Hydromagnetic waves in Earth's core and their influence on ...

<strong>Hydromagnetic</strong> <strong>waves</strong> <strong>in</strong> Earth’s s <strong>core</strong> 

<strong>and</strong> <strong>their</strong> <strong>in</strong>fluence on 

geomagnetic secular variation 

PhD Thesis submitted May 2005 

Christopher F<strong>in</strong>lay 

School of Earth <strong>and</strong> Environment 

University of Leeds 

Talk Outl<strong>in</strong>e 

0. Overview of geomagnetic field 

1. Analyisis of observations 

-space-time spectral analysis method 

-historical field evolution 

-archeomagnetic field evolution 

-geodynamo model field evolution 

2. Modell<strong>in</strong>g hydromagnetic <strong>waves</strong> 

-review of previous work & analytic model 

-numerical model (l<strong>in</strong>ear spherical case) 

-k<strong>in</strong>ematic action of wave flows 

3. Conclusions: Comb<strong>in</strong><strong>in</strong>g theory <strong>and</strong> observ.

Motivation: Our magnetic shield 

• Earth’s s magnetic field shields the planet from the solar w<strong>in</strong>d 

• But the field is not steady, rather is fluctuat<strong>in</strong>g 

• Aim: to better underst<strong>and</strong> mechanisms of field variation

Geomagnetic field observations 

• Permanent observatories 

• Maritime observations 

• Repeat surveys 

• Satellites 

Portable fluxgate 

magnetometer 

CHAMP 

Hadley magnetic 

observatory

Geomagnetic field evolution 

• Radial magnetic field B r at the <strong>core</strong> surface 1590-1990

Field evolution mechanisms 

• How can <strong>core</strong> motions (<strong>in</strong>clud<strong>in</strong>g those <strong>in</strong>volved <strong>in</strong> the 

geodynamo mechanism) produce observed field changes 

• Advection of field l<strong>in</strong>es with flows close to the <strong>core</strong> surface 

(Example of a 

<strong>core</strong> flow 

<strong>in</strong>version, 

assum<strong>in</strong>g 

advection as the 

source of field 

evolution, 

from Jackson’97) 

• or flux expulsion <strong>and</strong> diffusion: new field at <strong>core</strong> surface 

• or hydromagnetic wave propagation

What are hydromagnetic <strong>waves</strong> 

• <strong>Hydromagnetic</strong> <strong>waves</strong> arise due to the 

elasticity given to fluids by the presence 

of a magnetic fields 

• When magnetic fields are distorted by 

flow, they respond via a Lorentz force 

act<strong>in</strong>g to oppose the distortion 

• In Earth’s s <strong>core</strong>, Coriolis <strong>and</strong> buoyancy 

forces are also <strong>in</strong>volved <strong>in</strong> the force 

balance <strong>and</strong> determ<strong>in</strong>e wave properties 

Observable 

Poloidal field 

Columnar 

wave flow 

• Slow <strong>waves</strong> with time scales of hundreds 

to thous<strong>and</strong>s of years are possible when 

Coriolis forces act to balance the Lorentz 

forces (MC <strong>waves</strong>) 

• Time scales ~ westward drift suggest a 

possible mechanism of field evolution 

Wave drifts 

azimuthally 

Toroidal field 

with<strong>in</strong> fluid <strong>core</strong> 

distorted by wave

Thesis aims 

To <strong>in</strong>vestigate of hydromagnetic <strong>waves</strong> as a source of 

geomagnetic field variation by: 

• Study<strong>in</strong>g observations of geomagnetic field evolution 

(historical <strong>and</strong> archeomagnetic) ) for evidence of <strong>waves</strong> 

• Study<strong>in</strong>g geodynamo models for evidence of <strong>waves</strong> <strong>and</strong> 

prob<strong>in</strong>g the underly<strong>in</strong>g mechanisms 

• Us<strong>in</strong>g hydromagnetic wave theory <strong>and</strong> numerical modell<strong>in</strong>g 

to better underst<strong>and</strong> wave properties <strong>and</strong> effect on B r 

