zotero-db/storage/ADRFMK7F/.zotero-ft-cache

Dynamo theory and GFD
Steve Childress
16 June 2008
1 Origins of the dynamo theory
Dynamo theory studies a conducting ﬂuid moving in a magnetic ﬁeld; the motion of the body through the ﬁeld acts to generate new magnetic ﬁeld, and the system is called a dynamo if the magnetic ﬁeld so produced is self-sustaining.
1.1 Early ideas point the way
In the distant past, there was the idea that the earth was a permanent magnet. In the 1830s Gauss analyzed the structure of the Earth’s magnetic ﬁeld using potential theory, decomposing the ﬁeld into harmonics. The strength of the dominant ﬁeld was later found to change with time.
In 1919 Sir Joseph Larmor drew on the induction of currents in a moving conductor, to suggest that sunspots are maintained by magnetic dynamo action. P. M. Blackett proposed that magnetic ﬁelds should be produced by the rotation of ﬂuid bodies.
The ‘current consensus’ is that the Earth’s magnetic ﬁeld is the result of a regenerating dynamo action in the ﬂuid core. The mechanism of generation of the ﬁeld is closely linked dynamically with the rotation of the Earth. Similar ideas are believed to apply to the solar magnetic ﬁeld, to other planetary ﬁelds, and perhaps to the magnetic ﬁeld permeating the cosmos.
1.2 Properties of the Earth and its Magnetic Field
The magnetic ﬁeld observed at the Earth’s surface changes polarity irregularly. The nondipole components of the surface ﬁeld also vary with time over many time scales greater than decades, and have a persistent drift to the west. The ﬂuid core of the Earth is a spherical annulus, bounded by the solid inner core and the mantle. It is believed that the motion of the inner and outer core are suﬃcient to drive the geodynamo.
2

2 The homogeneous kinematic dynamo

The pre-Maxwell equations for a homogeneous moving conductor are

∇×B = µJ

∇

×

E

=

−

∂B ∂t

J = σ (E + u × B)

(1)

∇·B=0

∇ · E = q/

Combined, these equations simplify to

∂B ∂t

−

∇

×

(u

×

B)

−

η∇2B

=

0

(2)

where η = (µσ)−1 is the magnetic diﬀusivity. This equation is called the magnetic induction equation.

2.1 Kinematic Dynamo Model of the Earth’s Core
Neglecting the inner core, the conducting ﬂuid is contained in a sphere of radius r = rc. The exterior is regarded as free space. The equations are to be solved with a prescribed divergence-free (the core ﬂuid is assumed incompressible) velocity ﬁeld which is independent of time. On r = rc, the magnetic ﬁeld is continuous owing to the absence of magnetic monopoles and a concentrated surface current layer. The magnetic ﬁeld on r > rc also matches with an external vacuum magnetic ﬁeld

Be = ∇φe.

(3)

The tangential component of

E = η∇ × B − u × B

(4)

is also continuous at r = rc. We take the external ﬁeld to decay like a potential dipole at r = ∞, i.e., B ≈ O(r−3).
We pass to a dimensionless form using a characteristic length scale L, velocity scale U and time scale L/U . Using the vector identity

∇ × (u × B) = B · ∇u + u ∇ · B − u · ∇B − B ∇ · u,

(5)

and since the velocity and magnetic ﬁelds are divergence free (∇ · u = ∇ · B = 0), the induction equation can be written in the form

DB Dt

−

1 R

∇2B

=

B

·

∇u

(6)

in terms of the material derivative

DB Dt

=

∂B ∂t

+

u

·

∇B.

(7)

3

Here R is the magnetic Reynolds number, R = U L/η = U Lσ µ. For R << 1, the magnetic ﬁeld diﬀuses easily through the conductor. For R >> 1, the magnetic ﬁeld is ‘frozen into the moving conductor’. In this case, distortion and stretching of the ﬁeld lines are caused by the term B·∇u. Notice that eq. (6) corresponds to the vorticity equation upon substitution of ∇ × u in place of B.

