Download Lecture 11: Solar and stellar dynamos I: basic concepts

Lecture 11: Solar and stellar
dynamos I: basic concepts
During Lecture 10 it became evident that the stable, non-oscillatory, rotation-independent
magnetic fields of early-type stars are most likely of fossil origin. It was also stated that
for late-type stars, fossil fields may contribute only to weak fields in radiative zones,
while a dynamo mechanism is needed to explain the overall magnetism of these stars.
The final proof for dynamo origin of the magnetic fields in late-type stars was, however,
postponed until the present lecture.
We begin with some important facts about convection discussed in Lecture 7: MLT
was used to derive an estimate of the turbulent diffusivities νT and ηT ≈ 13 ul ≈ 108 −
109 m2 s−1 in the solar convection zone. Let us now compare these values to the molecular
diffusivities estimated for solar temperatures in Lecture 10 (1 − 103 m2 s−1 ). If we now
recall the definition of the diffusion timescale τd (Eq. 1), we can immediately conclude
that an increase of magnetic diffusivity by 5 orders of magnitude results in a decrease of
the diffusion timescale from the 1010 years down to 105 years. Evidently any magnetic
field entering a stellar convection zone will be destroyed by turbulence in a short timescale
compared to the stellar lifetime (even for the most massive stars with short lifetimes).
Cowling’s antidynamo theory, however, states that no axisymmetric velocity field
(like the global flow pattern of the Sun) can act as a dynamo for axisymmetric magnetic
field (such as the solar global magnetic field); let us now look into Parker’s (1955) idea
of cyclonic convection, and how it can act as a remedy for solar dynamo.
Cyclonic convection
Convection in the solar or stellar convective zones occurs in the form of hot, rising,
diverging bubbles, and cooler, converging, decending ones that dive towards the bottom.
The diverging and converging motions are due to density stratification, and thanks to
it, purely radial motion is transformed into latitudinal and azimuthal velocities. Let us
first investigate what happens in the situation that gravity and rotation are parallel, i.e.
in the poles of the stars. Let us now assume that we sit on the North pole of the star,
where rotation is anti-clockwise and gravity directed antiparallel to the rotation vector
(Fig. 7). Density is, therefore, decreasing along the rotation vector; the Coriolis force,
2Ω × u is acting on the diverging flow producing an clockwise twist on the fluid. Let us
now add in a toroidal magnetic field, and assume that the Reynolds number of the fluid
is high so that the magnetic field can be considered to be frozen in the fluid. The rising
bubble will, therefore, carry the toroidal field with it forming a loop that becomes twisted
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Kuva 7: Loop of magnetic field created by a cyclonic convective event from Parker (1970),
Fig. 1.
clockwise due to the cyclonic motion. For a twist angle between zero and π, a current
parallel to the magnetic field will be created, for angles between π and 2π, the current is
antiparallel. In the situation after twist by π/2, depicted in Fig. 7, the magnetic field of
the loop lies exactly in the meridional plane, i.e. the magnetic field of the loop is purely
poloidal. Let us now recall Lecture 10, where we derived the time evolution equation of
the toroidal (B) and poloidal (A) fields separately in the presence of axisymmetric nonuniform rotation and/or meridional flow pattern, and concluded that it is impossible
to generate poloidal field from the underlying toroidal one replenished by differential
rotation. Cyclonic convection, however, seems to be capable of acting as such a source.
Cyclonic convection is one example of flows that are helical. Helicity is defined as
H=
Z
u · (∇ × u) d3 r,
(14)
V
measuring the deviation of the motion from mirror symmetry. For the cyclone of Fig. 7,
helicity is negative. An incompressible flow at high Reynolds number conserves kinetic
helicity analogously to ideal MHD conserving magnetic helicity (for details, see Lecture
9).
Exercise 1: Visualise helicity by mirroring the flow. Compare helicity produced by cyclonic flow at different poles of a star.
A large amount of such loops, taken that their lifetime collectively is short enough,
will evidently create a large quantity of parallel currents and poloidal magnetic field loops
on the scales of the loops themselves. Convection is obviously a small-scale phenomenon,
so the cyclonic action generates small-scale poloidal field. Only if there is enough of
turbulence in the surroundings of the generated loops, can they be reconnected to form
larger and larger loops, finally arriving at the same length scale as the toroidal magnetic
field has.
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Kuva 8: This image visualises the helicity ω · u in a small region of a 5363 simulation of
helical rotating turbulence. The alignment of the two vectors are represented with different colour: green means negative and blue-magenta positive alignment). The Reynolds
number for the simulation is 5600 and the Rossby number is 0.06. Image courtesy of
Pablo Mininni, NCAR.