• Comb<strong>in</strong><strong>in</strong>g observations <strong>and</strong> theory: : are they compatible








2.Modell<strong>in</strong>g hydromagnetic <strong>waves</strong> 




3. Conclusions

Time-longitude plots 

• For azimuthal (east-west) field motions, choose a 

latitude of <strong>in</strong>terest, to construct time-longitude plots 

• used <strong>in</strong> meteorology <strong>and</strong> 

oceanography to study <strong>waves</strong> 

• gradients give speeds of features

Frequency-wavenumber 

spectra 

Taper ends of time series 

Zero pad time series 

Take FFT <strong>in</strong> time 

Take FFT <strong>in</strong> longitude 

Plot square of amplitudes

Speed determ<strong>in</strong>ation <strong>and</strong> use of 

the Radon transform 

= -30 o 

• The Radon transform is the projection of a 2D image along the direction 

normal to the l<strong>in</strong>e def<strong>in</strong>ed by the angle q 

• High amplitudes of the Radon transform occur where field features <strong>in</strong> 

the TL plot move coherently at an angle 90 + q to the x axis

Speed determ<strong>in</strong>ation <strong>and</strong> use of 

the Radon transform (II)

Field Process<strong>in</strong>g (I): Removal of 

time-averaged 

axisymmetric field 

= -30 o

Field Process<strong>in</strong>g (II): High pass 

filter<strong>in</strong>g 

Filter threshold 

t c =400 years 

= -30 o













Historical field evolution: gufm1 

(Jackson, Jonkers <strong>and</strong> Walker 2000) 

• 365 694 observations from 1590 to 1990 

• Maritime, observatory, survey <strong>and</strong> satellite data <strong>in</strong>cluded 

• Time-dependent field model (spherical harmonic <strong>and</strong> cubic B-spl<strong>in</strong>e) 

• Suitable for study<strong>in</strong>g field at <strong>core</strong> surface: regularised <strong>in</strong> space & time 

• Number of observations <strong>and</strong> spatial coverage varies with time 

• Accuracy of model is best <strong>in</strong> 20 th century, but still good at earlier times

gufm1: Processed B r with t c =400yrs (II) 

Equator 

20 degrees North

gufm1: Processed B r with t c =400yrs (III) 

- 

-

m=7 

T=125yrs 

to 333yrs 

gufm1: FK filter<strong>in</strong>g to isolate modes 

m=5 

T=125yrs 

to 333yrs 

m=3 

T=125yrs 

to 333yrs

gufm1: Dispersion 

• Difficult: need same source, so restrict to s<strong>in</strong>gle latitude: equator 

• Then only significant power over small spread of wavenumbers 

• No strong evidence for dispersion, ignor<strong>in</strong>g weak signals, hypothesis 

of no dispersion can expla<strong>in</strong> observated field evolution

gufm1: Geographical trends <strong>in</strong> speeds 

• Apply<strong>in</strong>g Radon transform speed determ<strong>in</strong>ation to longitude 

sub-w<strong>in</strong>dows 

• User def<strong>in</strong>ed threshold; only strongest signals shown 

• Speeds generally faster under south America 

• No clear trends as a function of latitude or longitude

gufm1: Temporal evolution 

1750 1850 1910 

• Can use Radon transform method with sub-w<strong>in</strong>dows <strong>in</strong> time to 

<strong>in</strong>vestigate temporal evolution, but lose velocity resolution 

• Strong equatorial signal present throughout record 

• In 20 th century strong westward signal at mid-latitudes <strong>in</strong> southern 

hemisphere 

• Mechanism of field evolution not stationary, but evolve over centuries

Summary of study of historical field 

• Spatially <strong>and</strong> temporally coherent (wave-like) patterns of 

field evolution exist 

• Strongest, most persistent, signal is m=5, T=250yrs, 

travell<strong>in</strong>g at ~17kmyr -1 westwards at the equator 

• Evidence for m=7 <strong>and</strong> m=3 signals <strong>and</strong> a weak 

eastward signal at mid-latiudes 

• Mechanisms produc<strong>in</strong>g field evolution are not stationary 

• No strong evidence for either dispersion or simple 

latitud<strong>in</strong>al trends <strong>in</strong> azimuthal speeds













Archeomagnetic field evolution: 

CALS7K.1 (Korte 

<strong>and</strong> Constable 2005) 

• 32 653 observations from 5000 B.C to 1950 A. D. 

