Planetary interiors:
Magnetic fields, Convection and Dynamo Theory
3. How planetary magnetic fields are generated
Chris Jones, Department of Applied Mathematics
University of Leeds UK

FDEPS Lecture 3, Kyoto, 29th November 2017

Section 3.
3. How planetary magnetic fields are generated
3.1 Fundamentals of MHD and Maxwell’s equations(3.1) Fundamentals of MHD and Maxwell’s equations

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Pre-Maxwell equations 1.
Maxwell’s equations are the basis of electromagnetic theory.
Foundation of dynamo theory. They are (3.1.1)-(3.1.4).


1 ∂E
∂B
,
∇ × B = µj + 2(3.1.1, 3.1.2)
∇×E=−
∂t
c ∂t
(3.1.1) is the differential form of Faraday’s law of induction. If themagnetic field varies with time an electric field is produced. In anelectrically conducting body, electric field drives a current. Basis ofdynamo action.
(3.1.2) is Ampère’s law, which relates the electric current to themagnetic field it produces.
Maxwell added the term in red, but it is not needed in MHD,
because fluid flows are much slower than the speed of light.
E electric field, B the magnetic field, j is the current density, µ is
the permeability.
(3.1) Fundamentals of MHD and Maxwell’s equations

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Pre-Maxwell equations 2.

ρc(3.1.3, 3.1.4)
(3.1.3) expresses that there are no magnetic monopoles.
∇ · B = 0,∇·E=(3.1.4) says that electric field is generated by charges. However,
we eliminate E in MHD.
E electric field, B the magnetic field,
ρc is the charge density,  is the permittivity.
These equations are valid in a frame at rest.

(3.1) Fundamentals of MHD and Maxwell’s equations

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Ohm’s law

In a moving frame E must be replaced by E + u × B while j stays
the same. So in MHD Ohm’s law is
j = σ(E + u × B).

(3.1.5)The SI unit of electrical conductivity is Siemens/metre.
The term u × B is critical. Flow interacting with magnetic field
generates current, which in turn gives field. With the rightconfiguration, flow can sustain field.
Define magnetic diffusivity
1,(3.1.6)η=µσdimensions metre2 /second.

(3.1) Fundamentals of MHD and Maxwell’s equations

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Induction equation

Dividing Ohm’s law (3.1.5) by σ and taking the curl∂Bj+ ∇ × (u × B), (3.1.7)∇ × ( ) = ∇ × E + ∇ × (u × B) = −σ∂t
and using (3.1.2) to eliminate j,∂B= ∇ × (u × B) − ∇ × η(∇ × B),∂t(3.1.8)remembering (3.1.6). (3.1.8) is the induction equation.

(3.1) Fundamentals of MHD and Maxwell’s equations

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Alternative forms of the induction equation

If the conductivity is constant we can use the vector identity
curl curl = grad div -del2and (3.1.8) to write the constant conductivity induction equationas∂B= ∇ × (u × B) + η∇2 B.
(3.1.9)∂t
An alternative form of the constant diffusivity induction equation
for incompressible flow, ∇ · u = 0 is∂B
+ u · ∇B = B · ∇u + η∇2 B.
∂t(3.1) Fundamentals of MHD and Maxwell’s equations(3.1.10)

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Magnetic diffusion
If fluid velocity is zero∂B= η∇2 B.
∂t

(3.1.11)This is diffusion equation, so if field is zero at infinity, it justdiffuses away. Otherwise it comes to a uniform value.
Timescale: B = (B0 sin kx + B1 , 0, 0) in Cartesian coordinates at
t = 0, field evolves as
B = (B0 sin kx exp(−ηk 2 t) + B1 , 0, 0)(3.1.12)so e-folding time is 1/k 2 η. If k = π/L, e-folding time is L2 /π 2 η.
η ≈ 1 m2 /s in the Earth’s core. If L is ten metres, field decays by afactor e in about 10 seconds. L = 3.5 × 106 metres, time is 20,000
years.
(3.1) Fundamentals of MHD and Maxwell’s equations

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Perfect Conductor
The induction equation is∂B= ∇ × (u × B),∂t(3.1.13)in the zero diffusion limit. Frozen flux limit. The flux through aclosed curve C in the fluid, enclosing area A, isZ
B · ds.
A

Alfvén’s theorem says flux through any closed loop C is constantas the loop moves with the fluid.
Note that the size of the loop varies in time. If it shrinks, thismeans the field strength must be increasing, since magnetic flux =area times field strength.
Stretching therefore increases field intensity. Stretching is key todynamo action.
(3.1) Fundamentals of MHD and Maxwell’s equations

