Dynamos in Astrophysics

1
Dynamos in Astrophysics

• See Chapter 13 of Kulsrud for more details

2
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.

3
Decay and Generation Times

• Dominant, resistive decay time for B
• Td 4?L2 / ?c
• Since L is large in astrophysical systems and ?
  usually isnt, 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 earths
  field must be regenerated
• Also, simple production times are also Td so
  generating stellar and ISM fields could be very
  time consuming!

4
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

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
5
The Earths Dynamo

• The outer part of the earths 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?

6
Earths Differentiated Interior
7
Interior Density and Temperature
8
Cowlings 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 Ohms law and the induction
  equation

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
earths axis.
9
Heuristic Proof of Cowlings 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 Ohms Law at N, where B 0

Thus E? cant be 0 at N. But the induction eqn
says 2?rN 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.
10
Dynamos Need 3-D Velocity Fields

• Parker (1955) was the first to produce
  quasi-realistic non-axisymmetric velocity
  distributions with qualitative solutions for the
  earths B field
• The ?-? mean field dynamo theory was introduced
  by Steenback, Krause Radler (1966) and
  solutions of these equations supported Parkers
  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.

11
Fundamental Types of Dynamos

• SLOW
• Field is sustained against resistive decay, with
  Td lt Tlife
• Best example is the earths dynamo, which creates
  a field on the decay timescale, sustain it and
  also reverses it irregularly on timescales Td

• FAST
• Create B field on time short w.r.t. Td
• Also reverse it quickly
• Cant create net flux, but fluid motions can
  stretch field lines so twice original in one
  direction and negative original in opposite
• Expel backward flux ? doubled original!

12
Features of Fast/Slow Dynamos

• Fast can work in absence of resistivity
• Believed to create and amplify the galactic B
  field
• Solar is less clear it could be considered
  slow, as reconnection may reduce the effective
  decay time to as little as the 11 year polarity
  reversal
• But, the long decay time could be OK in the sun
  and solar dynamo must then be fast
• Mean-field theory distinguishes between these
  cases only via boundary conditions
• The physics questions are viability of
  reconnection in convection zone (?slow) or
  explusion of flux in coronal mass ejections and
  flares (?fast)

13
Outline of Parkers Model for Earths Dynamo

• 3-d Coriolis forces act on convection flows in
  outer core and drive dynamo
• Conservation of angular momentum means rising
  convective cells must have smaller angular
  velocities as they get further from earths axis
• Variable rotational velocity yields toroidal
  field, BT
• Starting w/ BT 0, we see one must develop via

Cylindrical coord ?
14
Parkers Model, 2

• Easy part in N hemisphere, B?lt0 and since ? ?1/?
  there must be an eastward toroidal field
• Develops over Td L/? so BT ? Bp ??? Td
• In S hemisphere, B?gt 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

15
Summary of Parkers Model

• Start w/ poloidal field (e.g., earths 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.

16
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 gt 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 earths
  dynamo suggest chaotic timescales 105 yr, which
  is in accord with magnetic reversals seen in
  rocks laid down near mid-oceanic ridges.

17
Plate Tectonics Seafloor Spreading
18
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

19
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.

20
Solar Prominences
UV image from SOHO Cooler (dark) and hotter
(bright) emissions from TRACE. The big prominence
is over 100,000 km long
21
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)

22
Solar Flare Movie
23
Coronal Mass Ejections Coronal HolesSOHO
Yohkoh
24
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.

25
Sunspot Group and Closeup
26
Sunspot Cycle
27
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.

28
Magnetically Linked Spots
29
Formation of Sunspots Magnetic Field Gets Wound
Up Amplified
30
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.

31
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 cant be
  changed by a velocity field w/ statistics
  invariant under reflection)

32
Mean-Field Dynamo Theory Outline

• Incompressible fluid at neighboring points r and
  r at times t and t.
• Ensemble average tensor product of vv(r) and
  vv(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

Since the correlation is obviously even in ?, A
B are even in ?, while C is odd.
33
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 Parkers 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 Parkers theory, C varies slowly w/
  position, since motions at the bottom of the
  convective cell have the opposite sense to those
  at its top

34
MDF Initial Physical Results

• So, in N hemisphere, for an upward moving cell
  (along x) we find that at its bottom the average
  of ?yvz- ?zvy gt0, representing
  counterclockwise cyclonic motion. Since vxgt0
  this implies Cgt0.
• At top of that cell ?yvz- ?zvy lt0 but vxgt0 still,
  so Clt0
• For downward moving cell, Clt0 at top and Cgt0 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

35
MFD Mathematical Results

• Derivation is fairly messy, just quote result

Here V is the mean (basically rotational)
velocity and
Physically, ? is the turbulent mixing term often
called turbulent resistivity convection cells
mix up and - lines of force, reducing the mean
field.
36
MFD Physical Interpretation

• Note that turbulent mixing cant actually destroy
  magnetic energy and if theres 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 ? gtgt ?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 shouldnt be a problem on
  large scales, such as the galactic disk

37
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

? is related to the kinetic helicity
? to a random walk for fluid elements
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