Download Literature Review on the Formation of Active Regions

Literature Review on the Formation of Active Regions
Sarah Jabbari
Department of Astronomy, Stockholm University
Supervisor: Prof. Axel Brandenburg
June 3, 2013, Revision: 1.65
Abstract
In this literature review, I start with the history of flux emergence studies and
in this regard, I have explained two different mechanisms that can lead to flux
concentration on the solar surface. The conventional theory is based on flux tubes
rising due to magneto-buoyancy and an alternative one is the negative effective
magnetic pressure instability (NEMPI). In this text, I describe both theories and
compare them. I then focus on the second idea, which is also of interest for my PhD
project. One of the applications of NEMPI is to use it to explain the formation
of active regions on the solar surface. I review the papers that are related to this
theory and I explain how this idea developed over the years. The most recent work
was the investigation of NEMPI with an α2 dynamo in spherical geometry. The
results shows that dynamo and NEMPI can work at the same time and they affect
each other in a way that the system changes to a coupled system. Another way to
study this new instability is to consider helioseismic signatures of NEMPI and to
find similarities between results from simulations and helioseismology.
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Contents
1 Introduction
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2 Early work on flux emergence
2.1 Early ideas about Ω loops . . . . . . . . . . . . . .
2.2 Convective collapse . . . . . . . . . . . . . . . . . .
2.3 Clustered versus monolithic sunspot models . . . .
2.4 Flux concentrations in deep convection simulations
2.5 Flux concentrations from mean-field effects . . . . .
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3 Mean-field approach in dynamo theory
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3.1 Two-scale assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Mean-field equations and α2 dynamo . . . . . . . . . . . . . . . . . . . . . 10
4 NEMPI
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4.1 Negative effective magnetic pressure . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Results from DNS and MFS . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 NEMPI versus flux tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Helioseismic signatures of flux concentrations
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5.1 Methods of local helioseismology . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 Flow field near active regions . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.3 Deep focussed time-distance helioseismology . . . . . . . . . . . . . . . . . 20
6 NEMPI and spherical α2 dynamos
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6.1 Outline of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Summary
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2
1
Introduction
There are number of phenomena on the solar surface which have direct relation with solar
activity. One of the most known ones are sunspots. Concentrations of magnetic field
cause dark areas on the solar surface which have radii between 2 to 20 Mm and life times
between one day to a few months. Their temperature is about 3000 to 4000 K, which is
cooler than their surrounding with a temperature of about 6000 K. So they look darker;
see Figure 1. In 1896, 200 years after introducing sunspots as a thermal phenomenon
by Hooke (as referred to by Ruzmaikin, 2001), Zeeman discovered the interaction of the
magnetic field with the electron angular moment (Mestel, 1999). This discovery was used
by Hale (1908) to measure the magnitude of the solar magnetic field. Since then, the
magnetic nature of sunspots unfolded. The Zeeman effect states that, if an ionized gas
is placed in the magnetic field, most of its spectral lines split into two. Therefore, as the
separation between lines is directly proportional to strength of the magnetic field, one
can measure the magnetic field at the solar surface using its spectrum of electromagnetic
radiation. Hale considered the spectrum of sunspots and compared it with that from a
portion of the Sun without sunspot (not very far from the spot). He showed that the
Zeeman effect exists in the spectrum of the solar surface (Hale, 1908); see Figure 2. Later
in 1919, Hale investigated the polarity of sunspot magnetic fields and showed that it
changes with the solar cycle (Hale, 1919); see Figure 3.
There are two different approaches proposed to explain magnetic field concentration
on the solar surface. One is the rising flux tube model and the other is the negative
effective magnetic pressure instability (NEMPI), or a similar instability based on effects
from the mean magnetic field. The flux tube models (monolithic and clustered models)
Figure 1: A full disk image of the sun which is taken by SDO/HMI at 9/01/2013
(a)continuum, the dark spots are sunspots (b) magnetogram,the black and white color
shows the opposite polarity of magnetic field in active regions and sunspots.
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will be discussed in the next section. In this context, also convective collapse of the
flux tube, which was proposed by Spruit (1979), has been reviewed (see Section 2). In
Section 2.5 the history of the second approach, NEMPI and the role of a mean magnetic
field on the formation of flux concentration is presented. Mean field theory of the dynamo
and its formulation has been explained in a separate section (see Section 3). In the same
section, the α2 dynamo also has been discussed. As NEMPI plays an important role in
my PhD project, I will explain in Section 4 its basics and review the work done so far.
