Solar Physics Winter School at Kodaikanal Solar Observatory,
December 10–22, 2006
Photospheric processes and magnetic flux tubes
Oskar Steiner
Kiepenheuer-Institut für Sonnenphysik, Freiburg i.Br.
steiner@kis.uni-freiburg.de
http://www.uni-freiburg.de/ ˜steiner
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Solar Physics Winter School at Kodaikanal Solar Observatory,
December 10–22, 2006
The magnetic fine structure of the quiet solar photosphere
Oskar Steiner
Kiepenheuer-Institut für Sonnenphysik, Freiburg i.Br.
steiner@kis.uni-freiburg.de
http://www.uni-freiburg.de/ ˜steiner
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Houses I and II of the Kiepenheuer-Institut in Freiburg
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§ 1 The concept of magnetic flux tubes
A magnetic flux tube or magnetic flux bundle is defined by the surface
generated by the set of field lines that intersect a simple closed curve.
Flux tubes are the building blocks of a magnetic configuration, but they must
not be thought of as independent isolated structures.
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The concept of magnetic flux tubes (cont.)
The magnetic flux, crossing a section S of the flux tube is given by
F =
Z
S2
B · dS
S
F2
S1
F1
−F1
Z
V
zZ }| { Z
z }| { Z
0
z }| {
∇ · B dx3 = − B ds + B ds + B · n̂ dσ = 0
S1
As a consequence of ∇ · B
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F2
ref
= 0:
S2
F1 = F2 .
tube
surface
The concept of magnetic flux tubes : Examples of magnetic flux tubes: Coronal loop
Coronal flux tubes in emission at Fe IX, 17.1 nm, corresponding to gas in the
temperature range 600 000 to 1 200 000 K. From the TRACE homepage.
Active regions observed on August
Active region seen on May 19, 1998
19, 1998 at 171 Å
at 171 Å
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The concept of magnetic flux tubes : Examples of magnetic flux tubes: Sunspot
Sunspots as examples of photospheric magnetic flux tubes
Scharmer & Langhans, Swedish Solar Telescope, SST, La Palma
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The concept of magnetic flux tubes : Examples of magnetic flux tubes: Sunspot
Courtesy,
N. Bello
Gonzales
German Vacuum Tower Telescope (VTT), Tenerife
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→Evershed flow
The concept of magnetic flux tubes :
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The concept of magnetic flux tubes : Examples of magnetic flux tubes: Pore
Pores have no penumbra and are much smaller than sunspots. Their size is
that of one or several granules.
11:08:30
20
arcsec
15
10
5
0
White-light image from the
0
5
10
15
arcsec
20
Swedish vacuum tower
Pore (left) and corresponding Dopplergram
telescope. M. Sobotka et
(right). A downdraft exists at the periphery of
al. (1999)
the pore (dark blue). Kashia Mikurda (KIS)
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The concept of magnetic flux tubes : Examples of magnetic flux tubes: Sunspot &
magnetic element
70
60
seconds of arc
50
40
30
20
10
0
0
20
40
seconds of arc
60
80
Sunspot and network magnetic elements. Pit Sütterlin, Dutch Open Telescope,
DOT
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§ 2 The discovery of small-scale magnetic flux concentrations
The advent of the magnetograph around 1950 (H.W. Babcock, K.O.
Kiepenheuer) enabeled researchers to investigate ever smaller magnetic
structures than pores, today generally called magnetic elements.
→
Zeeman effect → Stokes Polarimetry
– In 1968 Beckers and Schröter (SP, 4, 142) discover small patches of magnetic field
with a strength of 600 – 1400 Gauß and a diameter of around 1.3”, which they
named magnetic knots. With a contrast of 0.88 in the continuum, they are barly
visible in white light. They are located in active regions.
→ magnetic knot
– Magnetic elements even smaller than knots exist, which cannot be resolved with a
magnetograph. Their field strength must be determined by indirect methods, or by
inversion techniques.
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→ filling factor
→ line ratio
§ 2a Observations of magnetic flux concentrations at 0.1”
2 hours movie captured with the Swedish 1-m Solar Telescope (SST) on La
Palma by Luc Rouppe van der Voort & Michiel van Noort, University of Oslo.
“G-band continuum” with a 1 nm bandwidth filter centered at 436.4 nm. Trailing
part of an active region.
Rouppe van
der Voort et al.
2005, A&A
435, 327
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Observations of magnetic flux concentrations at 0.1” (cont.)
“Ribbon-like” structure. Berger, Rouppe van der Voort, Löfdahl et al. 2005,
A&A 428, 613 with the new 1 m Swedish Solar Telescope on La Palma
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Observations of magnetic flux concentrations at 0.1” (cont.)
Contrast profile of a
Observed filtergram with contrast profile of a “ribbon”
medium-sized KGB-model.
structure.
Knölker & Schüssler 1988,
A&A, 202, 275
ρe
pe
→ more
ρi
pi+p mag
flux tube
boundary
τ c =1 ‘solar surface’
→ isothermes
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200 km
1000 km
Observations of magnetic flux concentrations at 0.1” (cont.)
“Loops” and “flowers”.
Berger, Rouppe van der Voort,
Löfdahl et al. A&A 428, 613
with the Swedish Solar
Telescope on La Palma.
→ more
→ 0.01” Magnetograms ?
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Observations of magnetic flux concentrations at 0.1” (cont.)
G-band movie of a time span of 2 h 35” taken with from the HINODE (Solar-B)
satellite
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§ 3 What confines a magnetic flux tube?
Consider an isolated magnetic flux tube, embedded in
s^
flux tube
boundary
strength at the flux-tube boundary can be described with
Bi
internal
θ(x) with θ(x) = 0 for x < 0 and
θ(x) = 1 for x > 0. In the coordinates of the local
frame given by ŝ and n̂, where ŝ is tangential to the magnetic field of the flux-tube surface, B is given by
the step function
Be
ξ
a field-free medium. The discontinuity in magnetic field
n^
external
B = (0, 0, Bi − [Bi − Be ]θ(ξ)).
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§ 3 What confines a magnetic flux tube?
Consider an isolated magnetic flux tube, embedded in
s^
Bi
flux tube
boundary
a field-free medium. The discontinuity in magnetic field
strength at the flux-tube boundary can be described with
θ(x) with θ(x) = 0 for x < 0 and
θ(x) = 1 for x > 0. In the coordinates of the local
ξ
n^
internal
frame given by ŝ and n̂, where ŝ is tangential to the magexternal
netic field of the flux-tube surface, B is given by
c
j and
B = (0, 0, Bi − [Bi − Be ]θ(ξ)). Applying Ampère’s law: ∇ × B =
4π
′
using θ (ξ) = δ(ξ) (Dirac’s δ -distribution) we get:
c
(0, [B]δ(ξ), 0), where [B] = Bi − Be .
j=
4π
Be
the step function
Integration over an ε-range and letting ε
→ 0, leads to the
sheet current which flows perpendicular to the ŝ-n̂ plane.
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c
j =
[B]
4π
∗
What confines a magnetic flux tube? (cont.)
Lorentz force
z }| {
1
dv
= −∇p + (j × B) +ρg⊙
From the equation of motion
ρ
dt
c
c
and Ampère’s law: ∇ × B =
j , we obtain in the static case
4π
1
∇p = 4π ((∇ × B) × B) + ρg and further using the vector idendity
1
(∇ × B) × B = (B · ∇)B − ∇(B · B)
2
1
B2
) + ρg +
(B · ∇)B
∇p = −∇(
8π
4π
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What confines a magnetic flux tube? (cont.)
Lorentz force
z }| {
1
dv
= −∇p + (j × B) +ρg⊙
From the equation of motion
ρ
dt
c
c
and Ampère’s law: ∇ × B =
j , we obtain in the static case
4π
1
∇p = 4π ((∇ × B) × B) + ρg and further using the vector idendity
1
(∇ × B) × B = (B · ∇)B − ∇(B · B)
2
1
B2
) + ρg +
(B · ∇)B
∇p = −∇(
8π
4π
We decompose the last term into a component parallel (ŝ) and perpens^
boundary
ξ
n^
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= B ŝ follows:
∂
∂
B · ∇ = B · ŝ · ∇ = B
and further (B · ∇)B = (B
)B =
∂s
∂s
2
∂B
∂ B
∂
2 ∂ŝ
2 n̂
ŝ + B
=
(
)ŝ + B
,
B (Bŝ) = B
∂s
∂s
∂s
∂s 2
Rc
where Rc is the curvature radius of the field line. Thus, we obtain:
dicular (main normal n̂) to a surface field line: From B
What confines a magnetic flux tube? (cont.)
B2
∂ B2
B 2 n̂
∇p = −∇(
) + ρg +
(
) · ŝ +
8π
∂s 8π
4π Rc
Multiplication of this equation with n̂ yields the force balance perpendicular to the tube
∂p
∂ B2
B2 1
surface:
=− (
) + ρg · n̂ +
∂n
∂n 8π
4π Rc
and integration over a small intervall
[−ǫ, ǫ] across the
tube surface gives:
s^
flux tube
boundary
Bi
pi Be
pe
^
n
ξ
ε
ε
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Zpe
2
Be
/8π
dp
pi
| {z }
pe −pi
Z
=
B̄ 2
B2
) +ρg · n̂2ǫ +
2ǫ
d(
8π
4πRc
Bi2 /8π
|
{z
1 (B 2 −B 2 )
− 8π
e
i
from which we obtain:
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}
Be2
Bi2
pe +
= pi +
8π
8π
§ 4 A microscopic picture of the sheet current
Stronger magnetic field, smaller
radius of curvature
+
gradient
dB0
dy
Electrons and ions in a magnetic
field with a transverse gradient,
Magnetic field
out of page
−
showing gradinet
B drift motion.
After Krall & Trivelpiece, 1973
Weaker magnetic field, larger
radius of curvature
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§ 4 A microscopic picture of the sheet current
Stronger magnetic field, smaller
radius of curvature
+
gradient
dB0
dy
Electrons and ions in a magnetic
field with a transverse gradient,
Magnetic field
out of page
showing gradinet
−
B drift motion.
After Krall & Trivelpiece, 1973
Weaker magnetic field, larger
radius of curvature
Magnetic field
out of plain
Higher density
of plasma
Net current
Lower density
of plasma
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Drift current in a plasma with a density gradient.
1111
0000
0000
1111
0000
1111
Due to the inbal-
ance of gyrating particles, a current
results without a net transport of
charges. After Krall & Trivelpiece,
1986
A microscopic picture of the sheet current (cont.)
Higher density
of plasma
−
−
−
Field−free plasma
−
−
−
Lower density
of plasma
net current
Magnetic field
out of page
At the boundary of an isolated flux tube we have both effects: Drift current because of a
sharp gradient in magnetic field. This drift current is not cancelled by gyrating particles
within the flux tube because of the reduced particle number-density there.
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§ 5 The equations for a hydrostatic flux tube
In this paragraph, we derive the equations for computing the magnetic structure of a
vertical, axisymmetric flux tube without twist.
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The equations for a hydrostatic flux tube (cont.)
Consider a vertical axisymmetric flux tube without twist, embedded in an external
medium of pressure pe . The magnetohydrostatic system of equations to be solved is:
0 = −∇p + ρg + j × B ,
(1)
∇ × B = 4πj ,
(2)
∇·B=0.
(3)
We decompose (1) in components parallel and perpendicular to the magnetic field.
Thus, multiplying (1) by B gives
B · (∇p − ρg) = 0 ,
(1-a)
and taking the cross product of (1) with B gives
j=
1
B × (∇p − ρg) ,
2
B
where we used that B × (j × B)
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= B 2 j − (B · j) · B = B 2 j
(1-b)
The equations for a hydrostatic flux tube (cont.)
Using cylindrical„
coordinates and the solenoidality of the magnetic field
«
1 ∂Az
∂Aφ ∂Ar
∂Az 1 ∂
∂Ar
B=∇ × A=
−
,
−
, ( (rAφ ) −
) ,
r ∂φ
∂z
∂z
∂r
r
∂r
∂φ
«
„
1 ∂
∂Aφ
, 0,
(rAφ ) for an axisymmetric flux tube
which reduces to B = −
∂z
r ∂r
«
„
1 ∂Ψ
1 ∂Ψ
and further
without twist. With Ψ := rAφ ⇒ B = −
, 0,
r ∂z « r ∂r
„
1 ∂2Ψ
1 ∂Ψ
1 ∂2Ψ
∇×B= 0, −
+ 2
, 0 so that Ampères law for the
−
2
2
r ∂z
r ∂r
r ∂r
φ-component (the r- and z -components being zero), becomes (Grad-Shafranov):
∂2Ψ
1 ∂Ψ
∂2Ψ
−
= −4πrjφ
+
∂r2
r ∂r
∂z 2
The magnetic field components can be recovered from the scalar potential Ψ by:
1 ∂Ψ
1 ∂Ψ
Br = −
,
Bz =
.
r ∂z
r ∂r
Note that curves of Ψ = const. describe field lines of the system.
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The equations for a hydrostatic flux tube (cont.)
If gravity acts along the negative z -axis and s measures the
δz
s^
θ
distance along magnetic field lines, inclined at an angle θ to
δs
n^
the vertical:
g
B · (∇p − ρg) = 0
⇒
dp
ds
+ ρg cos θ = 0
Ψ
p = nkB T = (ρ/m̄)kB T ⇒ ρ = m̄p/(kB T ), thus,
dp
m̄g
dp
m̄g
1
m̄g
+
p=0 ⇒
=−
dz With
:=
and
dz
kB T
p
kB T
H
kB T
integration from a reference height 0 to z we obtain

p = p0 (Ψ) exp −
ZΨ,z
Ψ,0
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dz ′
H(T (Ψ, z ′ ))


The equations for a hydrostatic flux tube (cont.)
In component notation B
Using it in Eq. (1-b):
= (Br , 0 , Bz ) and ∇p = (∂p/∂r , 0 , ∂p/∂z).
1
jφ = 2
B
From
»
–
∂p
∂p
p
Bz
− Br (
+ ) .
∂r
∂z
H
2
p = p0 (Ψ) exp 4−
ZΨ,z
dz
H(T (Ψ, z ′ ))
Ψ,0
˛
∂p
p
∂p ˛˛ ∂Ψ
we obtain
=− +
∂z
H
∂Ψ ˛z ∂z
above equation for jφ reduces to
and
˛
∂p ˛˛
jφ = r
∂Ψ ˛z
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′
3
5 ,
˛
∂p
∂p ˛˛ ∂Ψ
=
so that the
˛
∂r
∂Ψ z ∂r
The equations for a hydrostatic flux tube (cont.)
At the surface of the tube the magnetic field will, in general, be discontinuous,
thus resulting in the existence of a sheet current there. Using
1
1
2
2
pe − pi =
(Bi − Be ) =
(Bi − Be )(Bi + Be )
8π
8π
and from Ampère’s law
c
(Bi − Be )
j =
4π
∗
we finally get
jφ∗
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2(pe − pi )
=
Bi + Be
The equations for a hydrostatic flux tube: Solution procedure
1. Specify initial magnetic configuration
2. Calculate pressure distribution consisten with this field configuration
2
p = p0 (Ψ) exp 4−
ZΨ,z
dz
′
H(T (Ψ, z ′ ))
Ψ,0
3
5
3. Evaluate the volume and sheet current respectively
˛
∂p ˛˛
jφ = r
∂Ψ ˛z
and
jφ∗ =
2(pe − pi )
Bi + Be
4. Integrate Ampère’s equation
∂2Ψ
1 ∂Ψ
∂2Ψ
−
= −4πrjφ
+
2
2
∂r
r ∂r
∂z
and go back to step 2 and iterate 2–4.
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The equations for a hydrostatic flux tube (cont.)
Boundary conditions for solving Ampère’s equation:
–
0
W
Ψ=0
L
Ψ = r B0 z (r) dr
merging with
neighbouring
flux tube
Ψ
=0
z
W is given by the magnetic filling
factor f :
p
W = R0 / f
– Free boundary problem
z
r
0
0
R0
r
Ψ = r B0 z (r) dr
0
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W
The equations for a hydrostatic flux tube (cont.)
Pizzo (1990) used a “body-fitted” nonorthogonal coordinate system to map the physical
domain into a unit square computational domain. A multigrid elliptic solver is used at
each iteration stage for solving Ampère’s equation. Fiedler & Cally (1991) use a
similar mesh in which contours of
constant Ψ (field lines) constitute one
coordinate, the normalized arc length
along field lines the second one.
Computational meshes for a model
sunspot. Left: Constant arc length
collocation. Right: Mesh with auxiliary
“internal gridding surface”. From Pizzo,
1990
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The equations for a hydrostatic flux tube (cont.)
Jahn (1989) and Jahn & Schmidt (1994) use a similar method for the construction of
sunspot models.
From Jahn & Schmidt (1994)
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The equations for a hydrostatic flux tube (cont.)
In case of horizontal temperature equilibrium, Ti (z)
= Te (z) = T (z), we have
(neglegting any horizontal variation in ionization degree) Hi (z) = He (z) = H(z).
Then,
2
pi (z) = p0i exp 4−
Zz
dz
′
H(T (z ′ ))
0
from which follows that pi (z)
3
2
5 pe (z) = p0e exp 4−
Zz
dz
′
H(T (z ′ ))
0
< pe (z) ∀z assuming that pi does not depend on
radius (thin tube approximation) because p0e
− p0i = B02 /8π > 0. Since
Ti (z) = Te (z) it follows for the densities that ρi (z) < ρe (z) ∀z . Since above
conditions can be expected to hold very well in the photosphere we can say that
photospheric flux tubes are rarefied , one also says partially evacuated .
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3
5,
§ 6 The magnetic structure of a hydrostatic flux tube
Consider the most simple case of a constant axial field strength and gas pressure at the
base level, z0 , of the flux tube, where the radius is R0 . Then
8
< Bz0 r2 /2 for r ≤ R0
Ψ(r, z0 ) =
: Bz0 R02 /2 for r ≥ R0
from
Bz =
1 ∂Ψ
r ∂r
and from
pe − pi =
, pi0
2
2
(R0 )
+ Br0
Bz0
= pe0 −
8π
1
(Bi2 − Be2 ). Note, that the final
8π
Br0 (R0 ) is not known from the very beginning so that the boundary condition for the
pressure needs to be adjusted in the course of the iteration.
If we further assume that the temperature at a given hight level is constant,
then the gas
˛
∂p ˛˛
=0
pressure is constant too, so that the volume current is jφ = r
˛
∂Ψ z
and we
have a potential field inside the flux tube. However, the sheet current at the tube
surface,
toc
jφ∗
2(pe − pi )
=
, remains.
Bi + Be
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The magnetic structure of a hydrostatic flux tube (cont.)
Flux tube with a field strength of 1500
Gauß and a radius of 100 km at the base of
the photosphere (τc
= 1). Superimposed
are plots of the radial variation of Br
(dashed curve) and of Bz (solid curve) at
different heights, both normalized to the
value of Bz at the axis, indicated in Gauß.
The flux tube merges with neighbouring
fluxtubes at a height of ≈
500 km. The
filling factor is f = (R0 /W )2 = 0.1.
From Steiner, Pneuman & Stenflo (1986)
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The magnetic structure of a hydrostatic flux tube (cont.)
Left: A flux tube with non-uniform base pressure and magnetic field strength. p0 (r)
increases parabolically from its value at the axis to three times that value at the surface.
