(iii) Theoretical Geomagnetism
Lecture 9:
Self- Exciting Dynamos: Kinematic Theory
9.0 What is a self-exciting dynamo?
Dynamo = A device that converts kinetic energy
into electromagnetic energy.
• Dynamos use motion of an electrical conductor through a magnetic
field to induce electrical currents.
• These electrical currents also generate (secondary) induced
magnetic fields that are sustained as long as energy is supplied.
Self-exciting dynamo = No external fields or currents are needed
to sustain the dynamo, aside from a weak
seed magnetic field to get started.
i.e. self-exciting dynamo is self-sustaining
9.0 What is a self-exciting dynamo?
• To generate and sustain magnetic fields self-exciting dynamos
require:
(i) A weak initial ‘seed’ magnetic field (which can be removed later).
(ii) A suitable arrangement of the motions of the electrical conductor,
electric current pathways and resulting magnetic fields.
(iii) A continuous supply of energy to drive the electrical conductor
sufficiently fast for self-excitation to be possible.
• Many natural magnetic fields are thought to be produced by a
self-exciting dynamo mechanism.
e.g. Galactic magnetic fields, solar magnetic field, geomagnetic field.
Lecture 9: Self- exciting dynamos
9.1 Simple example 1: The disk dynamo
9.2 The induction equation and magnetic energy evolution
9.3 Simple example 2: Stretch-twist-fold dynamo
9.4 The Alpha-effect and the Omega-effect
9.5 Mean field dynamos
9.6 3D spherical dynamos
9.7 Experimental dynamos
9.8 Summary
68 Theory of the Geodynamo
9.1 Simple example 1: The disk dynamo
(a)
(b)
B
ω
B0
–– A
–––
–––
–
N–
–
–
u
u × B0
0
D
I S2
E
I
D
+ +++ + + ++ ++ +++
– –
P
– –
N– –
–
– – A′
I
R
W
(From Roberts, 2007)
• Disk is rotated in an applied magnetic field causing an induced
EMF, but no closed circuit for current to flow (yet).
(c)
ω
S1
f the Geodynamo
A
9.1 Simple example 1: The disk dynamo
Siemen
(b)
B0
ω
B
68 Theory of the Geodynamo 0
field that
I S2
A
current. S
I
(a) D
betw
I (b)
B0
ω route
ω
B0
E
I
u D
D
A and in the
A
R
R
–
–
+
B0
S2
+
–––
I
++ ++ +
W – –– ––S1
almost one
N–
P
I
I
D
–
toDthe two
E
I
u
u
D
R
R
A′
A′
+ +++ + + ++ ++ +++
winding
is
W
S
– –
1
u• ×Now
B0 let a conducting
Pwire join A to A’, being maintained
by sliding
– –
produced
–
N– –
contacts (‘brushes’).
A current now flows and can be used to light
– magnetic
A′
the bulb, but–the
field due to current I doesn’t reinforce
B.
A′applied
m
fields are p
ω
B
A9 ! A. U
9.1
SimpleBexample 1: The disk dynamo
(c)
S2 A
ω
form acros
B
B
W
S2 A
the motio
B D
u
that u " B
W
JI
J×B
1
B D
u
tion. If !
J
S
J
×
B
1
I
I
W
magnetic
I
I
W
A′
will contin
B
A′
B
B
by the em
B
create a m
• Instead, let wire W loop round the disk. The induced field B produced
omopolar
dynamo
(see
text
forof the disk.
velocity g
by I now
B in the
plane
If rotation
rate exceeds
Figure
1 reinforces
The homopolar
dynamo
(see
text for
some critical value B can be removed => self-excited dynamo.
explanation).
has becom
–
–
–
(From Roberts, 2007)
0
(From Roberts, 2007)
0
0
Lecture 9: Self- exciting dynamos
9.1 Simple example 1: The disk dynamo
9.2 The induction equation and magnetic energy evolution
9.3 Simple example 2: Stretch-twist-fold dynamo
9.4 The Alpha-effect and the Omega-effect
9.5 Mean field dynamos
9.6 3D spherical dynamos
9.7 Experimental dynamos
9.8 Summary
9.2.1 The magnetic induction equation
• Recall that in a moving, electrically conducting medium Ohm’s law
has a extra term:
J = σ(E + u × B)
(1)
i.e. motion across a magnetic field can
generate a perpendicular current.
• Combining this with the Maxwell equations describing no magnetic
monopoles, EM induction and Ampere’s law yields:
∂B
= ∇ × (u × B) + η∇2 B
∂t
(2)
• So, field changes can be produced by induction due to fluid motions
or by Ohmic dissipation (magnetic diffusion) due to finite conductivity.
9.2.2 Magnetic energy evolution
• Taking the dot product of B/µ0 with (2) and integrating over the
over the conducting volume V we obtain the following terms:
!
dEm
=
dt
!
DJ =
V
Wu =
!
V
V
!
1
B
·(η∇2 B) dV =
µ0
σ0
B ∂B
·
dV =
µ0 ∂t
!
V
∂
∂t
"
B2
2µ0
#
dV
2
−(∇×B) +∇·[B×(∇×B)] dV = −
V
V
1
B
·[∇×(u×B)] dV =
µ0
µ0
!
!
B·[(B·∇)u−(u·∇)B] dV =
V
1
µ0
!
!
J2
(J × B)
dV −
dS
σ
S
0 σ
B·(B·∇)u dV −
V
!
1
B 2 u dS
2µ0 S
• Surface terms can be neglected if surface is rigid (no flow across it) and there
is no electrical currents flowing at the boundary.
9.2.2 Magnetic energy evolution
• Taking the dot product of B/µ0 with (2) and integrating over the
over the conducting volume V we obtain:
dEm
=
dt
Wu =
V
B ∂B
·
dV =
µ0 ∂t
!
V
1
B
·[∇×(u×B)] dV =
µ0
µ0
!
1
B0
·(η∇2 B) dV = 0
µ
σ
V
!
"
∂
∂t
!
V
DJ =
!
!
B2
2µ0
#
dV
B·[(B·∇)u−(u·∇)B] dV =
V
1
µ0
!
B·(B·∇)u dV −
V
−(∇×B)2 +∇·[B×(∇×B)] dV = −
V
V
dEm
= Wu + DJ
dt
Change in
magnetic energy
!
=
Work done by flow
on magnetic field
10
2µ
!
B 2 u dS
S
!
J2
(J × B)
dV −
dS
σ
S
0 σ
(3)
+
Ohmic Dissipation
(always -ive)
• So, for the magnetic energy to grow, enough work must be done on
the field by the fluid motions to overcome Ohmic dissipation.
relax
back
to initial
its
initial
illustrates
Hjj!
