Dynamos in Astrophysics
See Chapter 13 of Kulsrud for
more details
Some Fundamental Questions
• Where do magnetic fields come from?
• How can seed fields be amplified?
• How can fields avoid decaying?
• B fields are everywhere:
planets, stars, interstellar medium,
galaxies, intracluster medium, and
maybe in the intergalactic medium too.
Decay and Generation Times
• Dominant, resistive decay time for B:
• Td = 4L2 / c
• Since L is large in astrophysical systems and usually
isn’t, it takes a long time for field decay: 1010 yr for stars,
1026 yr for galactic disk
• So, if fields are primordial, they can last the age of the
star or the universe
• However, Td, = 105 yr, so somehow the earth’s field
must be regenerated
• Also, simple production times are also ~ Td so
generating stellar and ISM fields could be very time
consuming!
Velocities can Drive Dynamos
• Shearing velocities can yield smaller zones and
shorter timescales for field lifetime (or generation)
• Solar polarity reversal shows that even for stars the
effective L is smaller than the whole radius
• Resistive MHD eq. for B: B c 2 B
(v B)
t 4
For the earth, the last term has Td ~105 yr, so the first
term is the dynamo term and requires:
v ≈ R/Td ≈ 3x108cm/3x1012s ≈ 10-4 cm/s
to create something close to a steady state
The Earth’s Dynamo
• The outer part of the earth’s core is molten iron/nickel
while the inner part is solid: total dimension ~ 3500 km
• Heat that keeps the outer core liquid comes from:
1) Phase transition at solid/liquid boundary
2) Radioactive decay, mostly from U and Th
3) Remnant heat of formation: collision of
planetesimals
This provides enough energy to drive a dynamo with
velocities of 10-4 cm/s through convection in outer core
How can one find a velocity field that causes the two
terms in MHD eq. for B to balance?
Earth’s Differentiated Interior
Interior Density and Temperature
Cowling’s Anti-Dynamo Theorem
Says: one cannot find a 2-d velocity field that produces
a steady-state (Cowling, 1934). Doing so is
equivalent to finding time independent solutions of
both Ohm’s law and the induction equation:
vB 1 B
E j E
c c t
If the terrestrial magnetic field is symmetric then there has
to be a place where the poloidal field vanishes.
Ex: If B is up-down symmetric about the equator then
radial component, Br = 0, in equatorial plane.
E.g: If B points up in vacuum outside sphere, because no
net flux can cross plane, B must be down near earth’s
axis.
Heuristic Proof of Cowling’s Theorem
• I.e., at some point N, Br = 0 and therefore Bz must
change sign.
• Hence poloidal field lines must surround the point N.
• This implies the existence of a toroidal current, j
• Toroidal component of Ohm’s Law at N, where B = 0:
E (1/c)(v B) E j 0
Thus E can’t be 0 at N.
But the induction eqn says 2rN E is the time rate of
change of the poloidal flux threading the axisymmetric
circle through N.
However, since B is time independent, the flux is also:
A CONTRADICTION.
Dynamos Need 3-D Velocity Fields
• Parker (1955) was the first to produce quasi-realistic
non-axisymmetric velocity distributions with
qualitative solutions for the earth’s B field
• The - mean field dynamo theory was introduced
by Steenback, Krause & Radler (1966) and solutions
of these equations supported Parker’s picture.
• This allowed models of solar-dynamo driven in its
outer convective zone and for dynamos in galactic
disks that could generate fields on astrophysically
sensible timescales.
Fundamental Types of Dynamos
• SLOW • FAST
• Field is sustained • Create B field on time
against resistive decay, short w.r.t. Td
with Td 0 ; westward BT
• HARD PART: getting Bp from BT
• Parker showed rotating convection cells can do this:
• Assume pure toroidal field toward E (as in N hemisphere)
• As flow converges toward axis at bottom of a cell, fluid
must rotate faster, i.e., in same direction as earth,
counterclockwise
• But horizontal inward flow in cell goes to right of cell axis
and upward
• Loops of flux form as viewed in the upward-north plane
because of twisted loop
Summary of Parker’s Model
• Start w/ poloidal field (e.g., earth’s dipole), upward
outside and downward inside the core
• Differential rotation velocity streches the field to make
toroidal field to the east (N hemi) and to west (S)
• Combination of convection and Coriolis force twists
the toroidal field into poloidal loops: each of them w/
same sign and reinforce the original poloidal field
• So, if convective flow velocities are of the right
amount then the poloidal field can be reinforced at
the right rate to counteract resistive decay, producing
a steady-state.
B Generation, Stabilization & Oscillation
• But, if vconv is too big, then B would keep growing
• This self-limits, however, because B would get strong
enough to affect the convective flows and patterns:
they would slow to the point where steady-state is OK
• If convection cells + Coriolis forces produced a cell
rotn > 180 deg then the original field would be
weakened, rather than reinforced.
• This should lead to an irregular field reversal because
the feedback of the field on the convection is so
complex. Models of the earth’s dynamo suggest
chaotic timescales ~105 yr, which is in accord with
magnetic reversals seen in rocks laid down near mid-
oceanic ridges.
Plate Tectonics: Seafloor Spreading
Magnetic Reversals
Date Oceanic Crust
Magnetic Fields are Frozen
in rocks: as N and S
poles move and switch,
these fields can date
when rocks solidified
from magma.
Reversals occur every few
hundred thousand years
but are not regular
SOLAR ACTIVITY: Powerful
• Spectacular activity: PROMINENCES, FLARES and
CORONAL MASS EJECTIONS
• These can extend to 100,000 km or more into the corona.
• Typically large amounts of matter following magnetic field
lines.
