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: 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?
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: 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: 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.
SLOW Field is sustained against resistive decay, with Td < Tlife Best example is the earth’s 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 Can’t 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! Fundamental Types of Dynamos
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)
Outline of Parker’s Model for Earth’s 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 earth’s axis • Variable rotational velocity yields toroidal field, BT • Starting w/ BT = 0, we see one must develop via: Cylindrical coord:
Parker’s Model, 2 • Easy part: in N hemisphere, B<0 and since 1/ there must be an eastward toroidal field • Develops over Td ≈ L/ so BT ≈ Bp Td • In S hemisphere, B> 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.
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)
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 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.
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: 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 but vx>0 still, so C<0 • For downward moving cell, C<0 at top and C>0 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: 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.
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: 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