the 3 main parts of the course 1. structure and physical...

The 3 main parts of the course

1. Structure and physical properties of Earth’s core

2. Observational geomagnetism and inverse problem ingeomagnetism

3. Earth’s magnetic field and rotational magnetohydrodynamics

1

http://jupiter.ethz.ch/~stijn/teaching.html

Key words

• Earth’s liquid core is composed mainly with iron • Above ~ 1000 K, iron is not magnetic • But it is a good electrical conductor

Composition of the core:

• Rotation of the planet • Thermal convection

Flow:

• At first order: axial magnetic dipole (~aligned with the axis of rotation)

Geometry of the magnetic field:

DYNAMO: movement u

mag. field B

self-sustainment

2

1. Fundamentals of Electromagnetism and Induction equation

2. Self-Exciting dynamos: kinematic theory

3. Core Dynamics I

4. Core Dynamics II

5. Numerical modeling of the Geodynamo

6. Reproducing a dynamo in the laboratory

Outline of the part

3

Theoretical Geomagnetism

Lecture 1:

Fundamentals of Electromagnetism and the Induction equation

4

1.1 Induction by a moving conductor

1.2 Maxwell’s equations of Electromagnetism

1.3 Approximations and moving reference frames

1.4 The Induction Equation

1.5 Summary

Appendix: Ohm’s law from microscopic considerations

Lecture 1: Fundamentals of Electromagnetism and the Induction equation

5


• 3 effects result:

(1) An electrical current is induced in the conductor.

(2) This current causes a magnetic field that adds to the original field, such that the conductor appears to drag the field along with it.

(3) The combined magnetic field interacts with the current resulting in a Lorentz force that acts on conductor, opposing its motion.

(From Davidson, 2001)

• To begin, let us consider what happens when an electrical conductor is pulled through a magnetic field:

6

http://web.mit.edu/hml/ncfmf.html(from 2:48)

http://web.mit.edu/hml/ncfmf.html






1.5 Summary

Appendix Ohm’s law from microscopic considerations

7

1.2.0 Mathematical preliminary

8

• For any sufficiently smooth scalar function and vector field the following relations hold:

f(r, ,, t)F (r, ,, t)

r r f= 0 r · r F = 0

• Any sufficiently smooth vector field can always be decomposed as follows:

F (r, ,, t)

F = rg +rG

1.2.1 Basic Assumptions

i.e. It is assumed that these fields are well defined and smoothly varying everywhere in space and individual particles are ignored.

(N.B. Good approximation if considering length scales much larger than the size and mean free path length of the particles in the medium.)

• Our mathematical description of electromagnetism will be in terms of macroscopic, continuum, physical quantities measured in SI units:

B(r, θ,φ, t) Magnetic Flux Density (often called Magnetic Field) in Tesla (T).

E(r, θ,φ, t)

J(r, θ,φ, t)

Electric Field in Volts per meter (Vm-1).

Electric Current Density in Amperes per cubic meter (Am-3)

ρe(r, θ,φ, t) Electric Charge Density in Coulombs per cubic meter (Cm-3)

9

1.2.2 Maxwell’s equations: An Overview• The relationship between magnetic fields, electric fields and electrical currents are encapsulated in just 4 simple equations:

∇ · E =

ρe

ϵ0Gauss’s Law: E- fields produced by charge density

∇ · B = 0 Lack of monopole sources of B-field so B-field is solenoidal.

∇× E = −∂B

∂tFaraday’s Law of Induction: E-field induced by changing B-field

∇× B = µ0J + ϵ0µ0

∂E

∂tAmpere-Maxwell law: B-field produced by currents or by changing E-field(displacement currents)

• Note, this is the form of Maxwell’s equations for materials with no permanent magnetization or electric polarization and whereϵ0

µ0

= electrical permittivity of free space = 8.85 x10 -12 C2 N-1 m-2

= magnetic permeability of free space = 4π x 10-7 N A-2(ϵ0µ0)

−1/2 = c = 3 × 108ms−1and

10

1.2.3 Maxwell’s equations: Gauss’s law of electrostatics

∇ · E =

ρe

ϵ0

• Consider the Gauss’s law

• If there is no time-varying magnetic field, then by Faraday’s Law

∇× E = 0 E = −∇φand

• The electric potential is then defined by Poisson’s equation,∇

2φ = −ρe

ϵ0

with solutions of the form:

• Thus and hence at any point can be obtained by integration over all the sources of electric field (the charge density).