!jujujjjwjback
wj ¼
j ¼to ately
relax
itsately
initial
state.
illustrates
theThis
lear fromsphere.
that
jHclear
j !from
jujjw[23a]
j ¼ that
relax
back
toThis
its
state.state.
This
illustrates
the the
sphere.It
Itis isclear
from
[23a]
thatately
jjH
![23a]
!2 2 " "
crucial
importance
of
magnetic
reconnection
; a; a‘maximally
helical
onein
inwhich
whichTheory
U U=L=L
crucial
of magnetic
reconnection
as a as a
importance
of magnetic
reconnection
a
ofimportance
the Geodynamo
77 as
helical flow’
flow’crucial
isis one
O
aximally O
helical
flow’
is‘maximally
one in which
means
of
‘locking-in’
amplified
magnetic
means
‘locking-in’
the the
amplified
field. field
jH jjH¼j ¼
jujjjuwjjj.w j.
means of ‘locking-in’
theof amplified
magnetic
field.magnetic
Said
another
way,
magnetic
reconnection
provides
Said
another
way,
magnetic
reconnection
the the
The
dynamo
– Said
STF.another
Obtainway,
The
stretch-twist-fold
dynamo
STF.
Obtain
anan magnetic
reconnection
provides
the provides
h-twist-fold
–stretch-twist-fold
STF.
Obtain
an reader
d’ helicity,elastic
Hdynamo
<
0,
in
its
Northern
Hemilets
go
of
the
band
at
any
stage,
it
will
immedicrucial
element
of irreversibility
in theindynamo.
The
crucial
element
of irreversibility
elastic
(rubber)band,
band,loosely
loosely corresponding
corresponding
toto
the
(rubber)
theflux
fluxof irreversibility
crucial
element
in the
dynamo.
The the dynamo. The
band,from
loosely
corresponding
to
the
flux
ately
relax
back
to
its
initial
state.
This
illustrates
the
clear
[23a]
that
H
!
u
w
¼
j
j
j
j
j
j
process
of
reconnection
in
the
flux
tube
occurs
near
process of reconnection in the flux tube occurs near
tube
Figure6(a),
6(a), and
and use
use its tension
totoof
roughly
tube
in inFigure
tension
roughly
process
reconnection
in the
flux‘R’tube
occurs
near the field gra6(a), andhelical
use
its
tension
to
roughly
the
point
marked
in
crucial
importance
of
magnetic
reconnection
as‘c’, awhere
maximally
flow’
is
one
in
which
the
point
marked
‘R’panel
in panel
‘c’, where the field grasimulate
tensionofofthe
the field
field lines
lines in
the
flux
tube.
simulate
thethetension
in
the
flux
tube.
the point
marked ‘R’
in panel
‘c’, where
the field.
field
gradients
are
greatest.
The
result
of
the
two two
nsion of the The
field tension
lines in inthethe
fluxband
tube.maymeans
of
‘locking-in’
the
amplified
magnetic
dients are greatest. The result ofreconnection,
the reconnection,
be
The tension in the band may dients
be systematically
systematically
are
greatest.
The
result
of
the
reconnection,
two
separate
rings,
is shown
in panel
(d). (d).
Said another
way, magnetic
reconnection
provides
the
n the band
may
be
systematically
increased
three-step,
stretch-twist-fold
process:
separate
rings,
is shown
in panel
h-twist-fold
dynamo
–byby
STF.
Obtain
an
increased
a athree-step,
stretch-twist-fold
process:
9.1
Simple
example
1:
The
disk
dynamo
The
STF
dynamo,
originally
proposed by
separate
rings,
is
shown
in
panel
(d).
of irreversibility
the dynamo.
(1) ‘stretch’ the
rubber
band tocrucial
doubleelement
its length;
The in
STF
dynamo,The
originally proposed by
hree-step,
stretch-twist-fold
process:
band, loosely
corresponding
the flux
(1)
‘stretch’ theto
rubber
band to double its length; Vainshtein
andtube
Zeldovich
(1972),
has become the
The
STF dynamo,
proposed
by
(2)
‘twist’
it into
a length;
figure-8; and
process
of reconnection
in the originally
flux
occurs
near(1972),
Vainshtein
and Zeldovich
has become the
rubber
band
to
double
ethe
6(a),
and use
its
tension
toits
roughly
(2)
‘twist’
it
into
a
figure-8;
and
9.2
The
induction
equation
and
magnetic
energy
evolution
paradigm
for
‘fast
dynamos’,
which
are
defined as
(3) ‘fold’ the loops of the 8 onthe
top
of
one
another
Vainshtein
and‘R’Zeldovich
hasdynamos’,
become
point marked
in panel
‘c’,(1972),
where
the
field
gra- the
paradigm
for
‘fast
which
defined as
nsion
the field
lines the
in the
fluxoftube.
into aoffigure-8;
and
(3) ‘fold’
loops
the
8
on
top
of
one
another
processes of field amplification that operateare
on timeto form two linked loops dients
ofparadigm
the are
same
size
as
greatest.
The
result
of
the
reconnection,
two
for ‘fast
dynamos’,
which
are
defined that
as where
processes
of
fieldofamplification
operate
the of
band
beoriginal
systematically
einloops
the
8may
onSimple
top
of
onelinked
another
to
form
two
loops of2:
theStretch-twist-fold
same size
as scales
independent
T ! in systems
Rmon
is time9.3
example
dynamo
the
band.
separate
rings,
is
shown
in
panel
(d).
processes of field amplification
that 8.03.3.3
operate
timescales
of below
T on
systems
where
three-step,
stretch-twist-fold
process:
! in
two linked
loops
oforiginal
thearesame
large
(seeindependent
Section
and
Childress
and Rm is
thesteps
band.size as
These
illustrated
in Figures 6(a)–6(c), in
The
STF
dynamo,
originally
proposed
by
scales
independent
of
T !(1995)).
in Section
systems
where
Rm
isandnecessarily
large
(see
8.03.3.3
Childress and
the band.
rubber
band
to
double
its length;
Clearly,
a fastbelow
dynamo
These
are
illustrated
in Figures
6(a)–6(c),
inal
which
panel
(a)
depicts
the initial
flux
tube,
panel Omega-effect
(b)in Gilbert
9.4steps
The
Alpha-effect
and
the
Vainshtein
and
Zeldovich
(1972),
has
become
the
Gilbertfast
(1995)).
fast
dynamo
reconnection,
inaorder
to lock innecessarily
the
large
(see
Section
8.03.3.3
below
andClearly,
Childress
and
t into
a figure-8;
and
sketches
it when
it is stretched
twisted,
and
panel
which
(a) depicts
the initial
tube,
panel
(b) requires
re
illustrated
inpanel
Figures
6(a)–6(c),
in andflux
paradigm
for
‘fast
dynamos’,
which
are
defined
as
dynamo-generated
field
as
fast
as
it
is
produced.
requires
fast dynamo
reconnection,
in order to lock in the
edepicts
loops ofthe
the
8shows
onMean
oftube,
one
(c)
it after
itisisanother
subsequently
The
energy
Gilbert
(1995)).
a fast
necessarily
sketches
ittop
when
itfield
stretched
twisted,
and
panel Clearly,
initial
flux
panel
(b) andfolded.