2.2 Kinematic Dynamo as an Eigenvalue Problem

For a given velocity ﬁeld u, eq. (6), is linear for the magnetic ﬁeld B, and we may separate

variables

B = eλt b(x), r < 1

B = eλt ∇φ, r > 1

(8)

Upon change of variable (8), the induction equation becomes

Lb

=

1 R

∇2b

−

u

·

∇b

+

b

·

∇u

=

λb

(9)

Here, length dimension L is L = rc. The values of λ allowing acceptable solutions b are the eigenvalues, and depend on R. We say u(x) is a (steady) kinematic dynamo if for some R > 0 there exists an eigenvalue λ(R) such that the real part of λ is greater than zero. Elsasser, Bullard, and others developed a theory of the spherical dynamo in this setting.
Note that although the kinematic equation is linear in B, the dynamo problem has a nonlinear character, since the ‘correct’ velocity ﬁeld u is not known a priori. Mathematically, we are trying to ﬁnd the right function u(x) determining L, so that the dynamo property is realized. More generally, we might consider a time periodic x and an analogous Floquet problem.

2.3 Expansion in eigenfunctions

Consider the space of complex-valued divergence-free vector ﬁelds b(x) in r < 1, matching continuously with a potential ﬁeld in r > 1. The eigenfunctions bn satisfy

L bn = λn bn, r < 1

(10)

bn = ∇φn, r > 1

and are continuous on r = 1. The eigenvalues are known to be discrete and countable, and the eigenfunctions are complete in the above space. If the eigenfunctions were known to be mutually orthogonal, in the inner product

(bm, bn) =

b∗m bn dV

(11)

R3

then, we would have
∞

|B| =

e2 (λn)||bn||2

R3

n=1

(12)

Thus, if (λn) = 0 for all n the magnetic energy would be bounded by its initial value. This is the case for a normal operator on a Hilbert space, one which commutes with its adjoint.

4

However, the operator L is not normal so the energy is not a simple sum of decaying non-negative terms. In fact, substantial transient growth of ﬁeld energy is possible even if u is not a dynamo. This complicates numerical proofs of the dynamo property. The dynamo problem is one of the best examples of non-normality, and the eﬀects become especially pronounced in the limit of large magnetic Reynolds number R. In the case of the Earth, R is of the order 103.

2.4 Free-decay modes for a rigid spherical core

In can be shown that in R3, the following decomposition of a divergence-free vector ﬁeld A

is possible

A = ∇ × (T r) + ∇ × ∇ × (P r)

(13)

where T = T (r) and P = P (r) are scalar functions called the toroidal and poloidal com-

ponents of A, respectively (ﬁgure 1). Consider this decomposition for u = 0, so that we

have

1 R

∇2bn

=

λn

bn

(14)

The eigenfunctions bn are decomposed as follows:

bn = ∇ × (Tn r) + ∇ × ∇ × (Pn r)

(15)

In the case of an axis-symmetric problem, one can express the eigenfunctions in terms of their components in spherical coordinates as follows:

bn

=

1 r

L2P

er

+

1 r

∂ ∂θ

∂ ∂r

rP

eθ

+

∂ ∂θ

T

eφ

(16)

where L2 is the surface Laplacian. If bn is expanded in spherical harmonics

∞l

bn =

bml f (r)Ylm(θ) eimφ

l=1 k=−l

(17)

the surface operator has the property

L2 bn = −l(l + 1) bn

(18)

The exterior of the dynamo is current free, and with the pre-Maxwell equation ∇ × B = µJ

one obtains ∇ × B = 0. In terms of eigenfunctions, the curl of the magnetic ﬁeld can be

written

∇ × bn = ∇ × (−∇2(Pn r) + ∇∇ · (Pn r)) + ∇ × ∇ × (Tn r) = ∇ × (−∇2(Pn r)) + ∇ × ∇ × (Tn r)

(19)

where we have used the identity ∇ × ∇ × A = ∇∇ · A − ∇2A. We reckon that the toroidal
component of the magnetic ﬁeld is the poloidal component of the curl of the ﬁeld, and similarly −∇2Pn becomes the toroidal component of the curl of the ﬁeld:

(∇ × bn)T = −∇2Pn

(20)

(∇ × bn)P = Tn

5

Figure 1: Toroidal and poloidal ﬁeld lines for a sphere rotating about the z axis.