Right: The electric current shows a volume component in the interior and a sheet
current at the surface of the flux tube. From Steiner, Pneuman & Stenflo (1986)
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The magnetic structure of a hydrostatic flux tube (cont.)
Introducing the scalar function G
= rBφ one can treat the case of an axisymmetric
flux tube with twist. Ampère’s law then becomes
„
If G
2
2
∂G ∂ Ψ
1 ∂Ψ
∂G
∂ Ψ
−
, −
,
+
∂z
∂r2
r ∂r
∂z 2
∂r
«
= −4πrj .
= G(Ψ) (torque-free condition) we need only solve the φ-component, for which:
˛
1
∂p ˛˛
∂G
+
jφ = r
G
.
˛
∂Ψ z 4πr ∂Ψ
In the absence of an external magnetic field, the magnitude of the sheet current
2(pe − pi )
, directed perpendicular to the field
Bi
lines at the surface and, hence, no longer purely azimuthal. The φ-component of the
remains the same as before:
|j∗ | =
sheet current is:
jφ∗ =
2(pe − pi )
Bi
r
1−(
The solution procedure remains the same as before.
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Bφ 2
) .
Bi
The magnetic structure of a hydrostatic flux tube (cont.)
Uniformly twisted flux tube with
B0 τ0
r for r ≤ R0 and
B0φ =
R0
τ0 = 0.3. Right: Distribution of
Bz and Bφ . Left: 8 field lines 45◦
apart on the flux-tube surface.
From Steiner, Pneuman & Stenflo
(1986)
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The magnetic structure of a hydrostatic flux tube (cont.)
Curiously, there is a maximum twist that can be applied to such a flux tube. For the
region far enough above the merging height the radial component of the magnetic field
vanishes and the gas pressure becomes negligible so that Ampères law reduces to
1 ∂Ψ
∂G
∂2Ψ
−
+
G
=0.
2
∂r
r ∂r
∂Ψ
For the case of a uniform twist at the flux-tube base, when G(Ψ)
=
2τ0
Ψ, the
R0
solution of this equation is
2τ0
Ψ = C1 rJ1 (
r) ,
R0
which has a maximum at
rmax =
with λ
λR0
,
2τ0
≈ 2.4 being the first zero oft the zeroth order Bessel function J0 . rmax may not
be surpassed by the maximal tube radius, W , as long as we have have no return flux.
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The magnetic structure of a hydrostatic flux tube (cont.)
Hence, for a given allowed flux tube expansion defined by the filling factor
f = (R0 /W )2 , we must have
λp
τ0 ≤
f.
2
Physically, this limit can be understood in terms of the force balance between a point on
the axis and a point where the maximum twist occurs. At the axis there is no twist and
Bz2 /(8π) alone must be in force balance with the magnetic pressure excerted by
Bφ max .
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§ 7 The magnetic canopy
The expansion rate of hydrostatic flux tubes critically depends
on the combination of external to internal atmosphere.
The figure to the left shows the expansion if we arbitrarily increase
the internal temperature for z
(above τc
≥0
= 1) by a factor, indicated
for each curve. Likewise, the figure to
the right shows expansion rates for
various plasma β and two different
temperature factors. Note the
horizontal spreading of the flux-tube
surface at low altitudes in the most
extreme cases.
From Steiner & Pizzo (1989)
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The magnetic canopy (cont.)
A critical hight at which the flux tube must expand in horizontal direction can be derived
from the expressions for gas pressure close to the tube surface along a magnetic field
line, pi , and in the surrounding
atmosphere,
0
1 pe :
pi,e (z) = p0i,e exp @−
B 02
ln
8π
0
ln p
p
0e
i
e
z crit
Zz
0
dz ′ A
.
′
Hi,e (z )
If
Te < Ti ⇒ He < Hi and the pressure differ-
ence, pe −pi , and with it the surface field strength, decreases rapidly with height. Correspondingly, the flux
tube expands until a critical height, zcrit , at which the
field assumes a horizontal direction.
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The magnetic canopy (cont.)
A critical hight at which the flux tube must expand in horizontal direction can be derived
from the expressions for gas pressure close to the tube surface along a magnetic field
line, pi , and in the surrounding
atmosphere,
0
1 pe :
pi,e (z) = p0i,e exp @−
zZcrit
Zz
0
dz ′ A
. With pe (zcrit ) = pi (zcrit )
′
Hi,e (z )
(β0 +1)/β0
1
1
p0e
(
−
)dz = ln
He
Hi
p0i
0
B 02
ln
8π
0
ln p
p
0e
i
e
z crit
If
H=const.
⇒
zcrit
z}|{
Hi He
p0e
=
ln
Hi − He
p0i
Te < Ti ⇒ He < Hi and the pressure differ-
ence, pe −pi , and with it the surface field strength, decreases rapidly with height. Correspondingly, the flux
tube expands until a critical height, zcrit , at which the
field assumes a horizontal direction.
toc
ref
The magnetic canopy (cont.)
A strongly expanding flux tube with a low critical
height ensues when embedded in a cool atmosphere
with a temperature that decreases beyond the traditional temperature minimum (COOLC or RE in the figure on the right). Such atmospheres were suggested
on the basis of infrared observations in lines of CO.
The figure to the left shows the gas pressure of the
RE-atmosphere (dot-dashed) in combination with the
′
gas pressure of the flux-tube atmosphere (C of the
above figure), for various base field-strengths. Critical heights of 780, 860, and 1000 km correspond to
base field-strengths of 1300, 1500, and 1700 Gauß.
toc
ref
The magnetic canopy (cont.)
′
Flux tube with a C -atmosphere embedded in the cool RE-atmosphere. The base
field-strength is 1500 Gauß. The fieldlines spread into a horizontally extending canopy
field at a height of 900 km and merge with the field of neighboring flux tubes.
toc
ref
The magnetic canopy (cont.)
Chromospheric magnetograms of a unipolar
network region show near the limb a fringe
pattern in polarity. They show a polarity inversion across a line that coincides with the
limbward edge of a unipolar magnetic network field. The figure shows an example
magnetogram taken in the line of Mg I b2
(5173 Å). In the vicinity of “C” one can see
a fringe pattern. In the photosphere, this region sows a unipolar enhanced network.
From R.G. Giovanelli: 1980, SP 68, 49
The fringe pattern can be understood in terms of the magnetic canopy. The canopy
“roots” in the network and overlies the internetwork region.
toc
ref
The magnetic canopy (cont.)
ff
θ
zc
ff
0
τ 50
limb
b
=1
a
From Steiner (2000)
Lines of sight passing through the network field receive a magnetic field component
toward the observer and so do lines of sight passing through the canopy field to the left
(disk center) side of the network (dashed line of sight). The canopy field to the right
(limbward) side of the network, however, gives rise to a line of sight component of
opposite direction (solid line of sight), which causes the fringe of opposite polarity.
toc
ref
The magnetic canopy (cont.)
The corresponding magnetogram of the magnetic network at the photospheric level
would be of exclusively positive polarity. Towards the limb, the network fields apparently
move closer together with increasing line-of-sight aspect angle θ , giving rise to an ever
narrower fringe pattern of alternating polarity in the magnetogram.
From observations, such as shown above, Giovanelly & Jones derived canopy heights
of 600–1000 km in quiet-Sun regions and as low as 200 km in active regions.
toc
ref
§ 8 The thin flux-tube approximation
The approximation of thin flux tubes is valid as long as all length scales along
the tube are large in comparison to the tube diameter. For an axisymmetric
vertical tube this means that
R/H << 1,
kR << 1 ,
where H is the scale height of any quantity, e.g., pressure in the tube, k is the
vertical wave number of any perturbation propagating along the tube, and R
the tube radius.
In this case we can expand all physical quantities in radial direction and fully
maintain their height dependency.
toc
ref
The thin flux-tube approximation (cont.)
Consider a magnetohydrostatic, axisymmetric, vertical magnetic flux tube and assume,
that we can neclect any variation of Bz in radial direction (zeroth order approximation).
If the temperature, and hence the pressure scale height Hp , is the same inside and
outside the tube, then
2
B0z
/8π
z }| {
z
Bz2
pe − pi = (pe0 − pi0 ) exp(−
)=
Hp
8π
from which we can compute Bz (z). Using the conservation of magnetic flux
Φ = 2π
R(z)
R
Bz (z)rdr = 2πR2 (z)Bz (z) = const = 2πR02 (z)Bz
0
we can compute the flux-tube radius as a function of height and finally obtain
z
z
) and Bz (r, z) = B0z exp(−
).
R(z) = R0 exp(
4Hp
2Hp
It takes about four pressure scale heights for the tube radius to expand by a factor of e.
toc
ref
The thin flux-tube approximation (cont.)
In order to satisfy solenoidality we can compute the first order term of the radial field
expansion. With Br (r)
= Br1 r we obtain
1 ∂
Bz (z)
∂Bz
1
∇·B=
(rBr ) +
= 0 = 2Br −
,
r ∂r
∂z
2Hp
hence,
Br (r, z) =
Bz (z)
r.
4Hp
However, note, that the boundary condition
Bz2
z
is zeroth-order accurate only.
)=
pe − pi = (pe0 − pi0 ) exp(−
Hp
8π
toc
ref
The thin flux-tube approximation (cont.)
Generally, we can expand the full system of the MHD-equation:
∇·B=0,
∂B
= ∇ × (v × B) ,
∂t
∂ρ
+ ∇ · (ρv) = 0 ,
∂t
∂v
1
ρ
+ (v · ∇)v = − ∇p + ρg +
(∇ × B) × B ,
∂t
4π
γp ∂ρ
∂p
+ v · ∇p =
+ v · ∇ρ .
∂t
ρ
∂t
toc
ref
The thin flux-tube approximation (cont.)
All pertaining quantities are expanded as a power series in r :
vr (r, z, t) = rvr1 (z, t) + r3 vr3 (z, t) + · · · ,
vφ (r, z, t) = rvφ1 (z, t) + r3 vφ3 (z, t) + · · · ,
vz (r, z, t) = vz0 (z, t) + r2 vz2 (z, t) + · · · ,
Br (r, z, t) = rBr1 (z, t) + r3 Br3 (z, t) + · · · ,
Bφ (r, z, t) = rBφ1 (z, t) + r3 Bφ3 (z, t) + · · · ,
Bz (r, z, t) = Bz0 (z, t) + r2 Bz2 (z, t) + · · · ,
pz (r, z, t) = p0 (z, t) + r2 p2 (z, t) + · · · ,
ρz (r, z, t) = ρ0 (z, t) + r2 ρ2 (z, t) + · · · .
toc
ref
The thin flux-tube approximation (cont.)
The fact that the radial and the azimuthal components of v and B are odd
series in r , while the axial components vz and Bz as well as the scalars p and
ρ are even series is a general property of scalars and vector fields of an
axisymmetric system. This was shown by Ferriz Mas & Schüssler, 1989.
Introducing this expansion into the system of MHD-equations and sorting the
different orders (k
= 0, 1, . . . , 2n), one obtains a system of 9n + 5
equations (where primes denote derivatives after z ):
toc
ref
The thin flux-tube approximation (cont.)
Even orders:
k = 0, 2, . . . , 2n
∂Bzk
−
=(k + 2)
∂t
∂ρk
−
=(k + 2)
∂t
X
i+j=k
=
−p′k
∂vzj
+
ρi
∂t
1
− gρk +
4π
X
(4 × (n + 1) equations)
X
i+j=k+1
X
(vri Bzj − Bri vzj ) ,
ρi vri +
i+j=k+1
′
ρi vzj vzl
i+j+l=k+1
(ρi vzj )′ ,
i+j=k
+
X
jρi vzj vrl
i+j+l=k+1
i+j+l=k
X
X
iBzi Brj
1 X
′
′
),
+ Bri Brj
−
(Bφi Bφj
4π
i+j=k
X
∂ρi
∂pj
−γ
pj ) +
(j − iγ)ρi pj vrl
(ρi
∂t
∂t
i+j+l=k+1
i+j=k
X
+
(ρi p′j − γρ′i pj )vzl = 0 .
X
i+j+l=k
toc
ref
The thin flux-tube approximation (cont.)
Odd orders:
k = 1, 3, . . . , 2n − 1
(4 × n equations)
X
∂Brk
−
=
[(vzi Brj )′ − (Bzi vrj )′ ] ,
∂t
i+j=k
X
X
∂Bφk
′
′
−
=
[(vzi Bφj ) − (Bzi vφj ) ] + (k + 1)
(vri Bφj − Bri vφj ) ,
∂t
i+j=k
X
i+j=k
∂vrj
ρi
+
∂t
i+j=k+1
X
′
′
vzl
ρi vrj
X
+
i+j+l=k+1
i+j+l=k
ρi (lvrj vrl − vφj vφl )
1 X
[jBzi Bzj + (1 + j)Bφi Bφj ] ,
= −(k + 1)pk+1 +
4π
i+j=k
X
i+j=k
∂vφj
ρi
+
∂t
X
′
ρi vzj vφl
i+j+l=k
1 X
1
′
=
Bzi Bφj +
4π
4π
i+j=k
toc
ref
+
X
(l + 1)ρi vrj vφl
i+j+l=k+1
X
i+j=k+1
(j + 1)Bri Bφj .
The thin flux-tube approximation (cont.)
These equations are complemented with the solenoidality condition:
Even orders:
k = 0, 2, . . . , 2n
′
+ (k + 2)Br(k+1) = 0 .
Bzk
However, these equations are not independent. Instead use the potential A for the
poloidal part od B:
∂A
1 ∂(rA)
Bp = (Br , 0, Bz ) = (−
, 0,
.
∂z
r ∂r
Then, the r - and z -components of the induction equation reduce to the same
expression:
−
vr ∂(rA)
∂A
= vz A′
.
∂t
r ∂r
The expansion of A is of the form
A(r, z, t) = rA1 (z, t) + r3 a3 (z, t) + . . .
toc
ref
The thin flux-tube approximation (cont.)
The solenoidality condition as well as the r -components of the induction equation drop
out and are replaced by the above equation for A, where
Br(2m+1) = −A′2m+1 , Bz(2m) = (2m + 2)A2m+1 m = 0, 1, 2, . . . , n .
The MHD equations are then reduced to a system of 7n + 4 first order partial
differential equations for 7n + 5 unknown functions (one more because of Bz2n that is
not given by the n odd order equations for A(r, z, t)). To close the system one has to
specify the kind of problem one wishes to consider with the corresponding boundary
conditions. These are, e.g., for a magnetic flux tube embedded in a field-free plasma:
˛
B ˛
p+
8π ˛
2˛
toc
ref
= pe (R) .
r=R
The thin flux-tube approximation (cont.)
Applying the recipe outlined above to leading order one obtains the following 5
equations for the 5 unknown functions ρ0 , p0 , A1 , vz0 , and vr1 :
′
) = −p′0 − ρ0 g ,
ρ0 (v̇z0 + vz0 vz0
(1)
ρ̇0 + (ρ0 vz0 )′ + 2ρ0 vr1 = 0 ,
(2)
Ȧ1 = −vz0 A′1 − 2vr1 A1 ,
p0 γ
′
ṗ0 + vz0 p0 =
(ρ˙0 + vz0 ρ′0 ) ,
ρ0
2 2
p0 +
A1 = pe .
4π
(3)
(4)
(5)
Note that even in the lowest order, the system will keep essential non-linearities of the
full MHD-equations.
toc
ref
The thin flux-tube approximation (cont.)
In the hydrostatic case, Eqs. (1)–(5) reduce to
−p′0 − ρ0 g = 0 ,
1 ′
p0 +
Bz0 = pe ,
8π
γp0
d p0
=
= c2 ,
dρ0
ρ0
which is, together with
Φ = 2πR2 (z)Bz0 = const
what we have derived at the beginning of this paragraph.
toc
ref
The thin flux-tube approximation (cont.)
This figure demonstrates for a vertical hydrostatic flux tube embedded in a standard
solar atmosphere the deviation of the zeroth-order flux-tube approximation from the full
(numerical) solution as a function of height and spatial scale . Each curve shows the
height at which the field strength at the tube wall deviates from the axial value by the
specified fractional amount D
= |Bz (r = 0) − Bz (r = R)|/Bz (r = 0) as a
function of the tube radius Rτe =1 .
toc
ref
§ 9 Magnetic flux tube in radiative equilibrium
There are two basic modes of energy transport in the solar photosphere and
convection zone: radiative and convective. When all of the energy is
transported by radiation, we have radiative equilibrium, conversly, pure
convective transport is called convective equilibrium. In a stationary transport
process, the frequency distribution of the radiation, or the partioning of energy
between the radiative and the convective mode of transfer, may be altered; but
the energy flux as a whole is rigorously conserved. Formally, this is expressed
by
∇ · Ftot = 0;
Ftot = Frad + Fconv .
In radiative equilibrium:
∇ · Frad = 0 ,
Fconv = 0 ,
In convective equilibrium:
∇ · Fconv = 0 ,
Frad = 0 .
toc
ref
Magnetic flux tube in radiative equilibrium (cont.)
In the solar photosphere radiative energy transfer by large prevails so that
∇ · Frad = 0 is a good approximation. With I(r, n̂, ν) being the radiative intensity
with frequency ν propagating in direction n̂ at location r in a multidimensional
coordinate frame, the total radiative flux is given by
Frad =
Z Z∞
I(r, n̂, ν)n̂ dν dω .
4π 0
I(r, n̂, ν) has dimension ergs cm−2 s−1 hz−1 sr−1 , correspondingly has Frad
dimension ergs cm
−2
−1
s
. The radiation field follows as a solution of the radiative
transfer equation
(n̂ · ∇)I(r, n̂, ν) = η(r, ν) − κ(r, ν)I(r, n̂, ν) .
The emissivity η(r, ν) is given by η(r, ν)
function.
toc
ref
= κ(r, ν)S(r, ν) with S being the source
Magnetic flux tube in radiative equilibrium (cont.)
From these equations we obtain:
Z Z∞
(n̂ · ∇)I(r, n̂, ν) dν dω
=
∇ · Frad
4π 0
Z Z∞
!
(κ(r, ν)S(r, ν) − κ(r, ν)I(r, n̂, ν) = 0 .
=
4π 0
1
With J(r, ν) =
4π
Z
I(r, n̂, ν) dω being the mean intensity, we finally obtain the
4π
constraint equation for radiative equilibrium:
Z∞
0
toc
ref
κ(r, ν)S(r, ν) dν =
Z∞
0
κ(r, ν)J(r, ν) dν .
Magnetic flux tube in radiative equilibrium (cont.)
In general we do not know the temperature distribution T (r) that satisfies radiative
equilibrium. If we start with a guess T
(0)
(r) for which we have calculated the correct
source function S(r, ν), we will find that the constraint equation for radiative
equilibrium is not satisfied. It is therefore necessary to iteratively adjust T (r) until the
requirement of radiative balance is satisfied.
toc
ref
Magnetic flux tube in radiative equilibrium (cont.)
Assume a given temperature distribution T (r) of the magnetohydrostatic configuration
for which we also know the pressure p(r) and density ρ(r). This allows us to derive
LTE-values of opacities κ(r, ν). We then are able to compute the radiation field at any
point r and any frequency ν by evaluation of the formal solution of the radiative transfer
equation
Jν (r) = Λν (r, r′ )Bν (r′ ) + Gν (r) ,
where Λν is the integral operator which adds the intensities at r caused by emision at
′
all the points r in the considered computational domain, and where Gν is the
transmitted mean intensity due to the incident radiation field into this domain. We also
use strict local thermodynamic equilibrium (LTE) (Sν
= Bν ), although a scattering
component could be included in the solution method that follows.
toc
ref
Magnetic flux tube in radiative equilibrium (cont.)