9.5
dynamos
processes
of
field
amplification
that
operate
on
timeAlfve
´n’s
twisted
kink
dynamo
–
ATK.
a piece
dynamo-generated
field
as
fast
as itTake
is produced.
within,
and
the
tension
of,
the
flux
tube
are
increased
by
of
the same
sizepanel
as
(c) loops
shows
it twisted,
after
it isand
subsequently
folded.
Thefast
energy
requires
reconnection, in order to lock in the
ntwo
it islinked
stretched
and
of
taut
rope
and
start
twisting
its
ends,
as ina piece
scales
T ! inAlfve
systems
where
is – ATK. Take
a factor
ofthe
4 intension
step (1).of,
Steps
can,
in independent
principle,
´n’sastwisted
kinkRm
dynamo
ginal
within,
and
the(2)–(3)
flux
tube
are
increasedbeby of field
dynamo-generated
as
fast
it
is
produced.
9.6
3D
spherical
dynamos
r it isband.
subsequently
folded.
The
energy
Figure
7(a).rope
As the
an instability,
it will as in
repeated
and
over
again,
the sense
twist
in stepbe8.03.3.3
large
(see
Section
below
and
Childress
of dynamo
taut
andresult
startofand
twisting
its ends,
re illustrated
in of
Figures
6(a)–6(c),
in
factor
4over
in step
(1).
Steps
(2)–(3)
can, of
in
principle,
Alfve
´n’s
twisted
kink
–
ATK.
Take
a
piece
tension
of,athe
flux
tube
are
increased
by
develop
kinks
like
those
often
seen
on
the
cord
joining
(2)
beingflux
always
thepanel
same. This
gives
the band
the senseClearly,
Gilbert
(1995)).
a fast 7(a).
dynamo
necessarily
As its
the
result as
of an instability, it will
) depicts the
initial
tube,
(b) the
repeated
over
over
again,
sense
of
twist
in stepand Figure
ofmotion,
taut
rope
start
twisting
9.7
Experimental
tep (1). Steps
(2)–(3)
can,and
inorprinciple,
a
telephone
handset
toends,
its
cradle in(Figure 7(b)).
of
‘handedness’
helicity ofbeadynamos
screw
and
the
requires
fast
reconnection,
in
order
to
lock
in
the
develop kinks like those often seen on the cord joining
en it is stretched
and
twisted,
(2) being
always
the and
same.panel
This gives Figure
the band the sense
As the
result offlux
an rope
instability,
willsame way will
A magnetic
twisted init the
broken
reflection
symmetry
that isdynamo-generated
crucial to 7(a).
the sucnd over again,
the sense
of twist
in step
field
astelephone
fast as it ishandset
produced.
a
to
its
cradle
(Figure 7(b))
er it is subsequently
folded.
The
energy
of ‘handedness’
or
helicity
of
a
screw
motion,
and
the
also develop
a kink
and, cord
because
of the large field
9.8gives
cess
of Summary
thisthe
dynamo.
The
tension
in the
band (analogous
develop
kinks
like
those
often
seen
on
the
joining
stension
the same.
This
band
the
sense
Alfveto
´n’sthe
twisted
dynamo
– ATK.
Take
piece in the same way will
A
magnetic
fluxcrossing
rope atwisted
of,broken
thetoflux
tube are
increased
bythatprogressively
reflection
symmetry
is crucial
suc- kink
gradients
nearcradle
the
point7(b)).
R in the figure, a
magnetic
tension)
increases
as steps
a
telephone
handset
to
its
(Figure
or
helicity
of
a
screw
motion,
and
the
taut
rope
and start
twisting
ends,and,
as because
in moveofaway
alsowill
develop
a(Figure
kink
the from
large field
tep (1). Steps
(2)–(3)
can,
in principle,
be
cess(1)–(3)
of thisare
dynamo.
Theand
tension
inof
the
band
tube
detachits
7(c))
and
repeated,
at some
stage
the(analogous
reader’s
A
magnetic
flux
rope
twisted
in
the
same
way
will
on
symmetry
that
is
crucial
to
the
sucFigure
7(a).asAs
annear
instability,
it the
willpoint
the crossing
a
and over again,
the sense
of twist
in step
to hands
magnetic
increases
progressively
steps
will tension)
tire,
making
it impossible
to increase
thethe result
thegradients
fluxof
rope.
Simultaneously,
torsionRof in
thethe
ropefigure,
is
also
develop
a
kink
and,
because
of
the
large
field
mo.
The
tension
in
the
band
(analogous
develop
kinks
like
those
often
seen
on
the
cord
joining
tube will
and move
away from
s the same.(1)–(3)
This
gives
the
band
the
sense
arefurther.
repeated,
and
at the
some
reader’s released,
tension
This
marks
endstage
of thethe
kinematic
but detach
increases(Figure
again as7(c))
the twisting
continues
gradients
near
the
crossing
point
R
in
the
figure,
a
nsion)
increases
progressively
as
steps
a
telephone
handset
to
its
cradle
(Figure
7(b)).
regime
(intire,
which
fluid
motion
is prescribed
a priori and
hands
making
itthe
impossible
to increase
the until
or helicity
of
a will
screw
motion,
and
thefurther
flux rope.
Simultaneously,
torsion
of thearope is
loops
detach. The fluxthe
rope
has become
tube
will
detach
(Figure
7(c))
and
move
away
from
eated,
andtension
at
some
stageThis
the
reader’s
the
field
and
start
of the
regime
Adynamic
magnetic
flux(inrope
twisted
in
the
same
way
will
machine
for
generating
flux
tubes.
A
sufficiently
further.
marks
end
of the
kinematic
on
symmetry
that
is grows)
crucial
to the
the
suc-the
released, but increases again as the twistinglarge
continues
which
the
field
quenches
its
own
further
growth).
If
the
twisting
ratethe
creates
fast
(Alfvén,
1950).
making
it impossible
to fluid
increase
theis prescribed
thedevelop
fluxa rope.
Simultaneously,
torsion
ofdynamo
the
rope
is rope
also
a and
kink and,
because
ofloops
thea detach.
large
field
regime
(inthe
which
motion
priori
amo.