One way to get information about the sub-layer of the solar surface is helioseismology. A
brief overview of these methods together with work done in relation to flux concentration,
is discussed in Section 5. Finally, the framework, model and the result of my first project
are illustrated in Section 6. In the last section a summary of the literature review has
been presented (see Section 7).
2
Early work on flux emergence
Most sunspots form in specific areas on the solar surface, which are known as active
regions. These areas have especially strong magnetic field and most solar surface activities
like sunspots, solar flares and coronal mass ejections (CMEs) frequently form in these
regions. Active regions appear bright in X-ray and ultraviolet images. There are different
theories about the formation and variation of these regions.
2.1
Early ideas about Ω loops
In an early paper of 1955, Parker presented an idea about sunspot formation which
could explain most properties of sunspots like their east-west orientation, bipolarity, their
position in low latitudes (Spörer’s law) and the change of polarity inversion with time and
latitude. He suggests that a large enough buoyant magnetic flux tube tends to rise and
can carry flux lines of the Sun’s toroidal field to the upper layers. As a flux tube pierces
the photosphere, it forms a pair of sunspots (Parker, 1955a). In this theory, active regions
are treated as a phenomenon that their roots embed in depth around 104 km. In a review,
Parker has discussed the magnetic origin of solar activity (Parker, 1977). The early ideas
of magnetic flux appearance in the solar photosphere, also have been described by Zwaan
(1978). Figure 4 shows in summary of how a rising flux tube can lead to the formation of
an active region. Zwaan reviewed these ideas in subsequent papers (Zwaan, 1985, 1987).
In 1978, Parker suggested that the interaction between a weak magnetic field and
convective processes in the small flux tubes leads to an amplification of stronger magnetic
field. In the quiet sun, there are small magnetic flux tubes, which exist in the junction of
super granule boundaries where there is a strong down draft. He presented a new effect in
small flux tubes, which leads to strong cooling and thus to magnetic field concentration.
This effect is different from the other theories, which suggest radiation as a main mechanism for cooling the photosphere. They proposed that this new mechanism is related
to the suppression of convective heat transfer by the magnetic field. In fact, the plasma
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Figure 2: The figure shows different component (violet and red) of doublets for both
northern and southern spot. Such doublet in Spectrum desires a very strong magnetic
field, which they referred that field to the sunspot (Hale, 1908).
Figure 3: This figure presents the polarity of sunspots, which were attained by Hale
(1919).
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Figure 4: The illustration of rising magnetic flux tube which reaches to the surface and
forms the active regions (Zwaan, 1978).
compresses the magnetic field into the downdraft mentioned a bow but the buoyancy
force compensates the downward flow in the flux tube. This phenomenon leads to cooling
inside the flux tube. He have shown theoretically that small decrease in temperature over
many scale height may lead to the reduced magnetic pressure in the solar surface, which
results in magnetic field concentration (Parker, 1978).
2.2
Convective collapse
Spruit (1979) used Parker’s idea regarding magnetic flux tubes to explain convective
collapse of flux tubes. In another work with Zweibel, they found a critical value for the
magnetic field strength needed to get such field concentrations. For a field stronger than
a certain critical value, the magnetic field will suppress convection. They computed this
critical value for the solar convection zone to be about 1270 G at the solar surface (Spruit
& Zweibel, 1979). In this case they divided flux tubes into two types, stable flux tubes
(with magnetic field bigger than the critical value) and unstable ones (with magnetic field
less than the critical value). For the second group of tubes, when the field strength is low
enough, the instability sets in and, according to Parker’s idea, leads to downward flow,
the temperature decreases, which results in magnetic field concentration in the upper
layers. But there is a limitation for this downward flow too. If the resulting magnetic
field is bigger than the critical value, the tube settles in a new equilibrium with the same
properties as the initial one but with a lower energy. This is what is called convective
collapse of flux tubes. Figure 5 shows a sketch of convective collapse of a magnetic flux
tube. On the other hand, since the value of the resulting magnetic field is small enough,
downward displacement in the tube continues and the tube vanishes on the surface and
sinks down to the lower layer.
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Figure 5: Schematic representation of a flux tube before (dashed line) and after (solid
line) convective collapse (Spruit, 1979).
2.3
Clustered versus monolithic sunspot models
Later in 1979, Parker suggested that magnetic field concentrations on the surface, which
lead to the formation of sunspots, are due to many small flux tubes. This is referred to as
the cluster model of sunspots, as opposed to the more traditional monolithic models where
the sunspot would have uniformly distributed magnetic flux. With his model he explained
how a group of separate magnetic flux tubes in the convection zone reach the surface
through magnetic buoyancy and there, they make a single big flux concentration with a
correspondingly larger magnetic field; see Figure 6. In this paper he has investigated the
instability and structure of sunspots using this new model. He shows that the flux tubes
of different size have the same depth.