Defining the integral operator K so that Kφ
:=
R∞
κν φdν , where φ is a scalar
0
function, we can now write the constraint equation for radiative equilibrium as:
KBν = KΛν Bν + KGν
However, with a given initial temperature distribution, T
(0)
(r), this equation will in
general not be satisfied but we can compute a correction ∆T (r) demanding that
KBν (T (0) + ∆T ) = KΛν Bν (T (0) ) + KGν
Expanding Bν (T
+ ∆T ) ≈ Bν (T ) + (∂Bν /∂T )|T ∆T , we can compute the
temperature correction:
∆T =
toc
ref
K(Λν − I)Bν (T ) + KGν
K(∂Bν /∂T )|T
Magnetic flux tube in radiative equilibrium (cont.)
For most practical purposes this iterative scheme is very slow. The next better scheme
to use is a Jacobi-like iteration, in radiative transfer known as accelerated Λ-iteration :
∆T =
K(Λν − I)Bν (T ) + KGν
,
∗
K(1 − λν )(∂Bν /∂T )|T
∗
where λν is the diagonal element of the matrix representing Λν (r). Note that
∆T = ∆T (r) and that the above equation must be evaluated for each r.
The formal solution of the transfer equation must be computed in each iteration. An
efficient way to do this consists in using the so called method of short characteristics
and exploiting the symmetry properties .
toc
ref
Magnetic flux tube in radiative equilibrium (cont.)
Two-dimensional slab atmosphere. The rectangle in the upper left corner represents a
magnetic flux sheet with a rarified atmosphere. The incident radiation on the bottom
and on the right side of the computational domain are computed from the undisturbed,
plane-parallel atmosphere with κ
= κe . The left side is an axis of symmetry of the
configuration. The top boundary has no incident radiation.
κ e = const
I in = I undisturbed
toc
ref
I in = I undisturbed
κ i = 0.2 κ e
I in = I out
axis of symmetry
I in = 0
Magnetic flux tube in radiative equilibrium (cont.)
Two-dimensional slab atmosphere. The rectangle in the upper left corner represents a
magnetic flux sheet with a rarified atmosphere. The incident radiation on the bottom
and on the right side of the computational domain are computed from the undisturbed,
plane-parallel atmosphere with κ
= κe . The left side is an axis of symmetry of the
configuration. The top boundary has no incident radiation. Right: Resulting isotherms.
κ e = const
I in = I undisturbed
toc
ref
I in = I undisturbed
κ i = 0.2 κ e
I in = I out
axis of symmetry
I in = 0
Magnetic flux tube in radiative equilibrium (cont.)
Magnetic flux tube with
Bz (r = 0, z = 0) =
1600 Gauß and a
radius of
z [km]
R(z = 0) = 100 km
in radiative equilibrium.
Contour lines of
4800
constant temperaure
and
6000
7200
8400
are
r [km]
toc
ref
indicated.
Magnetic flux tube in radiative equilibrium (cont.)
Magnetic flux tube with
Bz (r = 0, z = 0) =
1600 Gauß and a
radius of
z [km]
R(z = 0) = 100 km
log τ
in radiative equilibrium.
−2.0
Contour lines of
4800
−1.0
constant temperaure
and constant optical
−2.0
6000
0.0
7200
−1.0
8400
depth
1.0
are
r [km]
toc
ref
indicated.
Magnetic flux tube in radiative equilibrium (cont.)
Magnetic flux tube with
Bz (r = 0, z = 0) =
1600 Gauß and a
radius of
z [km]
R(z = 0) = 100 km
log τ
in radiative equilibrium.
−2.0
Contour lines of
4800
−1.0
constant temperaure
and constant optical
−2.0
6000
0.0
depth and the domain of
1.0
prescribed, fixed
7200
−1.0
8400
temperature are
r [km]
toc
ref
indicated.
Magnetic flux tube in radiative equilibrium (cont.)
Why is there a temperature elevation in the photospheric layers of a magnetic flux tube?
Consider two points P1 and P2 .
P1 receives a higher intensity of radiation from the flux
tube’s hot walls as compared to P2 . Both points receive no radiation from the top.
Therefore, the mean radiation is higher in P1 than in P2 , J1
radiative equilibrium J
=S=
R
> J2 . In
Bν dν ∝ T 4 and hence, T1 > T2 .
P1
P2
P3
P4
τ c =1
toc
ref
Magnetic flux tube in radiative equilibrium (cont.)
Temperature profile along the flux tube axis, left as a function of log τc , right as a
function of height. Dashed and dot-dashed curve correspond to the temperature along
the axis of flux tubes with a field strength of 1500 and 1600 Gauß at z
= 0,
respectively. The dotted curve is from the semi-empirical model of Keller et al. 1989.
The solid curve represents the surrounding atmosphere.
toc
ref
Magnetic flux tube in radiative equilibrium (cont.)
Semi-empirical atmospheres for plage flux-tubes (dottet) and network flux-tubes
(dashed) by Solanki & Brigljević (1992). The solid line corresponds to model C (average
quiet Sun) of Fontenla et al. (1999).
toc
ref
§ 10 The physics of faculae
The magnetic elements of the network are hardly visible at disk center, except in
potospheric spectral lines or in the wings of chromospheric lines, or in special
wavelength ranges like the G-band. They become increasingly brighter towards
the limb for heliographic angles
µ = cos(θ) larger than about 0.6, where
they appear as extended bright areas,
called faculae.
Facular points of an area of 20 × 27
2
arcsec at disk center. Photograp by
Mehltretter (1974 !) with the R. B. Dunn
Telescope on Sacramento Peak at
393.4 ± 0.8 nm
toc
ref
The physics of faculae (cont.)
Speckle reconstructed
image of facular region
taken with the 1 m Swedish
Solar Telescope in the
continuum at 487.5 nm.
Field of view approximately
80′′ × 80′′ .
From Hirzberger & Wiehr
(2005), A&A 438, 1059
toc
ref
The physics of faculae (cont.)
There exists a long list of center to limb measurements of the continuum
contrast of faculae. The measurements are contradictory because they are
spatial resolution dependent and because of selection effects. There exists an
equally long list of models most notable the “hot wall” model of Spruit (1976).
From Spruit (1976), Sol. Phys. 50, 269
toc
ref
The physics of faculae (cont.)
Center-to-limb variation (CLV)
of continuum contrast (500 nm)
Continuum contrast (575 nm) of 880 net-
of a flux tube in radiative equi-
work bright points as a function of helio-
librium as shown in § 9.8 along
graphic angle. Squares represent the con-
with observed (dashed and
trast of granulation. From Auffret and Muller
dot-dashed) values.
(1991), Observatoire du Pic du Midi
toc
ref
The physics of faculae (cont.)
Faculae at θ
= 61◦ in the continuum at 587.5 nm (left) and in the G-band (right). Solar
′′
limb is right. Tickmarks indicate 1 distances. The facular brightening occurs on the
disk-center side of granules limbward of a dark “facular lane”.
From Hirzberger & Wiehr (2005), A&A 438, 1059
toc
ref
The physics of faculae (cont.)
Mean spatial scan through faculae
at θ
= 61◦ in the 587.5 nm
continuum (top) and in the G-band
(bottom). Note the flat limbward
decrease and the centerward dark
“facular lane”.
From Hirzberger & Wiehr (2005),
A&A 438, 1059
toc
ref
The physics of faculae (cont.)
- From a location at the solar surface and lateral to the flux sheet one “sees” a more
transparent sky in the direction to the flux sheet compared to a direction under
equal zenith angle but away from it.
τ c =1
- Correspondingly, from a wide area surrounding the magnetic flux sheet or flux tube,
radiation escapes more easily in the direction of the flux sheet/tube.
- A single flux sheet/tube impacts the radiative escape in a cross-sectional area
(“radiative cross section” ) that is much wider than the magnetic field concentration
proper.
toc
ref
The physics of faculae (cont.)
From Steiner (2005) A&A 430, 691
• Double humped contrast profile at disk
center • Sharp disk-center side increase
due to “‘hot wall” • Gentle limbward decline
at µ = cos θ = 0.5 due to lines of sight
• profiles with 30◦ ≤ θ ≤ 60◦ show
within a narrow region a “dark lane”
centerward (in front of) the facular
that traverse flux sheet in photospheric
brightening • Lines of sight of dark lane
layers (left of LOS) • Contrast enhancement
travel through internal atmosphere of low
wider at µ
temperature gradient • The dark lane is the
= 0.5 than at disk center •
Smooth distribution of polarization signal
toc
ref
manifestation of the “cool bottom” of faculae
The physics of faculae (cont.)
c) Equivalent width of total polarization
of FeI 630.25 nm
b) Continuum-contrast profiles for
θ = ±45◦
a) Two surfaces of constant optical
τc = 1 and zenith angle
θ = ±45◦ . Lines of sight mark
depth
region of dark facular lane
Results:
Part of the dark lane phenomenon is caused by the cool deep layers of the flux sheet
interior but also the downflow at the flux-sheet interface contributes to it. Close to the
flux sheet the downflow is seen, which is dark like an intergranular lane.
toc
ref
The physics of faculae (cont.)
Appearance of faculae in 3-D
MHD-simulations:
Center to limb variation of the
G-band intensity emanating
from a simulation box of
6 × 6 Mm at
cos θ = µ = 1.0, 0.8, 0.6,
and 0.4.
Right: Observation at
µ = 0.63.
From Carlsson, Stein, and
Nordlund (2004) ApJL, 610,
L137
toc
ref
The physics of faculae (cont.)
Appearance of faculae in
3-D MHD-simulations:
Comparison of observed
faculae (top) with faculae from
the simulation box of
6 × 6 Mm at µ = 0.5 with
hBi = 400 G (middle) and
200 G (bottom).
Right: Contrast profiles of
facula of the middle panel (top)
and the bottom panel (bottom).
Keller, Schüssler, Vögler, and
Zakharov (2004) A&A 607, L59
toc
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The physics of faculae (cont.)
Rapid temporal variability of faculae. De Pontieu et al. (2006), ApJ 646, 1405
“Dark bands”
in observations and
in simulations
Dark bands form naturally in the course of the evolution of a granule. They
correspond to the dark lane that forms when a granule is about to split, similar
to the central dark region of an exploding granule.
toc
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§ 10a Faculae and irradiance variability
The enhanced radiation output from magnetic elements is believed to determine the
long-term solar-irradiance variability – the variability of the “solar constant” with the
solar cycle. The center-to-limb variation of the facular contrast influences the precise
behaviour of this variability and the relation between solar irradiance and solar
luminosity.
A question in the physics of solar irradiance variability is: What is the connection
between the variation in solar brightness and the evolution of the magnetic field at the
solar surface?
See also the Nature review of Foukal, Fröhlich, Spruit, and Wigley (2006) for a recent
view on this topic.
toc
ref
Faculae and irradiance variability (cont.)
Composite from solar irradiance
measurements. From the Website
of the World Radiation Center
http://www.pmodwrc.ch/
toc
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Faculae and irradiance variability (cont.)
In an attempt to model solar irradiance variability from magnetograms, Fligge, Solanki
and Unruh (2000) use model atmospheres of faculae, sunspots, and the quiet Sun to
compute corresponding spectra as a function of disk position. They assign each pixel
(i, j) on the solar disk a facular filling-factor, αfi,j (Φ, t), and a spot filling-factor ,
αsi,j (Φ, t), depending on the magnetic flux Φ and time t to obtain
tot
(λ) = (1 − αsi,j (Φ, t) − αfi,j (Φ, t)) · I q (µ(i, j), λ)
Ii,j
+ αsi,j (Φ, t) · I s (µ(i, j), λ)
+ αfi,j (Φ, t) · I f (µ(i, j), λ)
Pixel (i, j) is considered to lie within an active region if its magnetic flux density
surpasses a center-to-limb dependent threshold value Φth (µ(i, j)). If the continuum
intensity of this pixel is more than 10σ lower than that of the quiet Sun at equal µ it
belongs to a sunspot, otherwise to a facular region.
toc
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Faculae and irradiance variability (cont.)
Top: Magnetogram (left) and
corresponding white-light image (right) from MDI on SOHO.
Bottom: Extracted maps for
faculae (left) and sunspots
(right). From Fligge et al. 2000
toc
ref
Faculae and irradiance variability (cont.)
Measured (dashed) and modelled (solid) solar total
and spectral irradiance variations from the time
between 15 August (day 228) and 11 September
(day 255) 1996. The model is able to reproduce the
double humped structure originating from the CLV of
facular contrast. From Fligge et al. (2000)
toc
ref
Faculae and irradiance variability (cont.)
Reconstruction (solid) of the total solar irradiance composed by Fröhlich & Lean (1998).
The long-term behaviour, i.e., the difference between maximum and minimum (a) as
well as shorter-term variations (b) are well reproduced. From Fligge et al. (1998)
toc
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§ 11 The interchange instability of magnetic flux tubes
Consider a section perpendicular to the axis of a straight flux-tube. It can be shown,
that a perturbation of the flux-tube boundary in such a way that the volumes V1 and V2
are equal, does not change the total energy of the configuration.
unperturbed
boundary
V1
perturbed
boundary
V2
unperturbed
boundary
The instability that evolves from this perturbation is called interchange instability, as the
magnetic field and gas of volume 1 is interchanged with the magnetic field and gas of
volume 2. It is also called flute instability because of the shape of the perturbed surface
(like the vertical parallel grooves on a classical architectural column, called flute).
toc
ref
The interchange instability of magnetic flux tubes (cont.)
Consider a small flux-tube section with a small
perturbation ξ . The grey shaded area, Vi , be the
magnetic flux tube, Ve is field-free.
Vi
ξ
n
n is the sur-
S
Ve
face normal pointing out of the field-free plasma.
In order that there is a net restoring force on the displaced surface we must, for the
indicated displacement ξ , have
p0itot + δpitot > p0e + δpe
since for the equilibrium configuration p0itot
⇒
δpitot > δpe
= p0e . From this follows the condition for
stability:
B2
).
|ξ · n|n · ∇pe < |ξ · n|n · ∇(pi +
8π
(1)
This criterion also follows from a more general energy principle due to Bernstein et al.
(1958). It is both, necessary and sufficient for stability.
toc
ref
The interchange instability of magnetic flux tubes (cont.)
Within the flux tube we have
B2
1
∇pi + ∇(
) = ρi g +
(B · ∇)B
8π
4π
as was derived from the momentum equation in § 3. In the external atmosphere
∇pe = ρe g .
Using these equations in the previously derived stability criterion (1) we obtain:
n·
»
–
1
(B · ∇)B − (ρe − ρi )g > 0 .
4π
From the first two equations we have
B2
1
(B · ∇)B − (ρe − ρi )g = ∇(pi +
− pe )
4π
8π
|
{z
}
=0
on S
which means that the bracketed vector in (2) is parallel to n on the surface S .
toc
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(2)
The interchange instability of magnetic flux tubes (cont.)
Therefore, gravity can be eliminated taking the horizontal component of this vector. If h
is a horizontal vector pointing out of the fluxtub into the field-free plasma we get
h · [(B · ∇)B] < 0 .
(3)
Along any field line in S the magnitude of the component of B in any fixed outward
horizontal direction must decrease upwards. For an untwisted axisymmetric flux tube
this criterion can be expressed in cylinder coordinates as:
˛
dBr ˛˛
<0.
˛
dz S
(4)
B2
From n · (B · ∇)B = −
(see derivation in § 3), where Rc is the curvature radius
Rc
of the surface in axial direction, results another useful form of (2) for an axisymmetric
untwisted flux tube (χ is the inclination of S w.r.t the vertical):
B2
−
+ (ρe − ρi )g sin χ > 0 .
4πRc
toc
ref
(5)
The interchange instability of magnetic flux tubes (cont.)
We could compute dBr /dz from the results obtained with the thin flux-tube
approximation in § 8. However, there we assumed a constant pressure scale height for
both, the internal and external atmosphere. When using a realistic model atmosphere,
Meyer et al. (1977) came to te conclusion that only flux tubes with Φ
> 1019 Maxwell
are stable against the flute instability. Sunspots and pores have flux in excess of 10
mx. With typical values of R
19
= 100 km and B = 1000 Gauß resulting in
Φ ≈ 3 · 1017 G, magnetic elements are liable to the flute instability!
Indeed, from G-band-bright-point movies (e.g., Rouppe van der Voort et al.) one gets
the impression that magnetic elements are subject to fluting. On the other hand their
typical life time of 6–8 minutes is still in excess of the crossing time for Alfvén or sound
waves of about 100 s. Also there are bright points that undergo continual fragmentation
and merging in a relatively stable location, persisting over several hours.
toc
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The interchange instability of magnetic flux tubes (cont.)
One remedy would consist in introducing twist, which would effectively supress the
intergange instability.
Another remedy was proposed by Schüssler (1984) who investigated the effect of
various flows within and around a flux tube.
Of all these, the most efficient in supressing the
z
B=0
Vi
n
flute instability is a whirl flow surrounding the flux
χ
tube. Such a flow may arise from the “bathtub
effect” in the intergranular downdrafts. The cor-
g
responding stability criterion now reads
r
B=0
Ve
toc
S
ve = vB eΦ
A
˛
vφ2
dBr ˛˛
1
− ρe
Bz
<0.
4π
dz ˛S
R
Br that grows with height destabilizes, while
vφ stabilizes.
ref
(6)
The interchange instability of magnetic flux tubes (cont.)
A typical stability diagram shows the maximal whirl flow velocity, vφ , needed to stabilize
a magnetic flux tube as a function flux Φ.
Three stability curves from Bünte et al.
(1993). The two, requiering fastes flow
velocities are derived from the model
atmosphere of Meyer et al. (1977) and
a from a modern model atmosphere,
both using the thin flux-tube
approximation. A model flux-tube from
a numerical solution requires a vφ max
of only 2 km/s. Magnetic tension forces
indirectly act to supress the
interchange instability.
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§ 12 The formation of flux tubes by flux expulsion
Consider a kinematic flow consisting of stationary rolls in a twodimensional box
of size L × L:
πx
πz
πx
πz
u = U [− sin( ) cos( ) , 0 , cos( ) sin( )]
L
L
L
L
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The formation of flux tubes by flux expulsion (cont.)
Imposing an initial vertical magnetic field
and that the field must be vertical at all
boundaries leads to the sequence
opposite (Galloway & Weiss, 1981). The
clockwise flow first advects the field but
later, when boundary layers have
evolved, gradients are large enough so
that the diffusion term dominates in the
induction equation. The magnetic flux is
thus expelled from the cell interior and
concentrates in flux sheets near the
boundary.
toc
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The formation of flux tubes by flux expulsion (cont.)
We can estimate the width, d, of the flux sheet. From the resistive induction equation
∂B
= ∇ × (v × B) + η∇2 B .
∂t
The time scale of field diffusion is τd
= d2 /η , while the time scale for field advection is
τad = L/v . In the steady state field decay must be balanced by field advection.
Therefore τd = τad from where
√
Rm = vL/η .
d = L/ Rm
We may also estimate the field strength in the flux sheet assuming that the whole flux
across the box is advected into flux sheet:
B0 L = Bd
For a flux tube we would get B
toc
ref
⇒
= Rm B0 .
