THE SOLAR DYNAMO
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October 27, 2010
Introduction:
The astronomes assume there is a dynamo eect but they can-
not fully explain the activity of the sun. The unexplained phenomena are:
ˆ
the cycle of 11 years of the sun
ˆ
the maunder minimun
ˆ
the buttery diagram
ˆ
the Joyle's law
Jospeh Larmor,in 1919 was the rst physicst who suggested the sunspot of the
sun was created by a dynamo.(he suggested thereafter to assume this is the
same process who created the Earth's magnetic eld).
The solar dynamo is the phenomenon by which the magnetic elds of the
sun are generated through the interaction of the eld with convection and rotation.There are two mechanism assiociated in dynamo activity: the omega eect
and the alpha eect. When the uid stresses dominate the magnetic stresses,
the shear ow stretches the magnetic eld line in the direction of the shear: this
is the omega eect. The alpha eect occur when the helical ow lift and twist
eld lines into orthogonales planes.
The precise mechanism by which the sun generated its eld remain poorly
understood despite the fact there is a lot of theories about this mechanism.
First we will see the buttery diagram and uid dynamic properties,then we
will present the dynamo theory and the two eects involved.
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1 The buttery diagram and uid dynamic properties.
1.1 The Buttery diagram.
The observation of the sunspot (and solar activity) show that solar activity
waxes and wanes with an 11-year Cycle. The number of sunspot rise rapidly
from the minima (near to zero) to maxima 3 or 4 year later.The activity of
the suns is slower during the lastest years of the cycle. Most measures of solar
activity show this asymmetric rise and decline but exhibit large variations from
one cycle to the next.
There is a minimun called Maunder Minimun of 1645 to 1715. During this
minimun the sunsport cycle seems to have ceased entirely. This non-linear (and
chaotic) behavior suggest that the dynamo is not a simple oscillatory process.
If the activity seems to be a chaotic phenomenom, the sunspots do not
appear radomly over the surface of the sun. In fact they are concentrated in
two latitude bands.
This observation is illustrated by a buttery diagram (gure 1).This diagramm marks the latitude at which sunspots appear each 27-day rotation of the
sun from May 1874 to June 1994.
At the beginning of the cycle the sunspot
seems to appear only in the lattitude near 30°. Then, when the cycle progresses,
the latitude bands widen and come closer to the equator where they disappear
at the next minimun.
This phenomemon is know as the Spörer's Law.
This
law suggests a presence of a ow or a wavelike propagation for the source of the
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activity.
The group of sunspots tend to occur along a mostly east-west line.These
groups usually are tilted so that the preceding spots are closer to the equator
than the following spots. This is the Joy's Law. There is another process which
suggest there is a dynamo eect.This phenomenom is not show by the buttery
diagram.
The magnetic measurement of the sun revealed the Hale's Polarity Laws.
The preceding spots in one hemisphere have one polarity while the following
spots in the other hemisphere are of opposite the polarity.The polarities reverse from one 11-year sunspot cycle to the next to produce a 22 year cycle for
magnetic activity.
These observation suggest that there is a dynamo eect.
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2 Dynamo Theory
2.1 Basic equation
2.1.1 Equation of the solar dynamo:
The basic equation of dynamo theory is the magnetic induction equation built
from Maxwell's and Ohm's law:
∂t B = ∇x(vxB) + η∇²B
(equation 1)
where:
ˆ
B is the magnetic induction
ˆ
v is the uid velocity
ˆ η
is the magnetic diusivity
The velocity and the magnetic induction are separated into mean and uctuation
part. An average of the induction equation gives the mean-eld equation that
contains a new induction term given by the average of the vectoriel product of
the uctuation velocity and magnetic induction.