Mean-field theory
The mathematical basis for describing Parker’s cyclonic convection idea was developed
during the 1960’s in Potsdam by Steenbeck, Krause & Rädler (1966) using the meanfield electrodynamics approach (in Lecture 8 the mean-field approach for Navier-Stokes
equations was described).
We divide both the velocity and magnetic field into the mean U , B and fluctuating
u, b components, and require that the mean fields vary slowly in time over a timescale L
whereas the fluctuating components having zero means vary over a much smaller length
scale l. In other words, there exists a significant separation of scales between the mean
and fluctuating fields. The averages are defined as volume and time (so called ensemble)
averages. Substituting these expressions to the induction equation, averaging (Reynolds
rules applicable), and manipulating gives us the equations for the mean and fluctuating
magnetic fields
∂B
∂t
∂b
∂t
= ∇ × (U × B + E − η∇ × B),
(15)
= ∇ × (U × b + u × B + G − η∇ × b),
(16)
where
(17)
E = u × b,
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(18)
G = u × b − u × b.
The first turbulent correlation is called as the electromotive force (emf ) playing a key role
in mean-field dynamo theory. The ultimate goal is to express this term in the simplest
possible realistic form from some turbulent closure or theory, not to have to solve for the
fluctuating quantities explicitly, but make it possible to deal only with the mean-field
quantities. After a brief examination of the form of the magnetic field equations, one can
anticipate that the expression for E must depend linearly on the mean magnetic field B,
as the equation for the fluctuating field, and thereby also u × b, is linear in B. Such a
relationship can be formally expressed with
E i = αij B j + βijk
∂B j
+ ...,
∂xk
(19)
where the dots stand for possible higher order terms. In the kinematic case (the velocity
field is given), the pseudotensors αij and βijk , usually termed as transport coefficients,
depend only on the statistical properties of the flow. As long as the assumption of the
scale separation holds, convergence of the series expansion can be expected. Moreover,
every one order higher derivative should be smaller than the order lower one by a factor
of l/L ≪ 1; therefore we should expect the series to converge quickly so that we should
care only about the first few terms in it.
The detailed derivation of the transport coefficients is presented in the magnetohydrodynamics course; here we merely state the results from the simplest possible turbulence
model, i.e. homogeneous and isotropic turbulence under the First-Order Smoothing Approximation, FOSA. Under this approximation, the term G in the equation for fluctuating field is neglected, after which an explicit expression for the emf can be obtained;
this is valid only if
Rm = ul/η ≪ 1, or St = uτ /l ≪ 1.
(20)
These conditions are not generally fullfilled neither under the solar nor stellar conditions,
but anyway, it is customary to proceed. For helical turbulence, the expression for the
emf reads
(21)
E = αB − β∇ × B,
where α and β are pure scalar fields given by
α = − 13
Z
Z
t
β=
1
3
t
−∞
−∞
u(t) · ω(t′ )dt′ ≡ − 31 τc u · ω.
u(t) · u(t′ )dt′ ≡ 13 τc u2 ,
(22)
(23)
The mean-field dynamo equation now reads
∂B
= ∇ × (U × B + αB − ηT ∇ × B),
∂t
(24)
where ηT = η + β, the total magnetic diffusivity resulting from both the molecular
and turbulent effects. The impact of convective turbulence is indeed two-fold: first, the
collective inductive action of convective cells is manifested through the α-effect. Secondly,
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the enhanced turbulent diffusion appears as the β-term, the diffusivity being of the order
of magnitude derived by MLT.
Exercise 2: Based on exercise 1, estimate the α-effect in the Northern vs.
Southern hemispheres of a star. What kind of latitude dependence could be
expected?
Solar-type αΩ-dynamos
Let us now re-examine the case of nonuniform but axisymmetric large-scale flow, such
as solar differential rotation, with the inclusion of the α-effect. The fluid is assumed to
be isotropic with α and ηT being constant. To simplify things even further, let us adopt
a Cartesian coordinate system, z corresponding to radial, y to azimuthal, and x to the
latitudinal direction in the spherical coordinate system. This is more or less identical
approach to the Parker (1970) model. For axisymmetric flows, ∂/∂y vanishes, and the
shearing flow can be written as uy = (0, uy (x, z), 0). The equations for the toroidal and
poloidal components read (derived in Exercise 3)
!
∂2A ∂2A
+
,
∂x2
∂z 2
∂A
∂t
= αB + ηT
∂B
∂t
∂2A ∂2A
= α
+
∂x2
∂z 2
!
(25)
+ BP · ∇uy + ηT
∂2B ∂2B
+
∂x2
∂z 2
!