Using eq. (16) and considering the poloidal and toroidal components of the curl of the ﬁeld,

we obtain

1 r

L2Tn

=

−

1 r

l(l

+

1)Tn

=

0

(21)

so the toroidal ﬁeld is zero at the exterior (Tn = 0, r > 1). Looking at the φ-component of

the ﬁeld rotational, we have

∂ ∂θ

(∇2Pn)

=

0

(22)

Using the matching condition on the core boundary, one obtains ∇2Pn = 0, r > 1.

It then follows that we may consider separately problems for Tn and Pn

∇2Tn = λnTn ∇2Pn = λnPn

r<1

(23)

Tn = 0 ∇2Pn = 0

r≥1

Here Tn, Pn, ∂Pn/∂r are continuous on r = 1, and Pn ∼ O(r−2) as r → ∞. The ﬁrst few solutions are

T1

=

sin πr r

,

λ1 = −π2,

(24)

where r = |x|, (this mode is not believed to be relevant to the Earth’s toroidal ﬁeld except perhaps during reversals);

P1 =

π

cos r

πr

−

sin πr r2

cos θ,

r < 1,

P1

=

−

π

cos r2

θ,

r > 1,

(25)

(this is the basic dipole component of the Earth’s ﬁeld with axis aligned with the rotation

axis);

T2 =

µ cos µr r

−

sin µr r2

cos θ,

λ2 = −µ2 ≈ −20.2,

(26)

6

where µ is the ﬁrst zero of the spherical Bessel function j1, given by tan µ = µ (this is a plausible dominant toroidal component of the Earth’s ﬁeld).

2.5 The Omega Eﬀect
An example of transient growth of magnetic energy in a non-dynamo is the so called Omega Eﬀect, in which diﬀerential rotation causes deformation of poloidal ﬁeld lines, which induce a toroidal ﬁeld (ﬁgure 2). Let u = U (r)eφ, where r = x2 + y2 in cylindrical polars, and represent the magnetic ﬁeld, taken as symmetric with respect to the z-axis, in the alternative poloidal-toroidal decomposition

B = B(r, z, t)eφ + ∇ × A(r, z, t)eφ.

(27)

Now assume A = rP determines the z-aligned dipole ﬁeld and is independent of time. Taking the eφ components of the induction equation, B is then found to satisfy

∂B ∂t

+

r

∂A ∂z

∂ ∂r

U (r) r

−

1 R

∇2

−

1 r2

B = 0.

(28)

The second term here represents the omega eﬀect - depending on U (r) it can act as a source term for the toroidal component B, and shows how the poloidal ﬁeld is coupled to the toroidal ﬁeld and can cause the latter to grow. This however is not enough to create a dynamo, as we shall see below. Note a velocity U (r)eφ is easily realized in a rotating sphere of ﬂuid as a geostrophic ﬂow.

3 Establishing the possibility of a homogeneous dynamo

3.1 A negative result: Cowling’s theorem
Cowling (1934) argued that no homogeneous dynamo can exist for axisymmetric u and B. This is a result of the failure of the poloidal ﬁeld to be maintained against ohmic dissipation. Taking

u = U (r, z, t)eφ + ∇ × ψ(r, z, t)eφ, B = B(r, z, t)eφ + ∇ × A(r, z, t)eφ, (29)

we ﬁnd

∂A ∂t

−

∂ψ ∂z

1 r

∂ ∂r

(rA)

+

∂A ∂z

1 r

∂ ∂r

(rψ)

−

1 R

∇2

−

1 r2

A = 0,

(30)

∂B ∂t

+

r

∂A ∂z

∂ ∂r

U r

−

1 r

∂ ∂r

(rA)

∂U ∂z

−

r

∂ψ ∂z

∂ ∂

B r

+

1 r

∂ ∂r

(rψ)

∂B ∂z

−

1 R

∇2

−

1 r2

B = 0.