In the second part, he has discussed aerodynamic properties of such flux tubes (Parker,
1979). He showed in his later work that, with some assumptions, it is possible to obtain the
depth where the hypothetical anchor points lies. By using this theory, one can estimate
the depth of origin of solar active regions and sunspots. They suggested that this depth
is roughly equal to the size of the active region. For a normal active region, this depth is
about 105 /km.
In his paper he has used the behavior of active region in the sun surface to explain
the dynamical behavior beneath the convection zone (Parker, 1984).
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Figure 6: A sketch of a group of flux tubes with in the first 1000 km under the surface
that pressed together to form an active region or a sunspot (Parker, 1979).
Figure 7: Separation of opposite polarity of magnetic field (magnetic field concentration)
on the upper layer due to magnetoconvection resulted by Stein & Nordlund (2012).
2.4
Flux concentrations in deep convection simulations
Although coherent flux tubes, which form deep in the convection zone, have been known
to have the potential to form active regions, it also has been shown that the convective motions are important in the formation of active regions by promoting the uplift of
the magnetic structures between supergranular downdrafts. Recently, Stein & Nordlund
(2012) have introduced magnetoconvection as a possible origin of magnetic flux emergence. This implies that it is possible to envisage active regions as shallow events. They
have shown by simulation that it is not necessary to have a coherent flux tube to form an
active region; see Figure 7. In fact, magnetoconvection itself gives rise to flux tubes and
leads to the formation of an active region.
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Spruit (2012) has confronted some ideas about the solar cycle with observations. He
suggests that the interaction between magnetic field and turbulent convection is not responsible for the solar cycle and that buoyant instability of the magnetic field itself results
in the solar cycle.
2.5
Flux concentrations from mean-field effects
A different idea, which is able to explain large-scale magnetic field concentrations, has
been proposed by Kleeorin et al. (1989, 1990). They suggested that the effective magnetic
pressure in turbulent plasma could have a negative value, which leads to large-scale magnetic instability. In fact, the turbulent pressure is suppressed by magnetic field such that
the turbulent contribution to the magnetic pressure becomes negative. This instability
occurs in the presence of strong density stratification; preferentially near the solar surface
on the scales encompassing those of many turbulent eddies. This instability is invoked
as an explanation for magnetic field concentrations in the upper layer of the convection
zone (Kleeorin et al., 1989, 1990). As NEMPI is the basic theory of our research, it will
be described in more detail in Section 2.3.
3
3.1
Mean-field approach in dynamo theory
Two-scale assumption
Dynamo theory of magnetic field starts from the fact that every one of toroidal and
poloidal field of a rotating object acts as power source for the another one. This early
idea of Parker in 1955 suggests that stretching of poloidal field due to differential rotation
of the object leads to the creation of the toroidal field and on the other hand, the effect
of helical turbulence on the toroidal field produces poloidal field (Parker, 1955b). One
approach to formulate this idea was presented by mean field theory, where one assumes
that every quantity is in the form of a mean and a fluctuating part, i.e.,
F = F + f.
(1)
The important point is that we do not impose any restriction on the strength of the fluctuating part, so this is different from perturbation theory. The two important equations
here are the momentum and induction equations:
ρ
1
DU
= −∇p + J × B + ρg + ρν ∇2 U + ∇∇ · U + S · ∇ ln ρ ,
Dt
3
(2)
∂B
= ∇ × U × B + λ∇2 B,
(3)
∂t
where ν and λ are kinematic viscosity and magnetic diffusivity, respectively, and both
of them are assumed constant. S is the traceless rate of strain tensor of the flow. By
applying mean field theory, expressions for the magnetic field and velocity in induction
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equation we able to consider the effect of turbulence on the magnetic field fluctuation by
introducing the mean electromotive force. We have
3.2
B =B+b
(4)
U = U + u,
(5)
Mean-field equations and α2 dynamo
By replacing relations (4) and (5) in (2) and (3) and taking averages of them by using
the Reynolds averaging rules we get:
∂B
= ∇ × U × B + λ∇2 B + ∇ × E,
∂t
(6)
where E = u × b. In the case of isotropic turbulence, the mean EMF is given by
E = αB − ηt ∇ × B.
(7)
The expression shows that in the case that α is not zero the mean magnetic field will be
produced by u × b . Substituting (7) to (6) we get
∂B
= ∇ × (U × B) + η∇2 B + ∇ × (αB) − ∇ × (ηt ∇ × B),
∂t
(8)
Now, to get α2 dynamo we should put a condition that there is no mean flow (U = 0).