√
B = Rm B0 .
The formation of flux tubes by flux expulsion (cont.)
As the field becomes stronger, it counteracts the motion and the kinematic regime
changes to the dynamic regime. In this regime the magnitude of the Lorentz force
becomes of the order of of the inertial force in the momentum equation:
1
(∇ × B) × B ≈ ρv · ∇v .
4π
From this the order of magnitude of the maximal attainable field strength is:
v2
1 B2
=ρ
4π d
d
For solar surface values of v
⇒
Bmax = Beqp
p
= 4πρ v .
= 2 km/s and ρ = 3 · 10−7 g cm−3 we obtain
Beqp = 388 Gauß. The equipartition field strength is considerabely smaller
than the observed kGauß fields.
However, Galloway, Proctor, & Weiss, (1977) derived in compressible, resistive, viscous
numerical simulations Bmax /Beqp
toc
ref
p
= ν/η = Pr mag .
§ 13 Flows in magnetic elements: Observations
In the seventies and the early eighties downdrafts in magnetic flux tubes with
velocities up to 2.2 km s−1 were reported. But Solanki & Stenflo (1986) noted
that a spurious zero-crossing shift is measured as a consequence of
insufficient spectral resolution.
V
Spectral smearing of an asymmetric
Stokes V profile with the blue lobe
being larger than the red one (which is
ab
Ab
usually the case for plage and network
Ar
vzc
toc
ref
λ
ar
magnetic elements at disk center)
results in a redshifted zero-crossing
wavelength.
§ 13 Flows in magnetic elements: Observations
In the seventies and the early eighties downdrafts in magnetic flux tubes with
velocities up to 2.2 km s−1 were reported. But Solanki & Stenflo (1986) noted
that a spurious zero-crossing shift is measured as a consequence of
insufficient spectral resolution.
V
Spectral smearing of an asymmetric
Stokes V profile with the blue lobe
being larger than the red one (which is
ab
Ab
usually the case for plage and network
Ar
vzc
toc
ref
λ
ar
magnetic elements at disk center)
results in a redshifted zero-crossing
wavelength.
Flows in magnetic elements: Observations (cont.)
Subsequently, high spectral resolution measurements, carried out with the Fourier
transform spectrometer at Kitt Peak resulted in no significant zero-crossing velocities.
“Epur si muove” – And yet it does move!
Grossmann-Doerth et al. (1996) and Sigwarth et al. (1999) measured a mean downflow
−1
velocity of 0.7 – 0.8 km s
. These mean velocities are averages over up to 93,000
magnetic elements with Stokes V amplitudes >
0.15%. Average over the weakest
−1
network elements (amplitudes below 1%) would result in a velocity of 1 km s
−1
however, that weak magnetic elements show a scatter of up to ±5 km s
. Note,
.
The difference with respect to older measurements resulting in no significant
zero-crossing shift is due to the high sensitivity of the employed polarimeters and to the
better spatial resolution compared to the FTS data. Magnetic elements with a Stokes V
signal (≥ 2%) show negligible zero-crossing shifts, also in these latest measurements.
toc
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§ 14 Flows in magnetic elements: Theory
Consider a slender, vertical flux tube with Br
≪ Bz and vr ≪ vz so that the
zeroth-order flux-tube approximation is applicable. We then need to consider motion in
vertical direction, only.
From mass conservation we have,
A ρv (z+ δz)
∂
∂
(Aρ) +
(Aρv) = 0
∂t
∂z
and from magnetic flux conservation
A(z+ δz)
A(z)
A ρv (z)
toc
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BA = Φ = const ,
which can be summarized to
∂ ρ
∂ ρv
( )+
( )=0.
∂t B
∂z B
(Walén’s equation in 1-D)
(1)
Flows in magnetic elements: Theory (cont.)
Furthermore, we have the momentum equation
ρ
dv
dt
=−
∂p
− ρg ,
∂z
(2)
and, quasi in lieu of the Lorentz force in the momentum equation,
B2
p+
= pe ,
8π
(3)
and the equation for isenthropic flow:
∂p
∂p
+v
= c2
∂t
∂z
which can be expressed as
dp
dρ
=
„
where c is the adiabatic speed of sound.
toc
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„
dp
∂ρ
∂ρ
+v
∂t
∂z
«
dρ s
= c2 ,
«
,
(4)
Flows in magnetic elements: Theory (cont.)
Equations (1) – (4) are perturbed around the static equilibrium,
p = p0 + p̃eiωt and similarly for ρ, v, and B ,
while v
= ṽ exp(iωt). Substitution into (1) – (4) and linearization leads, for example
for Eq. (3), to
p + δp +
1
1
(B + δB)2 = p + δp +
(B 2 + 2BδB + δB 2 ) = pe + δpe .
8π
8π
It is assumed that the perturbation has no effect on the external atmosphere, so that
δpe = 0. Thus,
1
B0 B̃ = 0 .
4π
2
A further simplification is that β = p/(B /(8π)) = const. for the equilibrium state.
p̃ +
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Flows in magnetic elements: Theory (cont.)
For linearizing the momentum equation we write
∂δv
∂(p + δp)
∂δv
+ δv
)=−
− (ρ + δρ)g .
(ρ + δρ)(
∂t
∂z
∂z
Using that for the unperturbed state −p
˙ + δρg
obtain ρδv
′
− ρg = 0 and neglecting 2nd-order terms we
+ δp′ = 0 . Proceeding in the same way with Walén’s equation and
the equation for isentropic flow leads to the following system:
˙ − ρδB
˙ + ρ′ δvB + ρδv ′ B − δvρB ′ = 0 ,
δρB
˙ + δρg + δp′ = 0 ,
ρδv
1
δp +
BδB = 0 ,
4π
˙ + ρ′ δv) = 0 .
˙ + p′ δv − γp (δρ
δp
ρ
toc
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Flows in magnetic elements: Theory (cont.)
Using the time separation Ansatz δq
= q̃eiωt for δq = δρ, δB, δv, δp, we eliminate
time derivatives and obtain the following system, equivalent to Roberts & Webb (1978):
iω ρ̃B − iωρB̃ + ρ′ ṽB + ρṽ ′ B − ρB ′ ṽ = 0 ,
iωρṽ + ρ̃g + p̃′ = 0 ,
1
p̃ +
B B̃ = 0 ,
4π
iω p̃ + ṽp′ − c2 (iω ρ̃ + ρ′ ṽ) = 0 .
We get
from (2)
from (3)
and from (4)
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1 ′
ρ̃ = − (p̃ + iωρṽ) ,
g
4π p̃
B̃ = −
,
B
iω p̃ + ṽp′
′
−
ρ
ṽ .
iω ρ̃ =
2
c
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Flows in magnetic elements: Theory (cont.)
In Eq. (1) we substitute B̃ from (6) and iω ρ̃ from (7) to obtain
ff

2
1
B ′
2 ′
′
p̃ =
p
)ṽ
−
ρB
ṽ
,
(ρB
B
−
2
B2
c
iω( c2 + 4πρ)
(8)
while (5) into (4) gives
iω(g p̃ + c2 p̃′ )
.
ṽ = 2 2
2
′
′
c ω ρ + gc ρ − gp
(9)
Furthermore we substitute in (1) ρ̃ and B̃ with help of (5) and (6) in order to obtain an
equation in terms of ṽ and p̃ (and derivatives) only:
iω ′
−
(p̃ + iωρṽ)B 2 + iω4πρp̃ + ρ′ B 2 ṽ + ρB 2 ṽ ′ − ρBB ′ ṽ = 0
g
into which we can either substitute (8) or (9) in order to obtain an ordinary
homogeneous second order differential equation for ṽ or p̃, respectively. Doing the
former we arrive after some algebra at the equation of
toc
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(10)
Flows in magnetic elements: Theory (cont.)
Spruit & Zweibel (1978) who have allowed for variation with p and T of the mean
molecular weight, µ, and the specific heats (cp , cv ), which they consider essential:
′′
ṽ +
„
−1
+
2Hp
γ ′ c2T
γc2
«
′
ṽ +

2
»
1+β
ω
γ
+
δ(∇ − ∇a ) +
c2T
2Hp2
p′0
=−
p0
«
„
∂ ln T
∂ ln T
−
∇ − ∇a =
∂ ln p
∂ ln p S
Hp−1
1
1
1
1 + γβ/2
=
+
=
c2T
c20
c2A0
c20
„
′
Hp c2T
γc2
–ff
ṽ = 0 ∗)
«
γp
∂p
1
=
ρ
∂ρ S 1 + βγ/2
«
„
∂ ln µ
δ =1 −
∂ ln T p
c2 =
β =8πp0 /B02
Derivatives are taken with respect to the vertical coordinate, z , counted positive in radial
direction away from the Sun.
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Flows in magnetic elements: Theory (cont.)
Equation (∗) must be complemented with boundary conditions. We are interested in the
region of strong superadiabaticity, i.e., where δ is largest. Spruit & Zweibel use
ṽ(z0 ) = ṽ(z1 ) = 0 ,
where z0 and z1 are sufficiently far away from this region, i.e., for the lower boundary
depth z1
= 5000 km, while the upper boundary is set to the temperature minimum
region at 500 km height.
Equation (∗) together with the boundary condition then defines an eigenvalue problem :
O ṽ = ω 2 ṽ .
For a given atmospheric model and a given value of β , there exist solutions only for
certain values of ω , ωi
i = 1, 2, 3, . . ..
The solution for ω strongly depends on the parameter β , which is a measure for the
magnetic field strength in the flux tube.
toc
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Flows in magnetic elements: Theory (cont.)
For large β , there always exist negative eigenvalues ω
2
⇒ For weak fields the
static state is unstable. Only if β is smaller than a critical value βc , are all eigenvalues
positive, which is necessary for stability. The result is
βc = 1.83
corresponding to a critical field strength of about 1000 Gauß.
From Spruit &
Zweibel (1979).
(η
toc
ref
2
= −ω 2 )
Flows in magnetic elements: Theory (cont.)
Sketch of the field intensification throught convective collapse (Spruit, 1979). The
initially unstable layer at z
= z0 undergoes convective collapse and moves a distance
ξ to an end state in which β < βc at z . Assuming the transition to evolve quasi static,
we can compute ξ .
dz 0
z0
B0 ρ0 p0
ξ
dz
z
B ρp
toc
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Flows in magnetic elements: Theory (cont.)
For an initial state of β0
< 2, the flux tube is stable and the final (βs ) is equal to the
initial state (slope +1). For an initial state with β0
> 2 (dashed line), the final state is
reached where the vertical crosses the solid line below the dashed one with the
corresponding final field strength βs .
From Spruit
(1979)
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Flows in magnetic elements: Theory (cont.)
Try this descriptive explanation: In the limit of very strong magnetic field (β
≪ 1) the
descending cool gas bubble can not further contract since it is “frozen into” the stiff
magnetic flux tube. In this limit the scale height of the radius is 4 pressure scale height
of the external atmosphere (see §8). Correspondingly the volume of the descending
bubble within the flux tube contracts more slowly than for a descending bubble
in the surrounding atmosphere.
Vi 1
=
Vi 2
Tblob = Tsurr
After some distance the density of
the bubble within the flux tube
reaches the density of the
Tblob < Tsurr
ρblob < ρsurr
Vf 1
>
Tblob < Tsurr
ρblob > ρsurr
Vf 2
surrounding material and becomes
even lighter at which point
buouancy takes over and stops
further descending.
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§ 15 The convective collapse: Numerical simulation
The process of field intensification in magnetic flux tubes through the
superadiabatic effect (Parker, 1978) was subject of numerous numerical
simulations.
Linear stability analyses, others than that by Spruit & Zweibel (1979) and
Spruit (1979), were also carried out by Webb & Roberts (1978), Unno & Ando
(1979), Webb & Roberts (1980), Venkatakrishnan (1986), and Hasan (1986).
Nonlinear numerical simulations in one spatial dimension were carried out by
Venkatakrishnan (1983) and Hasan(1984). Hasan (1985) and Venkatakrishnan
(1986) have taken non-adiabatic effects into account.
Hasan (1986) showed, that the end state of the convective collapse is not a
static flux tube, rather a state of overstable oscillation.
toc
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The convective collapse: Numerical simulation (cont.)
Variation of B (solid curve), v (dashed curve), and T (dotted curve) as a function of
time at a height of 50 km in the photosphere. Initially, β
= 7. The amplitude of the
oscillation increases with time from 300 G to 450 G, from 550 K to 650 K, and from 1.2
−1
km s
toc
−1
to 1.9 km s
ref
for B , T , and v , respectively. From Hasan, 1985
The convective collapse: Numerical simulation (cont.)
Radiative exchange permits the oscillation to extract energy from the
surrounding medium and as a result the amplitude grows in time. However, if
radiative exchange occurs too rapidly it can lead to a dispersal of the magnetic
flux. Thus, for the convective collapse to be effective, the time scale for
convective instability, tcon , should be less than the time scale for radiative
exchange, trad , in the layers driving the instability.
toc
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The convective collapse: Numerical simulation (cont.)
A very careful linear stability analysis of the radiative effects on the onset of
convective collapse and on the overstable oscillatory solution was carried out
by Rajaguru & Hasan (2000). Among other results they found a critical flux
limit of approx. 1 × 1018 Mx that demarcates small-scale flux-tubes into two
groups:
– Flux concentrations above this limit collapse unhindered by radiative
effects to a field strength >
1160 G, and
– flux concentrations below this limit are subject to the inhibiting action of
radiation. They exhibit a flux-flux density relation.
Strong tubes (high flux density) of large enough sizes (>
300 km) are
radiatively damped, hence are not subject to the overstability.
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The convective collapse: Numerical simulation (cont.)
Flux/field-strength relation for tubes with a
Flux/field-strength relation from infrared
1.3 × 1018 Mx separating the un-
observations. From Solanki et al. (1996)
flux <
stable (to the left) from the stable region
(to the right). From Rajaguru & Hasan
(2000) ApJ 544, 522
toc
ref
A&A 310, L33
The convective collapse: Numerical simulation (cont.)
Growth rates (solid curves) of unstable
Growth rates of convective and over-
and frequencies (dotted curves) of the
stable modes (fundamental mode) as a
overstable mode for closed bottom (cb)
function of surface (photosphere) radius
and open bottom (ob) boundaries. From
a0 of flux tube. From Rajaguru & Hasan
Rajaguru & Hasan (2000)
(2000)
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The convective collapse: Numerical simulation (cont.)
Numerical simulation of the convective intensification of surface magnetic fields
in 2D. Boundary condition for thermal variables and velocity:
∂T
=0; v=0
∂y
”wave absorbing layer”:
addition of a diffusion
term ε · d2 q/dy 2 to
each equation
2400 x 1400 km
periodic
periodic
y
x
pg = pg (t) ; inflow: s = sin (t)
∂(ρv)
∂s
= 0 ; outflow:
=0
∂y
∂y
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The convective collapse: Numerical simulation (cont.)
Boundary condition for the magnetic field:
∂By
Bx = 0 ;
=0
∂y
periodic
periodic
∂By
Bx = 0 ;
=0
∂y
toc
ref
The convective collapse: Numerical simulation (cont.)
0s
toc
50 s
ref
Binit = 400 G
100 s
The convective collapse: Numerical simulation (cont.)
0s
toc
50 s
ref
Binit = 400 G
100 s
The convective collapse: Numerical simulation (cont.)
0s
toc
50 s
ref
Binit = 400 G
100 s
The convective collapse: Numerical simulation (cont.)
150 s
Binit = 400 G
250 s
Density (top), magnetic
lines of force (middle),
and velocity (bottom). In
panel e (top) the density
is replaced by a plot of
isothermals. The
horizontally running curve
corresponds to optical
depth one. From
Grossmann-Doerth,
Schüssler, & Steiner
(1998) A&A 337, 928
toc
ref
The convective collapse: Numerical simulation (cont.)
150 s
Binit = 400 G
250 s
Density (top), magnetic
lines of force (middle),
and velocity (bottom). In
panel e (top) the density
is replaced by a plot of
isothermals. The
horizontally running curve
corresponds to optical
depth one. From
Grossmann-Doerth,
Schüssler, & Steiner
(1998) A&A 337, 928
toc
ref
The convective collapse: Numerical simulation (cont.)
t = 50 s
t = 150 s
t = 100 s
t = 250 s
Left: Plasma beta as a function of height. Right: Vertical velocity as a function
of height along the central field line of the flux sheet
toc
ref
The convective collapse: Numerical simulation (cont.)
Temporal evolution of the magnetic
field strength at two continuum
optical depths (left) and of the
continuum contrast (right) between
1600 km ≤ x ≤ 2400 km. Times
in s in the left-hand margin. Note
that the peak in the magnetic field
is considerabely wider than the
corresponding intensity signal
toc
ref
The convective collapse: Numerical simulation (cont.)
Binit = 100 G
Binit = 100 G, else same initial condition as in 400 G run. a – d correspond
to 60 s, 115 s, 175 s, and 230 s, respectively. After instant b the flux sheet
extends through the open bottom of the computational domain
toc
ref
The convective collapse: Numerical simulation (cont.)
525
612
551
Sequence of Stokes V (left)
1521
and Stokes I (right) profiles of
1443
Fe I 1.5648 µm. Times are
indicated in the left margin of
each panel. The numbers to
1661
the left give the field strength
1356
as derived from the peak
separation of Stokes V
toc
ref
The convective collapse: Numerical simulation (cont.)
Stokes V zero-crossing shift,
represented as Doppler
velocity in km s−1 for three
different spectral lines. Positive
shift indicates downflow
toc
ref
The convective collapse: Numerical simulation (cont.)
Stability diagrams from 1D
convective collapse simulations: St
= stable, Us = unsable, Sh = shock
formation. From Takeuchi (1999)
Tubes of strong enough initial field strength are stable. Sufficiently weak flux
tubes of lare enough radius produce a rebound shock.
toc
ref
§ 16 The convective collapse: Observation
In a search for the “convective collapse”, Bellot Rubio et al. (2001) obseved with the
Teneriffe Infrared Polarimeter (TIP) at the VTT a quiet Sun region, where they expected
to find most easily isolated magnetic elements. The data consist of spectra of two Fe I
lines at 1.565 µm over a spatial distance of 34” taken with a cadence of 5.6 s over a
time period of 1h. The effective spatial and temporal resolution is 28 s and 0.8”,
respectively. They found one event which possibly is a convective collapse.
The image shows the
temporal evolution of
the Stokes V profile
of this event. Time
goes from bottom to
top, dispersion in the
horizontal direction
toc
ref
The convective collapse: Observation (cont.)
Bellot Rubio et al. distinguish three phases:
1.)
≈ 9 min, moderate redshift, slightly increasing
magnetic signal,
2.)
≈ 4 min, strong redshift and increase of magnetic
signal,
3.)
toc
ref
≈ 2 min, large blueshift and disappearence of
Stokes-V signal.
The convective collapse: Observation (cont.)
Stokes-V profiles of
the third phase,
showing the
appearance of a new,
blue-shifted
component, while the
initial, red-shifted
profile weakens and
disappeares. From
Bellot Rubio et
al. (2001)
toc
ref
The convective collapse: Observation (cont.)
Temporal evolution of
the internal velocity,
the external velocity,
the temperature, and
the field strength at
two different heights,
0 and 100 km, in the
atmosphere. Results
from “Stokes
inversion” analysis.
