance.
Caused by spontaneous self-
focussing of radiated energy
onto rotation axis
Numerical simulations of rotating turbulence from NCAR, US
Columnar vortices (cyclones, anti-cyclones) common in rotating turbulence,
created by Inertial waves.
Iso-surfaces of helicity. Red is negative, green positive.
(h > 0 means right-handed spirals, h < 0 means left-handed)
Wave packets spatially segregate helicity (perfect for dynamo!)
Inertial wave packets are helical – ideal for dynamo
Remember the strong equatorial jet
Could this be generating the asymmetric
helicity pattern?
Dispersion pattern of inertial wave packets
from a buoyant blob
Note pairing of cyclone and anti-cyclone above
and below
(Davidson, GJI, 2014)
Buoyancy field
Energy surfaces coloured by helicity
Note helicity is negative in the ‘north’
(blue) and positive in the ‘south’ (red)
(Davidson & Ranjan, GJI, 2015)
Velocity iso-surfaces
Numerical simulation: wave-packets emerging form a layer of random buoyant blobs
Numerical simulation
Surfaces of axial velocity
(positive is red, negative is blue)
Compare!
Note alternating cyclones-
anticyclones
If we include the dynamic influence of the
mean magnetic field, the wave packets
become anisotropic.
Modified inertial waves, magnetostrophic waves … Both helical.
24~
C
T
p R
Q
c
g
Speculative scaling for Helical-wave α2 Dynamo
Input: • α effect (modelled as helical wave packets of maximum helicity)
drive mean current that supports global field via Ampere’s law
• Curl (buoyancy) ~ Curl (Coriolis)
• Joule dissipation ~ rate of working of buoyancy force
Driving force:
Rate of working of buoyancy force:
Key parameters:
3/1
Cp RV
Prediction:
,,, rmsAP BVVu
:scaleVelocity
22
3 ~~ uV
V aP
scaleplumetransverse
In numerical dynamos viscosity sets δ:
?...setswhatBut
3/1Ek~CR
But what sets δ in the planets?
Comparison OK…
(Davidson, GJI, 2016)
Comparison of predicted scaling with numerical dynamos 1
CC
rmsB
C
Pp
R
u
R
B
R
V
Ro,,
3
3/12/12/1EkPr~ mPB
6/12/1Ek~Ro P
,
,
(Davidson, GJI, 2016)
Comparison of predicted scaling with numerical dynamos 2
4.0Ro uInertial waves cease to propagate for
Suggests loss of dipolar field at 1~EkRa2 Q
(Davidson, GJI, 2016)
Speculative scaling for Helical-wave α2 Dynamo Cont.
Dimensionless dependant parameters:
Predictions:
pVu
pVu ~
,P
rms
V
B
P
P
rms V
V
B~
Earth
Jupiter
Saturn
Measured
13 x 10-5
5 x 10-5
2 x 10-5
Predicted
8 x 10-5
15 x 10-5
11 x 10-5
C
rms
R
B
But what sets δ in the planets?
Hypothesis: dynamo saturates at minimum magnetic energy compatible with given
Convective heat flux.
Elsasser ~ 1 would have the gas giants multipolar
Thank You
References
Self-focussing of inertial-wave radiation to give quasi-geostrophy
Davidson, Staplehurst, Dalziel, JFM, 2006
Helicity generation/segregation and α-effect via inertial wave-packets
launched from equatorial regions
Davidson, GJI, 2014
Dynamics of a sea of inertial wave-packets launched from equatorial
regions
Davidson & Ranjan, GJI, 2015
Scaling laws for helical wave dynamos
Davidson, GJI, 2016
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