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Magnetic Reynolds number
Non-dimensionalise induction equation. We choose typical lengthscale L∗ and a typical fluid velocity U∗ . Introduce scaled ˜variables
t=

L∗
t̃,
U∗

x = L∗ x̃,u = U∗ ũ(3.1.14)˜ ∗ , and (3.1.9) becomesso that ∇ = ∇/L∂B−1 ˜ 2˜= ∇×(ũ×B)+Rm
∇ B,
∂ t̃

Rm =

U∗ L∗
, (3.1.15, 3.1.16)
η

Rm being the dimensionless magnetic Reynolds number.
Large Rm means induction dominates over diffusion. In
astrophysics and geophysics Rm is almost always large, but inlaboratory experiments it is usually small, though values up to∼ 50 can be reached in large liquid sodium facilities.
(3.1) Fundamentals of MHD and Maxwell’s equations

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Flux expulsion
y
0
x
t=0.0

d
t=0.5

t=2.0

An imposed flow that resembles convection rolls
πx
πy
πx
πy
u = (−U sincos, U cos
sin
, 0)
d
d
d
d
is imposed. The induction equation

t=20.0(3.1.17)∂B= ∇ × (u × B) + Rm−1 ∇2 B,(3.1.18)∂twith Rm = 1000 is solved numerically.
The field is initially vertical, and the field at multiples of theturn-over time d/U is shown. An initially uniform field stirred byconvection quickly expels the magnetic field out of the roll.
(3.1) Fundamentals of MHD and Maxwell’s equations

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Stretch Twist Fold dynamo

A loop of flux is first stretched to twice its length, reducing
cross-section by half. By Alfvéns theorem, the field strength must
double. Now twist the loop to get to (b), and then fold to get to(c). Apply small diffusion at X to reconnect. We have doubled thetotal flux.
(3.1) Fundamentals of MHD and Maxwell’s equations

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3.2 Kinematic dynamo problem(3.2) Kinematic dynamo problem

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Kinematic Dynamo problem
Velocity u is a given function of space and possibly time. Dynamic,
or self-consistent, dynamo problem is when u is solved using the
momentum equation. In kinematic dynamos, only the induction
equation (3.1.9) is solved.
Kinematic dynamo problem linear in B. If u independent of time,
B = B0 (x, y , z)e pt ,B0 → 0 as x → ∞.

(3.2.1)

Usually infinite set of eigenmodes B0 each with a complex
eigenvaluep = σ + iω.
(3.2.2)σ is the growth rate, and ω the frequency. If one or more modes
have σ positive, we have a dynamo. If ω = 0, steady dynamo. If
ω 6= 0 growing dynamo waves.
(3.2) Kinematic dynamo problem

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Anti-Dynamo theorem 1.

In Cartesian coordinates (x, y , z) no field independent of z which
vanishes at infinity can be maintained by dynamo action. So its
impossible to generate a 2D dynamo field.
Because ∇ · B = 0, any 2D field can be written
B = Bẑ + ∇ × Aẑ.

(3.2.3)Insert this into the induction equation (3.1.10) and we get two
equations∂A+ (u · ∇)A = η∇2 A,(3.2.4)∂t∂B+ (u · ∇)B = η∇2 B + BH · ∇uz .
(3.2.5)∂t(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Anti-Dynamo theorem 1 continued

Now multiply (3.2.4) by A and integrate over the whole volume,
Z
Z
Z
∂
1 2
1
A dv + ∇ · uA2 dv = −η (∇A)2 dv .
(3.2.6)∂t2
2
Divergence term vanishes, because the fields is small at large
distance.
The term on the right is negative definite, so the integral of A2
continually decays.
It will only stop decaying if A is constant, in which case there is nofield.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Decay of B

Once A has decayed to zero, BH is zero, so there is no source termin (3.2.5). We can then apply the same argument to show B
decays to zero. This shows that no nontrivial field 2D can be
maintained as a steady (or oscillatory) dynamo.
Note that if A has very long wavelength components, it may take avery long time for A to decay to zero, and in that time B mightgrow quite large as a result of the driving by the last term in(3.2.5). But ultimately it must decay.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Anti-Dynamo theorem 2
No dynamo can be maintained by a planar flowu = (ux (x, y , z, t), uy (x, y , z, t), 0)
No restriction is placed on whether the field is 2D or not in thistheorem.
The z-component of (3.1.10) is∂Bz+ u · ∇Bz = η∇2 Bz ,∂t