(31)

It is immediately clear from the equation for A that there is no coupling from the toroidal

ﬁeld to the poloidal ﬁeld, and that the poloidal ﬁeld will inevitably decay. Multiplying this

equation for A by r2A and integrating by parts, one ﬁnds

d dt

r<1

1 2

(rA)2

dV

=

−

1 R

|∇(rA)|2 dV ≤ 0,
R2

(32)

7

Figure 2: The Ω eﬀect: diﬀerential rotation causes the twisting of poloidal ﬁeld lines which generates toroidal ﬁeld.

and this implies that the poloidal ﬁeld must necessarily decay over time. There are many such anti-dynamo theorems. The upshot is that although the velocity
ﬁeld may be simple, the magnetic ﬁeld must be fully three-dimensional. It is thus signiﬁcant that the observed planetry and stellar magnetic ﬁelds exhibit non-axisymmetric components.

3.2 A necessary condition for dynamo action

Note that the existence or non-existence of dynamo action will depend upon the value of R = U Lσµ. The magnetic energy is

Em =

V

1 2µ

|B|2

dV,

(33)

and its rate of change is

µ

dEm dt

=

V

B

·

∂B ∂t

dV

= − B · ∇ × E dV

V

(34)

= − E · ∇ × B dV

V

= u · (B × ∇ × B) − 1 |∇ × B|2 dV.

V

σµ

8

For a dynamo to exist we need this rate of change of magnetic energy to be non-negative. Now if Um = maxV |u|, the we can write

u · (B × ∇ × B) dV ≤ Um |B||∇ × B| dV

V

V 1/2

1/2

(35)

≤ Um |B|2 dV

|∇ × B|2 dV ,

V

V

using the Cauchy-Schwarz inequality. It is known that for a bounded homogeneous conductor V surrounded by vacuum there is a length LV such that, for any B in the space of realizable magnetic ﬁelds,

V

|∇ × B|2 dV

≥

1 L2V

|B|2 dV.
R3

(36)

Thus we have

u · (B × ∇ × B) dV ≤ UmLV |∇ × B|2 dV,

(37)

V

V

so

µ

dEm dt

=

UmLV

−

1 σµ

|∇ × B|2 dV.
V

(38)

This is negative if UmLV σµ < 1 in which case a dynamo cannot exist. Thus a necessary condition for dynamo action is that UmLV σµ ≥ 1. In particular dynamo action occurs as a bifurcation from the state of no magnetic ﬁeld, as the magnetic Reynolds number is

increased.

3.3 The Bullard-Gellman computation
There was an early attempt to solve the eigenvalue problem utilizing direct expansion in the models of free decay of a spherical conductor. A steady velocity ﬁeld was chosen as representative of rotating convection. The truncated system of modal equations produced a positive result, but in 1969 Gibbons and Roberts re-examined this choice of u and established that it was not in fact a kinematic dynamo. The original result was a numerical artifact.

