# The FIP and Inverse FIP Effects in Solar and Stellar Coronae

J. Martin Laming
Code 7684 Naval Research Laboratory
Washington DC 20375, USA
email<EMAIL_ADDRESS>

###### Abstract

We review our state of knowledge of coronal element abundance anomalies in the
Sun and stars. We concentrate on the first ionization potential (FIP) effect
observed in the solar corona and slow-speed wind, and in the coronae of solar-
like dwarf stars, and the “inverse FIP” effect seen in the corona of stars of
later spectral type; specifically M dwarfs. These effects relate to the
enhancement or depletion, respectively, in coronal abundance with respect to
photospheric values of elements with FIP below about 10 eV. They are
interpreted in terms of the ponderomotive force due to the propagation and/or
reflection of magnetohydrodynamic waves in the chromosphere. This acts on
chromospheric ions, but not neutrals, and so can lead to ion-neutral
fractionation.

A detailed description of the model applied to closed magnetic loops, and to
open field regions is given, accounting for the observed difference in solar
FIP fractionation between the slow and fast wind. It is shown that such a
model can also account for the observed depletion of helium in the solar wind.
The helium depletion is sensitive to the chromospheric altitude where ion-
neutral separation occurs, and the behavior of the helium abundance in the
closed magnetic loop strongly suggests that the waves have a coronal origin.
This, and other similar inferences may be expected to have a strong bearing on
theories of solar coronal heating.

Chromospheric waves originating from below as acoustic waves mode convert,
mainly to fast mode waves, can also give rise to ion-neutral separation.
Depending on the geometry of the magnetic field, this can result in FIP or
Inverse FIP effects. We argue that such configurations are more likely to
occur in later-type stars (known to have stronger field in any case), and that
this explains the occurrence of the Inverse FIP effect in M dwarfs. We
conclude with a discussion of possible directions for future work.

##  1 Introduction

Working during the early years of solar UV and X-ray spectroscopy, Pottasch
(1963) found evidence for significantly higher abundances of Mg, Si, and Fe in
the low solar corona than in the photosphere, and concluded, somewhat
reluctantly that “the chemical composition in the solar atmosphere differs
from the photosphere to the corona”. These elements, Mg, Si, and Fe, are all
elements with first ionization potential (FIP) less than 10 eV, now known to
be routinely enhanced in abundance in the corona with respect to photospheric
values, a phenomenon that has become known as the “FIP effect”. High FIP
elements such as O, Ne, and He, have much smaller abundance enhancements, or
even abundance depletions in the corona. Although the possibility of elemental
fractionation between the solar photosphere and corona only really began to be
taken seriously in the mid 1980s, with the publication of influential reviews
by Meyer (1985a, b), recognizing the work of Pottasch (1963) almost fifty
years ago makes the problem of understanding the FIP effect nearly as old as
that of coronal heating. In fact, modern models of the effect to be discussed
in detail below, in which the fractionation is driven by the ponderomotive
force of Alfvén waves, make an intimate connection between the abundance
anomaly and coronal heating mechanisms, such that the FIP effect may yield
several important insights into the nature of the latter. This is a theme we
will develop throughout this review.

The solar FIP effect manifests itself in several modes of observation.
Spectroscopy, as pioneered by Pottasch (1963) and extensively reviewed at
intervals over the last 20 years (e.g., Feldman, 1992; Feldman and Laming,
2000; Feldman and Widing, 2003; Saba, 1995), reveals the composition primarily
of the “closed loop” corona. Meyer (1985a, b) also considered elemental
abundances measured in situ in the solar wind, and in solar energetic particle
events, which flow out along open magnetic field lines. Observations with
Ulysses, the first mission to fly over the solar polar regions (Wenzel et al.,
1992) revealed FIP fractionation varying with wind speed (Zurbuchen et al.,
1999; von Steiger et al., 2000). Slow speed solar wind had abundances
resembling those in the closed loop solar corona, whereas high speed wind from
polar coronal holes had a much lower level of FIP fractionation. With the
advent of the Solar and Heliospheric Observatory (SOHO), a more coherent
observational picture began to emerge. Coronal holes, the source of the
relatively unfractionated fast solar wind, were themselves shown to have
similar abundances to the wind emanating from them. The lower latitude closed
field corona was also shown to have FIP fractionated plasma, similar to the
slow speed solar wind. Schmelz et al. (2012) give a modern view of coronal
element abundances derived from these various solar physics sources, coronal
spectroscopy, solar wind and solar energetic particles.