(26)
As can be seen, the inductive effect of convective turbulence enters both equations, whereas the differential rotation can only generate toroidal field from the poloidal one (customary to call this effect as the Ω-effect). For strong differential rotation (e.g. solar),
however, the α-term in the toroidal field equation is small in comparison to the term
describing the Ω-effect; therefore it is customary to neglect the toroidal α-term in solar dynamo models. This is not valid, however, in the rapid rotation regime, where the
strength of the differential rotation becomes diminished with respect to the strength of
the α-effect (α2 Ω-dynamos). The dynamo can, in fact, work perfectly well in the total
absence of the Ω-effect (α2 -dynamos).
Exercise 3: Derive the mean-field dynamo equations for the vector potential
of the poloidal field (A) and toroidal field (B) using the FOSA-emf from Eq.
(21).
Let us begin by investigating the αΩ-mechanism in more detail, further assuming
that the rotational velocity depends linearly on the radial coordinate. Substituting this
into the equations with only poloidal α source term, gives
!
∂2A ∂2A
+
,
∂x2
∂z 2
∂A
∂t
= αB + ηT
∂B
∂t
∂A
= −Ω
+ ηT
∂x
∂2B ∂2B
+
∂x2
∂z 2
(27)
!
.
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(28)
These are PDEs with constant coefficients, therefore it is instructive to seek for planewave solutions of the form e(ik·x+λt) , where λ = σ + iω is the eigenvalue vector, σ is
the growthrate, and ω is the oscillation frequency of the solution. If axisymmetric solutions are sought for, then the wavevector lies in the radial-latitudinal plane. A detailed
description of the method of linear stability analysis and solving eigenvalue problems are
presented in the MHD-course; here we merely give the dispersion relations and interpret
the meaning of them and their eigenvalues. The dispersion relation for αΩ-dynamo mechanism reads
(λ + ηT k2 )2 + iαΩk = 0,
(29)
and the eigenvalues
λ± = −ηT k2 ± (−iαΩk)1/2 .
(30)
The real and imaginary parts can be separated
Reλ± = σ = −ηT k2 ± | 12 αΩk|1/2 ,
(31)
Imλ± = ω = ±| 21 αΩk|1/2 .
(32)
Growth of the magnetic field occurs for σ > 0; these solutions are supercritical. For
subcritical decaying solutions σ < 0; marginal solution has σ = 0. The solutions are imaginary and therefore oscillatory; the oscillation frequency is given by ωcyc ≡ ±| 21 αΩk|1/2 .
The dynamo wave travels in the direction
(33)
s = α∇Ω × êy ,
i.e. along the isocontours of shear, the α-effect vs. Ω-effect determining the sign. This is
called the Parker-Yoshimura sign rule.
Exercise 4: In Exercise 2 you were asked to deduce the sign of the α-effect
on different hemispheres of a star. Apply the Parker-Yoshimura sign rule to
determine the needed radial angular velocity distribution to produce equatorward dynamo wave.
Solar kinematic dynamo models
In addition to nonuniform rotation, the rotation law derived in detail from helioseismology, the same observational technique has revealed that meridional circulation pattern also
co-exist in the solar convection zone. The circulation amplitude is roughly 10–20 ms−1
close to the surface, directed polewards. The circulation pattern most likely consists of
one cell, the flow being directed towards the equator at the bottom of the convection
zone.
Let us now try to measure the magnitudes of the different solar effects by nondimensionalising the relevant axisymmetric equations. Let us measure length with the radius
of the star, [x] = R, let [ηT ] = η0 be a typical value of the magnetic diffusivity, [B] = B0
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for the magnetic field, [u] = u0 for the velocity, and [α] = α0 (having the dimension of velocity). The unit of time can be derived from the units of length and magnetic diffusivity:
[t] = R2 /ηT . Substituting and cleaning up gives the nondimensional equations
∂A
∂t
∂B
∂t
1
Rm
uP · ∇ (sA) + Cα αB + ηT ∇2 − 2 A,
s
s
1
B
= −Rm s∇ ·
uP + CΩ sBP · ∇Ω + ηT ∇2 − 2 B.
s
s
= −
(34)
(35)
with the control parameters
Cα =
Ω0 R 2
u0 R
α0 R
, CΩ =
, and Rm =
,
η0
η0
η0
(36)
describing the magnitude of the α-effect, Ω-effect, and meridional circulation with respect
to diffusion. The angular velocity of the Sun is the best known of these numbers: Ω0 ≈
2.6×10−6 s−1 . The meridional flow amplitude is quite well-established at the solar surface
(roughly 10-20 ms−1 ), but deeper down, the flow amplitude relies on estimates from
mass conservation (≈1-2ms−1 ). The ratio Rm/CΩ ≈ 10−2 . The least known quantities
are the turbulent transport coefficients. MLT gives roughly ηT = 5 × 108 m2 s−1 for the
magnetic diffusivity. Local numerical convection models (e.g. Käpylä et al. 2006) give
the magnitude of α0 ≈ 2 − 3ms−1 . This yields the ratio Cα /CΩ ≈ 10−3 , indicating that
the solar dynamo is obviously working in the αΩ-regime.