The tension
in
band
(analogous
until
further
The
flux
has become a
near
point
inboundary
the figure,
a
the field
grows)
dynamic
regime
(incrossing
This marks
theprogressively
end
of and
the the
kinematic
nsion)
increases
as start
stepsof thegradients
released,
but the
increases
again
the
twisting
continues
machine
generating
flux
tubes.
A[2]
sufficiently
When
! 6¼ as
0,forR
the
conditions
simplify. large
which
the
field
itsand
own
growth).
Ifloops
the
tube
will
detach
(Figure
7(c))
and
move
from
peated,
at some
stagequenches
thea priori
reader’s
h fluid and
motion
is(a)
prescribed
twisting
rate
creates
a fastbecome
1950).
until
further
detach.
flux
ropeaway
has
a (Alfvén,
(b) further
From
a The
physical
standpoint,
itdynamo
is clear
that an infiB
B
,and
making
it impossible
to increase
the(in themachine
flux rope.
theconcentration
torsion
theofrope
islarge
nitelyflux
thin
current,
even if it could
the start
of the dynamic
regime
forSimultaneously,
generating
A of
sufficiently
When !tubes.
6¼ 0, the
boundary conditions
[2] simplify
be
set
up
initially,
would
instantly
diffuse
into a layer
.quenches
This marks
the
end
of
the
kinematic
released,
but
increases
the twisting
continues
its own
If the
twisting
rate
createsagain
aFrom
fast
(Alfvén,
1950).
• Special
arrangements
ofasdynamo
solid
conductors
(b)
(a)B further growth).
B
amoving
physical
standpoint,
it is clear that an infi-
••
Lecture 9: Self- exciting dynamos
•
•
•
•
9.3 Simple example 2:
Stretch-twist-fold dynamo
of finite
which
J (though
possibly large)
B loops detach.
ch fluid motion is prescribed a priori and B until further
Thethickness,
flux ropeinhas
become
a
and wires
can
produce
a self-exciting
dynamo.
nitely
thin
concentration
of surface
current,
even ifthat
it could
is
finite.
This
layer
replaces
the
current,
When
!
¼
6
0,
the
boundary
conditions
[2]
simplify.
) and the start of the dynamic regime (in
machine for generating flux tubes. A sufficiently large
be Xset
up initially,
would
instantly
is,
Thus,
6¼ 0, theinto
con-a layer
0 instantaneously.
(b)
From rate
a physical
it(Alfvén,
is clear
that
anwhen
infi-! diffuse
quenches its own
growth).
twisting
creates astandpoint,
fastCdynamo
1950).
B further
B
BB If the
of finite
thickness,
in the
whichsame?
J (though possibly large)
[2]
But what
of fluidditions
motions
cantodo
nitelytype
thin concentration
of reduce
current,
even
if it could
is finite.
This layer
the surface current, that
When
6¼ 0,initially,
the boundary
conditions
[2] replaces
simplify.
(c)
(d)
be set!up
would instantly
diffuse
into
a layer
•
Consider
‘Stretch-Twist-Fold’
thought
experiment
is,
C
instantaneously.
Thus,
X
0
(b)
From a physical standpoint, it is clear that an infi- when ! 6¼ 0, the conB
B
B
of finite thickness, in(a)
which [2]
J (though
(b) possibly large)
(c)
ditions
reduce
dynamo
and
Zeldovich
nitely of
thinVainshein
concentration
of current,
even to
if(1975):
it could
is
finite.
This
layer
replaces
the
surface
current,
that
R
B
B
B
B
be set up initially,
would instantly diffuse into a layer
(c)
(d)
is,
C
instantaneously.
! 6¼ 0,large)
the conB
X
0
of finite
in which
(though
(1)
Flowthickness,
‘stretches’
fluxJThus,
tubewhen
topossibly
double
its length
R
(a)
(b)
(c)
ditions
[2]
reduce
to
is finite. This layer replaces the surface current, that
R
Bis, Flow
B
Bis
Figure
6 BStretch-twist-fold.
closed
Figure
7 Alfvén’s
kink condynamo. (a) A flux rope is
‘twists’
tube
into
a figure-8.
Cflux
instantaneously.
Thus,
when
!twisted
6¼ 0, the
* (a) A (2)
X 0tube
(d)
(b) stretched, twisted, and (c) folded over
onto
itself,
so
ditions [2] reduce
to twisted and (b) becomes unstable to the formation of kinks.
R
The large
nearanother
R allows
kink to detach
(b)
(c)
R theto
(3) Flow ‘folds’ loops
on field
topgradients
of one
from the rope, as indicated in (c).
form 2 linked loops the same size as original.
creating large field gradients near R where
one loop
(a)
detaches
from
the
other,
as
indicated
in
(d).
(d)
B
B
Figure 6 Stretch-twist-fold. (a) A closed
Figure 7 Alfvén’s
(a) flux tube is
(b)
(c) twisted kink dynamo. (a) A flux rope is
(b) stretched, twisted, and (c) folded
overMagnetic
onto itself, sodiffusion
twisted
andat
(b) becomes
unstable
the formation of kinks.
(4)
acts
* where
thereto are
R field gradients near R allows the kink to detach
R
B
creating B
large field gradients
near R where one loop
The large
large field gradients.
detaches from the other, as indicated in (d).
from the rope, as indicated in (c).
ch-twist-fold. (a) A closed flux tube is
Figure
7
Alfvén’s
twisted
dynamo.
(a) A flux
Rkinkvery
(5) Left with 2 flux tubes
similar
torope
theis
sted, and (c) folded over onto itself, so
ld
gradients near
where
one
loop
(From
Roberts,
2007)
ch-twist-fold.
(a) ARclosed
flux
tube
is
e other,
in (d).
isted,
andas(c)indicated
folded over
onto itself, so
twisted and (b) becomes unstable to the formation of kinks.
original.....
process
now
repeat.....
The
large
field gradients
nearcan
R allows
kinkrope
to detach
Figure
7 Alfvén’s
twisted kink
dynamo.
(a)the
A flux
is
from the
as indicated
in (c).
twisted
andrope,
(b) becomes
unstable
to the formation of kinks.
eld gradients near R where one loop
he other, as indicated in (d).
The large field gradients near R allows the kink to detach
from the rope, as indicated in (c).