From Bellot Rubio et al. (2001)
toc
ref
The convective collapse: Observation (cont.)
Top: Internal velocity as a function of height at
= 13.1 min. Middle: Location of the
discontinuity as function of time. Bottom:
Velocity difference across the discontinuity as
function of time. From Bellot Rubio et al. (2001)
toc
ref
The convective collapse: Observation (cont.)
The observations show all the characteristics of the convective collapse seen
in MHD simulations. However, quantitatively, there remain differences.
In the observation the field intensifies from about 400 Gauß to only maximal
600 Gauß. Convective collapses in numerical simulations typically lead to
kG-flux tubes. There are other questions, e.g.: Why is the velocity in the
external medium so much similar to the flow within the magnetic element? How
often are events like this only one observed?
toc
ref
§ 17 Interaction of magnetic flux tubes with convective motion
G-band filtergram of an area of 32 × 22 arcs of the solar surface. The red line drawn
across an elongated bright structure corresponds to the width of the two-dimensional
simulation domain of 2400 km. The red box shows the size of the three-dimensional
simulation domain of 4800 × 4800 km. Image by K. Mikurda, F. Wöger, and O. von der
Lühe with the German Vacuum Tower Telescope (VTT) at Tenerife
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
τ=1
surface
computational domain
1.4 Mm
τ=1
2.8 Mm
1.4 Mm
convection zone
4.8 Mm
4.8 Mm
convection zone base
Typical size of a three-dimensional computational box (left) on scale with the
convection zone boundaries (right)
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
The ideal MHD-equations can be written in conservative form as:
∂U
+∇·F =S,
∂t
where the vector of conserved variables U , the source term S due to gravity and
radiation, and the flux tensor F are
U = (ρ, ρv, B, E) ,
0
S = (0, ρg, 0, ρg · v + qrad ) ,
ρv
B
“
”
B
·B
B
ρvv + p + B8π
I − BB
4π
B
F =B
B
vB − Bv
B
@ “
”
·B
1
E + p + B8π
v − 4π
(v · B) B
The tensor product of two vectors a and b is the tensor ab
toc
ref
1
C
C
C
C
C.
C
C
A
= C with elements cmn = am bn .
Interaction of magnetic flux tubes with convective motion (cont.)
The total energy E is given by
E = ρǫ + ρ
v·v
B ·B
+
,
2
8π
where ǫ is the thermal energy per unit mass. The additional solenoidality constraint,
∇ · B = 0,
must also be fulfilled. The MHD equations must be closed by an equation of state which
gives the gas pressure as a function of the density and the thermal energy per unit mass
p = p(ρ, ǫ) ,
usually available to the program in tabulated form. The radiative source term is given by
qrad = 4πρ
1
Jν (r) =
4π
toc
ref
I
Iν (r, n)dΩ ,
Z
κν (Jν − Bν )dν ,
I(r, n) = I0 e−τ0 +
Z
0
τ0
“σ
π
”
T 4 (τ ) e−τ dτ
Interaction of magnetic flux tubes with convective motion (cont.)
Likewise as done in § 9 we could formally write
Jν (r) = Λν (r, r′ )Bν (r′ ) + Gν (r) ,
where Λν is the integral operator which adds the intensities at r caused by emision at
′
all the points r in the considered computational domain, and where Gν is the
transmitted mean intensity due to the incident radiation field into this domain. In the
formula
I(r, n) = I0 e−τ0 +
Z
0
τ0
“σ
π
”
T 4 (τ ) e−τ dτ
τ0 is the optical depth from the boundary to location r along direction n and
dτ = κ ds, where κ is the total opacity and s the spatial distance along a line through
location r in direction n. Here, the index ν has been dropped for the quantities I0 , τ0 ,
τ , and κ. In practice the frequency integration is either replaced by using frequency
mean quantities (Rosseland mean opacity) or it is approximated by a method of multiple
frequency bands. For a detailed description of the latter method see Ludwig (1992).
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
In practice it is not the ideal MHD-equations that are solved but rather some kind of a
viscous and resistive form of the equations with flux tensor
0
ρv
B
«
„
B
BB
B
·
B
B
I−
−σ
ρvv + p +
B
8π
4π
F=B
B
Bv − vB − η[∇B + (∇B)T ]
B
B „
«
@
B·B
1
E+p+
v−
(v · B) B + η(j × B) − σv + qturb
8π
4π
where σ
1
C
C
C
C
C,
C
C
C
A
= νρ[(∇v) + (∇v)T − (2/3)(∇ · v)I] is the viscous stress tensor,
η = (ν/Prm ) = 1/(4πσ) the magnetic diffusivity with σ being the electric
conductivity, and η(j
number.
q turb is a turbulent diffusive heat flux, which would typically be proportional to
the entropy gradient:
toc
× B) = (η/4π)(∇ × B) × B . Prm is the magnetic Prandtl
ref
q turb = −(1/Pr)νρT ∇s, where Pr is the Prandtl number.
Interaction of magnetic flux tubes with convective motion (cont.)
Typically, ν is not taken to be the molecular viscosity coefficient but rather some
turbulent value that takes care of the dissipative processes that cannot be resolved by
the computational grid. Such subgrid-scale viscosities should only act where velocity
gradients are strong causing srong turbulence. Therefore, they typically depend on
velocity gradients like in the Smagorinsky-type of turbulent viscosity where
( "„
«2 „
«2 „
«2 #
∂vx
∂vy
∂vz
t
+
+
+
ν =c 2
∂x
∂y
∂z
„
∂vy
∂vx
+
∂y
∂x
«2
+
„
∂vx
∂vz
+
∂z
∂x
«2
+
„
∂vy
∂vz
+
∂z
∂y
«2 )1/2
,
where c is a free parameter. This parameter is normally chosen as small as possible
just in order to keep the numerical integration stable and smooth, but otherwise having
no effect on large scales.
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
Some numerical (high resolution) schemes feature an inherent dissipation that acts like
the explicite dissipative terms shown in the flux tensor above. This artificial viscosity is
made as small as possible but just large enough in order to keep the numerical scheme
stable. One then only has to program the ideal equations. Of course, in this case it is
difficult to quote the actual Reynolds and Prandtl numbers because they change form
grid cell to grid cell depending on the flow. Therefore, for some applications it might be
preferable to explicitly include the dissipative terms in the equations using constant
dissipation coefficients, which then allows for well defined dimensionless numbers.
However one integrates the ideal equations on a discrete computational grid one is
locked with a discretization error that normally assumes a form similar to the dissipative
terms in the non-ideal equations.
See LeVeque, Mihalas, Dorfi, & Müller (1998) for more on computational methods for
astrophysical fluid flow.
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
Typical boundary conditions for thermal variables and velocity:
∂vx,y
∂z
= 0 ; vz = 0 ;
∂ǫ
∂z
= 0 or
∂ 2ǫ
∂z 2
=0
periodic
z
y
periodic
x
∂vx,y
∂z
Z
= 0 ; ρvz dσ = 0 ; outflow:
∂s
∂z
=0
inflow: s = const
Example of a simulation of solar granulation. The emergent intensity over an area of
12 × 12 Mm is shown. The computation was carried out with the CO5 BOLD-code in a
domain of 400 × 400 × 165 grid cells. The granular contrast is 16.65% at 620 nm.
Courtesy, M. Steffen, AIP
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
Typical boundary conditions for the magnetic field:
Bx,y = 0 ;
∂Bz
∂z
=0
periodic
z
y
periodic
=0
periodic
z
periodic
y
x
x
Bx,y = 0 ;
∂Bx,y,z
∂z
∂Bz
∂z
=0
outflow:
∂Bx,y,z
∂z
=0
inflow: By = Bz = 0, Bz = const.
Example of a simulation of magneto-convection in the solar photosphere by Vögler,
Shelyag, Schüssler, Cattaneo, Emonet, and Linde, A&A 429, 355. 1 h movies of the
continuum intensity, the vertical magnetic field, the temperature, and the
vertical velocity
toc
ref
→ PDFs
Interaction of magnetic flux tubes with convective motion (cont.)
Edge-on view of a thin magnetic sheet. Note the spreading of field lines in the
photosphere. At a depth of ≈
300 km below the surface the sheet is disrupted.
From Vögler et al. (2005) A&A 429, 335
toc
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Interaction of magnetic flux tubes with convective motion (cont.)
Good agreement with the thin tube
approximation.
hpgas ii (solid), hpgas ie (dashed),
hpgas ii + hpmag ii (dotted).
Top:
h|B|ii (solid), h|B|ie (dotted),
p
2µ(hpgas ie − hpgas ii ) (dashed)
Bottom:
→ more
toc
ref
Interaction of magnetic flux tubes with convective motion (cont.)
Two-dimensional simulation of the interaction of a magnetic flux sheet with convective
motion (Steiner, Grossmann-Doerth, Knölker, and Schüssler, 1998).
Initial condition:
- Magnetic flux tube embedded in a standard model atmosphere
-
T = T (y), density reduced within flux sheet so that β = pmag /pgas = 1
- Smooth transition between flux sheet and surrounding plasma
- Small velocity perturbation
Methods of numerical integration:
- Numerical integration with flux corrected transport (FCT) scheme of Zalesak (1979)
- Integration of the induction equation with the FCT scheme of DeVore (1991)
- Constraint transport (CT) for maintaining ∇ · B
=0
- Adaptive mesh refinement of Berger & Oliger (1984), Berger & Colella (1989)
toc
ref
Interaction of magnetic flux tubes with convective motion: Snapshot
toc
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Interaction of magnetic flux tubes with convective motion: Animations
Evolution of temperature and magnetic field: QuickTime movie. From t
≃ 2 : 00 to
5 : 00 min, a growing “granule” pushes the flux sheet to the left. Shock fronts in the
non-magnetic atmosphere are well visible between 4:30 and 7:30 min. From 8:00 to
9:00 min a strong shock propagates vertically within the flux sheet. The bending of the
flux sheet to the right reaches a maximum at about 10:00 min, followed by a fast motion
to the left. Note the bow shock associated with this swaying. Strong shocks in the flux
sheets are again visible between 15:00 and 17:00 min.
Tracer particles follow plasma motion: QuickTime movie. Note the persistent downflows
on both sides of the flux sheet. Within the flux sheet, oscillatory motion along the field
lines and the propagation of shocks can be seen; shocks are reflected at the top due to
the closed boundary condition. For the same reason, the downflows are forced to bend
sideways near the bottom of the box.
toc
ref
Interaction of magnetic flux tubes with convective motion: Animations
Stokes I , V , and Q of the spectral line Fe I 5250.2: QuickTime movie.
Horizontally-averaged Stokes profiles for vertically incident line of sight, normalized to
the average continuum intensity, are given in three panels placed above the
temperature/fieldline panel.
At t
≃ 8 : 00 min and 15:50 min Stokes V takes on a complex shape due to the
passage of a shock wave in the flux sheet (“shock signature”). At 10:20 min the Q
signal becomes large caused by the sizeable horizontal field component of the strongly
inclined flux sheet.
The full movie, published in the accompanying video to volume 495 of ApJ, is available
here as QuickTime movie.
toc
ref
Interaction of magnetic flux tubes with convective motion: Continuum contrast
Normalized continuum intensity for
vertical incidence as a function of
horizontal position. The eight curves are
separated in time by about 100 s each.
Each curve shows a local maximum at the
position of the magnetic flux sheet, which
is embedded in darker downflow regions.
A strong inclination of the flux sheet
exposes the “hot wall” of the flux sheet at
a favourable angle toward the observer,
leading to the strong and extended
brightening visible in curve e for
1400 km <
toc
ref
x < 1900 km
Interaction of magnetic flux tubes with convective motion: Shock front
Upward propagating shock wave
in a magnetic flux sheet. Height
profiles of the vertical velocity
(vy , lower curve) and
temperature (T , upper curve)
are given at horizontal location
x = 1180 km at time
t = 16.5 min. The shock front
located at a height of 360 km has
heated the post-shock material
by about 1600 K. It rapidly cools
off due to expansion and
radiation
toc
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Interaction of magnetic flux tubes with convective motion: Shock transit
Time sequence of Stokes I (left) and Stokes
V (right) of Fe I 5250.2, reflecting the transit
of a shock front. Time increases from bottom
to top and consecutive profiles are separated
by 10 seconds each. The superposition of a
redshifted pre-shock and a blueshifted
post-shock profile leads to the complex
V -profiles i–o during the shock transit. The
instanteneous height positions (in km) of the
shock front are given to the right of the
corresponding Stokes V profiles. Stokes I
originates mainly in the field free region
outside the flux sheet and it gets only weakly
affected in the far blue wing.
toc
ref
Interaction of magnetic flux tubes with convective motion: Downflow jet
Left: Vertical velocity component in two heights (solid:
y = 0 km, dashed: y = −200
km) as function of the horizontal coordinate (x) in the central part of the flux sheet at
t = 14.5 min. Downflow jets adjacent to the flux sheet are visible. Right: Horizontal
velocity at height y
on the flux sheet
toc
ref
= 0. The downflows are maintained by horizontal flows impinging
Interaction of magnetic flux tubes with convective motion: Red wing
The downflow jets drag some fluid
within the magnetic flux sheet with
them downwards, leading to a
downflow of magnetized material.
This results in a redshifted
component in the synthetic Stokes
profile. Observed Stokes V
profiles show an extended “tail” in
the red wing, too. The figure
shows a comparison between
observed Stokes V profiles of Fe I
5250.2 and 5250.6 and synthetic
profiles of these lines from the simulation run. The “red tails” of the synthetic profiles are
in qualitative agreement with the corresponding features in the observed profiles.
toc
ref
§ 18 Stokes analysis of the simulation result
Stokes V profile asymmetries provide a useful diagnostic for the gas motion within
magnetic elements and in their close surroundings.
V
ab
Ab − Ar
δA :=
Ab + Ar
Ab
Ar
vzc
λ
δa :=
ar
ab − ar
ab + ar
A relation between the sign of the area asymmetry and the gradients of the absolute
magnetic field strength and the velocity along a line of sight is given by:
sign(δA)
= −sign(
d|B| dv(τ )
·
)
dτ
dτ
(Solanki & Pahlke, 1988; Sanchez Almeida et al., 1989)
toc
ref
(1)
Stokes analysis of the simulation result (cont.)
The simulation shows often an accelerating downflow or upflow within the flux tube.
According to the Eq. (1) and, as illustrated in the following figure, this situation
corresponds to a negative area asymmetry, which is at variance with the observed
positive mean value of δA
= 6% of Sigwarth et al. (1999).
v
d|B|
dτ > 0
dv(τ )
dτ > 0





v
⇒ δA < 0
d|B|
dτ > 0
dv(τ )
dτ > 0





⇒ δA < 0
Sign of the Stokes V area asymmetry formed along the flux-tube axis with an
accelerating downflow (left) or upflow (right)
toc
ref
Stokes analysis of the simulation result (cont.)
The situation changes completely for lines of sight that pass through the expanding
boundary of the flux tube as sketched in the figure. Then, the area asymmetry is
positive.
LOS
v=0
v
d|B|
dτ < 0
dv(τ )
dτ > 0





⇒ δA > 0
Sign of the Stokes V area asymmetry formed along lines of sight that pass through the
expanding flux-tube boundary
How realistic are these sketches? We have no means to answer this question
observationally, because it addresses the internal structure of magnetic elements that is
far beyond the spatial resolution capabilities of present time solar telescopes, but we
may attempt an answer by examination of snapshots from a simulation.
toc
ref
Stokes analysis of the simulation result (cont.)
+
–
+
+
+
–
+
+
+
–
+
+
–
+
–
–
+
–
–
+
–
+
+
–
+
+
–
+
+
–
+
+ -
+
Stokes V profiles that emanate from vertical lines of sight distributed over the horizontal
interval between 600 and 1380 km of the simulation snapshot. Positive area
asymmetries result in the two regions labeled with “+”, while the lines of sight in the
middle region labeled with “−” contribute negative area asymmetries, exactly as
expected from the simple sketches above. From Steiner (1999)
toc
ref
Stokes analysis of the simulation result (cont.)
We conclude that the measured, spatially unresolved Stokes V area asymmetry of
network elements results from a delicate balance of negative and positive contributions
due to the internal structure of the elements. The simulations also suggest that the
amplitude asymmetry, δa, is not subject to a similar balance and therefore shows larger
and almost always positive values.
A statistical analysis similar to the ones made by Sigwarth et al. (1999) has been carried
out with synthetic profiles of the simulation series. Considering the individual snapshots
to be independent single magnetic elements allows for the determination of mean
values of δA, δa, and vzc . The result is compiled in the table, where the first two rows
are derived from synthetic profiles of simulation models, the rest from observations.
toc
ref
Stokes analysis of the simulation result (cont.)
Ref.
Instrument
Region
FeI
δa
δA
vzc
1
Simulation
temporal distrib.
6302.5
14.9
0.4
0.74
1
Simulation
temporal distrib.
5250.2
14.1
-2.6
0.67
2
ASP
Quiet Sun average
6302.5
15.0
6.0
0.73
2
ASP
Active region
6302.5
9.4
0.7
0.48
3
Zimpol
Quiet Sun average
5250.2
14.8
3.9
1.0
4
ASP
Plage average
6302.5
10.0
3.0
0.2
5
FTS
Network
5250.2
22.4
7.4
<0.3
5
FTS
Plage
5250.2
16.2
6.6
<0.3
1 Steiner et al.: 1998, ApJ 495, 468;
2 Sigwarth et al.: 1999, A&A, 349, 941;
3 Grossmann-Doerth et al.: 1996, A&A 315, 610;
ApJ 474, 810;
toc
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4 Martı́nez Pillet et al.: 1997,
5 Solanki S.K.: 1993, Space Science Review 63, 1
§ 19 MHD-simulation from the convection zone to the
chromosphere
At the Kiepenheuer-Institut, Schaffenberger, Wedemeyer-Böhm, Steiner &
Freytag have carried out a three-dimensional MHD-simulation that
encompasses the integral layers from the top of the convection zone to the
mid-chromosphere.
We use CO5 BOLD, a finite volume code for solving the hydrodynamic
equations in two or three spatial dimensions. It is based on Riemann solvers
and higher order reconstruction schemes.
For MHD we use a constrained transport scheme for the magnetic field and a
2nd-order accurate HLL Riemann solver .
toc
ref
MHD-simulation from the convection zone to the chromosphere (cont.)
z = 60 km
z = 1300 km
4000
4000
3000
3000
y [km]
y [km]
z = -1210 km
2000
2000
1000
1000
1000
0
2000
3000
x [km]
1
log |B| (G)
4000
2
1000
0
2000
3000
x [km]
1
log |B| (G)
4000
2
1000
0.2
0.4
2000
3000
x [km]
0.6
0.8
log |B| (G)
4000
1.0
1000
2000 3000
x [km]
4000
1.2
Horizontal sections through 3-D computational domain. Color coding displays log |B| with
individual scaling for each panel. Left: Bottom layer at a depth of 1210 km. Middle: Layer 60 km
above optical depth τc
= 1. Right: Top, chromospheric layer in a height of 1300 km. White
arrows indicate the horizontal velocity on a common scaling. Longest arrows in the panels from left
to right correspond to 4.5, 8.8, and 25.2 km/s, respectively. Rightmost: Emergent intensity .
toc
ref
MHD-simulation from the convection zone to the chromosphere (cont.)