• Big flares yield lots of COSMIC RAYS (mostly protons)
moving close to the speed of light.
• Cosmic Rays can penetrate to the earth's atmosphere,
yielding spectacular auroral displays, power grid failures
and disrupted communications.
Solar
Prominences
UV image from
SOHO
Cooler (dark) and
hotter (bright)
emissions from
TRACE.
The big
prominence is
over 100,000 km
long
Solar Flares
• More powerful than
prominences, flares
are explosions that
only take a few
minutes to erupt; gas
escapes from
magnetic confinement
• Spots (visible)
+photosphere (UV)
+magnetic loops
(EUV)
Solar Flare Movie
QuickTime™ an d a
Sorenson Video deco mpressor
are need ed to see this p icture .
Coronal Mass
Ejections & Coronal
Holes
SOHO Yohkoh
Mundane Activity: SUNSPOTS
• These proved that the Sun rotates differentially (faster at
equator), and is therefore a fluid.
• Mean sidereal Period for the Sun is about 26 days.
• Sunspot number fluctuates, reaching a maximum every 11
years.
• At minima, spots are further from the equator, and get
closer during maxima.
Sunspot Group and Closeup
Sunspot Cycle
Sunspot Properties
• Magnetic polarities of spots reverse every 11 years so that the
FULL SOLAR CYCLE is 22 years long.
If N hemisphere leading spots now are N poles,
the N hemisphere trailing spots are S poles,
the S hemisphere leading spots now are S poles,
the S hemisphere trailing spots are N poles,
but 11 years from now the polarities are opposite.
• Sunspots are dark because they are cooler (roughly 4000 K
instead of 5760 for the rest of the photsphere).
This means their powers (proportional to T4) are roughly a
quarter as large so they are dark only in comparison to the
surrounding bright surface.
• Sunspots are cooler because their strong magnetic fields
(typically 300 Gauss vs 1 Gauss in the rest of the photosphere)
inhibit convection.
Magnetically Linked Spots
Formation of Sunspots: Magnetic
Field Gets Wound Up & Amplified
Production of Magnetic Fields
Require
• Rotation (and, almost always, convection too)
• Fluid (liquid, gas, plasma)
• with magnetic properties:
ionized hydrogen for Sun,
metallic hydrogen zone for Jupiter and Saturn,
molten iron (outer core) for Earth.
Idea of Mean-Field Dynamo Theory
• Get time development of magnetic field from statistics
of velocity field
• Key assumptions:
– Turbulent scales small compared to large scale B
– Turbulent velocities have short correlation time
– Simplify to statistically isotropic velocities and
incompressible fluids
– Allow statistics to be noninvariant under
reflections; this means cyclonic flows are OK
(needed as B is pseudo-vector and can’t be
changed by a velocity field w/ statistics invariant
under reflection)
Mean-Field Dynamo Theory: Outline
• Incompressible fluid at neighboring points r and r’ at
times t and t’.
• Ensemble average tensor product of v=v(r) and
v’=v’(r’) over all positions differing by =r-r’ and =t-t’
• This velocity correlation function depends only on
these differences and is invariant under all rotations
but not under reflections
• The most general form of such a correlation is:
v A(, )I B(, ) C(, ) I
v
Since the correlation is obviously even in , A & B are
even in , while C is odd.
Mean-Field Dynamo: Physical Meaning
• Assuming A, B, & C depend only on and
• But only true locally and usually vary w/ position on
larger scales
• A & B represent Parker’s convection cells
• C gives the rotation of the cells via Coriolis force;
• E.g., C represents the cyclonic feature of convection
• The extent to which a poloidal field is generated is
the extent to which cyclonic rotations exceed anti-
cyclonic ones
• Also, in Parker’s theory, C varies slowly w/ position,
since motions at the bottom of the convective cell
have the opposite sense to those at its top
MDF: Initial Physical Results
• So, in N hemisphere, for an upward moving cell
(along x) we find that at its bottom the average of
yv’z- zv’y >0, representing counterclockwise cyclonic
motion. Since vx>0 this implies C>0.
• At top of that cell yvz- zvy 0 still, so C0 at
bottom (still). [In S hemisphere C has opposite sign]
• Key point: C must change sign to allow poloidal flux
generation
• This is correct parity to produce the net toroidal field,
since it reverses between the hemispheres
MFD: Mathematical Results
• Derivation is fairly messy, just quote result:
B c 2
(V B) ( B) ( ) B
t 4
Here V is the mean (basically rotational) velocity
and
2 C(0, )d A(0, )d
0 0
Physically, is the turbulent mixing term; often called
turbulent resistivity: convection cells mix up + and -
lines of force, reducing the mean field.
MFD: Physical Interpretation
• Note that turbulent mixing can’t actually destroy
magnetic energy and if there’s enough resistivity the
fluctuations will be destroyed: the slow dynamo case
• But, if is small the term can produce a big random
field deviating from the mean field
• Fast dynamo: >> c/4 so can neglect the c/4
term in the MFD eqn and we are dealing w/ an ideal
fluid so flux must be conserved by MFD theory
• Conceivable that if flux if mixed very finely magnetic
reconnection can further merge + and - fields, thus
destroying magnetic energy
• But this reconnection shouldn’t be a problem on large
scales, such as the galactic disk
MFD: Final Slide!
• There are more physical meanings for and than
their expressions in terms of integrals of correlation
function pieces. Let c be an effective correlation time
for A defined via: 0
A(0, )d A(0,0) c
1
is related to the kinetic helicity: c v ( v)
3
1
& to a random walk for fluid elements:
c v2
3
This is because x2=(vx c )2 (t/ c) = (1/3)v2t= t
and is related to the
amount of rotation multiplied
by the height of a convective cell: z= t