Lecture 9: Self- exciting dynamos
9.1 Simple example 1: The disk dynamo
9.2 The induction equation and magnetic energy evolution
9.3 Simple example 2: Stretch-twist-fold dynamo
9.4 The Alpha-effect and the Omega-effect
9.5 Mean field dynamos
9.6 3D spherical dynamos
9.7 Experimental dynamos
9.8 Summary
9.4.1 Toroidal and Poloidal fields
• Toroidal field (or flows) stay
entirely on spherical surfaces:
Axisymmertic examples of
Toroidal and Poloidal fields
(Credit: Graeme Sarson)
• Poloidal field (or flows) cut
across spherical surfaces:
74 Theory of the Geodynamo
9.4.2 The Omega-effect
(a)
(b)
(From Roberts, 2007)
Figure 2 The !-effect. (a) a field line in a meridian plane
(single arrow) is sheared by a zonal flow (double arrows);
(b) as
result of the
shearing,
the field
line is bent, so
• Conversion
ofapoloidal
field
to toroidal
field.
creating a $–component of B.
• Caused by differential rotation (shear) of toroidal flows.
The intermittent character of MHD turbulence is
reflected by the existence of flux tubes in which B is
more intense than in their surroundings. Frequently, B
is visualized as being discontinuous across the surface
76 Theory ofM
theof
Geodynamo
the9.4.3
tube, which
therefore a current sheet, by
Theis Alpha-effect
[2a]. Because of the field line tension, a closed flux
(a)
(a)
tube will (b)
tend to collapse in much
the same way as a
stretched elastic band contracts when released.u If there
is no compensating stretching process, the collapse
. B has subsided to the level of the
. will continue until
• Regeneration
of poloidal field
field field.
in its environment. Similarly,
a bent flux tube
(b)
from toroidal
tends to straighten, returning some of its magnetic
energy
to the
energy of the fluid frozenRonto
(can also sometimes
bekinetic
used
to
B = B0 + b
B0
produce toroidal
field from
it; an example
is given in the next subsection.
Figure 4 Flux expulsion. (a) A conducting sphere
poloidal
embedded
in a field)
similarly conducting, stationary medium lies in
When " 6¼ 0
the conductor w
U " ¼ "/L , wher
field gradient (
given below.
When thinki
helpful to rega
Wherever Rm #
theorem and the
in picturing MH
ever, likely to b
Rm $ 1). The fo
and reconnectio
The !-eff
•
described in the
process that halt
!-effect. As B!$
increasingly acts
in Figure 2(b).
and less, driftin
velocity U " of
where $ is the f
!M
of order RmB
B
B!$ ceases, the s
equal but oppos
A similar ex
8.03.3.5: Suppose
Bin a spherical co
vanishes, the co
a uniform magnetic field B and is set in rotation. (b) The field is
ω
prevented; resis
ultimately almost completely expelled from the sphere.
8.03.2.6
The
Imperfect
Conductor:
lines inwards.
(c)
• Caused by upwelling poloidal
Reconnection
The Sweet–
fluxflows
sheets. possessing
It might seem at vorticity.
first sight that this can
only be detrimental to dynamo action but this is not
simplest and mo
speaking,
a afield
every
j
necessarily the Strictly
case. Consider
induction by
steady line passes through
flow. An X-typepoint
stagnation
point maybut
existonly
between
in space,
a finite number of lines can Breconnection, is
one circulation and its neighbor. (An example is given
U , of a weakly
be shown
in sketches
of surfield line topology, where the
in Section 8.03.4.3
below.) The
expelled flux
Figure 5 indicates
The alpha-effect
mechanism.
flux rope bent
together t
rounds this point
and, since the
streamlines of
separate
(FromaRoberts,
2007)(a) Aforces
crowding
together
field lines
more
by the velocity u of a cyclonic eddy, (b) is twisted by the
there, the field lines almost frozen to them are ‘expo%B (Figure
3(a
the eddy,
creating
large field gradients
near R
intense field. In these sketches,vorticity
fieldw oflines
may
appear
nentially stretched’. This is an efficient way of
where the flux loop detaches as indicated in (c).
field is approxi
to disconnect
reconnect
transforming kinetic
energy into and
magnetic
energy. in a new topology as the
0
•
9.4.3 Simple example 3: Alpha-Omega dynamo
Omega-effect: Poloidal field -> Toroidal field
(From Love, 1999)
Alpha-effect: Toroidal field -> Poloidal field
Lecture 9: Self- exciting dynamos
9.1 Simple example 1: The disk dynamo
9.2 The induction equation and magnetic energy evolution
9.3 Simple example 2: Stretch-twist-fold dynamo
9.4 The Alpha-effect and the Omega-effect
9.5 Mean field dynamos
9.6 3D spherical dynamos
9.7 Experimental dynamos
9.8 Summary
9.5.1 Mean-field electrodynamics
• We begin with the magnetic induction equation:
∂B
= ∇ × (u × B) + η∇2 B
∂t
(2)
• We separate the velocity and magnetic field into mean and fluctuating
parts:
u(r, t) = U0 (r, t) + u! (r, t),
with
< u! >= 0
B(r, t) = B0 (r, t) + b! (r, t),
with
< b! >= 0
< B(r, t) >= B0 (r, t)
(4)
and < u(r, t) >= U0 (r, t)
• This separation is valid if the fields are characterised, for example, by
2 lengthscales: a large scale L and a small scale l0 and the averaging
is carried out over an intermediate lengthscale a where l0 <<a<< L
!
such that:
3
< ψ(r, t) >=
ψ(r + ξ, t) d3 ξ
(5)
3
4πa
|ξ|<a
• Small scale fluctuations are assumed random so they average to zero.
9.5.1 Mean-field electrodynamics
• Substituting from (4) into (2) and carrying out the averaging operation
on each term we obtain the the mean field induction equation:
∂B0
= ∇ × (U0 × B0 ) + ∇ × E + η∇2 B0
∂t
(6)
where E =< u! × b! > is a non-zero mean EMF arising from the
product of random, small scale fluctuations of the field and flow.
• Subtracting (6) from (4) we obtain an induction equation for the
evolution of the fluctuating part of the magnetic field:
∂b!
= ∇ × (U0 × b! ) + ∇ × (u! × B0 ) + ∇ × G + η∇2 b! (7)
∂t
where
G = u! × b! − < u! × b! > (‘Pain in the neck’ term!)
9.5.1 Mean-field electrodynamics
• In order to make progress in solving (6) we need to find a way to
express E in terms of the mean fields U0 and B0.
• We proceed assuming that fluctuations in the magnetic field
are driven solely by the large scale field B0.
b!
!
• Under this assumption we can further assume that b and B0 are
linearly related, so E and B0 will also be linearly related so we can
postulate an expansion for E of the form:
Ei = αij B0j + βijk
∂B0j
+ .....
∂xk
• αij and βijk are pseudo-tensors determined by the statistics of the
small scale motions (turbulence). In the simplest isotropic case:
αij = αδij and βijk = β$ijk
and so
E = αB0 − β(∇ × B0 ) + ....