(3.2.7)

because the B · ∇uz is zero because uz is zero. Multiplying (3.2.7)
by Bz and integrating, again the advection term gives a surface
integral vanishing at infinity, and so Bz decays.
If Bz = 0, then
∂By
∂Bx
+
= 0.
(3.2.8)∂x∂y(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Anti-Dynamo theorem 2 continued

which means
Bx =

∂A
,
∂y

By = −∂A∂x(3.2.9)for some A, and then the z-component of the curl of the induction
equation gives∂∇2H A+ ∇2H (u · ∇A) = η∇2H ∇2 A,
∂t

∇2H =

∂2
∂2
+
. (3.2.10)∂x 2 ∂y 2

Fourier transform in x and y , the ∇2H = kx2 + ky2 which then can becancelled out, so (3.2.10) is just (3.2.4) again, which on
multiplying through by A leads to the decay of A again. So thewhole field decays if the flow is planar.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Axisymmetric field decomposition

An axisymmetric field and flow can be decomposed into
B = B φ̂ + BP = B φ̂ + ∇ × Aφ̂,u = sΩφ̂ + uP = sΩφ̂ + ∇ ×

s = r sin θ.

ψ
φ̂,
s

(3.2.11)

The induction equation now becomes quite simple,1∂A 1+ (uP · ∇)(sA) = η(∇2 − 2 )A,
∂t
s
s
∂B
B
1
+ s(uP · ∇)( ) = η(∇2 − 2 )B + sBP · ∇Ω.
∂tss
Gives important insight into the dynamo process.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems(3.2.12)(3.2.13)

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Anti-Dynamo theorem 3, Cowling’s theorem

An axisymmetric magnetic field vanishing at infinity cannot be
maintained by dynamo action.
Polar coordinate version of theorem 1. Multiplying (3.2.12) by s 2 A
and integrating, and eliminating the divergence terms by convertingthem to surface integrals which vanish at infinity, we get
Z
Z
∂
1 2 2
s A dv = −η |∇(sA)|2 dv(3.2.14)∂t2
which shows that sA decays.
Note that sA = constant would give diverging A at s = 0 which is
not allowed.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Cowling’s theorem continued

Once A has decayed, BP = 0 in (3.2.14), and now multiplying(3.2.14) by B/s 2 givesZZ
1 −2 2∂Bs B dv = −η |∇( )|2 dv(3.2.15)∂t2s
and since we don’t allow B proportional to s, which doesn’t vanish
at infinity, this shows that B must decay also. So there can be no
steady axisymmetric dynamo.
Note this theorem disallows axisymmetric B, not axisymmetric u.
The Ponomarenko dynamo and the Dudley and James dynamos(see section 2 below) are working dynamos with axisymmetric u
but with nonaxisymmetric B.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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Anti-Dynamo theorem 4, Toroidal flow theorem
A purely toroidal flow, that is one with u = ∇ × T r for some scalarfunction T , cannot maintain a dynamo. Note that this means that
there is no radial motion, ur = 0.
This is the polar coordinate version of theorem 2. First we show
the radial component of field decays, because∂(r · B) + u · ∇(r · B) = η∇2 (r · B),∂t(3.2.16)

so multiplying through by (r · B) and integrating does the job.
Then a similar argument to that used to prove theorem 2 showsthat the toroidal field has no source term and so decays. For
details see Gilbert (2003) p 380. It is not necessary to assume
either flow or field is axisymmetric for this theorem.

(3.2) Kinematic dynamo problem: Anti-dynamo theorems

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3.3 Working kinematic dynamos(3.3) Working kinematic dynamos

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Kinematic Dynamos

We look at dynamos with a prescribed velocity field, mostly atime-independent velocity
Ponomarenko dynamo: simplest known kinematic dynamo, modelfor Riga dynamo experiment
G.O.Roberts dynamo: model for the Karlsruhe experiment. Also
has roll geometry, and a field on scale larger than roll size
Spherical dynamos: the Dudley-James dynamo.