Astrophysical EUV and X-ray spectroscopy, made possible by the 1990’s launches
of the Extreme Ultraviolet Explorer (EUVE), the Advanced Satellite for
Cosmology and Astrophysics (ASCA), Chandra and XMM-Newton, allowed coronal
abundances in stars to be measured for the first time. Again, nearly two
decades after the first such observations, various observations seem to be
falling into place. In this review, we will attempt to synthesize these two
strands of observations, solar and stellar, into one complete picture of
coronal element abundance anomalies.

Following this introduction, in Section 2 we briefly review recent
developments in solar photospheric abundances, where improved spectroscopic
data and the application of 3D radiation transfer calculations have ushered in
a revised solar composition. Section 3 describes the various facets of the
solar FIP effect, while Section 4 surveys stellar FIP and Inverse FIP effects.
Section 5 describes early attempts to model the solar FIP effect. Section 6
lays out the model advocated here, where the ponderomotive force due to Alfvén
waves propagating through, or reflecting from, the chromosphere accelerates
ions up or down, while leaving neutral atoms unaffected. Section 7 describes
the results of such a model, and discusses its important implications. Section
8 concludes with suggestion for future research directions.

##  2 Solar Photospheric Composition

###  2.1 Review

Any work on solar coronal abundance anomalies must begin with reviewing the
photospheric composition. Despite its long history, the composition of the
photosphere, also taken as a proxy for “cosmic abundances”, has undergone
significant revisions in recent years. We take as our starting point the
composition review of Anders and Grevesse (1989), this being the standard
solar composition in use for the early studies of the FIP effect, and for many
years the default abundance set implemented by spectral fitting software in
use in X-ray astronomy. This was updated by Grevesse and Sauval (1998), who
revised downwards by a small amounts N and O (0.13 and 0.10 dex, respectively)
and also Fe which moved from 7.67 (on a logarithmic scale where the abundance
of H is 12) to 7.50, in agreement with the meteoritic value. This last
modification stemmed from improvements in atomic data used to analyze solar
spectrum (e.g., Holweger et al., 1991; Biémont et al., 1991), and is largely
supported by more modern analyzes (Asplund et al., 2000a, b; Asplund, 2000;
Bellot Rubio and Borrero, 2002).

The next major revision came to the photospheric abundance of O by Allende
Prieto et al. (2001), who applied a three-dimensional time-dependent
hydrodynamical model solar atmosphere to the observed 6300 Å line. These
authors recognized that this line, attributed to a forbidden line of neutral
O, is also blended with Ni I, with the result that the O abundance of 8.83 in
Grevesse and Sauval (1998) (8.93 in Anders and Grevesse, 1989) was revised
down to $8.69\pm 0.05$. A number of subsequent papers analyzing other allowed
and forbidden lines in O I, and also molecular OH supported this change
(Asplund et al., 2004, 2005a; Meléndez, 2004; Socas-Navarro and Norton, 2007;
Meléndez and Asplund, 2008), though Ayres (2008) and Caffau et al. (2008)
offered more cautious views. Joining, O, C (Allende Prieto et al., 2002;
Asplund et al., 2005b; Caffau et al., 2010) and N (Caffau et al., 2009) also
underwent downward revisions in their abundances.

In Table 1 we collect the recommended solar photospheric abundances of various
commonly observed elements from Grevesse and Sauval (1998), Asplund et al.
(2009), (also given in Grevesse et al., 2010) Caffau et al. (2011), and Scott
et al. (2014b, a) and Grevesse et al. (2014) for comparison and reference. Not
given here are result from Lodders (2010), who for the elements of most
interest here appears to quote the average of Asplund et al. (2009) and Caffau
et al. (2011). The data sources for each element are all listed in these
reviews. For future reference we remark that aside from the revisions above,
the noble gases will be of interest to us, since their photospheric abundances
are often determined from coronal observations (e.g., Feldman and Widing,
1990; Young, 2005a), on the assumption that no fractionation occurs between
the photosphere and the corona for these elements. This is an assumption we
shall scrutinize. Only He is determined independently, from helioseismology
(Basu and Antia, 2004), in an analysis that includes the revised solar
metallicity.