2.5
500
2.0
0
1.5
-500
1.0
-1000
0.5
z [km]
1000
log |B| (G)
3.0
0.0
1000
2000
3000
x [km]
4000
Snapshot of a vertical section showing log |B| (color coded) and velocity vectors projected on the
vertical plane (white arrows). The b/w dashed curve shows optical depth unity and the dot-dashed
and solid black contours β
= 1 and 100, respectively.
movie with β
= 1 surface
Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag, 2005, in Chromospheric and Coronal Magnetic Fields,
Innes, Lagg, Solanki, & Danesy (eds.), ESA Publication SP-596
toc
ref
MHD-simulation from the convection zone to the chromosphere (cont.)
35
1300
30
30
800
20
600
10
400
200
400
600
z [km]
1200
1000
25
1100
20
1000
900
15
800
10
700
5
800 1000 1200 1400
x [km]
3400
3600
3800
4000
x [km]
4200
4400
Two instances of shock induced magnetic field compression. Absolute magnetic flux
density (colors) with velocity field (arrows), Mach
β = 1-contour (white solid).
toc
ref
= 1-contour (dashed) and
|B| [G]
40
|B| [G]
z [km]
1200
MHD-simulation from the convection zone to the chromosphere (cont.)
2
1
3.2
vs
vx
0.8
log ρ
p
Bz
cs
cA
β
toc
−10.4
8.6
12.6
6.0
4.0
1.31
ref
−3.8
−10.7
2.5
5.2
3.9
2.4
2.14
2.4
7.0
v2
v1
ρ2
v
B
≅ 2 ≅ 1
ρ1
v2
B1
2
2
2
v 1 > cs 1 + cA 1
MHD-simulation from the convection zone to the chromosphere (cont.)
2.5
500
2.0
0
1.5
-500
1.0
z [km]
1000
log |B| (G)
3.0
0.5
-1000
0.0
1000
2000
3000
x [km]
4000
Snapshot of a vertical section showing log |B| (color coded) and B projected on the vertical
plane (white arrows). The b/w dashed curve shows optical depth unity and the dot-dashed and
solid black contours β
= 1 and 100, respectively. Schaffenberger, Wedemeyer-Böhm, Steiner & Freytag,
2005, in Chromospheric and Coronal Magnetic Fields, ESA Publication SP-596
toc
ref
MHD-simulation from the convection zone to the chromosphere (cont.)
The formation of the small-scale canopy field proceeds by the action of the
expanding flow above granule centers. This flow transports “shells” of
horizontal magnetic field to the upper photosphere and lower chromosphere,
where layers of different field directions may come close together, leading to a
complicated meshwork of current sheets in a height range from approximately
400 to 900 km.
toc
ref
MHD-simulation from the convection zone to the chromosphere (cont.)
Logarithmic
density,
current
log |j|, in a
vertical cross section
(top panel) and in
four horizontal cross
sections in a depth
of 1180 km below,
and at heights of
90 km, 610 km, and
1310 km above the
average
height
of
optical depth unity
from
left
to
right,
respectively. The arrows in the top panel
indicate the magnetic
field
strength
direction.
toc
ref
and
MHD-simulation from the convection zone to the chromosphere (cont.)
Estimate of the ohmic dissipation of the current sheets:
The typical current density in the
height range from 400 to
4
1000 km is 10 statamp cm
i.e., 0.03 A m
−2
−2
,
. From
Schaffenberger,
Wedemeyer-Böhm, Steiner &
Freytag, 2006
The electrical conductivity in the photosphere and the lower chromosphere is about
10...100 A/Vm (Stix, The Sun). Using this value, the ohmic dissipation is
1 2
1
Pj = j ≈
0.032 ≈ 10−5 . . . 10−4 W m−3 .
σ
10 . . . 100
When integrating over a height range of 500 km this heat deposition leads to an energy
flux of 10
toc
−5
. . . 10−4 · 5 × 105 = 5 . . . 50 W m−2 .
ref
§ 20 Future directions in numerical simulations
Future numerical simulations on solar convection and the magnetic coupling to
the outer atmosphere will concentrate on:
§ 20.1. More physics: Dynamic hydrogen ionization
§ 20.2. More Space: Towards the simulation of supergranulation cells and
the magnetic coupling from the convection zone to the corona
§ 20.3. Boundary conditions: Advecting magnetic field across boundaries
§ 20.4. Numerical experiments
toc
ref
§ 20.1 More physics: Dynamic hydrogen ionization
Under the condition of the solar chromosphere the assumption of LTE (local
thermodynamic equilibrium) is not valid. Even the assumption of statistical
equilibrium in the rate equations is not valid. Kneer (1980) showed that the
relaxation timescale for the ionization of hydrogen varies from 100 s to 1000 s
in the middle to upper chromosphere.
In order to compute the time dependent hydrogen ionization in a
three-dimensional environnment, simplifications are needed. We employ the
method of fixed radiative rates. We solve the time-dependent rate equations
n
n
j6=i
j6=i
l
l
X
X
∂ni
Pij
+ ∇ · (ni v) =
nj Pji − ni
∂t
Pij = Cij + Rij .
toc
ref
Future directions (cont.): More physics: Dynamic hydrogen ionization
Effect of dynamic H-ionization in the upper part of a 2-D simulation. Left column: LTE
ionization degree and electron density. Right column: Corresponding time-dependent
NLTE quantities. Bottom left: Gas temperature, which is the same for the LTE and the
time-dependent case. Leenaarts & Wedemeyer-Böhm 2006
toc
ref
→3-D
Future directions (cont.): More physics: Dynamic hydrogen ionization
The non-equilibrium electron number density is needed, . e.g., for computing
synthetic maps at (sub-)millimetre wavelengths, as the opacity at these
wavelengths is due to thermal free-free transitions of hydrogen, including H− .
Chromospheric
dynamics as we hope to
see it with the
Large Milimeter Array
(ALMA). Courtesy, Sven
Wedemeyer-Böhm, KIS
Brightness temperature maps (upper row) and contribution function (lower row) for the
LTE case (left) and with non-equilibrium electron densities (right).
toc
ref
§ 20.2 Towards the simulation of supergranulation cells
Efforts are underway to increase the simulation box so as to accommodate a
supergranulation cell. Recently, Stein et al. started a simulation of
48 × 48 × 20 Mm using 5003 grid cells. With this simulation they hope to find
out more about the origin of supergranulation and to carry out
helioseismological experiments.
Courtesy,
R.F. Stein
toc
ref
Future directions (cont.)
Recently, Hansteen et al. have carried out simulations over a hight range from the top
layers of the convection zone to the corona. They seek to understand the formation of
jets such as dynamic fibrils, mottles, and spicules in the solar chromosphere, which is
one of the most important, but also most poorly understood, phenomena of the Sun’s
magnetized outer atmosphere.
Courtesy,
M. Carlsson
toc
ref
§ 20.3 Boundary conditions: Advecting magnetic field
across boundaries
Recently, we have relaxed the condition of vanishing horizontal magnetic field
components at the top and bottom boundary. Now, horizontal magnetic field of a
specified flux density may be advected across the bottom boundary into the box
toc
ref
§ 20.4 Numerical experiments
Time sequence of a two-dimensional simulation of magnetoconvection starting with an
initial homogeneous vertical magnetic field of 10 G.
This sequence was repeated with a plane parallel, oscillatory velocity perturbation at
the bottom with an amplitude of 50 m/s and a frequency of 100 mHz. The logarithm of
the absolute velocity difference of the two runs shows the propagation and perturbation
of plane waves through the presence of a magnetic flux concentration.
toc
ref
§ 21 Waves carried by magnetic flux tubes
The movie sequence of § 17 suggests that the interaction of a magnetic flux tube with
convective motion excites transverse tube waves. Consider a straight vertical tube and
let us first compute the restoring force for a small
z
transversal displacement, ξ(z, x), such that the
flux-tube axis remains in the x, z -plane.
^
l
tangent vector to the axis, n̂ the principal normal.
ξ (z,t)
n^
l̂ is the
From § 3.3 we know that the magnetic tension force
in the direction of the principal normal is
2π/ k
(B 2 /4π)n̂/Rc , where Rc is the curvature radius
for which (1/Rc )
= ξ ′′ , where ξ ′ = ∂ξ/∂z . The
buoyancy force component in direction of n̂ is
x
toc
ref
(ρi − ρe )gξ ′ , where ξ ′ = tan α ≈ α.
Waves carried by magnetic flux tubes (cont.)
From this, we find for the restoring force
2
B
Fn = (ρi − ρe )gξ ′ +
ξ ′′ ,
4π
Fdyn = (ρ + ρe )ξ̈ .
Here, we incorporate the effect of the external medium on the dynamics by adding ρe to
which must be oposed by the inertial force of the displacement:
the internal intensity, which alone would do it if there were no external medium, which,
however, adds to the inertia. Hence, the equation of the transverse tube wave is:
ρi
ρi − ρe ′
¨
gξ +
c2A ξ ′′ ,
ξ=
ρi + ρe
ρi + ρe
where cA
= B/(4πρ)1/2 is the Alfvén speed.
If the atmosphere is isothermal we have a constant pressure scale height, Hp , of same
size inside and outside the flux tube. Then, ρe /ρi
= pe /pi = 1 + 1/β and using
βc2A = 2gHp leads to a 2nd-order homogeneous ODE with constant coefficients,
which has the solution:
toc
ref
Waves carried by magnetic flux tubes (cont.)
ξ ∼ exp(iωt + ikz + z/4Hp ) ,
with the dispersion relation
gHp
ω =
β + 1/2
2
„
1
k +
16Hp2
2
«
,
from which we see that no transverse tube wave can propagate if ω is smaller than the
cutoff frequency, ωc ,
ωc2
g
1
=
.
8Hp 2β + 1
Using these equations, Choudhuri et al. (1993a) investigate the propagation of a kink
wave initiated by the movement of its footpoint according to
vx (z = 0, t) = v0 e
toc
ref
−bt2
.
Waves carried by magnetic flux tubes (cont.)
Displacement, ξ(z, t), of the magnetic flux tube
as a function of height (s
= z/(4Hp )) at
various instants (τ
= ωc t, where ωc is the
p
cutoff frequency) for (1/ωc ) b/π =:
λ = 0.5. Choudhuri et al. (1993a)
Choudhuri et al. compute the total energy that is injected into the system by footpoint
motion. They find for a velocity of v0
= 3 kms−1 with a total displacement of about
750 km and typical solar values for density, pressure scale-height, etc., values of
several times 10
26
erg. They stress the importance of occasional rapid footpoint
motions over slow motions. Choudhuri et al. (1993b) come to similar conclusions when
incorporating a temperature jump simulating transition to coronal temperatures.
toc
ref
Waves carried by magnetic flux tubes (cont.)
We now reformulate the equation of the transverse tube wave using the reduced
displacement
Qk (z, t) defined by
ξ(z, t) = ez/(4Hp ) · Qk (z, t) .
Substituting ξ in the equation of the transverse tube wave leads to a partial differential
equation of the type of the Klein-Gordon-equation in one spatial dimension, the
Klein-Gordon-equation for transversal tube waves:
∂ 2 Qk
1 ∂ 2 Qk
1
Qk = 0 ,
− 2
−
∂z 2
ckink ∂t2
16Hp2
c2kink =
toc
ref
β
2gHp
= c2A
2β + 1
2β + 1
and
k2 =
1
.
2
16Hp
(1)
Waves carried by magnetic flux tubes (cont.)
Turning now to longitudinal waves we may take advantage of the equation for the
longitudinal velocity perturbation in § 14.7. Assuming an isothermal atmosphere with
Hp = const. we also have γ = const.. Furthermore, we continue to use β = const.
Then, Eq. (∗) reduces to
 2
ff
1 ′
1 1
ω
ṽ ′′ −
+
( − 1)(1 + β) ṽ = 0 ,
ṽ +
2
2
2Hp
cT
2Hp γ
which, as for the transversal tube wave is a linear, homogeneous, 2nd-order ODE, so
that we can proceed as before. The general solution again is
v ∼ exp(iωt + ikz + z/4Hp ) ,
with the dispersion relation

ω 2 = c2T k2 +
toc
ref
1
2Hp2
„
«ff
1
1
+ (1 + β)(1 − )
,
8
γ
Waves carried by magnetic flux tubes (cont.)
from which we see that no longitudinal tube wave can propagate if ω is smaller than the
cutoff frequency, ωc ,
ωc2 =
With v̈
c2T
2Hp2

ff
1
1
+ (1 − )(1 + β) .
8
γ
= −ω 2 v we can rewrite the equation for longitudinal waves in a isothermal,
vertical flux tube:
1 ′
1 1
v̈
v −
( − 1)(1 + β)v = 0 .
v − 2 +
2
2Hp
cT
2Hp γ
′′
Again we define a reduced displacement, Qλ (z, t):
ξk (z, t) = ez/(4Hp ) · Qλ (z, t) , i.e., v(z, t) = ez/(4Hp ) · Q̇λ (z, t) ,
toc
ref
Waves carried by magnetic flux tubes (cont.)
to obtain the Klein-Gordon-equation for the longitudinal tube wave (Rae & Roberts,
1982):
∂ 2 Q̇λ
1 ∂ 2 Q̇λ
2
−
−
k
λ Q̇λ = 0 ,
2
2
2
∂z
cT ∂t
(2)
1
1 + γβ/2
1
1
=
=
,
+
2
2
2
2
cT
c
cA
c0
»
–
1
1
1
1
2
kλ = 2
+ (1 − )(1 + β) .
Hp 16
2
γ
Hasan & Kalkofen (1999) solve Eqs. (1) and (2) with an additional forcing term on the
right hand side resulting from an external motion of granules.
toc
ref
Waves carried by magnetic flux tubes (cont.)
Response of a flux tube (β
“interaction time” of τ
= 0.3) on an initial footpoint motion with v⊥ = 1 km s−1 over an
= 50 s. Left: Variation of the reduced velocity, Q̇, of the kink and
longitudinal wave at two different heights in the solar atmosphere. After passage of the primary
impulse, the atmosphere oscillates as a whole with the cutoff period associated with the wave.
Left: Wave-energy flux in the vertical direction, where f0 denotes the filling factor of magnetic
fluxtubes at z
= 0. With t → ∞ velocity and pressure are out of phase by 90◦ so that the wave
carries no enery in this limit. Transverse waves are more efficiently excited than longitudinal waves.
From Hasan & Kalkofen (1999)
toc
ref
Waves carried by magnetic flux tubes (cont.)
When taking the footpoint motion from actual observations of
G-band bright points, the corresponding vertical energy flux in
transverse waves takes place in
brief and intermittant bursts, quite
in contradiction with the observed
persitency of emission from the
network region (left). Therefore,
Hasan, Kalkofen, & van Ballegooijen (2000) argue that unobserved random footpoint motion
occuring on a time-scale of a few seconds and a displacement-scale of a few km must be present.
Adding such high-frequency motion results in a much larger and less intermittent energy flux
(right). They conclude, that for transverse waves to provide sustained chromospheric heating, the
main contribution must come from high-frequency motions with periods of 5–50 s.
toc
ref
Waves carried by magnetic flux tubes (cont.)
These calculations, however, are only valid in the linear regime. In the non-linerar
regime mode coupling can take place. As a transversal tube wave propagates upwards
in the vertical direction, the velocity amplitude increases because of the exponentially
decreasing density and energy conservation and as it becomes comperable to the tube
speed, coupling to the longitudinal magneto-acoustic wave can take place and transfer
energy to them. Since the longitudinal mode can form shocks, the coupling ensures an
ample source of energy for heating the atmosphere. Numerical simulation of mode
coupling have been carried out by Ulmschneider, Zähringer, & Musielak (1991).
Mode coupling does probably also contribute to the generation of the longitudinal flow
and associated shock waves within the flux sheet in the simulation shown in § 17.
toc
ref
An external magnetic field removes the degen-
+3
+2
+1
0
−1
−2
−3
J=3
'
&
The Zeeman effect
eracy of the energy levels of electrons in an
atom, which split into sublevels of distinct mag-
∆ M = −1
∆ M=0
∆ M = +1
netic quantum numbers. Transitions between
these levels are subject to selection rules with
+2
+1
0
−1
−2
J=2
σ
the consequence, that the emitted light has distinct polarization properties.
π
σ
λ
For example, light travelling in the direction of the
magnetic field is left or right circularly polarized,
depending on whether it stems from a ∆M
or a
λb
λ0
λr
= −1
∆M = +1 transition. Depending on the
selected transition it also has different wavelength
Zeeman σ -components
in emission
→ ∆λB
→ backto § 2
$
%
e
g ∗ λ2 B
∆λB =
4πcme
→ Stokes
'
&
The Zeeman splitting
→ backto § 2
$
%
'
&
Stokes V profile
Absorption spectral line (Stokes I ) together with the corresponding Stokes V
(schematic).
Due to the Zeeman-effect, left or right cir-
1
cular polarization prevails in the flanks of a
Stokes I
spectral line. Stokes
V is the difference of
right and left circularly polarized light.
0
Stokes V
0
λ0
λ
→ backto § 2
$
%
'
&
Magnetic knot
Example of a magnetic knot, recorded with the Teneriffe Infrared
Polarimeter, TIP, at the German Vacuum Tower Telescope by
R. Schlichenmaier, KIS.
0.17 T
→ backto § 2
$
%
'
&
Magnetic filling factor – solar microscopy
11111
00000
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
resolution
element A res
Twodimensional spectroscopy has a spatial
resolution of ≈
0.5 arcs at best. Therefore,
Ares ≫ Aft . We can define a filling factor:
flux tube A ft
Aft
α=
.
Ares
With polarimetric measurements the state of the plasma in Aft can be
explored without the need of resolving the flux tube, since only the light from
Aft is polarized – This is the idea of solar microscopy (Stenflo). Note, however,
that there might be additional unobservable magnetic flux in Ares with zero net
polarization signal because of opposite polarity.
→ backto § 2
H
H
$
%
'
&
Magnetic filling factor – solar microscopy (cont.)
I : I -profile from Ares (observable),
Iα : I -profile from Aft , assuming that Landé g = 0 (non-observable),
Iσ1,2 : the two σ -components of the I -profile from Aft (non-observable).
1
Iα
Iσ
I σ2
1
0
λ0
λ0 − ∆ λH
λ0
λ0 + ∆ λH
1
V = [Iα (λ + ∆λH ) − Iα (λ − ∆λH )]
2
stems from Aft (observable)
→ backto § 2
H
H
$
%
'
&
Magnetic filling factor – solar microscopy (cont.)
Expansion of the σ -components at λ
= λ0 :
∂Iα
1 ∂ 3 Iα
1 ∂ 2 Iα
2
3
Iα (λ + ∆λH ) = Iα (λ) +
+
∆λ
∆λ
∆λH +
H
H + ...
2
3
∂λ
2 ∂λ
6 ∂λ
∂Iα
1 ∂ 3 Iα
1 ∂ 2 Iα
2
3
Iα (λ − ∆λH ) = Iα (λ) −
−
∆λ
∆λ
∆λH +
H
H + ...
2
3
∂λ
2 ∂λ
6 ∂λ
= 4.67 · 10−13 · gλ2 B , [B] = Gauß , [λ] = Å , and
k = 4.67 · 10−13 · λ2 we write ∆λH = k · g · B ⇒
–
»
3
1 ∂ Iα
∂Iα
2
(kgB)
+ ...