And that the turbulence is homogeneous. To get an α2 , it simplifies matters if we assume
that there is no mean flow (U = 0). In other words, the turbulence is homogeneous.
This implies that α and ηt are constants. It is therefor, straightforward to write the mean
induction equation in the form of
∂B
= ηT ∇2 B + α∇ × B,
∂t
(9)
ηT = η + ηt .
(10)
where
is total magnetic diffusivity. One possible solution to (9) is the real part of an expression
in the form of
B(x) = B̂(k)e(ik·x+λt) .
(11)
Replacing this expression in mean induction equation, we obtain
λB̂ = αik × B̂ − ηT k 2 B̂.
(12)
By separating the three components, we can write a matrix form of an eigenvalue
equation that should have zero determinant. So the dispersion relation will be
(λ + ηT k 2 ) (λ + ηT k 2 )2 − α2 k 2 = 0.
10
(13)
The general mean momentum equation also have the form of
DU
= −c2s ∇ ln ρ + g + F M + F K ,
Dt
(14)
where F K = νt (∇2 U + 13 ∇∇ · U + 2S∇ ln ρ) is the viscous force from the mean flow and
is used in all mean-field and large eddy simulations, while F M is the mean Lorentz force
and can be expressed as
FM = J × B +
1
∇(qp0 B 2 )...,
2µ0
(15)
where dots refer to extra terms that have been neglected. Here the second term is one
of the most important turbulent contributions to the mean Lorentz force. This will be
the subject of Section 4. In nonlinear mean field simulations, one solves (6) together with
(14) for different equations of state with different conditions.
4
NEMPI
In a turbulent plasma there are different possible instabilities; convection instability, largescale dynamo instability, as well as small-scale dynamo instability. There is also another
intermediate-scale instability, which makes it possible to concentrate magnetic field from
a weak initial magnetic field in a stratified and turbulent plasma. In comparison with the
dynamo-generated magnetic field in the sun, the field resulting from NEMPI has smaller
scale and, as the instability happens in the upper layers of the Sun it is not out of sight
that one can use it to explain the formation of active regions on the solar surface. In order
to get to this aim, it is necessary to study NEMPI in more detail. In the next subsection
the theory of NEMPI has been explained.
4.1
Negative effective magnetic pressure
The idea of NEMPI started from the fact that the effective magnetic pressure can be
negative in the case of a turbulent plasma. The total pressure of the turbulent plasma is
ptot = pg + pmag + pt
(16)
where pg , pmag , and pt are the gas, magnetic (B 2 /8π) and turbulent pressures, respectively.
As the total energy of the turbulence is approximately conserved for a turbulent plasma
with a weak magnetic field, the turbulent pressure can be written in the form of
2
B
pt = pt (0) − qp ,
8π
(17)
where the first term is the turbulent pressure in the case that the large-scale magnetic field
is absent (the net effect of turbulence on plasma pressure) and the second term shows the
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role of the large-scale magnetic field in forming small-scale magnetic field fluctuations,
which leads to an extra term in the turbulent pressure expression. Furthermore, qp is
a function of the large-scale magnetic field that is expected to be positive. Finally, the
expression for total pressure will attain the form
2
ptot
B
= pg + pt (0) + (1 − qp ) .
8π
(18)
This relation indicates that for qp > 1, the effective magnetic pressure will be negative so
it will decrease the total pressure of the plasma. This will turn on the instability and as
we have a fluid with small-scale turbulence, the turbulent eddies act like a generator to
help this instability to develop.
In the following subsection a summary of the simulations have been presented, which
scientist has implemented to study NEMPI by both direct numerical simulation and mean
field simulation.
4.2
Results from DNS and MFS
In their paper, Kleeorin et al. (1989, 1990) have derived an expression for effective magnetic force, which has the form of
Fm
B2
B
= −∇ (1 − qp )
+ B · ∇ (1 − qs )
,
8π
4π
"
#
"
#
(19)
where qs is, similar to qp , another function of the large-scale magnetic field. It is possible
to show that in the case of negative effective magnetic pressure, this force will reverse the
sign. It is also explicit from the expression that this force is a nonlinear term in large-scale
magnetic field. In fact, functions qs and qp relate the sum of the Reynolds and Maxwell
stresses to the mean magnetic field. Another important point is that the growth rate of
instability is directly related to the large-scale magnetic field. Rogachevskii & Kleeorin
(2007) have considered the functions qp (B) and qs (B) in details. Figure 8 illustrate the
plots for these functions for different value of magnetic Reynolds numbers and also the
plots for effective mean magnetic pressure and effective mean magnetic tension, which are
defined by
B2
pm = 12 (1 − qp ) 2
(20)
Beq
σB = (1 − qs )
q
B2
2
Beq
(21)
where, Beq = 4πρu2rms is the equipartition value of magnetic field.