At the rst order, this term is proportional to the magnetic induction and
its curl so that (the primes denote uctuation quantities):
g0 × B 0 = αB̄ − β∇ × B̄
(equation 2)
where:
ˆ
alpha is proportional to the helicity in the uctuation velocity eld
ˆ
Beta is proportional to the eddy diusivity
If we use the spherical polar coordinates (r
θ φ)
equation 1 can be writtend in
Bφ
B ρ =(Br ,Bφ )=∇xA where A is the vector potential.
terms of the mean toroidal (azimutal) component of the magnetic induction,
and the poloidal component
Now we have a pair of coupled equation with:
ˆ ∂t Bφ + (U ρ.∇)Uφ = (Bρ. ∇)Uφ + α∇xBρ + β∇²Bφ
(equation 3)
and
ˆ ∂t Aφ + (Uρ .∇)Aφ = αBφ + β∇²Aφ
(equation 4)
where:
ˆ
U is the mean uid velocity of the meridional ow,Uρ and dierential
rotation
Uφ
If we dont consider for the moment the meridional ow, equation 4 show us
that the poloidal eld is produced by the nonaxisymmetruc helical motions.
The equation 3 show us that the toroidal eld is produced by both the
and by
ω -eect
α-eect
in which the poloidal eld is stretched out by the dierential
rotation. The relative strength of these dierent terms determines the nature
of the resulting dynamo.
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2.1.2
α-eect
and ω-eect.
ˆ α-eect
The a-eect is the fact that the lines of the magnetic elds are reversed and
twisted which is caused by solar rotation.
ˆ ω -eect
The magnetic eld inside the sun are stretched by the dierential solar rotation.
2.2 interpretation of equations.
ˆ
If U<<α-eect
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The omega term is eliminated from the equation 3 and we have and α² dynamo
which depend only on the nonlinear
α
eect. These kinds of dynamo tend to
produce steadily growing elds.
ˆ
If U>>α-eect
We obtain and
αω -dynamo.
These dynamo produce oscilattory wawes that
propagate at right angle to the shear ow. Their propagation toward the poles
or toward the equator depend on the sign of a and the direction of the velocity
shear.
ˆ
If
α
and
ω -eect
are equal
We obtain now anα²ω -dynamo. These dynamo also produce oscilattory behavior.
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3 Two model of the solar dynamo
3.1 αω-dynamo
The kinematic dynamos model has been constructed from these equations by
taking the specic rotation prols of the sun and a functional form for aplha.
The dynamo produced in the 1970s reporduced many characteristics of the solar cycle and specially dynamo repect the Spörer's Law. These were the
dynamo.
αω -
The dierential rotation of the sun takes a poloidal magnetic eld
and shears it to produce a stronger toroidal eld below the surface. Then this
toroidal eld is lifted by the alpha efect to produce a poloidal eld of reversed
polarity. The main problem with this model is to respect the 22-year period.
The eect produced by the dynamo must be diminished, if not very short cycle
result.
Moreover the rotation proles used don't agree with the helioseismic
proles.
3.2 Magnetohydrodynamic dynamo
This model was suggested in 1980s. This model takes the equation of motion
and the induction equation and then calculates numerically both the velocity
eld and the magnetic eld. In this model, the convection itsel produces both
the dierential rotation for the omega eect and the helicity for the alpha eect.
However the calculate eld do not aggree with the observation. While the alpha
eect has the expected sign, the dynamo waves propagate toward the poles
contrary to Spörer's Law. Moreover this dynamo also had short cycle periods
due to the large magnitude of the alpha eect.
3.3 Conclusion of the two models.
A big problem shared by both types of model is the nature of the internal rotation as determined by helioseismology. If the magnetohydrodynamical models
produce surface roation proles in agreemebt with observations, the internal
proles disagree. For the kinematic models, the internal proies assumed to be
present disagree with the observation.Ananother problem share buy this two
models is that the magnetic ux tubes should be buoyant and not remain in the
convection zone enough for the uid motion to work on them
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Conclusion:
There is no model who can satisfy the observation.The key of
the solar dynamo problem is the understanding of the convection zone.
The
soho mission gives a lot of information and observation to the scientist but we
still must understand the phenomena in the convection zone.
Humanity has began to observe the sun since 2 hundred years. If we compare this periode with the life time of the sun, this is in fact a very short period.
The sun can have very long super-cycle which cannot observe by humanity,but
recent advances suggest that the solution to this solar dynamo may be forthcoming.
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Sources:
"Sur la dynamo uide : étude de la dynamo de Ponomarenko"
par Pascal Lambert
http://www.oca.eu/vigouroux
www.astrosurf.com
www.wikipedia.fr
http://solarscience.msfc.nasa.gov
All the pictures com from astrosurf.com
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