3.4 Parker’s model: The α eﬀect
In 1955, Parker addressed the problem of producing poloidal ﬁeld from toroidal ﬁeld, to complement the omega eﬀect. As we have seen, this is impossible if both the velocity ﬁeld and the magnetic ﬁeld are axisymmetric, but Parker envisaged averaging of the eﬀects of small up-wellings within the core which could lift and twist poloidal ﬁeld lines. The local magnetic Reynold’s number was take to be large, so that the magnetic ﬁeld was ‘frozen’ into the ﬂuid as it was lifted and twisted (ﬁgure 3). Under averaging, the twisted loops produced a mean current, aligned with the local toroidal ﬁeld line. In this way the poloidal ﬁeld can be induced from the toroidal ﬁeld, thus completing the dynamo cycle and maintaining it against dissipation. This mechanism is now know as the ‘α eﬀect’. Although Parker’s supporting

9

Figure 3: Toroidal ﬁeld lines are lifted and twisted by small scale non-axisymmetric components of the velocity ﬁeld. This generates a mean current in the toroidal ﬁeld direction which induces a poloidal ﬁeld, and is termed the α eﬀect.

calculations left many gaps, these have since been ﬁlled in and the model provides a crucial element of the dynamo cycle, breaking the constraints of Cowling’s theorem.
If u = U (s)eφ, the induction equation in this model is written as

B = B(r, z, t)eφ + ∇ × A(r, z, t)eφ,

(39)

∂A ∂t

−

1 R

∇2

−

1 r2

A = αB,

(40)

∂B ∂t

+

r

∂A ∂z

∂ ∂r

U r

−

1 R

∇2

−

1 r2

B = 0.

(41)

The added αB term here is crucial and completes the ‘α − ω dynamo cycle’.

3.5 Early examples of dynamo action
The Herzenberg dynamo (1958), comprises two solid rotating spheres of radius a embedded inside a larger sphere of radius R, with a R and a also small compared to the distance between the two spheres. Each sphere produces an ω eﬀect determined by the local ﬁeld at that sphere. The resulting induced ﬁeld increases the local ﬁeld at the other sphere, and regenerative dynamo action can result. Herzenberg’s theoretical analysis of this system relies on the smallness of the embedded spheres and the geometrical decay of the induced ﬁeld components, leading to a spatial ﬁltering. Such a dynamo was realized experimentally in solid iron by Lowes and Wilkinson (1963).
Backus (1958) gave a proof of dynamo action for a kinematic dynamo in a homogeneous spherical ﬂuid conductor based upon temporal ﬁltering of magnetic decay modes. This involves applying a velocity ﬁeld u1(x) for time 0 < t < T1, and then setting u = 0 for time T1 < t < T2, during which time the magnetic ﬁeld decays, ﬁltering out modes with the faster decay. A new velocity ﬁeld u2(x) is then applied for T2 < t < T3 and then u = 0 again. This sequence of applying velocity ﬁelds and then setting u = 0 to ﬁlter out fast decaying modes is then continued.
If the ‘basic mode’ (involving the slower decaying poloidal and toroidal components) is D with ||D|| = 1, an initial magnetic ﬁeld involving this basic mode with amplitude A, plus

10

some arbitrary magnetic noise N(0), with ||N(0)|| ≤ for some > 0, is

B(x, 0) = AD(x) + N(0).

(42)

If it can then be shown that at time t = T2N the magnetic ﬁeld has the form

B(x, T2N ) = AN D(x) + N(T2N ),

(43)

where |AN | ≥ |A| and ||N(T2N )|| ≤ , then dynamo action occurs. Thus the noise gets no bigger, and the basic ﬁeld is at least maintained.

3.6 Smoothing Applied to the Kinematic Equation

The smoothing method consists in splitting the dynamo kinematic equation operator

LB

=

∂ ∂tB

−∇

×

(u

×

B)

−

η∇2B

=

0,

(44)

and the velocity and magnetic ﬁeld into some rough and smooth components:

u = uS + uR,

B = BR + BS,

L = LR + LS,

(45)

LR B = −∇ × (uR × B),

LS

B

=

∂ ∂tB

−

∇

×

(uS

×

B)

−

η∇2B.