In their next papers, they have investigated the energy transfer from small-scale to
large-scale magnetic field due to negative effective magnetic pressure instability (NEMPI)
and they have tried to explain solar oscillation and sunspot formation by this new mechanism (Kleeorin et al., 1993, 1996). In this theory, active regions are treated as a shallow
12
Figure 8: (a) The Function qp (B) for different values of the magnetic Reynolds numbers;
ReM = 103 (thin solid line), ReM = 106 (dashed-dotted line); ReM = 1010 (thick solid line)
for homogeneous turbulence and at ReM = 106 (dashed line) for convective turbulence.
(b) The effective mean magnetic pressure pm at ReM = 106 for homogeneous turbulence
(thick solid line), and for a convective turbulence for the horizontal field (dashed) and for
vertical field (thin solid line. (c) The function qs (B) for different values of the magnetic
Reynolds numbers; ReM = 103 (thin solid line), ReM = 106 (thin dashed-dotted line),
ReM = 1010 (thick solid) for a homogeneous turbulence, and at ReM = 106 (dashed line)
for a convective turbulence. (d). The effective mean magnetic tension σB at ReM = 106 for
homogeneous turbulence (thick solid line), and for a convective turbulence (Rogachevskii
& Kleeorin, 2007).
phenomenon. in 2011, the existence of large-scale magnetic flux concentration due to
NEMPI has been shown by both mean field simulations (MFS) and direct numerical simulations (DNS) in a highly stratified isothermal gas with large plasma beta (Brandenburg
et al., 2011; Kemel et al., 2012a). Figures 9 and 10 show some of the simulation results.
Since then it is of interest to investigate NEMPI and its interaction with turbulent plasma.
Recently, the effect of forced turbulence on effective magnetic pressure has been studied
by direct numerical simulation (Brandenburg et al., 2012). Kemel et al. (2012b,c,d) has
considered NEMPI as a possible mechanism in the formation of active regions. They also
investigated the effect of non-uniformity of magnetic field on NEMPI. In their last paper
they increased the number of eddies to 30 in the study domain to get largescale separation
13
Figure 9: The first numerical demonstration of NEMPI in DNS that shows large-scale
magnetic field concentration resulting from NEMPI (Brandenburg et al., 2011).
Figure 10: Another demonstration of NEMPI with mean field modelling. Here it has been
shown how tension forces affect the magnetic field pattern (Kemel et al., 2012a).
(Kemel et al., 2012b,c,d).
In another work by Käpylä et al. (2012), the effects of turbulent convection on NEMPI
have been studied. They demonstrate that NEMPI still works if the entropy equation is
included, provided the background stratification is adiabatic, i.e., there is no stabilizing
force associated with Brunt-Väisälä oscillations. On the other hand, if the stratification
is isothermal, NEMPI only works if the cooling time is not too long, i.e., the stabilizing
effect from Brunt-Väisälä oscillations is weak.
Losada et al. (2012a,b) have used both MFS and DNS to investigate the effect of
rotation on NEMPI. They considered the development of NEMPI in the case of large
scale separation and in the presence of rotation. They found that even relatively slow
rotation, with Coriolis numbers around 0.1, suppresses NEMPI. Their results of mean
field simulations are compatible with direct numerical simulations, which again shows
that there is a very good agreement between DNS and MFS in the case of NEMPI.
The numerical investigation of functions, qp (β) and qs (β) has been done by Brandenburg et al. (2010). They have shown by direct numerical simulation of forced turbulence
with imposed field that these functions are positive and exceed unity for weak fields. β is
a parameter define by
B
β=
.
(22)
Beq
They have used this result to explain how the reduction happens on effective Lorentz
force, which leads to negative effective magnetic pressure. Their simulation admitted that
14
Figure 11: Right: time evolution of the meridional magnetic field and velocity vectors,
which has been resulted by 2D simulations. Left:The 3D simulation of magnetic field in
three different times. Here, again the field concentration due to NEMPI forms near the
surface (Brandenburg et al., 2010).
qp should be bigger that 2qs . They have investigated both solution of the forced turbulence
and mean field MHD on the large-scale Lorentz force in a density stratified layer. They
have shown in their simulations that the growth rate of instability is proportional to qs , qp ,
the strength of stratification and imposed field. In fact, the enhancing of any one of these
quantities increases the growth rate. They also have found out that increasing magnetic
diffusivity decreases the growth rate. Figure 11 shows their simulation result. In this
figure, the time evolution of magnetic field after saturation of instability has been shown
for two cases; 2D (right) and 3D (left) simulations. It can be seen from both plots how
NEMPI leads to the formation of magnetic structures near the surface. The interesting
thing about this figure is the bipolar magnetic field structures, which are formed on the
surface in the case of 3D simulations (Brandenburg et al., 2010). In the next subsection
comparison between NEMPI and the flux tube model has been demonstrated.
