Download FIP Enhancement by Alfvén Ionization

FIP Enhancement by Alfvén Ionization
D A Diver ([email protected]), L Fletcher
([email protected]) and H E Potts
([email protected])
Department of Physics and Astronomy, Kelvin Building, University of Glasgow,
Glasgow G12 8QQ, Scotland
Abstract. Alfvén ionization is offered as a possible mechanism underlying the
enhanced population of low First Ionization Potential(FIP) species in the solar
corona. In this process, the photospheric flow impinging on the magnetic structure
of a coronal flux tube collides and displaces ions in the magnetised plasma within
the flux tube. This leads to pockets of charge imbalance that persist due to the impeded electron transport perpendicular to the magnetic field. The localised electric
field then energises electrons to the impact ionization energy threshold of low FIP
components in the surface flow. Such species then remain trapped in the plasma,
and drift up the magnetic structure, causing a localised population enhancement
compared to photospheric levels.
1. Background motivation
Recent reviews( Feldman & Widing (2002), Peter (2002)) confirm that
there is yet no widely accepted single cause of the enhanced population in the solar corona of low first ionization potential (FIP) species
compared with their relative abundances in the photosphere. This enhancement is often referred to as the FIP effect. Low FIP species such
as atomic sodium, aluminium, calcium, chromium, nickel, magnesium
etc that are present in the solar atmosphere are observed to be at least
three times more abundant in the chromosphere and corona than in
the photosphere (Mohan et al (2000), Phillips et al (2003), Schwadron,
Fisk & Zurbuchen (1999)) with all the concomitant implications for
the solar wind. Table 1 presents a selected list of solar photospheric
elements, using data from Asplund, Grevesse & Sauval (2005). Notice
that the elements are ordered in terms of their critical Alfvén velocity,
rather than first ionization potential. The significance of this ordering
will become clear in the following section.
Despite the lack of a single cause of low FIP enhancement, there
are a variety of sophisticated physical processes that have been identified by various experts in the field, each of which may contribute to
the overall FIP effect. For example, Laming (2004) most recently suggests that, under simplified assumptions of geometry and wave energy
density, field gradients arising from chromospheric Alfén waves may
c 2005 Kluwer Academic Publishers. Printed in the Netherlands.
°
fipsfinalrev.tex; 13/01/2005; 16:46; p.1
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cause fractionation consistent with the FIP effect, neglecting ambipolar
forces. However, Wang (1996) uses only ambipolar flows generated by
transient coronal heating to recover the same FIP bias, based on the
collisional selective dragging of neutrals by protons. In Vauclair (1996),
a magnetic skimmer model uses an erupting magnetic field to elevate
neutral material, again based on proton drag.
We offer here a simple mechanism based on Alfvén ionization which
will complement those already existing, but which introduces the ionisation bias at a very early stage. The basic premise is that the surface
flow on the photosphere will be sufficient, near strong, perpendicular
magnetic features, to ionize preferentially components of the neutral gas
for which the ionization energy is equal to, or less than, the bulk kinetic
energy of the flow. In this way the lower FIP components are ionized
simply by the flow kinetics, creating automatically a fractionation of
the solar surface gas. The next section explains the physical mechanism
in more detail.
2. Alfvén Ionization
Where a constant stream of neutral gas flowing with speed v0 impinges
on a low-density stationary, magnetised plasma of similar chemical
composition, the gas will diffuse through, and be scattered by, the
plasma without significant alteration, unless the kinetic energy of the
neutrals (in the frame in which the plasma is seen to be stationary)
exceeds the ionization energy of the gas, in which case the gas will
be efficiently ionized (Alfvén, 1960). The mechanism at work can be
expressed in simple terms as follows: atoms from the impinging gas
collide with plasma ions and displace them; the local charge imbalance cannot be rectified immediately by electron motion because the
magnetic field, oriented perpendicular to the flow, inhibits electron
transport. The resulting electrostatic field continues to increase until
the potential difference limits the further escape of ions, meaning that
the potential barrier is now equivalent to the maximum energy of an
ion as a result of a collision, namely 12 mg v02 , where mg is the mass
of a neutral particle. The persistent electric field then accelerates the
local electron population to this energy. These pockets of energetic electrons will then ionize those incoming neutrals with an electron-impact
ionization threshold that is less than this energy. This mechanism
was verified experimentally as an efficient plasma ionization method
(Wilcox, 1959; Anderson et al, 1959; Fahleson, 1961), irrespective of
fipsfinalrev.tex; 13/01/2005; 16:46; p.2
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Table I. Table of selected components of the solar photosphere. The ionization
potential φI is given in electron volts, and the critical ionization speed vc is in
km s−1 using Eq. (1). Relative abundances are taken from Asplund, Grevesse &
Sauval, 2005.