• From depostional remanent magnetisation <strong>in</strong> lake sediments an 

thermal remanent magnetisation <strong>in</strong> lavas <strong>and</strong> archeological artefacts 

• Time-dependent field model (spherical harmonic <strong>and</strong> cubic B-spl<strong>in</strong>e) 

• Suitable for study<strong>in</strong>g field at <strong>core</strong> surface: regularised <strong>in</strong> space & time 

• Number of observations <strong>and</strong> spatial coverage varies with time 

• Accuracy of model is best <strong>in</strong> last 3000 years 

Lake 

sediment 

sites 

concentration 

Lavas 

(directional) 

Lavas 

(<strong>in</strong>tensity) 

(Decl<strong>in</strong>ation measurements <strong>in</strong> 

Europe as a function of time)

CALS7K.1: Comparison with gufm1 

• Can no longer use t c =400 years as CALS7K.1 conta<strong>in</strong>s little power 

on this time scale. Instead t c =2500yrs 

• Lack of data <strong>in</strong> southern <strong>and</strong> Pacific hemispheres 

• Conta<strong>in</strong>s little power at spherical harmonic degrees > 5 

• Azimuthal direction of field evolution captured correctly 

gufm1: 

CALS7K.1:

CALS7K.1: wave-like field 

evolution 

At 40 0 North 

At Equator 

• Spatially <strong>and</strong> temporally coherently field evolution patterns 

• Episodes of both westwards <strong>and</strong> eastwards (esp. at northern mid lat.) motions 

• Coexistence of eastward <strong>and</strong> westward motions

Summary of study of 

archeomagnetic field 


field evolution exist 

• Difficult to accurately determ<strong>in</strong>e wavenumbers <strong>and</strong> periods 

due to limitations of spatial <strong>and</strong> temporal resolution 

• Evidence for eastward motions of field features, 

espcially at northern mid-latitudes from 0 to 1500 A.D.













Analysis of geodynamo models 

(model of Wicht 2002) 

• Apply same methods to dynamo output to study possible mechanisms 

• Ouput from convection driven geodynamo model MAGIC of Wicht ’02 

• 3D, fully non-l<strong>in</strong>ear, spherical shell model of Bouss<strong>in</strong>esq convection 

• B.C. are no slip, electrically <strong>in</strong>sulat<strong>in</strong>g <strong>and</strong> isothermal 

• Driven by applied temperature difference between <strong>in</strong>ner <strong>and</strong> outer <strong>core</strong> 

• Two models studied: DYN1- simple model to study mechanisms 

DYN2- more Earth-like but also more complex 

• Non-dimensional parameters compared to Earth:

DYN1: B r at outer boundary

DYN1: u r at outer boundary

DYN1: Time-longitude plots 

Processed B r (t c =0.065 τ η 

−1 

) 

0.077 τ η 

u ø 

u r 

14 0 N

DYN2: Time-longitude plots 

0.100 τ η 

B r 

Damped B r 

u r 

20 0 S

Summary of geodynamo study 


field evolution are also observed <strong>in</strong> geodynamo models 

• Azimuthally drift<strong>in</strong>g B r concentrations form at <strong>and</strong> move 

along with downwell<strong>in</strong>gs (positions of fluid convergence) 

<strong>in</strong> observed wave flows 

• Wave flow pattern is mov<strong>in</strong>g more slowly than speeds 

of flow with<strong>in</strong> the wave 

• Mean azimuthal flows are present but are not large 

to account for all the azimuthal motions, suggest<strong>in</strong>g 

wave propagation is also occurr<strong>in</strong>g 

• <strong>Hydromagnetic</strong> <strong>waves</strong> are present <strong>and</strong> contribute to the 

observed field evolution; need to consider these <strong>in</strong> detail













Excitation mechanisms 

• Observations suggest a small number of dom<strong>in</strong>ant 

wavenumbers => <strong>in</strong>stability driv<strong>in</strong>g <strong>waves</strong> 