15
4.3
NEMPI versus flux tubes
The reason that NEMPI leads to the field structures near the surface is the fact that
NEMPI works in highly stratified turbulent plasma. As the magnetic field concentration
formed by NEMPI happens very close to the surface (highly stratified area)it can directly
lead to the formation of active regions or even sunspots. Here the buoyant force also
accompanies NEMPI in the formation of magnetic structures near the surface. Magnetic
buoyancy is the mechanism which can be used for both a flux tube or a stratified continuous magnetic field without any flux tubes. In the case of NEMPI, the second situation
exists. The flux tube situation has been used by the old theories to explain the rising of
the flux tubes from the deep inside the convection zone to reach the surface and create
the active regions (Parker, 1955a; Zwaan, 1978). In this mechanism as the flux tube has
a magnetic field stronger than its surroundings, the magnetic pressure inside the tube is
bigger than the magnetic pressure outside. So, to have equal total (gas and magnetic)
pressure inside and outside the tube, the density of the inside of flux tube decreases.
Finally, the buoyant forces, due to density difference between inside and outside the tube,
will act and the flux rises up. One of the arguments about this model arises from the
magnitude of the magnetic field at the bottom of the convection zone. For a rising flux
tube to save the same orientation during its ascent, the magnetic field of 105 G is needed
(Choudhuri & D’Silva, 1990; D’Silva & Choudhuri, 1993). This magnetic field is more
than hundred times stronger than the equipartition value. It is here that the importance
of studying the surface field concentration as a shallow event shows itself.
The field concentration generated by NEMPI is not strong enough to form active
regions or sunspots. So scientists suggest that NEMPI may be accompanied by some
other mechanism. One possible mechanism was proposed by Kitchatinov & Mazur (2000).
In their model, the suppression of convection motions (heat flux) by a mean magnetic
field leads to a decrease in temperature and formation of magnetic field concentration.
They have taken to account the fluid motion on flux emergence using mean field model.
The instability they described is due to the fact that eddy diffusivity is quenched by strong
magnetic fields. They suggest that this new instability, physically is compatible with
convective collapse phenomena presented by Spruit (1979). In near surface layer, cooling
from surface due to radiation and heating from bellow due to convective motions are
balanced. The instability sets in when this balance is disturbed by reduced heat transfer
due to the fact that magnetic field has quenched diffusivity. This leads to further cooling
in the surface; structure sinks to compensate the heat loss, which helps to concentrate
mean magnetic field even further (Kitchatinov & Mazur, 2000). It is of interest to study
this instability further by both mean field simulations and direct numerical simulations.
The suppression of turbulent hydrodynamic pressure by mean magnetic field also
has been studied in direct numerical simulations. In the work done by Brandenburg et
al. (2011). They have simulated strongly stratified, isothermal turbulent plasma (large
Reynolds number) with imposed uniform magnetic field (smaller than equipartition value)
and proper scale separation. Their results proved that the ratio B0 Beq0 should be in
the suitable range for NEMPI to work. This is consistent with theory and mean field
16
Figure 12: The averaged magnetic field ratio for different value of B0 /Beq0 . The figure
shows that the intense field concentration starts from B0 /Beq0 = 0.05 Brandenburg et al.
(2011).
Figure 13: The figure shows the fit curves of the effective magnetic pressure for different
value of qp0 . It is explicit from the plots that the effective magnetic pressure can get
negative values Kemel et al. (2012a).
calculation (see Figure 12).
5
5.1
Helioseismic signatures of flux concentrations
Methods of local helioseismology
Helioseismology helps scientists to improve their knowledge about the interior of the sun.
For instance, studying solar rotation and its relation with solar radius and latitude are
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24 Apr 01
Latitude {degrees}
23 Apr 01
25 Apr 01
2 Mm
40
30
20
10
0
135
150
165
180 120
180 120
135
150
165
180
14 Mm
120
30 m s-1
135
150
165
135
150
165
180 120
180 120
Carrington Longitude {degrees}
135
150
165
180
Latitude {degrees}
120
135
150
165
40
30
20
10
0
Figure 14: Flow patterns together with magnetograms (color coded) for a large active
region at two different depths of 2 Mm and 14 Mm Hindman et al. (2009).
some of the results of helioseismology. Also by using global-mode frequencies, it became
possible to estimate the depth of the convection zone better than previous methods (by
means of stellar models that rely on accurate opacities). There are different helioseismic
methods, which have been used by scientists. Every one of them has its advantages and
difficulties and they are used for different problems.
