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