(3.3) Working kinematic dynamos

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Ponomarenko Dynamo(a) Ponomarenko flow,u = sΩφ̂ + Uẑ, s < a; u = 0, s > a(3.3.1)

Solid body screw motion inside cylinder s = a.
(b) Riga dynamo experiment configuration(3.3) Working kinematic dynamos: Ponomarenko dynamo

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Ponomarenko Dynamo 2.
Flow has helicity,
H =u·∇×u=u·ζ=U

1 ∂ 2s Ω = 2UΩ.
s ∂s

(3.3.2)

Discontinuity of u at s = a provides strong shearing.
Evades planar motion anti-dynamo theorem through U.
√
a U 2 + a 2 Ω2
Rm =η
based on maximum velocity.
Seek nonaxisymmetric field of form
B = b(s) exp[(s + iω)t + imφ + ikz]

(3.3.3)thus evading Cowling’s theorem. Induction equation, (3.1.10),∂B+ u · ∇B = B · ∇u + η∇2 B.
∂t(3.3) Working kinematic dynamos: Ponomarenko dynamo

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Ponomarenko Dynamo 3. Equations(u · ∇)B = (ikU + imΩ)B − ΩBφ ŝ + ΩBs φ̂,(B · ∇)u = −ΩBφ ŝ + ΩBs φ̂,(3.3.4)(3.3.5)so
p 2 bs = ∆m bs −

2im
bφ ,
s2

p 2 bφ = ∆m bφ +

∆m =

1
m2
1 ∂ ∂
s
− 2− 2s ∂s ∂s
s
s

where

2im
bs , (3.3.6, 3.3.7)
s2

Inside, s < a, p = pi , ηpi2 = σ + iω + imΩ + ikU + ηk 2 ,Outside, s > a, p = pe , ηpe2 = σ + iω + ηk 2 .
Defining b± = bs ± ibφ ,p 2 b± = ∆m±1 b± .
(3.3) Working kinematic dynamos: Ponomarenko dynamo(3.3.8)28/71

Ponomarenko Dynamo 4. Solutions
Solutions areb± = A±

Im±1 (pi s), s < a,Im±1 (pi a)A±

Km±1 (pi s), s > a.
Km±1 (pi a)(3.3.9)Im and Km are the modified Bessel functions (like sinh and cosh)that are zero at s = 0 and zero as s → ∞ respectively.
With this choice, the fields are continuous at s = a. Also need Ezcontinuous (1.4.8d), so η(∇ × B)z − uφ Bs has to be continuous,
using (3.1.5), giving


∂bφ
∂bφ
η
|s→a+ −
|s→a− = aΩbs (a)(3.3.10)∂s∂s
writing the jump as [.],


∂b±
2η
= ±iaΩ(b+ (a) + b− (a)).
∂s(3.3) Working kinematic dynamos: Ponomarenko dynamo(3.3.11)29/71

Ponomarenko Dynamo 5. Solutions
Defining
S± =00pi Im±1(pi a) pe Km±1
(pi a)−
Im±1 (pi a)
Km±1 (pe a)(3.3.12)the dispersion relation is2ηS+ S− = iaΩ(S+ − S− ).

(3.3.13)

Needs a simple MATLAB code to sort it out.
Non-dimensionalise on length scale a and timescale a2 /η anddimensionless parameters are growth-rate s/Ω, frequency ω/Ω,
pitch of spiral χ = U/aΩ, ka and m. The diffusion coefficientη/a2 Ω = (1 + χ2 )1/2 Rm−1 .
For marginal stability set s = 0. For given χ, ka and m adjust Rmand ω until real and imaginary parts of2ηS+ S− − iaΩ(S+ − S− ) = 0. Minimise Rm over m and ka to get
the critical mode, and over χ to get the optimum pitch angle.
(3.3) Working kinematic dynamos: Ponomarenko dynamo

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Ponomarenko Dynamo 6. Results
When all this is done, we find Rmcrit = 17.7221, ka = −0.3875,
m = 1, a2 ω/η = −0.4103 and χ = 1.3141. Poloidal and toroidal
flow similar.
This is a low value of Rm, which motivated the Riga dynamo
experiment. Magnetic field is strongest near s = a where it isgenerated by shear.
At large Rm, there is a significant simplification, because thenmΩ + kU is small, so pe = pi and the ηk 2 terms are small. Bessel
functions have asymptotic simplifications at large argument.
Fastest growing modes given by|m| = (6(1 + χ

−2

−3/4

))

 2 1/2a Ω, s = 6−3/2 Ω(1 + χ−2 )−1/2 .
2η
(3.3.14)(3.3) Working kinematic dynamos: Ponomarenko dynamo

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Ponomarenko Dynamo 7. Results

Magnetic field for the Ponomarenko dynamo at large Rm. Surfaceof constant B shows spiralling field following flow spiral, andlocated near the discontinuity.
(3.3) Working kinematic dynamos: Ponomarenko dynamo

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G.O.Roberts dynamo

The Ponomarenko dynamo has a single roll, and the field at low
Rm is on scale of the roll, smaller at high Rm.
G.O Roberts dynamo has a collection of rolls and field can becoherent across many rolls.
Special case of ABC flows (Arno’ld, Beltrami, Childress)u = (C sin z + B cos y , A sin x + C cos z, B sin y + A cos x) (3.3.15)
with A = B = 1, C = 0. These flows have ∇ × u = u, so vorticity= velocity.
Clearly ABC flows have helicity.