Let P be a projection onto smooth ﬁelds (P 2 = P ), such as

P B = BS

(46)

and assume P LS = LS P . Subtracting P L from L, one gets

−(LR − P LR)B = (LS − P LS)B = LSB − LSBS = LSBR

(47)

The smooth projection of (LR − P LR)B is

P (LR − P LR)B = P LR − P 2LRB = P LR − P LRB = 0

(48)

so (LR − P LR)B is a rough ﬁeld. Assuming that LS can be inverted on rough ﬁelds, we

have

BR = −L−S 1(LR − P LR)B ≡ M B

(49)

So, one has

B − BS = BR = M B

(50)

or, if I − M is invertible on smooth ﬁelds, we obtain B in terms of its smooth component,

B = (I − M )−1BS

(51)

11

Thus, one can write

P L B = P (LS + LR)B = LSBS + P LR(I − M )−1BS = [LS + P LR(I − M )−1]BS = 0

(52)

and it can be shown by substitution that if BS satisﬁes

[LS + P LR(I − M )−1]BS = 0

(53)

then B = (I − M )−1 BS solves LB = 0. The smoothing method can be used to treat analytically the creation of the α-eﬀect.

3.6.1 First-Order Smoothing

Assuming that LRBS is a rough ﬁeld such as P LRBS = 0, one can make the following approximations

0 = [LS + P LR(I − M )−1]BS ≈ [LS + P LR(I + M )]BS = [LS + P LR M ]BS. (54)

Here, we have

LS

BS

=

∂ ∂t BS

−

∇

×

(uS

×

BS )

−

1 R

∇2

BS

(55)

and

P LR M BS = −P [LRL−S 1LR] BS ≈ P LR BR

(56)

with BR a linear function of BS. This is the basis of mean ﬁeld electrodynamics developed by Krause, Radler and others in the 60’s.
For example, set uS = 0 and let uR be a real steady solenoidal ﬁeld of the form

uR =

ak eik.x

k∈K

(57)

where K is all 3-vectors excluding the zero vector. If BS is large scale relative to uR, then

∇

×

(uR

×

BS

)

=

−

1 R

∇2BR

(58)

implies

BR = R i

k−2 k · BS akeik·x,

(59)

k∈K

and

P [uR × BR] ≈ R i

k−2(a∗ × ak)BS · k ≡ α · BS.

(60)

k∈K

It can be shown, by using the substitution ak = k × bk, that α is real and symmetric. For isotropic uR, of this form we may assume αij = αδij where

α

=

1 3

R

i

k−2(a∗k × ak) · k.

(61)

k∈R

12

If all k have unit length we see that

α

=

R 3 uR

·

(∇

×

uR),

(62)

involving a spatial average of the helicity density uR · (∇ × uR) of the velocity ﬁeld. A ﬁeld satisfying all the above conditions is the Beltrami ﬁeld (in which u is parallel to ∇ × u)

uR = (sin z + cos y, sin x + cos z, sin y + cos x),

(63)

with α = R/3. The role of helicity is reminiscent of Parker’s ’upwellings’, which involved a rising, twisting ﬂow.

3.6.2 Spatially periodic kinetic dynamos

This is a setting where complete analysis of the smoothing method may be carried out

explicitly, with or without time dependence. Considering here only steady ﬁelds, the real

velocity ﬁeld has the form

u = uR =

akeik·x

(64)

k∈K

The admissible magnetic ﬁelds have the complex form

B = ein·x

Bkeik·x, |n| < 1

k∈K +(0,0,0)

(65)

so B0 = BS (the components for k = 0 are all rough). Kinematic dynamo action can be deﬁned by growth of ﬁeld if n is suﬃciently small. This is the palindromic ‘so many dynamos’ theorem (G. O. Roberts): almost all such u are kinematic dynamos.