Eφ

• Considering a single charge at the origin and noting that the force onanother charge is , gives Coulomb’s law of electrostatics.F = qE

φ(r) =1

4πϵ0

∫ρe(r′)

|r − r′|

d3r,

11

1.2.4 Maxwell’s equations: Divergence-free Magnetic fields

• Consider the Maxwell equation,

∇ · B = 0

• This expresses the observational fact that there are no pointsources/sinks of magnetic field (i.e. no magnetic monopoles)

• Therefore:(i) Magnetic field lines can never end, but are always closed(ii) Same number of field lines as enter must leave a volume

• The continuity equation for incompressible fluids ( ) takes the same form, so the same amount of fluid as enters a volume must leave a volume.

∇ · u = 0

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1.2.5 Maxwell’s equations: Faraday’s Law of EM Induction

• The differential form of Faraday’s Law may be integrated over asurface S perpendicular to the magnetic field,

• Then using Stokes Theorem the LHS can be re-written as a lineintegral round the curve C surrounding S such that,

∫S

(∇× E) · dS = −

∫S

∂B

∂t· dS

∮C

E · dl = −

dΦ

dtwhere Φ =

∫S

B · dS

• Therefore, the change in magnetic flux through S is equal to the induced electric field integrated round C.

13

Jerome

1.2.6 Maxwell’s equations: Ampere-Maxwell Law

• Assuming that changes in the electric field are slow then,

∇× B = µ0J

• Integrating this over a surface S perpendicular to the B-field,

∫S

(∇× B) · dS = µ0

∫S

J · dS

• Then using Stokes theorem on the LHS gives us Ampere’s law:

∫C

B · dl = µ0I where I =

∫S

J · dS

• If we instead take the curl of we obtain a vectorial Poisson equation:

∇× B = µ0J

∇2B = −µ0∇× J

• The solution of this equation is known as the Biot-Savart Law,

B(r) =µ0

4π∇×

∫J(r′)

|r − r′|d3r′ =

µ0

4π

∫J(r′) × (r − r′)

|r − r′|3d3r′

14

Jerome

Jerome





1.6 Summary



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1.3.1 Approximations: Neglect of the displacement current

• Consider the relative magnitude of the 2nd term on the RHS and the term on the LHS of the Ampere-Maxwell equation:

|ϵ0µ0∂E/∂t|

|∇× B|∼

ϵ0µ0E/τ

B/l

• From similar scale analysis of terms in the Faraday’s Law equation we have, therefore|∇× E| ∼ |− ∂B/∂t| so that E/l ∼ B/τ

|ϵ0µ0∂E/∂t|

|∇× B|∼

1

c2

l2

τ2

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1.3.1 Approximations: Neglect of the displacement current

• are the characteristic length and time scales associated with EM field changes we wish to study. For Earth’s core, the largest lengthscales are ~ 3481km and shortest timescales ~1 yr

l, τ

• Therefore can safely neglect the displacement current term:

Physically this represents filtering out EM waves from the system.

|ϵ0µ0∂E/∂t|

|∇× B|∼

1

c2

l2

τ2∼

(

0.1ms−1

3 × 108ms−1

)2

∼ 10−20-19

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1.3.2 Moving reference frames: The Lorentz transformations

• The observed electric and magnetic field depend on the frame of reference in which they are measured (Einstein, 1905).