(3.3) Working kinematic dynamos: G.O Roberts dynamo

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G.O.Roberts dynamo 2.

Flow is two dimensional, independent of z, but has a z-component.
u = (cos y , sin x, sin y + cos x).

(3.3.16)

Avoids the planar antidynamo theorem.
(3.3) Working kinematic dynamos: G.O Roberts dynamo

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G.O.Roberts dynamo 3.


u=


∂ψ ∂ψ
,− ,ψ ,∂y∂x

ψ = sin y + cos x.

(3.3.17)

Magnetic field has to be z-dependent (Anti-dynamo theorem 1)
B = b(x, y ) exp(pt + ikz).

(3.3.18)b(x, y ) is periodic in x and y , but it has a mean part independentof x and y which spirals in the z-direction.
To solve the problem, Roberts inserted the form of B into the
induction equation, using a double Fourier series expansion ofb(x, y ).
The coefficients then form a linear matrix eigenvalue problem for p.

(3.3) Working kinematic dynamos: G.O Roberts dynamo

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G.O.Roberts dynamo 4. ResultspkGrowth rate p as a function of z wavenumber, k for various
 = Rm−1 .
Solid lines: G.O. Roberts numerical results. Dashed lines, A.M.
Soward’s asymptotic large Rm theory.

(3.3) Working kinematic dynamos: G.O Roberts dynamo

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Large Rm G.O.Roberts dynamo

Figure rotated through 45◦ . At large Rm, generated field isexpelled into boundary layers.
This gives enhanced diffusion, leading to lower growth rates andulitimately to decay.
p → 0 as Rm → ∞ means dynamo is slow.

(3.3) Working kinematic dynamos: G.O Roberts dynamo

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Spherical dynamo models
Following Bullard and Gellman, 1954, velocity for kinematic
spherical dynamos written
X
m
u=
tm
(3.3.19)l + slimwhere tml and sl are the toroidal and poloidal componentsmmtm
l = ∇ ×r̂tl (r , t)Yl (θ, φ),mmsml = ∇ × ∇ ×r̂sl (r , t)Yl (θ, φ)(3.3.20)

where −l ≤ m ≤ l.
Bullard and Gellman used u = t01 + s22 with t10 (r ) = r 2 (1 − r ),s22 (r ) = r 3 (1 − r )2 . In their original calculations, they found
dynamo action, but subsequent high resolution computations
showed they were not dynamos. Warning: inadequate resolution
can lead to bogus dynamos!
(3.3) Working kinematic dynamos: Dudley and James dynamo

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Dudley and James dynamos

Dudley and James looked at 3 models, using the notation of(3.3.20),
u = t02 +s02 , (a);u = t01 +s02 , (b);u = t01 +s01 , (c) (3.3.21)

with
t10 = s10 = r sin πr ,t20 = s20 = r 2 sin πr .

All steady axisymmetric flows.

(3.3) Working kinematic dynamos: Dudley and James dynamo

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Dudley and James dynamos 2.

t components give azimuthal flow only, s components give
meridional flow.
All three models give dynamo action. Since the flow isaxisymmetric, the field has exp imφ dependence, and m = 1 ispreferred.
(a) with  = 0.14 has Rmcrit ≈ 54 (steady). (b) with  = 0.13 has
Rmcrit ≈ 95 (oscillatory). (c) with  = 0.17 has Rmcrit ≈ 155(oscillatory).
In all cases, the toroidal and meridional flows are of similar
magnitude. Field is basically an equatorial dipole, which inoscillatory cases rotates in time.

(3.3) Working kinematic dynamos: Dudley and James dynamo

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3.4 Field generation in geodynamo models(3.4) Field generation in geodynamo models

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Field generation in numerical dynamos
0.18

80.0

0.12

48.0

0.06
0.00
-0.06

16.0
-16.0

-0.12

-48.0

-0.18

-80.0

E = 5 × 10−5 , Pr = Pm = 1, Ra = 400 with stress-freeboundaries.
Left: radial magnetic field at the CMB. Right: radial velocity atr = ri + 0.8.
This is a standard dipolar solution which persists for all time. Flow
has the columns predicted by linear theory, though they are now
time-dependent.