• Possible sources of <strong>in</strong>stability <strong>in</strong> the <strong>core</strong> <strong>in</strong>clude: 

- Tidal forc<strong>in</strong>g (elliptical <strong>in</strong>stability) 

- Topographic forc<strong>in</strong>g 

- Shear <strong>in</strong>stability 

- Magnetic field <strong>in</strong>stability (due to strong, complex fields) 

- Convective (buoyancy) <strong>in</strong>stability 

• Other mechanisms cannot be ruled out, but convective 

<strong>in</strong>stability <strong>in</strong>volves fewest assumptions

A simplified model of <strong>core</strong> dynamics 

• To obta<strong>in</strong> an analytically tractable solution, simplified model of 

<strong>core</strong> dynamics (after Soward’79) is <strong>in</strong>vestigated 

• Includes magnetic field, variable Coriolis force, convection <strong>and</strong> 

viscous, thermal <strong>and</strong> magnetic diffusion

Govern<strong>in</strong>g equations 

• To f<strong>in</strong>d the characteristic <strong>waves</strong> supported by this system: 

(i) Envoke Quasi-geostrophy assumption of z <strong>in</strong>dependence 

(ii) Substitute <strong>in</strong> wave-like solutions ~ e i(kx + ky -ωt) for u, b, Θ 

(iii) Elim<strong>in</strong>ate between equation to obta<strong>in</strong> a dispersion relation

Dispersion relation 

• Balanc<strong>in</strong>g different comb<strong>in</strong>ations of terms (i.e. for different 

parameter regimes) different types of <strong>waves</strong> are possible: 

• Free Rossby <strong>waves</strong>: 

• Free MC-Rossby <strong>waves</strong>: 

(Hide’s <strong>waves</strong>) 

• Thermal magneto-Rossby <strong>waves</strong>: 

(ignore <strong>in</strong>ertia <strong>and</strong> viscosity 

<strong>in</strong> low Pr m liquid metal limit) 

- Involve a balance between magnetic <strong>and</strong> Coriolis terms <strong>and</strong> 

are driven by buoyancy. Dissipation is primarily Ohmic 

- Dispersive <strong>and</strong> have phase speed dependent on latitude













Numerical model <strong>in</strong> full spherical 

geometry 

• Full spherical representation of Coriolis force 

• Imposed magnetic field: toroidal only with B 0 ∝ r o s<strong>in</strong> θ 

• L<strong>in</strong>earised about a state of no background flow 

• Use poloidal-toroidal representation 

• Exp<strong>and</strong> as spherical harmonics <strong>and</strong> Chebyshev polynomials 

• Free slip, electrally <strong>in</strong>sulat<strong>in</strong>g, isothermal B.C. 

• Express variables as superposition of normal modes e i(mø-ωt) 

• Complex, generalised eigenvalue problem 

• Solve us<strong>in</strong>g ARPACK rout<strong>in</strong>es to f<strong>in</strong>d Ra, ω for marg<strong>in</strong>ally 

stable <strong>waves</strong> 

-> Parameter survey

Dispersion (ω( 

vs m) 

• Waves are dispersive, with larger m faster for Λ ≥ 1 

• For larger Λ <strong>waves</strong> are travel <strong>in</strong>creas<strong>in</strong>gly fast westwards

Equatorial structure of thermal 

magneto-Rossby 

wave (m=5, Λ=1, Pr =0.1,Pr m =10 - 6 ) 

u r 

b r 

u ø 

b ø



wave (m=5, Λ=1, Pr =0.1,Pr m =10 - 6 ) 

u r 

b r 

u ø 

b ø



wave (m=5, Λ=1, Pr =0.1,Pr m =10 - 6 ) 

u r 

b r 

u ø 

b ø

Summary of numerical 

modell<strong>in</strong>g of <strong>waves</strong> 

• Thermal magneto-Rossby <strong>waves</strong>, with m=2 to 7, excited at Λ~1 

• Such <strong>waves</strong> travel slowly (thermal diffusion time scale) 