3.7 Braginskii’s theory for nearly axisymmetric dynamos (1964)

Braginskii used a smoothing technique for nearly axisymmetric ﬂows to derive the α eﬀect. Here, the ’smooth’ parts of the variables are the axisymmetric components, and the rough parts are the non-axisymmetric components. The smoothing operation is therefore an angle average over φ in cylindrical polar coordinates. Motivated by the fact that the magnetic Reynolds number R is large for the Earth and that the ﬁeld is dominated by a dipole aligned with the rotation axis, Braginskii considers expanding the ﬁelds in powers of = R−1/2;

u = U (r, z)eφ + u + 2uP ,

(66)

B = B(r, z, t)eφ + B + 2BP ,

(67)

where < u >=< B >= 0 where < · > means an angle average. The relevant timescale is shorter, so writing τ = 2t, the order terms in the induction equation give

∂B ∂τ

+

ruP

·

∇

B r

= rBP · ∇

U r

+ (∇ × E)φ +

∇2

−

1 r2

B = 0,

(68)

13

where E =< u × B >. If BP = ∇ × A(r, z, t)eφ, we also ﬁnd

∂A ∂τ

+

ruP

·

∇

A r

= Eφ +

∇2

−

1 r2

A.

(69)

Braginskii found that the contributions from E can be mostly absorbed into ‘eﬀective’ variables, denoted with a superscript e:

∂B ∂τ

+

rueP

·

∇

B r

= rBP · ∇

U r

+

∇2

−

1 r2

B = 0,

(70)

∂Ae ∂τ

+

rueP

·

∇

Ae r

= αB +

∇2

−

1 r2

Ae.

(71)

The term which cannot be absorbed gives rise to the α term, and in eﬀective variables these equations are identical to the Parker model, the toroidal ﬁeld exhibiting the α eﬀect.

3.8 Soward’s Lagrangian analysis of Braginskii’s problem
The same situation of a nearly axisymmetric ﬂow was analysed in a slightly diﬀerent way by Soward (1972). The key idea of this method is to make use of an invariance of the induction equation for the diﬀusionless case R = ∞;

Bt + u · ∇B = B · ∇u.

(72)

Consider two (prescribed) velocity ﬁelds u(x, t) and u˜(x˜, t), and the Lagrangian variables x(a, t) and x˜(a, t) for these ﬂows deﬁned by

dx dt

a

=

u(x, t),

dx˜ dt

a

=

u˜(x, t).

(73)

Then

u˜i

=

∂x˜i ∂t

+
x

∂x˜i ∂xj

uj

.

(74)

The solutions to the induction equation for the two ﬂows are

Bi(x,

t)

=

Bj (a,

0)

∂xi ∂aj

,

B˜i(x˜,

t)

=

B˜j

(a,

0)

∂x˜i ∂aj

.

(75)

Using the chain rule

∂x˜i ∂aj

=

∂x˜i ∂xk ∂xk ∂aj

(76)

we obtain

B˜i

=

∂x˜i ∂xk

Bk

(77)

Now the idea is to consider one of the velocity ﬁelds as including the ‘rough’ components

while the other is the smooth velocity ﬁeld without these rough components. For the Bragin-

skii problem this means taking the tilde variables to include non-axisymmetric components

of order , while the x variables do not include these. The x variables will therefore not be

14

physical. We seek a near identity transformation x(x˜, t) = x˜ + O( ) of R3 which eliminates the O( ) in the velocity, so that

u(x, t) = U (r, x)(1 + O( 2))eφ + 2ueP (x) + 2u (x, t) + o( 2).

(78)

The equations in x space can be angle averaged, and this results in Braginskii’s result (70) and (71) with A(x, τ ) in place of Ae(x˜, τ ). The α eﬀect is missing - it comes from the noninvariant diﬀusive term. The eﬀective variable Ae occurs when we transform the unphysical x variables back into the real physical space x˜.
The key point is that, by including the non-invariant diﬀusive term in this analysis, one
obtains the α eﬀect term in (71), along with terms which can be absorbed into eﬀective
variables.

4 Some examples of analytically accessible dynamos

There are some simple velocity ﬁelds for which dynamo action is known to occur. The Ponomarenko dynamo is a ﬂow in a cylinder in which the ﬂuid travels axially while rotating,

u = U (r)ez + rΩ(r)eφ

(79)

The magnetic ﬁeld is periodic in z,

B = b(r)eλt+imφ+ikz .