(3.4) Field generation in geodynamo models

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Flow in the convection columns
Anticyclone Cyclone

Anticyclone Cyclone↑z
Equatorial
plane

φ→
Red is primary flow, which convects the heat out. Blue issecondary flow which provides helicity u · ζ, where ζ is thevorticity. Helicity is important for dynamo action.
(3.4) Field generation in geodynamo models

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Bφ from dynamo modelAzimuthal average of Bφ for amoderately supercritical dynamo
Antisymmetric about theequator, so Bφ = 0 on theequator. Note Bz > 0 neartangent cylinder, usually gives
Bφ > 0 in N. hemisphere.
Field generally increases with snear ICB.

(3.4) Field generation in geodynamo models

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Mechanism for generating Bz from BφStart with Bφ > 0 in northern hemisphere where uz > 0 in
anticyclones, A, uz < 0 in cyclones, C .
>

↑
z

↓

B >0
z

>

↑

>
↓

B <0
z

B >0
z

>

↑

φ →
↑
ICB

Bz > 0

∨

s
↓
CMB
↓

C

∧

∧

Bz > 0
A

∨

∨

C

∧

Bz < 0
φ →

∧

A

∨

Green magnetic field line initiallyin φ-direction is displaced parallel
to rotation axis z by secondary
flow.
Primary flow sweeps positive Bz
towards the ICB, negative Bz
towards the CMB.

Net effect is to create positive Bz inside, negative Bz outside, justas in a dipolar field. In S. hemisphere Bφ and uz are reversed, so
effect is same.

(3.4) Field generation in geodynamo models

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Mechanism for generating Bφ from Bz
∨

∧

∧ ∧

∧

∨ ∨

∧

∧
ICB
Bφ<0

Bφ<0
↑
z

>
∧
>

Bφ>0
A

Bφ>0

∧

Bφ<0

Bφ<0

<
<

>
∧
>

C

Bφ<0

s
↓

∧

A

∨

∨

C

∧

Bφ>0
Bφ>0
φ →

∧

∧

∨ ∨

∧

∧ ∧

∧

∨

φ →

Northern Hemisphere. uz > 0 in anticyclones, A, uz < 0 in
cyclones, C .
Left: uφ flow resulting from uz profile stretches out red positive Bz
to give Bφ .
Right: constant z section in Northern Hemisphere viewed fromabove.
Positive Bφ is moved outward in radius, negative Bφ moved inward
in radius by the vortex circulations. This reinforces the original Bφconfiguration and allows the magnetic field to grow.
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3.5 Fast and slow dynamos(3.5) Fast and slow dynamos

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Fast and Slow dynamos
If magnetic diffusion time  turn-over time, (3.1.10) becomes∂B+ u · ∇B = B · ∇u + ∇2 B.
∂t(3.5.1)

where  = Rm−1 is small. Time scaled on turnover time L/U.
For steady flow B ∼ e σt , and if γ = Re(σ) flow is a fast dynamo if
γ0 = lim γ() > 0.
→0(3.5.2)

Flow is a slow dynamo if
γ0 = lim γ() ≤ 0.
→0

(3.5.3)

Fast dynamos grow on the turnover time (months in the Sun) not
the magnetic diffusion time (millions of years in the Sun).
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ABC flow

Typical Poincaré section shows chaotic regions and orderedregions. Ordered regions called KAM regions, or KAM tori.
‘Normal’ in chaotic ODEs. The ABC = 1 flow is unusual in having
rather large KAM regions and small chaotic regions.
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ABC dynamo results

Dynamo growth rate γ against log10 Rm.
Because the chaotic regions which give stretching are small, quitedifficult to show its a fast dynamo numerically.
However, γ seems to be increasing gradually rather thandecreasing at large Rm.

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Stretching properties

Take a point a and a small vector v origin at a. Now integrate theparticle path ODE’s with initial conditions x1 = a and x2 = a + v,
and monitor d = |x1 − x2 |.
If stretching is occuring, d will grow exponentially.
Liapunov exponent isΛ(a) = max lim sup
v

t→∞

ln d
t

(3.5.4)The maximum Liapunov exponent is found by taking thesupremum over all a.
Can be computed, but expensive! A practical definition of chaos isthat the Liapunov exponent is positive. In a given chaotic region,
Λ is usually the same, but it is zero in KAM regions.