In the next lecture we cover in more detail the capabilities of mean-field αω dynamo
models to describe the solar cycle, and some alternative approaches, such as the fluxtransport (also known as the Babcock-Leighton models) models.
Rapid rotators: α2 -dynamos
As discussed in length in Lecture 8, in the rapid rotation regime, differential rotation
is hard to maintain. From the dynamo point-of-view this means that for rapid rotators,
as the magnitude of the Ω-effect is decreasing with respect to the α-effect, the dynamo
approaches the α-dominated regime, governed by the equations
!
∂2A ∂2A
+
,
∂x2
∂z 2
∂A
∂t
= αB + ηT
∂B
∂t
∂2A ∂2A
= α
+
∂x2
∂z 2
!
+ ηT
(37)
∂2B ∂2B
+
∂x2
∂z 2
!
(38)
The dispersion relation for this system reads
(λ + ηT k2 )2 − α2 k2 = 0,
(39)
with purely real eigenmodes
λ± = −ηT k2 ± |αk|.
(40)
The instant conclusion is that α2 -dynamos are nonoscillatory. The dynamo instability is
possible for λ > 0, which is fullfilled for 0 < αk < ηT k2 . For positive αs, this corresponds
to the regime
(41)
0 < k < α/ηT ≡ kcrit ,
21
while for negative αs, kcrit < k < 0. This means that the magnetic field growth occurs
for small values of k (corresponding to large length scales).
The observations imply that rapidly rotating stars show cyclic activity, although the
activity patterns look fairly different from the solar ones (large spots at high latitudes,
flip-flops). In any case, in the light of kinematic and linear dynamo theory, oscillatory
modes are rather unexpected. Stellar dynamo theory is much less developed than the
solar one; in the next lecture the key results shall be discussed.
Nonlinearity
So far we have used the kinematic approach: the velocity fields have been assumed
to be known, and the back-reaction of the growing magnetic field on the flow has been
totally neglected. The final choice of the preferred mode and its saturation level, however,
occurs in the nonlinear regime, and sometimes the resulting nonlinear dynamo solution
is completely different than that of indicated by the linear growth rates. Dynamical
approach requires the solution of the full set of MHD equations; due to the increased
computational challenge, this regime is not yet well understood.
It is evident, however, that the magnetic field cannot grow to be infinitely large
due to the dynamo instability; the magnetic field will react back on the flow inhibiting
the its own generation mechanisms, both on large-scales (Malkus-Proctor effect) and
small-scales (α-quenching). In the solar case, the latter is more prohibitive than the
former, while for stellar dynamos, both of these mechanisms might be relevant. During
the recent years it has become understood that the suppression of the α-effect occurs
via the magnetic helicity conservation constraint (discussed in detail during the MHDcourse), causing the α-effect being very strongly suppressed in closed domains; in these
systems, dynamo action is very inefficient, and the saturation of the magnetic field occurs
very slowly. For systems allowing the small-scale magnetic helicity to escape, the α-effect
is less strongly quenched, and the saturation levels close to the equipartition strength
with turbulence are again possible in a realistic timespan. The most commonly used
algebraic α-quenching formula
αK
,
(42)
α(B) =
2
2
1 + B /Beq
therefore, remains as a valid approximate tool in the mean-field dynamo models. A more
accurate description would be to solve simultaneously the evolution equation for the
α-effect, resulting from the helicity conservation principle
dα
= −2ηt kf2
dt
2
α − αK
αhB i − ηt µ0 hJ · Bi + ∇ · F
+
2
Beq
Rm
!
.
(43)
Here αK is the kinematic value of the α-effect, Beq is the equipartition strength with
respect to turbulence, kf is the typical wavenumber of turbulence, Rm = ηt /η is the
modified magnetic Reynolds number, and F describes the helicity flux.
Bibliography
Käpylä, P. J., Korpi, M. J., Ossendrijver, M., Stix, M., 2006, A&A, 455, 401
22
Mestel, L., 1999, Stellar magnetism, Oxford University Press
Parker, E.G. 1955, ApJ, 122, 293
Parker, E.G. 1970, ApJ, 162, 665
Steenbeck, M., Krause, F., and Rädler, K.-H., 1966, Zs. f. Naturforschung, 21a, 369
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