Scaling laws for planetary dynamos driven by helical waves

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Scaling laws for planetary dynamos driven by helical waves P. A. Davidson A. Ranjan Cambridge

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The spontaneous conversion of a turbulent cloud into columnar wavesWhat keeps planetary magnetic fields alive? (Earth, Mercury, Gas giants)
• Two ingredients of the early theories: -effect, α-effect
• -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
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
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
Note the Earth is a 105 times critical !
Numerical simulations are getting something correct but much that is wrong !
Alternating cyclones &
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
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 ) and magnitude
Suggests similar dynamo mechanisms despite
different interior structures…?
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
5.2 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
Rotation-dominated flow: pressure gradient balances Coriolis force
Ωu 2p Geostrophic balance
How does the fluid know to move with the object ?
Incompressible rotating fluids can sustain
internal wave motion (Coriolis force
provides restoring force) called
Reason: angular momentum conservation.
of zero-frequency inertial wave packets
Davidson et. al (JFM, 2006)
Spontaneous formation of
created by Inertial waves.
(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?
from a buoyant blob
and below
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
mean magnetic field, the wave packets
become anisotropic.
24 ~
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)
Driving force:
Key parameters:
Comparison OK…
4.0Ro uInertial waves cease to propagate for
Suggests loss of dipolar field at 1~EkRa2 Q
(Davidson, GJI, 2016)
Dimensionless dependant parameters:
Hypothesis: dynamo saturates at minimum magnetic energy compatible with given
Convective heat flux.
Thank You
Davidson, Staplehurst, Dalziel, JFM, 2006
Helicity generation/segregation and α-effect via inertial wave-packets
launched from equatorial regions
Dynamics of a sea of inertial wave-packets launched from equatorial
Davidson, GJI, 2016