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 B Right-handed helical flow induces current anti-parallel to B Current Left-handed helical flow induces current parallel to B Self-sustaining α 2 dynamo needs sign of helicity opposite in north and south of core Why should the flow in the core be like that?
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
5 The α 2 dynamo This is a zero-order model of most numerical dynamos
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 10 9 - underpowered by a factor of 10 3 Typical results of dynamo simulations Weakly forced 10 times critical flow B Alternating cyclones & anti-cyclones is seat of dynamo action. Helicity is observed to be ve in the north, +ve in the south Moderately forced 50 times critical Note the Earth is a 10 5 times critical! Numerical simulations are getting something correct but much that is wrong!
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
More realistic model of helicity generation should be: Independent of viscosity Internally driven (independent of interior structures) Physically robust but dynamically random 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 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 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
Results from numerical simulations (Sakuruba & Roberts, 2009) A clue? Axial vorticity, equatorial slice Note the strong equatorial jet Temperature Can the equatorial plumes fuel the required asymmetric helicity distribution?
An old idea recycled (G I Taylor, 1921) Rotation-dominated flow: pressure gradient balances Coriolis force p 2uΩ Geostrophic balance requires u z 0 2D flow
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
Spontaneous formation of columnar eddy from a localised disturbance. Caused by spontaneous selffocussing of radiated energy onto rotation axis Reason: angular momentum conservation. Eddy grows and propagates at the group velocity of zero-frequency inertial wave packets Davidson et. al (JFM, 2006)
Columnar vortices (cyclones, anti-cyclones) common in rotating turbulence, created by Inertial waves. Numerical simulations of rotating turbulence from NCAR, US
Inertial wave packets are helical ideal for dynamo 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!)
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)
Numerical simulation: wave-packets emerging form a layer of random buoyant blobs Buoyancy field Velocity iso-surfaces Energy surfaces coloured by helicity Note helicity is negative in the north (blue) and positive in the south (red) (Davidson & Ranjan, GJI, 2015)
Numerical simulation Surfaces of axial velocity (positive is red, negative is blue) Note alternating cyclonesanticyclones Compare!
If we include the dynamic influence of the mean magnetic field, the wave packets become anisotropic. Modified inertial waves, magnetostrophic waves Both helical.
Speculative scaling for Helical-wave α 2 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: ~ g Q c 4R p T 2 C Velocity scale : V 1/ 3 p R C Key parameters: u, V, V B, P A rms transverse plumescale Prediction: V 3 P 2 Va ~ ~ u 2 (Davidson, GJI, 2016) But what sets...? In numerical dynamos viscosity sets δ: R C 1/ 3 ~ Ek Comparison OK But what sets δ in the planets?
Comparison of predicted scaling with numerical dynamos 1 p VP R C 3, B B rms R, C Ro u R C Ro ~ 1/ 2 P Ek 1/ 6 B ~ 1/ 2 P Pr 1/ 2 m Ek 1/ 3,, (Davidson, GJI, 2016)
Comparison of predicted scaling with numerical dynamos 2 Inertial waves cease to propagate for Ro u 0.4 Suggests loss of dipolar field at Ra Ek ~ 1 2 Q (Davidson, GJI, 2016)
Speculative scaling for Helical-wave α 2 Dynamo Cont. But what sets δ in the planets? Elsasser ~ 1 would have the gas giants multipolar Hypothesis: dynamo saturates at minimum magnetic energy compatible with given Convective heat flux. Dimensionless dependant parameters: Predictions: B rms R C B rms V P, u V p Earth Jupiter Saturn u ~ V p Measured 13 x 10-5 5 x 10-5 2 x 10-5 B rms V P ~ V P Predicted 8 x 10-5 15 x 10-5 11 x 10-5
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 Thank You