element
symbol
Xenon
Potassium
Chromium
Nickel
Vanadium
Cobalt
Manganese
Iron
Titanium
Scandium
Calcium
Krypton
Aluminium
Sodium
Silicon
Magnesium
Sulphur
Phosphorus
Chlorine
Argon
Boron
Lithium
Oxygen
Fluorine
Carbon
Beryllium
Nitrogen
Neon
Helium
Hydrogen
Xe
K
Cr
Ni
V
Co
Mn
Fe
Ti
Sc
Ca
Kr
Al
Na
Si
Mg
S
P
Cl
Ar
B
Li
O
F
C
Be
N
Ne
He
H
Z
atomic weight
Rel abundance
φI
vc
54
19
24
28
23
27
25
26
22
21
20
36
13
11
14
12
16
15
17
18
5
3
8
9
6
4
7
10
2
1
131.3
39.1
52
58.69
50.94
58.93
54.94
55.85
47.88
44.96
40.08
83.8
26.98
22.99
28.09
24.3
32.07
30.97
35.45
39.95
10.81
6.94
16
19
12.01
9.01
14.01
20.18
4.0026
1.0079
1.86E-10
1.20E-07
4.37E-07
1.70E-06
1.00E-08
8.32E-08
2.45E-07
2.82E-05
7.94E-08
1.12E-09
2.04E-06
1.91E-09
2.34E-06
1.48E-06
3.24E-05
3.39E-05
1.38E-05
2.29E-07
3.16E-07
1.51E-06
5.01E-10
1.12E-11
4.57E-04
3.63E-08
2.45E-04
2.40E-11
6.03E-05
6.92E-05
8.51E-02
1.00E+00
12.13
4.34
6.77
7.64
6.75
7.88
7.43
7.9
6.83
6.56
6.11
14.0
5.99
5.14
8.15
7.65
10.4
10.5
13
15.8
8.3
5.39
13.6
17.4
11.26
9.32
14.5
21.6
24.6
13.6
4.22
4.63
5.01
5.01
5.06
5.08
5.11
5.22
5.25
5.31
5.42
5.68
6.54
6.57
7.48
7.79
7.91
8.09
8.41
8.74
12.17
12.24
12.81
13.29
13.45
14.13
14.13
14.37
34.43
51.02
fipsfinalrev.tex; 13/01/2005; 16:46; p.3
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whether the moving neutral gas encountered stationary magnetised
plasma, or a moving plasma encountered stationary neutrals; it is only
the relative speed that counts.
The conditions required are that: (i) the stationary plasma is strongly
magnetized, so that transport perpendicular to the magnetic field is
significantly hampered and (ii) the neutral gas speed reaches vc , given
by
Ã
vc =
2eφI
mg
!1/2
(1)
where φI is the ionization potential of a neutral gas particle. Values of
vc for low FIP photospheric elements are given in the last column of
Table 1.
Clearly there are restrictions on the flow and magnetic field strengths
that will allow such a mechanism to operate. The neutral gas flow must
establish the local charge imbalance faster than the electron transport
can neutralise it. Suppose that plasma ions are displaced, by collisions
with the flowing neutrals, over a scale length equal to the ion Larmor
radius RL . Then the typical set-up time τs for collisions to establish a
charge imbalance across RL is given by
τs ≈
RL
v0
(2)
The characteristic time τe for electron transport to eliminate this charge
imbalance must depend on the electron drift-diffusion speed vde for
perpendicular transport across the magnetic field:
τe ≈
RL
vde
(3)
If Alfvén ionization is to work, then condition (i) means that
νe ¿ ωce
(4)
where νe is the collision frequency for electron-neutral particle encounters, and ωce is the electron cyclotron frequency. This ensures that the
electron diffusion coefficient perpendicular to the magnetic field is much
smaller than that parallel to the field.
Consider now the ratio of timescales
vde
µ⊥e E
τs
≈
=
τe
v0
v0
(5)
fipsfinalrev.tex; 13/01/2005; 16:46; p.4
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where
µ⊥e =
≈
eνe
2 )
me (νe2 + ωce
(6)
eνe
2
me ωce
(7)
is the component of electron mobility perpendicular to the magnetic
field that is independent of E × B drift, and E is the magnitude of the
electric field. (Recall that E × B drift is charge-independent, and so
cannot rectify charge imbalance.)
Using Poisson’s equation to balance the maximum gas particle energy with the electric potential caused by ejection of ions,
mg v02
2eRL
(8)
v0 ≈ RL ωi
(9)
τs
¿1
τe
(10)
E≈
and taking
yields eventually
thus ensuring that a potential barrier can be set up faster than the
charge equilibration time from electron motion. Stated another way,
(10) means that
RLe ¿ λmfp
(11)
where RLe is the electron Larmor radius, and λmfp is the mean free
path for electron-neutral collisions.