(8)
9.5.1 Mean-field electrodynamics
• Keeping only the 2 terms given on the RHS of (8) and substituting
back into (6) gives
∂B0
= ∇ × [(U0 × B0 ) + αB0 ] + (η + β)∇2 B0
∂t
• This is the mean field induction equation for isotropic turbulence.
• α is now seen to be a regenerative term which can help the large
scale magnetic field to grow.
•
β
is seen to be a turbulent diffusivity
• But, what mathematical form does α take and what does it
physically represent? This is a complicated and controversial
issue.
See the appendix for one answer in a very simple scenario...
(9)
9.5.2 Simple example 4: alpha-effect dynamo
• Let us consider a simple mean field dynamo that arises due to
the α term. If U0 =0 and defining an effective diffusivity ηe = η + β
(9) can be written as
∂B0
= α(∇ × B0 ) + ηe ∇2 B0
∂t
• We seek solutions that have
the special ‘Beltrami’ property
Consequently,
∇ × B0 = KB0
∇2 B0 = −∇ × (∇ × B0 ) = −K 2 B0
∂B0
= (αK − ηe K 2 )B0
∂t
So the induction
equation simplifies to:
9.5.2 Simple example 4: alpha-effect dynamo
From the induction eqn:
Separating the variables:
Integrating:
i.e.
• Thus, if
!
∂B0
= (αK − ηe K 2 )B0
∂t
1
dB0 =
B0
!
(αK − ηe K 2 )dt
ln B0 (r, t) − ln B0 (r, 0) = (αK − ηe K 2)t
B0 (r, t) = B0 (r, 0)e(αK−ηe K
2
)t
α > ηe |K|
=> Mean field will grow exponentially and
we have a self-exciting dynamo!!
• Since K ∼ L−1 this will happen if the length scale of the mean
field is sufficiently large.
9.5.3 Shortcomings of mean field
dynamo theory
• Mean field dynamo theory is very useful to help develop intuition
concerning how self-exciting dynamos can come about and allows
analytic/simple numerical solutions to be found.
• However, it involves many major assumptions and can’t be rigorously
justified when applied to geophysical and astrophysical systems.
• Major weaknesses include:
(1) Difficulty in deriving form of α and β for realistic scenarios.
Instead,these are often picked in an ad-hoc manner to fit observations.
(2) Theory of how α and β are modified when magnetic field become
strong enough to affect flows is unclear.
• To accurately model the geodynamo, we must leave behind mean
field theory and consider numerical solution of the full induction eqn.
Lecture 9: Self- exciting dynamos
9.1 Simple example 1: The disk dynamo
9.2 The induction equation and magnetic energy evolution
9.3 Simple example 2: Stretch-twist-fold dynamo
9.4 The Alpha-effect and the Omega-effect
9.5 Mean field dynamos
9.6 3D spherical dynamos
9.7 Experimental dynamos
9.8 Summary
9.6.1 Kinematic dynamo theory:
Solution as an eigenvalue problem
• Reminder: We are considering the KINEMATIC dynamo problem.
i.e. Assume the flow is known and ignore any back-reaction of the
generated field on the flow
• Once again we start
from the induction eqn:
∂B
= ∇ × (u × B) + η∇2 B
∂t
• We then seek normal mode
solutions of the form:
pt
!
B(r, t) = Re[B(r)e
]
• This transforms the induction eqn
into a linear eigenvalue problem:
!
! = ∇ × (u × B)
! + η∇2 B
pB
9.6.1 Kinematic dynamo theory:
Solution as an eigenvalue problem
• The solutions of the Linear eqn:
!
! = ∇ × (u × B)
! + η∇2 B
pB
• Define a sequence of complex Re[p ] ≥ Re[p ] ≥ Re[p ] ≥ Re[p ] ≥ ...
1
2
3
4
eigenvalues that can be ordered:
! 1 (r)ep1 t ]
• If Re[p1 ] > 0 then the corresponding field B(r, t) = Re[B
will exhibit exponential growth, which is oscillatory if Im[p1 ] != 0 .
• Challenge: To solve this eigenvalue problem given any 3D flow.
u = (sin 2y, sin 2x, sin(x + y))
(2.4
which has zero mean helicity. Nevertheless, dynamo action can occur! However growth rates mu
smaller than in the ABC case.
‘Helicity is not essential for dynamo ation, but it helps’
9.6.2 Numerical solutions for 3D flows
2.5 Spherical Dynamos
• Requires numerical solution, for example, by expanding the field and
Following Bullard and Gellman, 1954, velocity for kinematic spherical dynamos written
flows into of toroidal and poloidal parts,
each represented as a sum of
#
m
vector spherical harmonics:
u=
tm
l + sl
(2.5
i
m
where tm
l and sl are the toroidal and poloidal components
tm
r̂tm
t)Ylm (θ, spherical
φ),
sm
∇ × ∇ × but
r̂sm
t)Ylm (θ, look
φ)
(2.5.2a,
ley-James flow is probably
simplest
dynamo,
doesn’t
like convectiv
l = ∇ ×the
l (r,
l =
l (r,
which are non-axisymmetric. The Kumar-Roberts flow is more complex,
• Gubbins and co-workers have carried out
17 extensive tests of the
steady flows: u = !0 t0 + !1 s0 + !2 s2c + !3 s2s
(2.5.5
1
2
2
2
82
D. Gubbins
Meridional
Spiralling
Convection
c means cos 2φ and 2sDifferential
meansRotation
sin 2φ. The
last Flow
two terms
make
theCells
flow nonaxisymmetric, s
for the simpler, axisymme
ke a convective flow.
tical with the αω equation
bins et al.Flow
studied these flows for a range of ! values. Various radial dependence
also Roberts
considered
et al. 1966;
1972b
Streamlines:
This
choice
of
flow
has
efine the relative energy in the flow as D + M + C = 1, where D is differential rotation energ
of
, determined by !0 , M is energy of meridional circulation, measured by ! 1 dynamo:
and C the
is symmetry
convectio
are present). The present g
rom other two terms. They then vary D and M to see which effects give dynamos
at there
any isRm
symmetry, and
pal
ngly, there are large areas
where
no
dynamo
occurs
at
any
Rm.
Due
to
flux
expulsion.
(a) Section
(c) Section
Meridional
Meridional(b)Section
Equatorial
same symmetry has persis
" i are independent becaus
Figure 1. Fluid flow defined by eq. (5). (a) contours of v φ , controlled by paformations φ → φ + π /2 a
rameter " 0 , in meridian section (b) streamlines of meridional circulation (" 1 )
four combinations:
in meridian section (c) streamlines18
of convection rolls (" 2 , " 3 ) in equatorial
section.