• Let and be the fields in one frame of reference and , be those measured in a second inertial frame moving at velocity relative to the first. The transformation btw the frames is:

E E′

B′

B

u

E′ = γ(E + u × B) + (1 − γ)

(u · E)u

u2

B′ = γ(B + ϵ0µ0(u × E)) + (1 − γ)

(u · B)u

u2

where γ = (1 − u2/c2)−1/2

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1.3.2 Moving reference frames: The Lorentz transformations E

′ = γ(E + u × B) + (1 − γ)(u · E)u

u2

B′ = γ(B + ϵ0µ0(u × E)) + (1 − γ)

(u · B)u

u2

E′= E + u × B and B

′= B

• Thus moving to a reference frame travelling along with the core fluid: (i) the magnetic field is unchanged, (ii) the electric field is modified by effect of magnetic field on a moving conductor.

• Assuming motions in Earth’s core are non-relativistic i.e. then ~ 1 and the transformations simplify toγ = (1 − u2/c2)−1/2

u2

<< c2

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1.3.3 Ohm’s law in a moving reference frame

• Ohm’s law is an empirical law stating the experimentally observed relation between the electric fields and electric current density.

• For stationary conductors it takes the form:

• When considering a moving electrical conductor, the effective electric field in the frame moving with the conductor must be used:

where = electrical conductivity ( )J = σE σ Ω−1

m−1

J = σE′ = σ(E + u × B)

• For Earth’s core (predominantly liquid Fe at high P,T) the electrical conductivity is thought to be large ~ 1.5x106 Ω−1

m−1

• Again, note, this a non-relativistic approximation ( ).u2

<< c2

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1.3.4 Summary of Equations describing Electrodynamics of the Core

∇ · B = 0

∇× E = −∂B

∂t

∇× B = µ0J

J = σ(E + u × B)

• N.B.1: Only time-derivative of the magnetic field remains.• N.B.2: Neglect of the displacement current means we no longer need to consider the Gauss’s electrostatic eqn (decoupled).

Often referred to as: “the magnetohydrodynamic (MHD) approximation of electrodynamics”

• For moving conductors (where ) and considering slow changes in the EM fields ( ) then the evolution of the magnetic fields and electric currents are specified by:

u2

<< c2

(l/)2/c2 1

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1.5 Summary

Appendix: Ohm’s law from macroscopic considerations


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1.5.1 The Magnetic Induction Equation

• Substituting from (4) into (3) gives

• Recall the equations governing electrodynamics under the MHD approximation:

∇ · B = 0

∇× E = −∂B

∂t

∇× B = µ0J

J = σ(E + u × B) (4)

(3)

(2)

(1)

(5)1

µ0σ(∇× B) = E + u × B

• Take the curl of this and using the magnetic diffusivity η =1

µ0σ

∇× (η∇× B) = ∇× E + ∇× (u × B) (6)23

1.5.1 The Magnetic Induction Equation• Next, substituting from (2) into (6) and rearranging,

∂B

∂t= ∇× (u × B) −∇× (η∇× B) (7)

• If = constant then, we can use a standard vector calculus identity together with (1) to re-write the last term as

η

∇× (η∇× B) = η∇× (∇× B) = η(∇(∇ · B) −∇2B) = −η∇

2B

• Substituting this into (7) we arrive at the Magnetic Induction eqn:

• Thus, under the MHD approximation, if we know the motion of the conductor and the present magnetic field, we can calculate how the field evolves in time.

u

• This single equation describes the electrodynamics of Earth’s core !

∂B

∂t= ∇× (u × B) + η∇2

B (8)

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1.5.2 Magnetic Reynolds Number

∂B

∂t= ∇× (u × B) + η∇2

B

U• Assume that the velocity field has a characteristic magnitude

• Assume that the magnetic field has a characteristic magnitude

• Assume that the lengthscale over which both fields change is L

B

• Then the ratio of the magnitudes of the terms on the RHS will be:

|∇× (u × B)|

|η∇2B|∼

UB/L

ηB/L2∼

UL

η= Rm

• is known as the magnetic Reynolds number.Rm

• For global motions in : Earth’s core

Rm =

UL

η= ULσµ0

25

3 · 104 m

s 3.481 · 106 m 1.5 · 106A

2 s3

mkg 4 · 107 mkg

s2 A2

2000

1.5.3 Perfect Conductivity: Frozen Flux

• Using the standard vector calculus relation,

∇× (u × B) = (B ·∇)u − (u ·∇)B since ∇ · u = 0 and ∇ · B = 0.