Table 1: Recommended Solar Photospheric Abundances of Common Elements Element | Grevesse and Sauval (1998) | Asplund et al. (2009) | Caffau et al. (2011) | Scott et al. (2014b, a)
---|---|---|---|---
|  |  |  | Grevesse et al. (2014)
H | 12.00 | 12.00 |  |
He | $10.93\pm 0.004$ | $10.93\pm 0.01$ |  |
C | $8.52\pm 0.06$ | $8.43\pm 0.05$ | $8.50\pm 0.06$ |
N | $7.92\pm 0.06$ | $7.83\pm 0.05$ | $7.86\pm 0.12$ |
O | $8.83\pm 0.06$ | $8.69\pm 0.05$ | $8.76\pm 0.07$ |
Ne | $8.08\pm 0.06$ | $7.93\pm 0.10$ |  |
Na | $6.33\pm 0.03$ | $6.24\pm 0.04$ |  | $6.21\pm 0.04$
Mg | $7.58\pm 0.05$ | $7.60\pm 0.04$ |  | $7.59\pm 0.04$
Al | $6.47\pm 0.07$ | $6.45\pm 0.03$ |  | $6.43\pm 0.04$
Si | $7.55\pm 0.05$ | $7.51\pm 0.03$ |  | $7.51\pm 0.03$
P | $5.45\pm 0.04$ | $5.41\pm 0.03$ | $5.46\pm 0.04$ | $5.41\pm 0.03$
S | $7.33\pm 0.11$ | $7.12\pm 0.03$ | $7.16\pm 0.05$ | $7.13\pm 0.03$
Cl | $5.5\pm 0.3$ | $5.50\pm 0.30$ |  |
Ar | $6.40\pm 0.06$ | $6.40\pm 0.13$ |  |
K | $5.12\pm 0.13$ | $5.03\pm 0.09$ | $5.11\pm 0.09$ | $5.04\pm 0.05$
Ca | $6.36\pm 0.02$ | $6.34\pm 0.04$ |  | $6.32\pm 0.03$
Ti | $5.02\pm 0.06$ | $4.95\pm 0.05$ |  | $4.90\pm 0.04$
Cr | $5.67\pm 0.03$ | $5.64\pm 0.04$ |  | $5.62\pm 0.04$
Fe | $7.50\pm 0.05$ | $7.50\pm 0.04$ | $7.52\pm 0.06$ | $7.47\pm 0.04$
Ni | $6.25\pm 0.04$ | $6.22\pm 0.04$ |  | $6.20\pm 0.04$
Kr | $3.31\pm 0.08$ | $3.25\pm 0.06$ |  | $3.23\pm 0.06$

###  2.2 Helioseismology

The re-evaluation of the solar composition above has revised the solar
metallicity down from 0.0170 (Grevesse and Sauval, 1998) to the range 0.0153
(Caffau et al., 2011) to 0.0134 (Asplund et al., 2009), with Asplund et al.
(2005c) giving a value as low as 0.0122, largely driven by improved analysis
of lines of C, N, and O. The lower metallicity decreases the sound speed at
the base of the solar convection zone, yielding now a larger disagreement
between observed and modeled sound speeds (see, e.g., Figure 1 in Guzik and
Mussack, 2010). The reduced abundances also decrease the depth of the
convection zone in solar models, again worsening agreement between models and
helioseismic inversions (e.g., Basu and Antia, 2004).