("0 "1 "2 "3 ), ("0 "1 −"2
Implication of kinematic dynamo studies for
9.6.3 Results of parameter studies(−"
0 "1 "2 − "3 ).
sphere, because they optimize the helicity. The wide variety of radialFor time-periodic flows the solutions are
(Dudley
& James
time. Provided
we integrate
for 1989).
long enoT
function emphasizes the arbitrary nature of theD kinematic
problem
(Axial dipole for lowest Rm)
mode
will
ultimately
dominate
(the
slowe
because
of
these
symmet
Q (Axial
quadrupole
for lowest Rm)
but it is, in fact, no more or less arbitrary than
the use
of vector
D (Equatorial dipole for
nates
if
there
are
no
growing
modes).
lowest
Rm)
sign under reflection in We
th
M= Fraction of flowharmonics, which have no better case for admission other than that
by
restarting
the time integration after eac
Energy in meridional
angle π about the coordina
they satisfy the solenoidal and continuity conditions. We seek rocirculation.
initial solution is time-stepped for one peri
by theirtoleading
poloidalintv
bust, generic results that should be independent of the particular
solution orthogonal
the first. Repeated
D= Fraction of flow
(led
l = 1, m = 0), axial
ization
eachby
cycle
choice of radial function, but this can never be guaranteed. Dudley
Energy in differential
White
lines: after
oscillatory
solns soon leads to a set
represents
the
fastest
(orm
slowest
rotation.
dipole
Dgrowing
=1
e (l = 1,
& James (1989) review most of these flows.
method is fast
and
requires
very
little
memo
m = 1) [notation of Holm
The most intensive study of a single class of flows has been conmust store a large banded matrix. This mat
the higher
wavenumbers
a
ducted for generalizations of the Kumar–Roberts dynamo (Kumar
method is described
further
in Willis & Gu
&
Busse
(1998)
and
Sarso
& Roberts 1975) itself a development of the failed Bullard–Gellman
on the case " 2 = " 3 to opt
dynamo. The flow is defined as:
3 R E S U LT S
Changing the sign of R m
2s
v = "0 t 01 + "1 s 02 + "2 s 2c
+
"
s
,
(5)
3
2
2
combines
3.1 Dynamo
action with R m → −R
" 1 " 2 " 3 ).have
If two
we opposing
set " 2 =
Dynamo practitioners
ex
with radial functions defined by
(From Gubbins,
2008)
values
andworks
still as
explore
th
obvious
choice
of
flow
a dynamo
1954; Lilley
or that
almost all suffic
t10 (r ) = r 2 (1 − r 2 ),
by1970),
the two
parameters
"0 a
act
as
dynamos
e.g.
Roberts
(1972a).
The
R
must
be
defined
con
0
6
2 3
m
s2 (r ) = r (1 − r ) ,
flows
were,!therefore, studied to determine
• In many parts
of
parameter
space
NO
DYNAMOS
are
found
tween different flows, par
2.4 Dynamo2 action
of two-parameter family of flows. Blank squares
dynamos. Over half of the flows with " 2 = "
2
)
=
r
(1
−
r
)
cos(nπr
),
s22c (rFigure
regime
with more
do not support dynamo action. Greyscale gives solution symmetry with
of one symmetry
or
another;
manybalance
generate
• In regions 2swhere
solutions
with axial/equatorial dipole symmetry
are
c
4cm . White 2lines
2 are where oscillatory solutions have been found
lowest
R
even 3 with different
Rm ; and a fewregime
generate
(6)
The Braginsky
s2 (r ) = r (1 − r ) sin(nπr ).
to oscillatory
solutions inare
the Braginsky
limit.where
After Gubbins
c
found, smallercorresponding
embedded
regions
found
oscillatory
solutions
with identical
R
—for
example
the
lyi
m
rotation,
so one hasones
to be
Gibbons (2002).
The &
first
harmonic represents differential rotation about the coordibetween zones in Fig. 2.
the small but essential ra
can occur.
of the diamond in Fig. 2 hav
nate axis (which is implicitly assumed to be also the rotation axis),The sidesflow.
the cannot
first comprehe
are axisymmetric.InThey
generate D
where K is a finite non-zero constant. Eq. (13) describes quadratics
the second meridional circulation, and the last two convective overof
this
class
of states
flowsthat
outs
in (D, M) parameter space (Fig. 2) passing through each of the limit
of Cowling’s theorem, which
no
turn. These three constituents of the flow are shown in Fig. 1; in
a
a
e
rotating bodies whose dynamics are largely determined by
natural dynamos in that planets and stars are usually rapidly
from the driving mechanism, experiments differ also from
Lecture 9: Self- exciting dynamos
a spherical container.
number. The
in Cadarache
usesdisk
a cylindrical
instead of
9.1 experiment
Simple example
1: The
dynamo
vored flow because it has the smallest critical magnetic Reynolds
9.2 The
inductionMadison,
equation
magnetic
energy
evolution
the experiments
in Cadarache,
andand
Maryland.
e) is the
faof approximately
. c,d,e) Different flows under consideration for
9.3 Simple example 2: Stretch-twist-fold dynamo
each cell. The entire vessel has a diameter of about
and a height
Karlsruhe9.4
experiment.
The arrows and
indicate
directions within
The Alpha-effect
theflow
Omega-effect
1:1.68:3:25. The diameter of the return flow pipe is
. b) The
Mean
field
dynamos
and outer9.5
pipes
and the
length
of the inner pipe stand in the ratio
The drawing
notspherical
to scale. Indynamos
reality, the radii of the inner, central
9.6 is3D
Fig. 1. Sketch of several dynamo experiments. a) the Riga dynamo.
9.7 Experimental dynamos
(c)
(d)
9.8 Summary
(e)
9.7.1 The Riga Dynamo
1800-2500rpm
(a)
(b)
100kW
100kW
mobile 3D
Hall sensor
120C - 200C
reference coil
fluxgate
pressure sensor
small coil or
Hall sensor
15 fixed coils or
3 mobile coils
Na
2qm
"#$%&'(%)*!+%*,$
!
!
!
0.6 - 0.85 qm/s
• 3 coaxial stainless steel cylinders, each 3m long.
Astron. Nachr./AN 323 (2002) 3/4
• Liquid Sodium accelerated downwards by a propeller, resulting in
helical flow, returns to top by vertical flow (Ponomarenko dynamo)
• Achieved self-excited (oscillating) dynamo in 2000 for Rm~20.
Na 120°C
The Karlsruhe
Dynamo
9.7.2 The Karlsruhe Dynamo
, Phil. Trans. Roc. Soc. L, 421 (1972)
hys. JRAS, 42, (1975)
s. Lett. A, 226 (1997)
, Study Geoph. Geod., 42 (1998)
ieglitz, Naturwissenschaften, 87 (2000)
• Consisted of an array of 52 stainless steel spin generators, each with
a central cylinder where flow is vertical and an outer cylinder where
flow is helical. Liquid Sodium is pumped through pipes.