∂B

∂t= ∇× (u × B) (9)

• So,

∂B

∂t+ (u ·∇)B = (B ·∇)u or

DB

Dt= (B ·∇)u

Advection of Magnetic Field along with flow

Stretching of magnetic field by shear of flow

(10)

• Consider the case in which so the second term on the right hand side is negligible (i.e. perfect conductivity), then

Rm = ∞

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1.5.3 Perfect Conductivity: Frozen Flux• Consider a short line element drawn into a fluid at some instant. dl

D(dl)

Dt= u(r + dl) − u(r) = (dl ·∇)u

dl• It subsequently moves with the fluid, so the rate of change of is where and are the position vectors at the ends of . The equation of evolution of is then:

u(r + dl) − u(r) r r + dl

dl dl

• The magnetic field lines can be thought of as being frozen into the fluid (i.e. fluid elements lying on a field line at some instant must continue to lie on the fluid elements at all later times).

• But this is identical in form to (10) therefore when , the magnetic field evolves just like the line element, moving along with the fluid.

Rm = ∞

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1.5.3 Perfect Conductivity: Frozen Flux• In the Frozen Flux approximation, we schematically imagine flow sweeping the magnetic field along with it, moving and stretching it.

(From Baumjohann and Treumann, 1997)

28

1.5.3 Perfect Conductivity: Frozen Flux• In the limit of a perfect conductor , there are also strong consequences for changes in the flux through material contours.

e.g. Consider a surface S bounded by a closed material contour C that moves along with the fluid.

=

∫

S

(

∂B

∂t−∇× (u × B)

)

· dS = 0

d

dt

∫S

B · dS =

∫S

∂B

∂t· dS +

∮C

B · u × dl =

∫S

∂B

∂t· dS −

∮C

u × B · dl

i.e. The total flux enclosed by a material surface cannot change with time even if the shape of the evolves : FLUX IS FROZEN !

c c

t=t1 t=t2

• The limit is often called the frozen flux approximation.Rm → ∞ 29

1.5.4 Magnetic diffusion• Next consider the opposite limit in when so the first term on the RHS is negligible (i.e.no flow), then

Rm = 0

∂B

∂t= η∇2

B

• This is a classic vector diffusion equation. Simple dimensional analysis yields,

B

τD

=

ηB

L2so τD =

L2

η

• Thus magnetic field features with larger spatial gradients (smaller ) will diffuse faster for an particular magnetic diffusivity .L η

• For global scale fields in Earth’s core :

( More precise calculations give ~ 80,000yrs for dipole part of geomagnetic field)30

D (3.481 · 106 m)2

0.5m2 s1 770000 yrs

1.5.4 Magnetic Diffusion• Magnetic diffusion can also allow magnetic field structures to merge as well as decay, for example

(From Baumjohann and Treumannn, 1997)31

1.5.4 Combination of Advection and Diffusion• First, in cartesian co-ordinates (x,y,z), consider an initially constant magnetic field only in the z direction. This is acted on by a constant flow in the x direction which has a shear in the z direction . Consider 1st only advection and use:

B0 = (0, 0, B0)

u = (u(z), 0, 0)

e.g.u

B0∂Bz

∂t= 0 => Bz = B0 for all t

∂Bx

∂t= B0

∂u(z)

∂z

∂By

∂t= 0 => By = 0 for all t

Integrating the last w.r.t time and using Bx =0 at t=0 gives, Bx = B0

du

dzt

• So this simple shear causes the field to grow linearly with time in the direction of flow (will be important for dynamo action.... ).

∇× (u × B) = (B ·∇)u − (u ·∇)B since ∇ · u = 0 and ∇ · B = 0.