Note that (10) allows a pocket of unbalanced charge to be created,
(1) places a further restriction on how large the associated potential
must be before ionization can take place.
3. Application to solar abundances
The solar photospheric flow of neutral gas encounters a strong, verticallyaligned magnetic structure containing predominantly ionized plasma.
Depending on the actual magnitude of the flow speed, some atomic
species in the flow may be moving faster than the critical speed for
ionization; others will not (in accordance with Table 1). Strong flows
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near concentrated magnetic fields could reach the photospheric sound
speed, approximately 7 km s−1 , and so satisfy the ionization threshold
for Alfvén’s process. Supersonic horizontal flows have been observed in
active regions; for example, Meunier and Kosovichev (2003) detected
long-lived supersonic horizontal flows of the order of 7.7 km s−1 , and
Yang et al (2003) also observed peak horizontal flow speeds of 5.68
km s−1 in pores. Chae et al (2000) report observations of horizontal
flows up to 20.6 km s−1 .
In order to have the plasma sufficiently magnetized, we need to
guarantee that (10) holds. From Vauclair, 1996, we can take the collision frequency ν for electrons with neutrals in the lower chromosphere
to be νe ≈ 1.1 × 106 s−1 . Comparing this with the electron cyclotron
frequency ωce ≈ 1.8 × 1010 × B s−1 where B is the magnitude of the
magnetic flux density in Tesla, we have
νe
6 × 10−5
≈
ωce
B
(12)
and so (10) is satisfied for field strengths B ≥ 10−2 T, which is reasonable at the base of magnetic loops.
Vauclair’s data pertains to mid-chromospheric conditions. To be sure
that (10) is satisfied at the upper photospheric levels, the electronneutral collision frequency νe can be modelled (Cravens, 1997) approximately by
νe ≈ ken nn ≈ (10−8 cm−3 s−1 ) × nn
(13)
where nn is the neutral number density, per cubic centimetre. Taking
nn ≈ 1016 cm−3 gives a value for νe that is 100 times larger than the
chromospheric parameter. However, (10) is still reasonable for magnetic
field strengths greater than 0.01 T.
Selective Alfvén ionization is also consistent with the evolution of
bias as a function of time in regions of emergent coronal loops (Widing
& Feldman, 2001), since the perpendicular magnetic field component
at the photospheric surface is also evolving, and the flow imposes a
natural time for the cumulative bias to appear. Even in the case of a
static magnetic structure, the photospheric flow will continue to deliver
low-FIP material to the footpoints where it may be ionized.
Supposing the flow continues to be above the critical threshold
for some species. Let us assume a very simple picture, ignoring all
subsequent collisional processes, in which the low-FIP neutral, having
been ionized by Alfvén ionization, is now caught in a magnetic trap
formed by the magnetic loop with highest field strengths at the base.
We can consider the characteristic time τb for one transit of the loop
fipsfinalrev.tex; 13/01/2005; 16:46; p.6
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(the bounce-time) as a useful indicative timescale for the FIP evolution.
Now
τb ≈
Z L
ds
0
vq (s)
ds
(14)
where L is the path length along the loop, parametrised by s, and vq
is the speed along the major axis of the loop. Treating the magnetic
structure as a trap, we have that
2
1
2 m(vq
2
+ v⊥
) = Wq + W⊥ = constant
W⊥
= constant
B
(15)
(16)
where k refers to the direction of the local magnetic field, ⊥ is orthogonal to k, and Ws is the kinetic energy. Assuming that the newly
ionized neutral has negligible parallel velocity, since it originates from
the photospheric flow, we can take its kinetic energy to be entirely in
the form of W⊥ , and use this as the mirror condition. This means we
can write τb as
τb ≈
v0−1
Z L
0
µ
2L
=
v0
(1 − B(s)/B0 )1/2 ds
R
R−1
(17)
¶1/2
(18)
where R = Bmin /B0 is the mirror ratio, that is, the ratio of the
magnetic field strength at the loop top, Bmin , (weakest) to that at
the footpoint, B0 , (strongest), and where we have taken a simple linear
behaviour
·
B(s) ≈ B0
¸
2(R − 1)
1−
s ,
RL
s ≤ L/2
(19)
With indicative values of L ≈ 2 × 108 m, R ≈ 4 (Nakariakov & Ofman,
2001) and v0 ≈ 7 × 103 ms−1 (modest in comparison with Winebargeret
al (2002) who show flows in the range 15-40 km s−1 in coronal loop
structures) we have
τb ≈ 6.6 × 104 s ≈ 43 day
(20)
The ionized low-FIP species are then trapped in the magnetic loop,
and their numbers continue to increase as more neutral flow at the
footpoint delivers a continuous supply of freshly ionized material. This
result compares favourably with the analysis of Widing & Feldman,
2001, in which FIP bias is observed to increase linearly with time,
fipsfinalrev.tex; 13/01/2005; 16:46; p.7
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taking approximately a day to augment the FIP bias by unity for Mg
in active region plasmas.