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Time dependent flow fields

Rather than have a fully 3D steady flow field, like A = B = C = 1,
we can choose a 2D flow but make it a time-dependent flow. Field
still has exp ikz dependence, which makes numerics a lot easier.
Galloway-Proctor CP flow:u = ∇×(ψ(x, y , t)ẑ)+γψ(x, y , t)ẑ,ψ = sin(y +sin t)+cos(x+cos t)(3.5.5)
Very like the G.O. Roberts flow, except the stagnation point
pattern rotates round in a circle.
Also an LP flow, with ψ = sin(y + cos t) + cos(x + cos t).
Flow is now non-integrable, and has positive Liapunov exponents.

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Galloway-Proctor dynamo

CP flow
Left: Liapunov exponents: blue regions have little or no stretching,green/red has order one stretching
Right: snapshot of the normal field Bz . Note the good correlation
between strong field and strong stretching.

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Galloway-Proctor dynamo results

CP flow.
Growth rate against wavenumber k, exp ikz, for Rm = 800,
Rm = 2, 000 and Rm = 10, 000.
Note that the growth rate continues to increase as Rm is
increased, unlike the G.O. Roberts flow results.

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3.6 Mean field dynamo theory(3.6) Mean field dynamo theory

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About Mean Field Dynamo Theory

Subject divides into two areas(i) Underlying theory of MFDT, conditions for its validity, its
relationship to turbulence theory and its extension to include
nonlinear effects(ii) The solutions of MFDT equations and the new types ofdynamos they create: dynamo waves, αω dynamos and α2
dynamos.
There is surprisingly little interaction between these two activities.
Vastly more papers have been written on (ii), almost all accepting
the MFDT equations as a useful model.

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Mean and Fluctuating parts. 1.
Basic idea is to split the magnetic field and the flow into mean andfluctuating parts,
B = B + B0 ,u = u + u0

(3.6.1)

Reynolds averaging rules: assume a linear averaging process
B1 + B2 = B1 + B2 ,u1 + u2 = u1 + u2

(3.6.2)and once its averaged it stays averaged, so
B = B,

u = u,(3.6.3)

So averaging
B0 = u0 = 0.

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Mean and Fluctuating parts. 2.
Also, assume averaging commutes with differentiating, so
∂B
∂
=
B,
∂t
∂t

∇B = ∇B.

(3.6.5)Now we average the induction equation∂B= ∇ × (u × B) + η∇2 B,∂t
Using the Reynolds averaging rules∂B= ∇ × (u × B) + η∇2 B.
∂t(3.6.6)The interesting term is (u × B).
u × B = (u + u0 ) × (B + B0 ) = u × B + u × B0 + u0 × B + u0 × B0
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Mean Field Induction equation
So we have∂B= ∇ × (u × B) + ∇ × E + η∇2 B,∂t

E = u0 × B0 . (3.6.7)

E is called the mean e.m.f. and it is a new term in the induction
equation. We usually think of the primed quantities as being small
scale turbulent fluctuations, and this new term comes aboutbecause the average mean e.m.f. can be nonzero if the turbulence
has suitable averaged properties.
No longer does Cowling’s theorem apply! With this new term, we
can have simple axisymmetric dynamos. Not surprisingly, mostauthors have included this term in their dynamo work, though
actually it can be hard to justify the new term in astrophysicalapplications.
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Evaluation of (u × B)

If we subtract the mean field equation from the full equation,∂B0= ∇×(u×B0 )+∇×(u0 ×B)+∇×G+η∇2 B0 , G = u0 ×B0 −u0 × B0 .
∂t
(3.6.8)00This is a linear equation for B , with a forcing term ∇ × (u × B).
B0 can therefore be thought of as the turbulent field generated by
the turbulent u0 acting on the mean B. We can therefore plausibly
write
∂Bj
Ei = aij Bj + bijk
+ ··· .
(3.6.9)∂xk
where the tensors aij and bijk depend on u0 and u.

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Mean Field Dynamo equations
We don’t know u0 and its unobservable, so we assume aij and bijk
are simple isotropic tensors
aij = α(x)δij ,bijk = −β(x)ijk .

(3.6.10)

We now have the mean field dynamo theory (MFDT) equations inusual form,∂B= ∇ × (u × B) + ∇ × αB − ∇ × (β∇ × B) + η∇2 B. (3.6.11)∂t
If β is constant, ∇ × (β∇ × B) = −β∇2 B so the β term acts likean enhanced diffusivity. Even if it isn’t constant, we recall from
(3.1.8) that the term has the same form as the molecular diffusion
term.
We can now justify taking a large diffusion, choosing it to give
agreement with observation.
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Parker loop mechanism

Mean field theory predicts an e.m.f. parallel to the mean magnetic
field.
∂B= ∇ × (u × B) + ∇ × αB + ηT ∇2 B.
∂t(3.6.12).