• Such <strong>waves</strong> are dispersive 

• Imposed toroidal field is distorted by wave flow produc<strong>in</strong>g 

spatially coherent (wave-like) patterns <strong>in</strong> B r at outer boundary 

• Spatially periodic flows with regions of flow convergence at 

downwell<strong>in</strong>gs alternat<strong>in</strong>g with regions of flow divergence at 

upwell<strong>in</strong>gs 

• BUT….. models reported neglect azimuthal flows: Fearn <strong>and</strong> 

Proctor ’83 suggest these determ<strong>in</strong>e speed <strong>and</strong> structure of 

magneto-Rossby <strong>waves</strong> -> ongo<strong>in</strong>g work led by N. Gillet













1D model of wave flow act<strong>in</strong>g on B r 

• Consider a generic, drift<strong>in</strong>g wave flow act<strong>in</strong>g normal to a 

<strong>in</strong>itially uniform magnetic field: 

c ph 

B 

U 

2 important speeds: 

U= magntiude of flow with<strong>in</strong> wave 

C ph = ω / k = phase speed of wave pattern 

• Neglect<strong>in</strong>g magnetic diffusion, evolution of the magnetic field 

is determ<strong>in</strong>ed by the 1D frozen-flux magnetic <strong>in</strong>duction eqn: 

• Problem can be solved analytically by mov<strong>in</strong>g <strong>in</strong>to a frame of 

reference mov<strong>in</strong>g along with the wave <strong>and</strong> us<strong>in</strong>g the method of 

characteristics

1D model: Possible regimes 

(i) Concentration regime (|c ph | |U|: 0.5c ph =U) 

In drift<strong>in</strong>g 

frame: 

In orig<strong>in</strong>al 

frame: 

• Less concentration <strong>and</strong> pulsation with period ~ (2π/k)/c ph

2D model of wave flow act<strong>in</strong>g on B r 

• Consider<strong>in</strong>g more realistic scenario of wave flow at a 

spherical boundary act<strong>in</strong>g on an axial dipole radial field 

• Include limited magnetic diffusion: 

• Wave flow employed is the most simple (equatorially sym) 

<strong>in</strong>ertial wave flow <strong>in</strong> a sphere, which is also a solution to the 

Malkus MC wave problem:

2D model: Axial dipole <strong>in</strong>itial field 

• Propagat<strong>in</strong>g E S flow (no mean azimuthal flow) act<strong>in</strong>g on an E A field gives 

rise to a drifit<strong>in</strong>g E A field perturbation with no distortion of magnetic equator

Summary field concentration 

mechanism 

• Propagat<strong>in</strong>g wave flows can cause the formation <strong>and</strong> 

azimuthal motion of concentrations of B r 

• 2 regimes are possible: concentration or pulsations 

• Field concentrations occur at po<strong>in</strong>ts of flow convergence 

<strong>in</strong> the drift<strong>in</strong>g reference frame 

• For dynamically favoured E S flows, E A (e.g. axial dipole) B r 

does not generate the observed E S field concentrations













Influence of hydromagnetic <strong>waves</strong> 

on patterns <strong>in</strong> radial magnetic field 

• Non-axisymmetric 

hydromagnetic <strong>waves</strong> can effect B r 

produc<strong>in</strong>g spatially periodic patterns <strong>in</strong> 2 ways: 