One early solution proposed was to increase the abundance of Ne to compensate
(Antia and Basu, 2005; Bahcall et al., 2005a), by bringing the solar
metallicity back to its prior value. A survey in nearby active stars had
previously yielded the abundance ratio Ne/O in the range 0.3 – 0.4 (Drake and
Testa, 2005), significantly higher than the solar coronal value and closer to
that suggested. Drake and Testa (2005) argue that the relative constancy of
this ratio in a sample of over 20 stars suggests that the “true” Ne/O
abundance ratio should be around 0.4, and that the variation in the solar
corona must be due to some unknown fractionation. More recent work suggests
than an increased Ne abundance (of about 0.5 – 0.67 dex) is not a complete fix
(Lin et al., 2007), but a more modest increase of 0.45 dex is still acceptable
(see introduction of Guzik and Mussack, 2010). Ne, and also possibly Ar, are
the focus of revisions to composition because having no photospheric
absorption lines, their abundances can only be measured in the solar corona,
or in astrophysical sources as proxies for the solar photosphere. Asplund et
al. (2009) argue for an Ne/O abundance ratio of $0.175\pm 0.031$ following the
measurements of Young (2005a) in the solar corona. Other authors have
suggested that increased opacity (Christensen-Dalsgaard et al., 2009;
Serenelli et al., 2009), beyond that implied by the increase due to data from
the Opacity Project replacing older OPAL radiative opacities (Bahcall et al.,
2005) might ease the problem, while variations to the solar evolutionary
history with extra episodes of mass loss of accretion have also been
considered (Guzik and Mussack, 2010; Serenelli et al., 2011). Villante et al.
(2014) provide a recent evaluation, accounting for helioseismic and solar
neutrino data, again favouring the older photospheric abundance set of
Grevesse and Sauval (1998). Even more recently, Shearer et al. (2014) study
the variation of Ne/O measured by Ulysses/SWICS and ACE/SWICS over the solar
cycle between 1998 and 2012. Their results also favour a low Ne/O abundance
ratio in the range 0.10 - 0.15, although with unexpected variation.

Bergemann and Serenelli (2014) review these and other potential solutions to
the “solar abundance problem”. Other ideas include that of Lopes and Silk
(2013), who revisit the helioseismology problem in the light of revised
photospheric abundance for the Sun and solar-like stars. Relative to solar
analogs without planetary systems, the Sun appears to be underabundant in
metals, more so in refractory elements than in volatiles. The mass of the
“missing” elements from the solar convection zone appears to be similar to the
combined mass of the inner terrestrial planets, Mercury, Venus, Earth and
Mars, leaving open the possibility that the metallicity of the solar interior,
specifically the radiation zone, is higher than that of the convection zone,
as appears to be required. Lopes and Silk (2013) explore several such models,
computing the solar neutrino spectra with a view to future observational
capabilities. Their models however appear to need high mass loss from the
young Sun, which might be problematic (cf. Wood, 2004, 2006). Zhang (2014)
considers the effect of a turbulent kinetic flux within the solar convection
zone. A negative flux, i.e. turbulence propagating from the convection zone to
the radiation zone, goes some way to restoring agreement. A downwards
turbulent kinetic flux requires larger outward radiative and convective energy
fluxes, which lead to a deeper boundary between the convection and radiation
zones.

Finally the solar helium abundance is most accurately determined from
helioseismology, from the anomaly produced in the sound speed at the depths in
the convection zone where helium ionizes. Villante et al. (2014) give the
value as $Y=0.2485\pm 0.0035$, apparently the mean of the results derived by
Basu and Antia (2004) using GONG or MDI data.

##  3 The Solar FIP Effect: Overview

As mentioned above, the observation of element abundances in the solar corona
and wind has a history spanning decades, and has been reviewed several times
during that period (Meyer, 1985a, b; Feldman, 1992; Feldman and Laming, 2000;
Feldman and Widing, 2002, 2003, 2007). Here, rather than provide an exhaustive
review, we attempt to summarize and update the status of solar coronal
abundances, and refer readers back to these reviews for extensive references.

In contrast to photospheric abundances which are generally measured and given
relative to H (i.e absolute abundances), with a few exceptions to be noted
below, most coronal abundance measurements and results are relative, in that
one minor ion is compared to another minor ion (often O) with no reference to
H. This arises due to the difficulties in observing H. H generally has no
observable emission lines from the solar corona in bandpasses in which other
elements are observed, and requires different in situ instruments for
detection to those for heavy ions. Both difficulties lead to cross calibration
issues. Exceptions occur in the UV where H emission lines can sometimes be
detected, or in the X-ray region where the thermal bremsstrahlung continuum
can be taken as an indicator of the H abundance. We will highlight these
features where appropriate below.