Contrast with u × B perpendicular to the e.m.f. With constant α,
the α-effect predicts growth of field parallel to current µ∇ × B.
Recalling that the α-effect depends on helicity, we can picture thisprocess.

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Parker loop mechanism 2.

A rising twisting element of fluid brings up magnetic field. A loopof flux is created, which then twists due to helicity. The loop
current is parallel to the original mean field. Poloidal field has beencreated out of azimuthal field.
Note that if too much twist, current in opposite direction. First
order smoothing assumes small twist.
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Joy’s law

A sunspot pair is created when an azimuthal loop rises throughsolar photosphere. The vertical field impedes convection producing
the spot.
Joy’s law says that sunspot pairs are systematically tilted, with theleading spot being nearer the equator. Assuming flux was createdas azimuthal flux deep down, suggests that loop has indeed twisted
through a few degrees as it rose.
Provides some evidence of the α-effect at work. Basis of manysolar dynamo models.

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3.7 Mean field α-effect dynamos(3.6) Mean field α-effect dynamos

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Axisymmetric mean field dynamos

Mean field spherical dynamo equations with isotropic α are∂A 1
1+ (uP · ∇)(sA) = αB + η(∇2 − 2 )A,∂tss(3.7.1)B
1∂B+s(uP ·∇)( ) = ∇×αBP +η(∇2 − 2 )B +sBP ·∇Ω. (3.7.2)∂tssThe α-effect term is the source for generating poloidal field from
azimuthal field, as envisaged by Parker, and Babcock & Leighton.
Two ways of generating azimuthal field from poloidal field: the
α-effect or the ω effect. If first dominates, its an α2 -dynamo. If
the second its an αω dynamo.
There are also α2 ω dynamos where both mechanisms operate.

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The Omega-effectAn initial loop of meridional field threads through the sphere.
Inside of sphere is rotating faster than outside: differential rotation.
The induction term sBP · ∇Ω generates azimuthal field bystretching. Opposite sign Bφ across equator as in Sun.

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Dynamo waves 1.

Cartesian geometry, independent of y .
B = (−∂A/∂z, B, ∂A/∂x),u = (−∂ψ/∂z, uy , ∂ψ/∂x),(3.7.3)∂A ∂(ψ, A)+= αB + η∇2 A,∂t∂(x, z)∂(A, uy )∂B∂(ψ, B)+=− ∇ · (α∇A) + η∇2 B.
∂t∂(x, z)∂(x, z)(3.7.4)(3.7.5)Set ψ = 0, α constant, uy = U 0 z, a constant shear, ignore α termin B equation (αω model) and set A = exp(σt + ik · x). Dispersion
relation is(σ + ηk 2 )2 = ikx αU 0 .
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Dynamo waves 2.
Giving1+i
σ = √ (αU 0 kx )1/2 − ηk 2
2

(3.7.7)

with suitable choice of signs. This gives growing dynamo waves if
αU 0 term overcomes diffusion.
If the wave is confined to a plane layer, kz = π/d gives the lowest
critical mode, and there a critical value of kx for dynamo action.
The dimensionless combination D = αU 0 d 3 /η 2 is called the
dynamo number,and in confined geometry there is a critical D for
onset.
Note fastest growing waves have kz = 0 so propagate
perpendicular to shear direction z. If αU 0 > 0, +ve kx give growing
modes with Im(σ) > 0, so propagate in -ve x-direction. -ve kx has
growing modes with Im(σ) < 0, so waves always propagate in -ve
x-direction. Direction waves travel in depends on sign of αU 0 .
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α2 dynamos

Now set ψ = uy = 0, α constant, A = exp(σt + ik · x) to getdispersion relation(σ + ηk 2 )2 = α2 k 2
σ = ±αk − ηk 2

(3.7.8)which has growing modes with zero frequency. No dynamo waves,
but a steady dynamo results.
In bounded geometry there is a critical α for dynamo action.

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Spherical αω dynamos

Dipolar oscillatory solution of axisymmetric αω-dynamo in a
sphere. (a)-(h) goes through one period. Right meridional field,
left azimuthal field. α = f (r ) cos θ, ω = ω(r ). B antisymmetric
about equator, A symmetric.
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