• Self-exciting dynamo achieved in 2000. Agrees well with prediction
of mean field theory.
9.7.3 The Von-Karman Sodium Experiment
• Cylinder of 100L of liquid sodium driven by rotating propellers: flow
less constrained- turbulent fluctuations important!
• Surrounded vessel by liquid sodium at rest and used Fe propellers.
• Achieved self-exciting dynamo in 2006 at Rm ~33.
• Depending on relative rotation rates of propellers, many dynamical
regimes are possible including, reverals, intermittency and bursting.
Velocity fields can be genera
to dynamo action
9.7.3 The Madison Dynamo Experiment
• 1m diameter sphere of liquid Sodium with flow driven by counter
rotating propellers.
a=0.5 m, !=107 m
• Transient field growth but no self-exciting dynamo (yet!)
9.7.4 The Maryland 3m Experiment
• 3m sphere of liquid sodium: flow driven by rotation of inner sphere.
• Still under construction, but promises Rm ~ 680!
Lecture 9: Self- exciting dynamos
9.1 Simple example 1: The disk dynamo
9.2 The induction equation and magnetic energy evolution
9.3 Simple example 2: Stretch-twist-fold dynamo
9.4 The Alpha-effect and the Omega-effect
9.5 Mean field dynamos
9.6 3D spherical dynamos
9.7 Experimental dynamos
9.8 Summary
9.8 Summary: self-assessment questions
(1) How does a self-exciting disk dynamo work?
(2) Can you describe the alpha effect and the omega effect?
(3) Can you derive the mean field dynamo equations and
solve them for simple cases?
(4) How does one determine numerically whether a 3D flow
will acts as a self-exciting dynamo?
(5) How is self-exciting dynamo action studied experimentally?
Note: Everything in this lecture ignored the influence of the
magnetic field on the fluid motions.... not fully self-consistent!
Next time:
Core dynamics: Rotation, Convection and the Lorentz force.
References
- Davidson, P. A., (2001) An introduction to magnetohydrodynamics, Cambridge
University Press.
- Gubbins, D., (2008) Implications of kinematic dynamo studies of the geodynamo.
Geophysical Journal International, Vol 173, 79-91.
- Love, J. J, (1999) Reversals and excursions of the geodynamo. Astronomy and
Geophysics, Vol 40, 6, 14-19.
- Moffatt, H.K., (1978) Magnetic field generation in electrically conducting
fluids, Cambridge University Press.
- Roberts, P.H., (2007) Theory of the geodynamo. In Treatise on Geophysics, Vol 8
Geomagnetism, Ed. P. Olson, Chapter 8.03, pp.67-102. (especially section 8.03.2).
Appendix A: The significance of alpha
• To illustrate the significance of α we shall derive its form for a simple
case when the small scale flow takes the form of a circularly
polarized wave travelling in the z direction:
(10)
u! = u!0 (sin(kz − ωt), cos(kz − ωt), 0)
• We consider that B0 is constant and // to z. This is possible since
is independent of B0 . Further, we assume U0 = 0. Then (7) is:
∂b!
= (B0 · ∇)u! + ∇ × [u! × b! − < u! × b! >] + η∇2 b!
∂t
α
(11)
• One can show (11) is satisfied with flow (10) provided
B0 u!0 k
(ηk 2 cos γ − ω sin γ, −ω cos γ − ηk 2 sin γ, 0) with γ = (kz − ωt)
b = 2
2
4
ω +η k
!
3
−B0 u!2
0 k η
u ×b = 2
(0, 0, 1)
• Therefore,
ω + η2 k4
3
u!2
!
0 k η
!
B0 so that
and E =< u × b >= − 2
ω + η2 k4
!
!
3
u!2
0 k η
α=− 2
ω + η2 k4
Appendix A: The significance of alpha
• To interpret this result, we note first that the helicity density
flow is
!
!
!2
H
of the
H = u · (∇ × u ) = ku0
• Therefore α is directly proportional to the helicity density:
k2 η
α=− 2
H
ω + η2 k4
!
!
• So, in this example, when u is a circularly polarized wave then b
is also a circularly polarised wave, but is phase shifted relative to u! .
• The resulting u! × b! is uniform and parallel to B0 . It is the
!
!
phase shift between u and b due to the diffusivity η which leads
to the non-zero value of α .
• Although this derivation is rather special, similar results hold for
a random superposition of such waves.
• It is however very difficult to derive the form of α in general and its
form is often chosen in an ad-hoc manner to match observations.
Appendix B: Axisymmetric Mean Field Dynamos
• Several simple and physically interesting mean field dynamos arise
in the case when the mean field is axisymmetric. In this case it is
convenient to work in cylindrical polar co-ords (z, s, φ) and write:
! + Bp (s, z) where Bp = ∇ × (A(s, z)φ)
!
B0 (s, z) = B(s, z)φ
! + Up (s, z)
U0 (s, z) = U (s, z)φ
• In this co-ordinate system the mean field induction equation for the
! direction is:
evolution of the B component in the φ
! "
!
"
! "
U
1
B
∂B
2
+ s(Bp · ∇)
+ (∇ × E)φ + η ∇ − 2 B
= −s(Up · ∇)
∂t
s
s
s
!
• The remaining component associated with Bp = ∇ × (A(s, z)φ)
can be uncurled to yield an equation for the evolution of A:
∂A
= −s(Up · ∇)
∂t
!
"
! "
1
A
+ E φ + η ∇2 − 2 A
s
s
where as before we assume:
E =< u! × b! >= αB0
Appendix B: Axisymmetric Mean Field Dynamos
∂B
= −s(Up · ∇)
∂t
!
B
s
"
+ s(Bp · ∇)
!
U
s
"
!
"
1
+ (∇ × E)φ + η ∇2 − 2 B
s
!
"
! "
1
A
∂A
+ E φ + η ∇2 − 2 A
= −s(Up · ∇)
∂t
s
s
Terms associated with alpha-effect
Terms associated with omega-effect
• To achieve self-exciting dynamo action must have continuous cycle of:
! to B φ
!.
(i) Magnetic Energy transferred from Bp = ∇ × (A(s, z)φ)
!.
! to Bp = ∇ × (A(s, z)φ)
(ii) Magnetic Energy transferred from B φ
• The most important terms in transferring energy can be used to
classify the form of dynamo action:
Eφ
(∇ × E)φ s(Bp · ∇)(U/s)
α2
αω
Yes
Yes
No
Yes
No
Yes
α2 ω
Yes
Yes
Yes