(80)

This can be analysed for large R. The dynamo mechanism here involves the Ω eﬀect

which maintains the toroidal ﬁeld from the poloidal ﬁeld, and diﬀusion which maintains the

poloidal ﬁeld from the toroidal ﬁeld.

Another simple dynamo is the Roberts cell, which like the Beltrami ﬂow is periodic in

two dimensions

u = (cos y, sin x, cos x + sin y).

(81)

By transforming the variables x → x − y, y → π/2 + x + y, the Roberts cell takes the form

√

u = (cos y sin x, − cos x sin y, 2 sin x sin y)

(82)

An illustration of this ﬂow and the generation of magnetic ﬁeld is shown in ﬁgure 4.

5 MHD dynamos

So far we have only talked about kinetic dynamo theory, when the velocity ﬁeld is somehow prescribed and unaﬀected by the magnetic ﬁeld. In magnetohydrodynamics the velocity and magnetic ﬁelds both interact with each other and some physical mechanism (such as convection) which drives the ﬂuid velocity must be included. The momentum equation for the ﬂuid in a rotating frame with angular velocity Ω includes the Coriolis force and the Lorentz force,

ρ

Du Dt

+

∇p

+

2ρΩ

×

u

+

1 µB

×

(∇

×

B)

−

ρν∇2u

=

cg.

(83)

15

Figure 4: The Roberts cell with a z periodic ﬁeld applied. The ﬂow causes the production of z periodic y-ﬁeld from x-ﬁeld, and vice versa.
16

Figure 5: Magnetic ﬁeld from a numerical MHD simulation of the Earth’s core by Glatzmaier and Roberts.

The motion here is assumed to be driven by convection of some scalar c (temperature or

composition, for instance),

Dc Dt

−

Dc∇2c

=

Q.

(84)

If the convective forces are absent, and the velocity is axisymmetric, but the alpha-eﬀect

is included as in Braginskii’s model, and the normal component of u vanishes at the core

boundary, it is found that such an axisymmetric dynamo cannot be sustained in MHD.

The microscale driving needed to produce the small scale ﬁelds associated with an alpha

eﬀect can not sustain the dynamo. The Earth’s dynamo is now believed to be a result of

larger scale driving as a result of thermal and compositional convection associated with the

formation of the inner core.

Figure 5 shows a numerical simulation by Glatzmaier and Roberts, of a convective model

of the Earth’s core. This simulation exhibits the observed full reversals of the magnetic ﬁeld

on timescales of geophysical relevance. The role of the inner core is established as a reservoir

of magnetic ﬁeld during these reversals.

References
[1] E. C. Bullard, The Origin of the Earth’s Magnetic Field, Observatory, 70 (1950), pp. 139–143.
[2] T. G. Cowling, The Magnetic Field of sunspots, Monthly Notices R. Astron. Soc., 94 (1933), pp. 39–48.

17

[3] T. G. Cowling, Magnetohydrodynamics, Interscience Publishers, London, 1957. [4] W. M. Elsasser, Hydromagnetism. I. A Review, Am. J. Phys., 23 (1955), pp. 590–
608. [5] M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric
Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag, New York, 1987. [6] H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cam-
bridge University Press, 1978. [7] E. N. Parker, Hydrodynamic Dynamo models, Astrophys. J., 122 (1955), pp. 293–314. [8] E. N. Parker, Cosmical Magnetic Fields, Calrendon Press, Oxford, 1979. [9] G. O. Roberts, Spatially periodic dynamos, Phil. Trans. R. Soc. A, 266 (1970),
pp. 535–558. [10] P. H. Roberts, An Introduction to Magnetohydrodynamics, Americal Elsevier Pub-
lishing Company, New York, 1967. [11] P. H. Roberts, Dynamo Theory, vol. 2, American Mathematical Society, Providence,
RI, 1971, pp. 129–206.
18