Time-distance helioseismology is one of them. In this method, spatial-temporal properties of the wave are used to measure the travel times of the sound waves between two
different points on the solar surface. This method is used mostly for the investigation
of supergranulation on the solar surface. Holography is another seismological technique,
which is very close to time-distance method but in this method they use forward and
backward waves. Direct modeling, ring diagram analysis (based on power spectrum) and
Fourier-Hankel method are also another helioseismic techniques. The complete explanation of all these methods and their applications has been discussed in a review by Gizon
et al. (2010).
5.2
Flow field near active regions
Using the ring diagram analysis, it also had been shown previously (see Figure 14 and
15) that there is mean inflow (20 to 30 m s−1 ) near the surface in active regions at depths
below 7 Mm (Hindman et al., 2009), which is consistent with presence of inflow in the
case of NEMPI (Kemel et al., 2012b). There are also outflows in the immediate proximity
of the spot, which is the well-known Evershed effect. The inflow at larger distances is
accompanied with a return flow at a depth of more than 10 Mm.
18
Figure 15: The right figure illustrates the schematic diagram of large-scale circular motion
of fluid with about 20 m s−1 mean flow around the active region. The left schematic
diagram shows the effect of Coriolis forces on both inflow into the active region and the
out flow from sunspots Hindman et al. (2009).
Figure 16: The rising flux detection by traveling acoustic waves (Ilonidis et al., 2011).
5.3
Deep focussed time-distance helioseismology
Recently, helioseismic precursors of magnetic activity of the sun have been found by
Ilonidis et al. (2011). They have used the local helioseismology techniques to study the
emerging flux tube from about 60 Mm depths in convection zone. Figure 16 shows how
the acoustic waves traveling through convection zone detect the emerging flux. In fact,
the travel time of the waves is affected by thermal fluctuations, magnetic fields and flows.
With their work, it is possible to predict the formation of an active region 1 or 2 days
before their appearance on the visible surface of the sun.
Figure 17 presents the result of helioseismic investigation for Active region (AR) 10488.
Their studies suggests that the highest signature of emerging flux is related to the sound
waves fluctuation in about 57 to 66 Mm below the surface. This depth is well below the
depth where NEMPI works Losada et al. (2012b). So it is of interest to investigate the
helioseismic signature of the magnetic field concentration due to NEMPI by means of
helioseismic methods.
19
Figure 17: (A) Mean travel-time perturbation map of AR 10488 at the depth of 42
to 75 Mm. (B) photospheric magnetic field at the same time as (A). (C) Photospheric
magnetic field at the same location as (A) but 24 hours later. (D) Total unsigned magnetic
flux (red line) and magnetic flux rate (green line) of AR 10488. The vertical blue line marks
the start of emergence. The pink line shows the temporal evolution of the perturbation
index (in units of 125 s Mm2 ), which is defined as the sum of travel-time perturbations
with values lower than −5.4, within the signature of (A) (Ilonidis et al., 2011).
6
NEMPI and spherical α2 dynamos
As mentioned before, NEMPI is a young subject, which still needs to be investigated.
In this regards, a new model has been suggested to combine NEMPI with dynamo in
spherical coordinates (Jabbari et al., 2013). The model has been described in the following
subsection and the results of this work has been presented and discussed in the next
subsection.
6.1
Outline of the model
In a recent work by Jabbari et al. (2013) using mean field simulations of NEMPI with
an α2 dynamo they have investigated NEMPI under more realistic conditions like global
geometry and dynamo generated magnetic field. In all previous studies of NEMPI, people
have used an imposed field to produce NEMPI. In the case of spherical geometry it is
more straightforward to use dynamo generated magnetic field. In this paper, the combined
20
effects of a dynamo and NEMPI in a turbulent highly stratified plasma with adiabatic
equation of state has been investigated. Their simulations show that these two work
together in a constructive manner. The same as what has been founded by previous
simulations, in highly stratified plasma when the value of magnetic filed is about a few
percent of equipartition value the NEMPI starts growing. The α quenching has been used
to achieve to a suitable magnitude of mean magnetic field. With axisymmetry assumption,
the perfect conductor boundary condition has been adopted (derivative of both r and φ
components of magnetic vector potential versus θ and φ component of A are equal to
zero in both θ = 0◦ and 90◦ ). The results of the simulations has been demonstrated in
the following subsection.