+
V = kgB
3
∂λ
6 ∂λ
With ∆λH
In a first approximation Iα
≈ αI , which would be correct if the atmosphere in Aft was
identical with the atmosphere in Ares . If the I -profile from Aft and Ares differ only in
the depression, not in the line shape, we may define a line-weakening factor w :
→ backto § 2
H
H
$
%
'
&
Magnetic filling factor – solar microscopy (cont.)
Iα
1−
Iα
| {z c}
rel. line depression
in mag. region
=w
„
«
I
1−
.
Ic
| {z }
rel. line depression
in non-mag. region
Ic and Iαc are the continuum from Ares and Aft , respectively. Then
„
«
Iα
I α = I αc − w I αc − I c .
Ic
We set Iαc
∂I
∂Iα
= wα
, so that
= αIc and obtain
∂λ
∂λ
–
»
3
1∂ I
∂I
2
(kgB)
+ ... ,
+
V = kgBwα
3
∂λ
6 ∂λ
which can be solved for B if α and w are known. The latter two drop out in the
line-ratio method.
→ backto § 2
$
%
From the expansion of the σ -profiles we obtain (see Magnetic filling factor):
V = kgBwα
»
3
–
'
&
The line ratio method
∂I
1∂ I
3
(kgB)
+ ... ,
+
3
∂λ
6 ∂λ
from which we can in principle compute B if α and w were known. Taking a
spectral-line ratio, we obtain
»
3
–
∂I1
1 ∂ I1
2
(kg
B)
+ ...
+
1
3
6 ∂λ
V1
w1 g1 ∂λ
–
»
=
3
V2
w2 g2 ∂I2
1 ∂ I2
2
(kg
B)
+ ...
+
2
3
∂λ
6 ∂λ
and get so rid of the filling factor α. Taking two lines with small g -factors we can neglect
higher-order terms and get
V1
∂I2
w1 g1 ∂I1
)/(
).
≈
(
V2
w2 g2 ∂λ
∂λ
→ backto § 2
H
H
$
%
'
&
The line ratio method (cont.)
From (w1 /w2 )(λ) we can derive the
g
χe
Fe 5247.06
2
0.09
eV
Fe 5250.65
1.5
2.2
eV
temperature structure of the flux-tube
atmosphere. This has been done with lines
of the “thermal ratio” (see table).
Choosing a line pair with similar line parameters except as of the Landé g-factor, we
have w1
≈ w2 , thus,
»
3
–
1 ∂ I1
∂I1
2
(kg
B)
+ ...
+
1
3
6 ∂λ
V1
g1 ∂λ
»
– ,
≈
3
V2
g2 ∂I2
1 ∂ I2
2
(kg
B)
+ ...
+
2
3
∂λ
6 ∂λ
from which expression we may derive B . A suitable choice for a “magnetic ratio” is
given in the following table:
g
χe
Fe 5247.06
2
0.09
eV
Fe 5250.22
3
0.12
eV
→ backto § 2
H
H
$
%
'
&
The line ratio method (cont.)
If the magnetic field were weak one would measure
V5247
g5247
2
≈
= .
V5250
g5250
3
Typically measured values are V5247 /V5250
≈ 1. This means that the higher-order
terms are important (“Zeeman saturation”), and the magnetic field is strong.
Scatter plot of apparent flux densities observed in the two lines Fe I 5247.06 and Fe I
5250.22. If the field were intrinsically weak,
(≤
500 G), the points would fall around the
◦
dashed 45 line.
→ backto § 2
$
%
'
&
Individual magnetic bright points adjacent to a “ribbon” structure. Berger,
Rouppe van der Voort, Löfdahl et al. A&A 428, 613 with the new Swedish
Telescope on La Palma
→ backto § 2a
$
%
'
&
Formation of the bright
and the dark ring of an
ideal flux tube
τc =1
T1
T3 T2
I/Iquiet
1
→ backto § 2a
$
%
'
&
More “flowers”. Berger, Rouppe van der Voort, Löfdahl et al. A&A 428, 613 with the new
Swedish Telescope on La Palma
→ backto § 2a
$
%
'
&
Magnetograms at 0.1” spatial resolution....
LOS
... maybe a disappointment.
High expectations are at stake fueled by movies like the ones of Cattaneo & Emonet or
Bz , at constant geometrical height, not
→ vector polarimetry
→ backto § 2a
Vögler et al. Be aware that these show
a magnetogram signal.
$
%
'
&
Field strength |B|, filling factor β , and velocity v from Stokes inversion. From Beck, Schmidt,
Bellot Rubio, Schlichenmaier & Sütterlin (2005) [A&A in press].
→ backto § 2a
H
H
$
%
–
+
+
–
+
+
–
+
+
–
+
–
–
+
–
–
+
–
+
+
–
+
+
–
+
+
–
+
'
+ -
+
&
+
+
+
Stokes V profiles that emanate from vertical lines of sight distributed over the horizontal
interval between 600 and 1380 km of the simulation snapshot. Positive area
asymmetries result in the two regions labeled with “+”, while the lines of sight in the
middle region labeled with “−” contribute negative area asymmetries, exactly as
expected from the simple sketches above. From Steiner (1999)
→ backto § 2a
$
%
'
&
The critical height – the canopy height
B 02
ln
8π
0
ln p
z crit
p
0e
i
e
→ backto § 7.1
$
%
'
&
Method of short characteristics
α
γ
P
a
d
b
O
g
M
e
h
f
c
β
δ
Short characteristic MO. The intensity incident on point O is a mini formal solution along
the short characteristic MO. The physical variables at M and P are obtained by
quadratic interpolation using points a . . . h (8 point stencil). Points α . . . δ are used
additionally for the evaluation of the intensity incident at the cell boundary at M (12 point
stencil). After Kunasz & Auer (1988)
→ backto § 9.6
H
H
$
%
'
&
Method of short characteristics
Formal solution of the radiative transfer equation
with short characteristics. For rays propagating
from the lower left to the upper right one starts
in the lower left corner cell and integrates along
short characteristics in the bottom row cells from
left to right, using the (blue) boundary intensities.
Then, next row from left to right and so on
→ backto § 9.6
$
%
'
&
Observation by Sigwarth et al. 1999, A&A 349, 941
→ backto § 13
$
%
'
&
Overstability
If at the onset of instability oscillatory motion prevails, then one says, following
Eddington, that one has the case of overstability.
Eddington: “In the usual kinds of instability, a slight displacement provokes
restoring forces tending away from equilibrium; in overstability it provokes
restoring forces so strong as to overshoot the corresponding position on the
other side of equilibrium.”
→ backto § 15
$
%
'
&
Net circulation of granular flow correlates with
vortical flow at z
= 550 km.
vhorizontal and grey-scale map of Bz at
z = 550 km.
Top:
Bottom: Same as top at z
= 100 km.
→ backto § 17
H
H
$
%
'
&
Radiative channeling
T [103 K]
4.0
6.0
8.0
10.0
12.0
0.2
[Mm]
0.0
-0.2
-0.4
0.0
0.2
0.4
0.6
0.8
1.0
[Mm]
→ backto § 17
$
%
'
&
Statistical properties of the magnetic
field in a layer of 100 km thickness
around optical depth unity.
→ backto § 17
$
%
'
&
What is the CO5 BOLD code?
5
CO BOLD stands for COnservative COde for the COmputation of COmpressible
COnvection in a BOx of L Dimensions with L=2,3.
5
CO BOLD is designed for simulating hydrodynamics and radiative transfer in the outer
and inner layers of stars. Additionally, it can treat magnetohydrodynamics,
non-equilibrium chemical reaction networks, dynamic hydrogen ionization, and dust
formation in stellar atmospheres.
→ backto § 19
H
H
$
%
(Courtesy Sven Wedemeyer-Böhm)
→ backto § 19
'
&
Application examples of CO5 BOLD
H
H
$
%
'
&
5
What is the CO BOLD code? (cont.)
Simulation of solar granulation with
Simulation of a red supergiant with
CO5 BOLD. 400 × 400 × 165 grid cells,
CO5 BOLD. 2353 grid cells,
11.2 × 11.2 Mm, Mean contrast at
λ ≈ 620 nm is 16.65%.
mstar = 12m⊙ , Teff = 3436 K,
Rstar = 875R⊙
Courtesy M. Steffen, AIP
Courtesy Bernd Freytag
→ backto § 19
H
H
$
%
5
CO BOLD works with
'
&
5
What is the CO BOLD code? (cont.)
- Cartesian (non-equidistant) grids,
- realistic equation of state,
- non-local, multidimensional radiation transport,
- realistic opacities,
- various boundary conditions
5
CO BOLD is programmed with
- FORTRAN 90,
- OpenMP directives,
5
The manual for CO BOLD can be found under
http://www.astro.uu.se/˜bf/co5bold main.html
Just type CO5BOLD in Google.
→ backto § 19
$
%
'
&
A shock capturing numerical scheme
Two-dimensional radiation-hydrodynamic simulation of surface convection including the
chromospheric layer. The dimensions of the computational domain are: Width,
5600 km; Height above the surface of τ
= 1, 1700 km; Depth below this surface level:
1400 km.
S. Wedemeyer et al. 2004, A&A 414, 1121
→ backto § 19
H
H
$
%
For a basic example of a shock capturing numerical scheme, consider a
'
&
A shock capturing numerical scheme (cont.)
piecewise constant reconstruction with discontinuities at cell interfaces
(S.K. Godunov, 1959).
q
1
0
0
1
q
1
0
0
1
1
0
0
1
1
0
0
1
0
1
1
0
0
1
1
0
0
1
0
1
1
0
1
0
0
1
1
0
0
1
x
11
00
00
11
00
11
1
0
0
1
q
xi
1
0
0
1
0
1
1
0
0
1
11
00
00
11
00
11
x i+1
1
0
0
1
0
1
1
0
1
0
0
1
1
0
0
1
x
→ backto § 19
H
H
$
%
The shock-tube problem
'
&
A shock capturing numerical scheme (cont.)
t0
ρ
ρ
l
ρ
r
x
p
p
l
p
r
x
v
vl = vr
= 0
x
→ backto § 19
$
%
The shock-tube problem
t1
t0
ρ
ρ
ρ
l
ρl
ρ*
l
ρ*r
ρ
r
p
p
l
ρ
r
x
x
p
'
&
A shock capturing numerical scheme (cont.)
p
l
p*
p
p
r
r
x
x
v*
v
v
vl = vr
vl
= 0
x
vr
x
→ backto § 19
$
%
'
&
A shock capturing numerical scheme (cont.)
The shock-tube problem
t1
t0
ρ
ρ
ρ
l
ρl
ρ*
l
ρ*r
ρ
r
p
p
l
r
x
x
p
ρ
p
t
l
q*
p*
p
p
r
r
x
x
v*
v
v
vl = vr
l
vl
= 0
x
ql
q *r
qr
x
vr
x
→ backto § 19
$
%
'
&
A shock capturing numerical scheme (cont.)
The shock-tube problem
t1
t0
ρ
ρ
ρ
l
ρl
ρ*
l
ρ*r
ρ
r
p
p
l
r
x
x
p
ρ
p
t
l
q*
p*
p
p
r
r
v*
v
v
vl
= 0
q *r
qr
ql
x
x
vl = vr
l
x
vr
x
x
q0
q =q
0
r
q = q*
0
r
q = q*
0
l
q =q
0
rf
q =q
0
l
→ backto § 19
$
%
§ 1 The concept of magnetic flux tubes : References
Kirk, J.G., Melrose, D.M., Priest, E.R.: 1994, Plasma Astrophysics, Saas-Fee Advanced
Course 24, Springer-Verlag
Priest, E.R. : 1982, Solar Magneto-Hydrodynamics, Reidel, Dordrecht
Stix, M.: 2002, The Sun, Springer-Verlag, 2nd revised and enlarged edition
toc
ref
§ 2 Small-scale magnetic flux tubes : References
Babcock, H.W.: 1953, ApJ 118, 387
Beckers, J.M., Schröter, E.H.: SP 4, 142
Berger, T.E. and Title, A.M.: 2001 ApJ 553
Keller, C.U.: 1992, Nature 359, 307
Keller, C.U.: 1992, Ph.D. Thesis, ETH No. 9953
Kiepenheuer, K.O.: 1953, ApJ 117,447
Stenflo, J.O.: 1989, A&A Rev 1, 3
Stenflo, J.O.: 1994, Solar Magnetic Fields, Kluwer, Dordrecht
toc
ref
§ 2a Small-scale magnetic flux tubes : References
Rouppe van der Voort, L. H. M., Hansteen, V. H., Carlsson, M., Fossum, A.,
Marthinussen, E., van Noort, M. J., & Berger, T. E.: 2005, A&A 435, 327-337
Berger, T. E., Rouppe van der Voort, L. H. M., Löfdahl, M. G., Carlsson, M., Fossum, A.,
Hansteen, V. H., Marthinussen, E., Title, A., & Scharmer, G.: 2004, A&A 428,
613-628
Knölker, M. and Schüssler, M.: 1988, A&A 202, 275
toc
ref
§ 3 What confines a magnetic flux tube? : References
toc
ref
§ 4 A microscopic picture of the sheet current : References
Krall N.A., Trivelpiece, A.W.: 1986, Principles of Plasma Physics, San Francisco Press,
San Francisco Press
Nicholson, D.R.: 1983, Introduction to Plasma Theory, John Wiley & Sons, New York
toc
ref
§ 5 The equations for a hydrostatic flux tube : References
Cally, P.S.: 1991, JCP 93, 411
Fiedler, R.A.S. and Cally, P.S.: 1990, SP 126, 69
Hennig, B.S. and Cally, P.S.: 2001, SP 201, 289
Jahn, K.: 1989, A&A 222, 264
Jahn, K. and Schmidt, H.U.: 1989, A&A 290, 295
Low, B.C.: 1975, ApJ 197, 251
Pizzo, V.J.: 1986, ApJ 302, 785
Pizzo, V.J.: 1990, ApJ 365, 764
Pneuman G.W. and Kopp, R.A.: 1971, SP 18, 258
toc
ref
§ 6 The magnetic structure of a hydrostatic flux tube : References
Brownig, P.K. and Priest, E.R.: 1982, GAFD 21, 237
Brownig, P.K. and Priest, E.R.: 1983, ApJ 266, 848
Parker, E.N.: 1974, ApJ 191, 245
Pizzo, V.J.: 1986, ApJ 302, 785
Steiner, O., Pneuman, G.W., and Stenflo, J.O.: 1986, A&A 170, 126
toc
ref
§ 7 The magnetic canopy : References
Anzer, U. and Galloway, D.J.: 1983, MNRAS 203, 637
Evans, D.J. and Roberts, B.: 1992, Nature 355, 230
Giovanelli, R.G.: 1980, SP 68, 49
Giovanelli, R.G.: 1982, SP 80, 21
Giovanelli, R.G. and Jones, H.P.: 1982, SP 79, 267
Jones, H.P. and Giovanelli, R.G.: 1983, SP 87, 37
Jones, H.P. and Giovanelli, R.G.: 1982, SP 79, 247
Jones, H.P.: 1984, in Chromospheric Diagnostics and Modelling, B.W. Lites (ed), NSO,
Sacramento Peak NM
Solanki, S.K., Finsterle, W., Rüedi, I., and Livingston, W.: 1999, A&A 347, L27
Solanki, S.K. and Steiner, O.: 1990, , Astron. Astrophys. 234, 519
Steiner, O. and Pizzo, V.J.: 1986, A&A 211, 447
Steiner, O.: 2000, Chromosphere: Magnetic canopy, in Encyclopedia of Astronomy and
Astrophysics, Nature Publishing Group, Macmillan Publishers Ltd., UK,
www.ency-astro.com
toc
ref
§ 8 The thin flux-tube approximation : References
Defouw, R.J.: 1976, ApJ 209, 266
Ferriz Mas A. and Schüssler, M: 1989, GAFD 48, 217
Ferriz Mas A., Schüssler, M and Anton, V.: 1989, A&A 210, 425
Pneuman, G.W., Solanki, S.K., and Stenflo, J.O.: 1986, A&A 154, 231
Spruit. H.C.: 1981, A&A 102, 129
Steiner, O. and Pizzo, V.J.: 1986, A&A 211, 447
toc
ref
§ 9 Magnetic flux tube in radiative equilibrium : References
Fabiani Bendicho, P., Kneer, F., and Trujillo Bueno, J.: 1992 A&A 264, 229
Ferrari, A., Massaglia, S., Kalkofen, W., and Bodo, G.: 1985, ApJ 298, 181
Hasan, S.S.: 1988, ApJ 332, 499
Hasan, S.S. and Kalkofen, W.: 1994, ApJ 436, 355
Hasan, S.S., Kalkofen, W., and Steiner, O.: 1999 in Solar polarization, K.N. Nagendra
and J.O. Stenflo (eds) Kluwer
Kalkofen, W., Bodo, G., Massaglia, S., and Rossi, P. in Solar and Stellar Granulation,
R.J. Rutten and G. Severino (eds.), Kluwer
Kalkofen, W., Rosner, R., Ferrari, A., Massaglia: 1986, ApJ 304, 519
Keller, C.U., Solanki, S.K., Steiner, O. and Stenflo, J.O.: 1990, A&A 233, 583
Massaglia, S., Bodo, G., Kalkofen, W., and Rosner, R.: 1988, ApJ 333, 925
Pizzo, V.J., MacGregor, K.B., and Kunasz, P.B.: 1993, ApJ 404, 788
Pizzo, V.J., MacGregor, K.B., and Kunasz, P.B.: 1993, ApJ 413, 764
toc
ref
§ 9 Magnetic flux tube in radiative equilibrium : References (cont.)
Solanki, S.K. and Brigljević: 1992, A&A 262, L29
Steiner, O. and Stenflo, J.O.: 1990, in Solar Photosphere: Structure, Convection and
Magnetic Fields, J.O. Stenflo (ed.), IAU-Symp. No. 138, Kluwer, Dordrecht, p. 181
Steiner, O.: 1991, , Astron. Astrophys. 242, 290
Steiner, O.: 1990, , Astron. Astrophys. 231, 278
toc
ref
§ 10 The physics of faculae : References
Akimov, L.A., Belkina, I.L., and Dyatel, N.P.: 1982, Sov. Astron. 26, 334
Akimov, L.A., Belkina, I.L., and Dyatel, N.P.: 1988, Sov. Astron. 32, 660
Akimov, L.A., Belkina, I.L., Dyatel, N.P. and Marchenko, G.P.: 1987, Sov. Astron. 31, 64
Auffret, H. and Muller, R.: 1991, A&A 246, 264
Badaljan, O.G.: 1968, Soln. Dannye, 5, 105
Caccin, B. and Severino, G.: 1979, ApJ, 232, 297
Carlsson, M., Stein, R. F., Nordlund, Å., & Scharmer, G. B.: 2004, ApJL, 610, L137
Chapman, G.A. and Meyer, A.D.: 1986, SP 103, 21
Chapman, G.A. and Klabunde, D.P.: 1982, ApJ 261, 387
Deinzer, W., Hensler, G., Schüssler, and Weisshaar, E.: 1984, A&A 139, 435
De Pontieu, B., Carlsson, M., Stein, R., Rouppe van der Voort, L., Löfdahl, M., van
Noort, M., Nordlund, Å., & Scharmer, G.: 2006, ApJ, 646, 1405-1420
Foukal, P.and Duvall, T. Jr.: 1985, ApJ 296, 739
Frazier, E.N.: 1971, S.P. 21, 42
toc
ref
§ 10 The physics of faculae : References (cont.)