(i) By distortion of toroidal field <strong>in</strong> 

the <strong>core</strong> by upwell<strong>in</strong>g <strong>and</strong> 

downwell<strong>in</strong>g of wave flows 

x 

x 

(ii) Modulation of exist<strong>in</strong>g B r by 

convergence <strong>and</strong> divergence 

• Prefer (i) for simplicity (must occur), but 

(ii) also possible 

Wave flow 

Magnetic field 

Core surface

Motion of field patterns: 

wave propagation or advection 

• No observational evidence for either dispersion or 

consistent latitud<strong>in</strong>al trends 

• Convection driven <strong>waves</strong> when Λ ~ 1 seem to propagate 

too slowly 

• Previous theoretical studies (Fearn <strong>and</strong> Proctor ’83) 

suggest that azimuthal speeds of <strong>waves</strong> are determ<strong>in</strong>ed 

by the background azimuthal flows present 

• The most consistent explanation is therefore azimuthal 

motions are caused primarily by advection by flow

A proposed scenario consistent with 

both theory <strong>and</strong> observations 

(1) <strong>Hydromagnetic</strong> <strong>waves</strong> exist <strong>in</strong> the <strong>core</strong>, are driven by 

convection, <strong>and</strong> <strong>in</strong>volve a balance btw magnetic <strong>and</strong> Coriolis 

forces (convection-driven, magneto-Rossby <strong>waves</strong>) 

(2) These <strong>waves</strong> distort the toroidal field <strong>in</strong> the <strong>core</strong> produc<strong>in</strong>g 

spatially periodic patterns of new B r 

(3) Spatially coherent patterns are subsequently advected by 

azimuthal flows close to the <strong>core</strong> surface 

But… cannot conclusively rule out the possibilities that: 

(i) conc. of B r by convergence of wave flows 

(ii) non convection-driven wave propagation=>azimuthal motion

Summary of work carried out 

• Development of tools for analysis of azimuthal evolution 

of scalar fields 

• Detection <strong>and</strong> quantification of spatially coherent (wavelike) 

patterns <strong>in</strong> B r at the <strong>core</strong> surface 

• Observation of episodes of eastward <strong>and</strong> westward motion 

of patterns <strong>in</strong> B r 

• Identification of hydromagnetic <strong>waves</strong> as the orig<strong>in</strong> of 

azimuthally drift<strong>in</strong>g field concentrations <strong>in</strong> geodynamo 

models 

• Review of theory of hydromagnetic <strong>waves</strong> <strong>in</strong> rotat<strong>in</strong>g fluids 

• Documentation of parameter dependence of simple 

convection driven hydromagnetic <strong>waves</strong> 

• Deduction of <strong>in</strong>fluence of hydromagnetic wave flows on B r 

• Proposal of a scenario consistent with theory <strong>and</strong> 

observations

Suggestions for future work 

• Modell<strong>in</strong>g of nonl<strong>in</strong>ear hydromagnetic <strong>waves</strong> with realistic 

background magnetic field <strong>and</strong> azimuthal flows (N. Gillet) 

• Investigation of other (non-convective) wave excitation 

mechanisms 

• Further study of force balances <strong>in</strong>volved <strong>in</strong> hydromagnetic 

<strong>waves</strong> <strong>in</strong> geodynamo models 

• Investigation of whether or not magnetic diffusion is 

important for <strong>in</strong> the field evolution associated with 

azimuthal motions of hydromagnetic wave patterns 

• Study<strong>in</strong>g high resolution, time-dependent, field models 

derived from satellite data, spatially filter<strong>in</strong>g to <strong>in</strong>vestigate 

the structure of suggested wave modes 

• In the long term, us<strong>in</strong>g improved archeomagnetic field 

models <strong>and</strong> observations of gravity anomalies <strong>in</strong> the <strong>core</strong> 

to test the proposed scenario

Acknowledgements 

• Many thanks to all who have helped me dur<strong>in</strong>g my time as 

a PhD student, especially: 

- Andy Jackson 

- David Gubb<strong>in</strong>s 

- Ashley Willis 

- Mathieu Dumberry 

- Nicolas Gillet 

- Phil Livermore 

- James While 

- Nicola Pressl<strong>in</strong>g 

- Stuart Borthwick 

- Nick Barber 

& the rest of the geophysics <strong>and</strong> tectonics group 

• Monika Korte for provid<strong>in</strong>g the CALSK7.1 model 

• Johannes Wicht for provid<strong>in</strong>g geodynamo model output 

• Chris Jones <strong>and</strong> Steve Worl<strong>and</strong> for provid<strong>in</strong>g a copy of 

<strong>their</strong> magnetoconvection code

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