We could improve the sophistication of our model by taking account
of weakly collisional processes in the loop. This would alter the loop
transit time, and change the FIP evolution timescale in a involved way,
since although diffusion will lengthen the transit time compared to τb ,
the density enhancement of newly ionized low-FIP material may well
increase faster as a result.
4. Concluding Discussion
In this article we present a simple physical process that may account for
the enhanced abundance of low-FIP elements in the upper solar atmosphere. Alfvén ionization has the merit of converting the kinetic energy
in the photospheric flow into ionizing energy selectively for the lowFIP elements where such a flow impinges on a pre-existing magnetised
plasma. In this way, we have a simple mechanism that delivers up to 10
eV (generated from modest photospheric flows) to selected neutrals in
order to ionize them, consistent in part with the ethos of McKenzie &
Feldman (1994), who conclude that a mechanism that could impart a
small amount of energy only to selected particles could account for the
discrimination, albeit they were discussing ions, rather than neutrals.
Having ionized our neutrals by Alfvén ionization, we invoke a magnetic
bottle description to show how the increased abundance evolves with
time, with the newly ionized low-FIP neutrals moving up the magnetic
loop as a result of a magnetic gradient near the footpoint. This is
consistent with the physical picture of Schwadron, Fisk & Zurbuchen
(1999), in which the greatest low-FIP bias is seen in the slow solar
wind, assumed to originate from material stored in large coronal loops
and released sporadically via reconnection.
The ‘magnetic skimmer’ model (Vauclair, 1996) uses the magnetic
field too, in that an erupting field raises ionised material from the
photosphere up into the chromospheric plateau where any neutrals
dragged up in the process can be ionized. Our Alfvén ionization model
is complementary to this, in that we use the vertical magnetic elements
to generate the FIP enhancement; however, if the initial loop eruption
is sufficiently fast, then the horizontal field could also ionize, since it is
only the relative motion between magnetized plasma and neutral gas
that is important. Vauclair also notes that small dense loops acquire a
FIP bias in their footpoints as the loops grow; this is consistent with
both our treatments.
fipsfinalrev.tex; 13/01/2005; 16:46; p.8
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FIP enhancement fraction
12
Feldman et al (1998)
Reames (1995)
Mohan et al (2000)
Laming (2004)
Wieler et al (1995)
10
8
6
4
2
0
0
5
10
15
20
25
30
35
40
45
50
Critical Alfven velocity /km s−1
12
FIP enhancement fraction
10
8
6
Xe
4
2
0
0
5
10
15
Ionisation energy /eV
20
25
Figure 1. Observational data on FIP bias for a variety of elements, from a range of
sources.Top graph shows the bias as a function of vc ; lower graph shows the same
data as a function of FIP. Multiple data points for the same value on the horizontal
axis reflect the diversity in observation for particular elements.
Mohan et al (2000) note that the FIP bias for K seems to be greater
than all the other low-FIP elements; in our treatment, K has the lowest
critical flow speed of all the low-FIP elements, and so is the most readily
ionized by photospheric flows. Their observation of spatial structure in
the FIP bias is also in agreement with our model, where the bias itself
is generated at the footpoints, and enhanced by storage in magnetic
loops.
Although comparisons between FIP bias observations are difficult,
the elements which show the greatest relative bias are in accord with
those having the smallest vc (as shown in Table 1). Notice that Xe has
the lowest vc of all the quoted elements, yet has an ionization potential
of 12.3 eV, and so should not fall into the category of low FIP bias.
fipsfinalrev.tex; 13/01/2005; 16:46; p.9
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Studies of solar noble gas abundance however (Wieler & Baur, 2001)
reveal that Xe abundance in the solar wind is greater than photospheric
abundance by a factor similar to the standard low FIP bias. This is
also partly true of Krypton, which also has a sufficiently small vc for
Alfvén ionization to be possible. A compendium of observational results
is presented in the graphs of Figure 1. The top graph shows relative
abundance as a function of FIP; the lower graph presents the same
data re-ordered to show bias as a function of vc . The second graph
accommodates the anomalous Xe result more satisfactorily than the
first. Multiple points for the same value on the horizontal axis reflect
the diversity of data for the same element; given that the data spread
partly reflects the source type, and partly the date of the observation,
it may be that FIP bias is also a function of the solar cycle.
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