6.2
Results
Figure 18 shows meridional cross-sections of B/Beq (color coded) together with magnetic
field lines of poloidal magnetic field for different values of qp0 and stratification for Q =
1000. The dashed lines indicate the latitudes 49, 61.5, 75.6, and 76.4. it can be seen from
the plot that just for the qp0 bigger than 60, field concentrations occur. Because of the
fact that the growth rate of the instability is inversely proportional to the pressure scale
height for strong stratification, one should expect intense field concentration; in other
words, for weaker stratification, field concentrations vanishes completely.
In Figure 19 another result has been shown, which is effect of quenching parameter
on where the field concentration occurs. for smaller quenching or in another words for
stronger mean magnetic filed the NEMPI occurs in lower latitude. Also for bigger quenching, smaller magnetic field the NEMPI is more pronounced. The interesting results are
gained when the initial mean magnetic field is very weak (Qα = 10000). In this case the
oscillatory pole-ward migration happens, which is due to effect of NEMPI on dynamo.
Frequency of oscillatory behavior is about ω = 11.3ηt /R2 2. This pole-ward migration
also has been observed in the case of NEMPI in the presence of rotation (Losada et al.,
2012a,b). So it is not out of sight that they may have the similar mechanism. Furthermore, the toroidal field is normalized by the local equipartition value, i.e., the colors
indicate B/Beq (r).
Figure 20 shows the comparison between the NEMPI growth rate and the dynamo
growth rate of this coupled system. Dependence of Brms (solid lines) and Urms (dashed
lines) on time for qp0 = 0 (black), 5 (blue), 10 (red), 20 (orange), 40 (yellow), and 100
(upper black line for Brms ). It has been shown that the results for Urms depend only
slightly on qp0 , and this only when the dynamo is saturated.
The three inverse length scales current helicity, magnetic helicity and kinetic helicity
as a function of time have been shown in Figure 21. At time t0, the value of qp0 has been
changed from 0 to 100. kM and kc increases after introducing NEMPI, but kK drops to
very small values. The increase is due to increase of gradients associated with the resulting
flux concentration. The decrease of kK might be consequence of increase in kinetic energy.
21
Figure 18: In the left plot, the effect of qp function on formation of magnetic field concentrations has been illustrated. In the right side, the effect of stratification on NEMPI
development has been shown Jabbari et al. (2013).
7
Summary
In this literature review, I discussed about the history of sunspot and active region studies
briefly. What is more important for us is to understand the origin of these phenomena
because with this knowledge we learn more about the interior of the sun and specially
convection zone. This outer layer of the suns ’interior is known to be the place that the
roots of the active regions and sunspots are there. But there is a question: where in the
convection zone the roots are, deep down or upper area close to the surface?
To answer this question, many scientists worked hard and tried to find the best answer using all available methods like simulations, observations and theoretical studies. I
discussed two models, rising flux tubes by magneto buoyancy and NEMPI, which one
treated active regions as deeply rooted phenomena and the other studies them as more
shallower event. In this work we concentrated on the second model, NEMPI. I explained
the idea behind this model and summarized the works has been done so far to study this
new instability.
In the projects I am working and I will work in the future, I try to go some step further
22
Figure 19: The plot in the left is meridional cross-sections of magnetic field for different
values of quenching parameter, Q, for r = 1.001 (highest stratification) and qp = 100.
The illustration of pole-ward migration in the case of very strong quenching, Qα = 10000
has been presented in the right plot Jabbari et al. (2013).
in improving the model and study it under more realistic conditions, which can lead to
strong magnetic field concentration or even formation of active regions and sunspots.
For instance, to investigate the effect of geometry on instability, to study the interaction between dynamo and NEMPI and searching for helioseismic signature of magnetic
field concentration due to NEMPI are the projects, which have been suggested to be done.
The section discussing dynamo and mean field makes the connection to my first research projects, which already has been presented in section 3. The aim of the project
was to investigate NEMPI in spherical geometry with dynamo generated magnetic field.
The result showed that NEMPI and dynamo work together at the same time and even
can modify each other. but understanding the details of this coupled system needs more
studies. For example the existence of pole ward migration in the case of strong quenching
parameter is not completely understood and it needs further investigation.
Finally, in the last project related to NEMPI, It is interesting to see that how NEMPI
and helioseismology suggest a strong similarity in the result.
23
Figure 20: Dynamo growth rate together with NEMPI growth rate has been plotted
versus dimensionless time Jabbari et al. (2013).
Figure 21: The three inverse length scales current helicity, magnetic helicity and kinetic
helicity as a function of time have been plottedJabbari et al. (2013).
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