32

1.5.4 Combination of Advection and Diffusion• But have ignored diffusion, if this were present the field growth would be slower since the x-component of the induction equation is:

∂Bx

∂t= B0

∂u

∂z+ η

∂2Bx

∂z2

• Diffusion will become more important as the field gradients grow, until eventually a steady state is reach where advection and diffusion are in balance:

B0

∂u

∂z= −η

∂2Bx

∂z2

• Integrating twice w.r.t. z the steady state profile for Bx can be found:

Bx =

z∫0

−

B0

ηu(z′)dz′ + cz + d

• So, even if diffusion is not initially important in the induction equation, it will often be a vital ingredient of the saturated state.

33





1.5 Summary



34

1.6 Summary: self-assessment questions

(1) Can you state and describe the physical meaning of each of Maxwell’s equations?

(2) Can you derive the magnetic Induction equation?

(3) Do you understand the consequences of the frozen flux approximation to the Induction equation?

(4) Can you estimate the magnetic diffusion time for a particular system?

Next time: Dynamo Theory

35

References

- Roberts, P.H., (2007) Theory of the geodynamo. In Treatise on Geophysics, Vol 5 Geomagnetism, Ed. M. Kono, Chapter 8.03, pp.67-102. (especially section 8.03.2)

- Baumjohann, W.A. and Treumann, R.A., (1997) Basic Space Plasma Physics, Imperial College Press. (Detailed derivation of Generalised Ohm’s law, Section 7.3).

- Davidson, P.A., (2001) An introduction to magnetohydrodynamics, Cambridge University Press. (Good for physical understanding, especially Chapters 1,2,4).

- Jackson, J.D., (1998) Classical Electrodynamics, John Wiley and Sons Inc. (A wealth of detail on electromagnetism in general).

36





1.5 Summary



37

Ohm’s Law: A microscopic derivation• In previous section we ignored all microscopic effects and stated Ohm’s law as an empirical result.

• But, it can also be derived from microscopic considerations of motions of electrons and ions in a conductor.

• The equations of conservation of momentum density for the electrons and ions are (e.g. Baumjohann and Treumann, 1997 p.140):

∂(neve)

∂t+ ∇ · (neveve) = −

1

me

∇ · P e −nee

me

(E + ve × B) +R

me

∂(nivi)

∂t+ ∇ · (nivivi) = −

1

mi

∇ · P i +nie

mi

(E + vi × B) +R

mi

EM Lorentz forceFluid Pressure tensorTemporal variation of flux density

Collisions btw ions and electrons

NL momentum density flux

ns

ms

vs

= number density of species

= mass of species

= bulk velocity of species 38

Ohm’s Law: A microscopic derivation• Multiply the electron eqn by and the ion equation by and subtract the equations.

mi me

• Then assuming and (quasi-neutral) and defining the macroscopic current as :

ne ∼ ni ∼ n

J = e(nivi − neve)me

e

∂J

∂t= ∇ · P e + ne(E + ve × B) − R

• Note the dependence of the current density only on the electron fluid pressure, electron velocity and electron-ion collisions.

• Again assuming and (quasi-neutral) then the bulk fluid velocity is approximately that of the ions so,

ne ∼ ni ∼ n

vi ∼ u and ve ∼ u −

J

ne

• This allows the Lorentz force on the electrons to be split into 2 terms, one involving the bulk velocity and one the current density:

me

e

∂J

∂t= ∇ · P e + ne(E + u × B) − J × B − R

me/mi 1

me/mi 1

39

Ohm’s Law: A microscopic derivation• Next, we note that we can write the collision term as

R = nmeωc(vi − ve) = ne

J

σ

where σ =ne

2

meωc

where is the frequency of collisions.ωc

• Using this leads to an expression of the Generalised Ohm’s law:me

ne2

∂J

∂t+

J

σ=

1

ne∇ · P e + (E + u × B) −

1

ne(J × B)

• When can neglect pressure gradient on RHS.pe

neU BLp

<< 1

• When can neglect current changes on LHS.meσ

ne2 τ<< 1

• When can neglect Lorentz (Hall) term on RHS.σB

ne<< 1

• These are all true in Earth’s core so finally we are left with,

J = σ(E + u × B)40

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