Hirayama, T.: 1987, PASJ 30,337
Hirayama, T. and Moriyama, F.: 1979, SP 63, 251
Hirzberger, J., Wiehr, E.: 2005, A&A 438, 1059
Keller, C. U., Schüssler, M., Vögler, A., & Zakharov, V.: 2004, ApJ, 607, L59
Knölker, M. and Schüssler, M.: 1988, A&A 202, 275
Knölker, M. and Schüssler, M.: 1989 in Flux tubes in the solar atmosphere, E. Leer and
P. Maltby (eds.), Institut of Theoretical Astrophysics, University of Oslo
Kuz’minykh, V.D.: 1963, Sov. Astron. 6, 751
Lawrence, J.K.: 1988, S.P. 116, 17
Lawrence, J.K. and Chapman, G.A.: 1988, ApJ 335, 996
Libbrecht, K.G. and Kuhn, J.R.: 1984, ApJ 277, 889
Mehltretter, J. P.: 1974, Sol. Phys. 38, 43
Minasyants, T.M. and Minasyants, G.S.: 1977 Soln. Dannye, 3, 86
Muller, R.: 1975, SP 45, 105
toc
ref
§ 10 The physics of faculae : References (cont.)
Musaev, M.M.: 1982, Soln. Dannye 9, 95
Richarson, R.S.: 1933, ApJ 78, 359
Rogerson, J.B. Jr.: 1961, ApJ 134, 331
Spruit, H.C.: 1976, SP 50, 269
Steiner, O.: 2005, A&A 430, 691
Steiner, O. and Stenflo, J.O.: 1990, in Solar Photosphere: Structure, Convection and
Magnetic Fields, J.O. Stenflo (ed.), IAU-Symp. No. 138, Kluwer, Dordrecht, p. 181
Stellmacher, G. and Wieher, E.: 1973, A&A 29, 13
Taylor, S.F., Varsik, J.R., Woodard, M.F., Libbrecht, K.G.: 1998, S.P. 178, 1
Waldmeier, M.: 1949, Zeitschrift für Astrophysik 26, 147
Wang ,H., Psirock, T., Goode, P.R., Lee, C., Zirin, H., and Kosonocky, W.: 1998, ApJ,
495, 957
Wang, H. and Zirin, H.: 1987, S.P. 110, 281
Yudina, I.V.: 1976, Soln. Dannye 2, 88
toc
ref
§ 10a The physics of faculae : References
Fligge, M., Solanki, S. K., & Unruh, Y. C.: 2000, A&A 353, 380-388
Fligge, M., Solanki, S. K., Unruh, Y. C., Froehlich, C., & Wehrli, C.: A&A 335, 709-718
Foukal, P., Fröhlich, C., Spruit, H., & Wigley, T. M. L.: 2006, Nature, 443, 161-166
toc
ref
§ 11 The interchange instability of magnetic flux tubes : References
Berger, T.E. and Title, A.M.: 1996, ApJ 463, 365
Bernstein, I.B., Frieman, E.A., Kruskal, M.D. and Kulsrud, R.M.: 1958,
Proc. R. Soc. London A 244, 17
Bünte: 1993, A&A 276, 236
Bünte: 1993, Ph.D. Thesis, ETH-Zürich, No. 10357
Bünte and Saar, S.: 1993, A&A 271, 167
Bünte, M., Steiner, O., and Pizzo, V.J.: 1993, A&A 268, 299
Bünte, M., Hasan, S.S., and Kalkofen, W.: A&A 273, 287 A&A 268, 299
Frieman, E.A. and Rotenberg, M.: 1960 Rev. Mod. Phys. 32, 898
Meyer, F., Schmidt, H.U., and Weiss, N.O.: 1977 MNRAS 179, 741
Piddington, J.H.: 1975, Astrophys. Space Sci. 34, 347
Schüssler, M.: 1984, A&A 140, 453
Steiner, O.: Ph.D. Thesis, ETH-Zürich, No. 9292
Parker, E.N.: 1974, ApJ 191, 245
Parker, E.N.: 1975, SP 40, 291
toc
ref
§ 12 The formation of flux tubes by flux expulsion : References
Cattaneo, F.: 1999, ApJ 515, L39
Cattaneo, F. and Vainshtein, I.: 1991, ApJ 376, L21
Emonet, T. and Cattaneo, F.: 2001, ApJ 560, L197
Galloway, D.J. and Weiss, N.O.: 1981, ApJ 243, 945
Galloway, D.J., Proctor, M.R.E., and Weiss, N.O.: 1977 Nature, 266, 686
Schüssler, M.: 1991, in The Sun – a Laboratory for Astrophysics, J.T. Schmelz and J.C.
Brown (eds.), Kluwer
Sánchez Almeida, J.: 2001, ApJ 556, 928
toc
ref
§ 13 Flows in magnetic elements: Observations : References
Frutiger C. and Solanki S.K., 1998, A&A 336, L65
Grossmann-Doerth U., Keller C.U., and Schüssler M., 1996, A&A 315, 610
Martı́nez Pillet V., Lites B.W., and Skumanich A., 1997, ApJ 474, 810
Sigwarth, M., Balasubramaniam, K.S., Knlker, M., and Schmidt, W.: 1999, A&A 349,
941
Sigwarth, M: 2001, ApJ 563, 1031
Solanki S.K. and Stenflo J.O.: 1986, A&A 170, 120
Solanki S.K.: 1993, Space Science Reviews 63, 1
toc
ref
§ 14 Flows in a magnetic elements: Theory : References
Roberts, B. and Webb, A.R.: 1978 SP 56, 5
Vertical motions in an intense magnetic flux tube,
Parker, E.N.: 1978 ApJ 221, 368
Hydraulic concentration of magnetic fields in the solar photosphere. VI. Adiabatic
cooling and concentration in downdrafts,
Unno, W. and Ando, H.: 1979, Geophys. Astrophys. Fluid Dynamics 12, 107
Instability of a thin magnetic tube in the solar atmosphere,
Spruit, H.C. and Zweibel, E.G.: 1979 SP 62, 15
Convective instability of thin flux tubes,
Webb, A.R. and Roberts, B.: 1978 SP 59, 249
Vertical motions in an intense magnetic flux tube. II. Convective Instability,
Spruit, H.C.: 1979 SP 61, 363
Convective collapse of flux tubes,
toc
ref
§ 14 Flows in a magnetic elements: Theory : References (cont.)
Webb, A.R.and Roberts, B.: 1980 SP 68, 71
Vertical motions in an intense magnetic flux tube. IV. Radiative relaxation in a
uniform medium,
Webb, A.R.and Roberts, B.: 1980 SP 68, 87
Vertical motions in an intense magnetic flux tube. V. Radiative relaxation in a
stratified medium,
Hasan, S.S.: 1986 MNRSA 219, 357
Oscillatory motions in intense flux tubes,
Venkatakrishnan, P.: 1986 Nature 322, 156
Inhibition of convective collapse of solar magnetic flux tubes by radiative diffusion
toc
ref
§ 15 The convective collapse: Numerical simulation : References
Venkatakrishnan, P.: 1983, J. Astrophys. Astr. 4, 135
Nonlinear development of convective instability within slender flux tubes. I.
Adiabatic flow,
Hasan, S.S.: 1983 in ”Solar and Magnetic Fields: Origins and Coronal Effects”,
J.O. Stenflo (ed.), IAU Symp. 102, p. 73
Time-dependent convective collapse of flux tubes,
Hasan, S.S.: 1984 ApJ 285, 851
Convective instability in a solar flux tube. I. Nonlinear calculations for an adiabatic
inviscid fluid,
Venkatakrishnan, P.: 1985, J. Astrophys. Astr. 6, 21
Nonlinear development of convective instability within slender flux tubes. II. The
effect of radiative heat transport,
toc
ref
§ 15 The convective collapse: Numerical simulation : References (cont.)
Hasan, S.S.: 1985 A&A 143, 39
Convective instability in a solar flux tube. II. Nonlinear calculations with horizontal
radiative heat transport and finite viscosity,
Massaglia, S., Bodo, G. and Rossi, P.: 1989 A&A 209, 399
Overstability of magnetic flux tubes in the Eddington approximation,
Hanami, H. and Tajima, T.: 1991 A&A 377, 694
Numerical study of compressible solar magnetoconvection with an open
transitional boundary,
Takeuchi, A.: 1993, Publ. Astron. Soc. Japan 45, 811
The nonlinear evolution of a convective instability in a solar magnetic flux tube,
Takeuchi, A.: 1995, Publ. Astron. Soc. Japan 47, 331
On the convective stability of solar photospheric flux tubes,
toc
ref
§ 15 The convective collapse: Numerical simulation : References (cont.)
Steiner, O.: 1996, in Solar and Galactic Magnetic Fields, D. Schmitt and H.-H. Voigt
(eds.), Nachrichten der Akademie der Wissenschaften in Göttingen, II.
Mathematisch-Physikalische Klasse, Vol. 4, Vandenhoeck & Rupprecht in
Göttingen, p. 15
Convective intensification of magnetic fields at the solar surface,
Grossmann-Doerth, U., Schüssler, M. and Steiner, O.: 1998, A&A 337, 928
Convective intensification of solar surface magnetic fields: Results of numerical
experiments
Takeuchi, A.: 1999, ApJ 522, 518
Properties of Convective Instability in a Vertical Photospheric Magnetic Flux Tube
Rajaguru, S.P., Kurucz, R.L., & Hasan, S.S.: 2002, ApJ 565, L101
Radiative Transfer Effects and the Dynamics of Small-Scale Magnetic Structures
on the Sun
toc
ref
§ 15 The convective collapse: Numerical simulation : References (cont.)
Solanki, S.K., Zufferey, D., Lin, H., Rüedi, I., & Kuhn, J.R.: 1996, A&A 310, L33
Infrared lines as probes of solar magnetic features. XII. Magnetic flux tubes:
evidence of convective collapse?
toc
ref
§ 16 The convective collapse: Observation : References
Bellot Rubio, L.R., Rodrı́guez Hidalgo, I., Collados, M., Komenko, E., and Ruiz Cobo, B.:
2001 ApJ 560, 1010
toc
ref
§ 17 Interaction of magnetic flux tubes with convective motion : References
Cattaneo, F.: 1999, ApJ 515, L39
Deinzer, W., Hensler, G., Schüssler, M. and Weisshaar, E.: 1984, A&A 139, 426
Deinzer, W., Hensler, G., Schüssler, M. and Weisshaar, E.: 1984, A&A 139, 435
Emonet T. and Cattaneo, F.: ApJ 560, L 197
LeVeque, R.J., Mihalas, D., Dorfi, E.A., and Müller, E.: 1998, Computational Methods for
Astrophysical Fluid Flow, Saas-Fee Advanced Course 27, O. Steiner and
A. Gautschy (eds.), Springer
Ludwig H.-G.: 1992, ‘Nichtgrauer Strahlungstransport in numerischen Simulationen
stellarer Konvektion’, PhD-thesis Christian-Albrechts-Universität, Kiel, Germany
Knölker, M., Schüssler, M. and Weisshaar, E.: 1988, A&A 194, 257
Knölker, M., Schüssler: 1988, A&A 202, 275
Smagorinsky, J.S.: 1963, Monthly Weather Rev. 91, 99 and Schüssler, M.: 1998, ApJ
495, 468
toc
ref
§ 17 Interaction of magnetic flux tubes with convective motion : References (cont.)
Steiner, O.: 1991, Theoretical models of magnetic flux tubes: structure and dynamics, in
Infrared Solar Physics, D. Rabin (ed.), IAU-Symp. No. 154, Kluwer, p. 407
Steiner, O., Grossmann-Doerth, U., Knölker, M., and Schüssler, M.: 1995, Simulation of
the interaction of convective flow with magnetic elements in the solar atmosphere
in Reviews in Modern Astronomy, Vol. 8, G. Klare (ed.), Astronomische
Gesellschaft, Hamburg, p. 81
Steiner, O., Grossmann-Doerth, U., Knölker, M., and Schüssler, M.: 1998, ApJ 495, 468
Steiner, O., Knölker, M., and Schüssler, M.: 1994, Dynamic interaction of convection
with magnetic flux sheets: first results of a new MHD code, in Solar Surface
Magnetism, R.J. Rutten and C.J. Schrijver (eds.), NATO ASI Series C, Vol. 433,
Kluwer, Dordrecht, p. 441
Steiner, O., Knölker, M., and Schüssler, M.: 1997, Numerical simulations of magnetic
flux sheets, in Solar Magnetic Fields, V.H. Hansteen (ed.), Institute of Theoretical
Astrophysics, University of Oslo, Oslo, p. 31
toc
ref
§ 17 Interaction of magnetic flux tubes with convective motion : References (cont.)
Vögler, A., Shelyag, S., Schüssler, M., Cattaneo, F., Emonet, T., & Linde, T.: 2005, A&A
429, 335-351
toc
ref
§ 18 Stokes analysis of the simulation result : References
Hasan, S.S., Kneer, F., and Kalkofen, W.: 1996, A&A 305, 660
Hasan, S.S., Kneer, F., and Kalkofen, W.: 1998, A&A 332, 1064
Knölker, M., Grossmann-Doerth, U., Schüssler, M., and Weisshaar, E.: 1991,
Adv. Space Res. 11, 285
Leka, K.D. and Steiner, O.: 2001, Understanding small solar magnetic structures:
Comparing numerical simulations to observations ApJ 552, 354
Sanchez Almeida, J., Collados, M., & del Toro Iniesta, J. C.: 1989, A&A 222, 311-315
Solanki, S. K. & Pahlke, K. D.: 1988, A&A 201, 143-152
Steiner, O., Grossmann-Doerth, U., Schüssler, M., and Knölker, M.: 1996, Polarized
Radiation Diagnostics of Magnetohydrodynamic Models of the Solar Atmosphere
SP 164, 223
Steiner, O.: 1999, Flux tube dynamics, in Magnetic Fields and Oscillations,
B. Schmieder, A. Hofmann, and J. Staude (eds.), 3rd Advances in Solar Physics
Euroconference, ASP Conf. Ser. 184, p. 38
toc
ref
§ 19 MHD-simulation from the convection zone to the chromosphere : References
Freytag, B. & Mizuno-Wiedner, M.: 2003, Modelling the entire atmosphere of
Betelgeuse with 3-D simulations, in Modelling of Stellar Atmospheres, IAU
Symp. 210, N. Piskunov, W.W. Weiss, & D.F. Gray (eds.), ASP-Series
Godunov, S.K.: 1959, A finite difference method for the computation of discontinuous
solutions of the equations of fluid dynamics, Mat. Sb., 47, 271
Laney, C.B.: 1998, Computational Gasdynamics, Cambridge University Press
LeVeque, R.J.: 2002, Finite Volume Methods for Hyperbolic Problems, Cambridge
University Press
LeVeque, R.J., Mihalas, D., Dorfi, E.A., and Müller, E.: 1998, Computational Methods for
Astrophysical Fluid Flow, Saas-Fee Advanced Course 27, O. Steiner and
A. Gautschy (eds.), Springer
Schaffenberger, W., Wedemeyer-Böhm, S., Steiner, O., and Freytag, B.: 2005,
Magnetohydrodynamic simulation from the convection zone to the chromosphere,
in Chromospheric and Coronal Magnetic Fields, ESA Publication, SP-596
toc
ref
§ 19 MHD-simulation from the convection zone to the chromosphere : References
Schaffenberger, W., Wedemeyer-Böhm, S., Steiner, O., and Freytag, B.: 2006, Holistic
MHD-simulation from the convection zone to the chromosphere, in Solar MHD:
Theory and Observations – a High Spatial Resolution Perspective, J. Leibacher,
H. Uitenbroek, & R.F. Stein (eds.), ASP Conference Series
Steiner, O.: 2005, Recent progresses in the physics of small-scale magnetic fields, in
The Dynamic Sun: Challenges for Theory and Observations, Proceedings of the
11th European Solar Physics Meeting, D. Danesy, S. Poedts, A. De Groof, &
J. Andries (eds.), ESA Publication SP-600 (CD-ROM), p. 10.1
Toro, E.F.: 1999, Riemann Solvers and Numerical Methods for Fluid Dynamics,
Springer
Wedemeyer, S., Freytag, B., Steffen, M., Ludwig, H.-G., & Holweger, H.: 2004,
Numerical simulation of the three-dimensional structure and dynamics of the
non-magnetic solar chromosphere A&A 414, 1121-1137
toc
ref
§ 20 Future directions in numerical simulations : References
Kneer, F.: 1980, On time-dependent ionization in stellar chromospheres, A&A 87,
229-235
Leenaarts, J. and Wedemeyer-Böhm, S.: 2007, Time-dependent hydrogen ionisation in
3D simulations of the solar chromosphere. Methods and first results, A&A 460,
301-307
Stein, R. F., Benson, D., and Nordlund, Å.: 2006, Solar supergranulation-scale
simulations in Highlights of Recent Progress in the Seismology of the Sun and
Sun-Like Stars, 26th meeting of the IAU, Joint Discussion 17
Hansteen, V. H., De Pontieu, B., Rouppe van der Voort, L., van Noort, M., and Carlsson,
M.: 2006, Dynamic Fibrils Are Driven by Magnetoacoustic Shocks, ApJL 647,
L73–L76
Stein, R. F. and Nordlund, Å.: 2006, Solar Small-Scale Magnetoconvection, ApJ 642,
1246–1255
toc
ref
§ 21 Waves carried by magnetic flux tubes : References
Stix, M.: 2002, The Sun, Springer-Verlag, 2nd revised and enlarged edition
Choudhuri, A.R., Auffret, H., and Priest, E.R.: 1993a, SP 143, 49
Choudhuri, A.R., Dikpati, M., and Banerjee, D.: 1993b, ApJ 413, 811
Rae, I.C. and Roberts, B.: 1982, ApJ 256, 761
Hasan, S.S. and Kalkofen, W.: 1999, ApJ 519, 899
Hasan, S.S., Kalkofen, W., and van Ballegooijen, A.A.: 2000, ApJ 535, L67
Ulmschneider, P., Zähringer, K., and Musielak, Z.E.: 1991, A&A, 241, 625
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Table of content
§ 1 The concept of magnetic flux tubes
§ 2 Small-scale magnetic flux tubes
§ 3 What confines a magnetic flux tube?
§ 4 A microscopic picture of the sheet current
§ 5 The equations for a hydrostatic flux tube
§ 6 The magnetic structure of a hydrostatic flux tube
§ 7 The magnetic canopy
§ 8 The thin flux-tube approximation
§ 9 Magnetic flux tube in radiative equilibrium
§ 10 The physics of faculae
§ 11 The interchange instability of magnetic flux tubes
H
H
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Table of content (cont.)
§ 12 The formation of flux tubes by flux expulsion
§ 13 Flows in magnetic elements: Observations
§ 14 Flows in magnetic elements: Theory
§ 15 The convective collapse: Numerical simulation
§ 16 The convective collapse: Observation
§ 17 Interaction of magnetic flux tubes with convective motion
§ 18 Stokes analysis of the simulation result
§ 19 MHD-simulation from the convection zone to the chromosphere
§ 20 Future directions in numerical simulations
§ 21 Waves carried by magnetic flux tubes
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