Theoretically, it will also turn out to be easier to discuss relative
abundances. Modeling the action of the ponderomotive force on minor ions in an
ambient H atmosphere is much more straightforward than the multifluid
calculation that would be necessary to treat the back reaction of the Alfvén
waves on the H fluid itself, and to incorporate those effects into the
predicted FIP fractionations. In fact in many cases, it turns out the H and O
behave similarly because of their strong charge exchange coupling, due to
their similar ionization potentials, and this provides some justification for
our approach.

In the quiet solar corona and slow speed solar wind, elements with first
ionization potential (FIP) below about 10 eV are enhanced in abundance by a
factor of about 3, with typical variations in the range 2 – 5. This is
established by both remotely sensed (i.e., spectroscopic) and in situ
measurements and at the time of writing, despite early controversy, is now
considered an established fact. Variations in the fractionation with solar
region exist. Coronal holes, and the fast solar wind emanating from them are
known to have a significantly smaller degree of FIP fractionation than the
quiet corona and slow wind (e.g., Bochsler, 2007a; Feldman, 1998b). Brooks and
Warren (2011) show that FIP fractionated abundances measured in an active
region with the Extreme Ultraviolet Imaging Spectrograph (EIS) on Hinode
(Culhane et al., 2007) match those detected a few days later when the solar
wind from this active region would have reached 1 AU. The Sun viewed “as a
star” (Laming et al., 1995) shows FIP effect only for temperatures above about
$10^{6}$ K. This is illustrated in Figure 1, taken from Laming et al. (1995),
where the full disk solar emission distribution is determined from the solar
spectrum of Malinovsky and Heroux (1973), based on emission lines from low FIP
(filled symbols) and high FIP (open symbols) elements. The two emission
measures coincide up to $\log T\simeq 6$, and thereafter diverge, indicating
overabundance of low FIP elements. This result is corroborated by Young
(2005b, c), Feldman and Widing (1993); Feldman (1998a), and Young and Mason
(1998), and leads to the conclusion that structures emitting at temperature
below $10^{6}$ K must be distinct entities from those responsible for the
higher temperature emission. Feldman and Laming (1994) elaborate this argument
based on earlier observations, most notably those in Feldman (1983, 1987). It
appears likely that the high temperature FIP fractionated plasma resides in
coronal loops fed by chromospheric evaporation (the response of chromospheric
plasma to heat released in the corona conducted downwards, see e.g. Bray et
al., 1991; Reale, 2014), while the lower temperate plasma, attributed to
“unresolved fine structures” by Feldman (1983, 1987), can now be identified
with the recently discovered Type II spicules (De Pontieu et al., 2011;
Martínez-Sykora et al., 2011), or other dynamic loop structures (Hansteen et
al., 2014) recently observed with the Interface Region Imaging Spectrograph
(IRIS). Individual features distinct from Type II spicules at transition
region temperatures ($10^{4}$K $<T<10^{6}$K) may show FIP fractionation (see
examples in e.g. Feldman, 1992; Feldman and Laming, 2000; Feldman and Widing,
2002, 2003, 2007). We emphasize again that the result of Laming et al. (1995)
applies to the “Sun as a star”.

Active regions and flares can also have different fractionation, often reduced
from that observed in the quiet Sun. This is most clearly seen in spectra
acquired of plasma above a sunspot (Feldman and Widing, 1990). Phillips et al.
(1994) also measured a maximum Fe enhancement of a factor of 2 between
photosphere and corona, by comparing the photospheric Fe K $\beta$ line
excited by flourescence with the collisionally excited coronal Fe XXV
resonance line observed during flares by the YOHKOH Bragg Crystal Spectrometer
(Culhane et al., 1991). Later work with RHESSI (Lin et al., 2004) found much
stronger enhancements in the abundance of Fe relative to H from line to
continuum measurements (Phillips et al., 2006). Abundances of other elements
in flares relative to H have been measured by RESIK (Sylwester et al., 2005).
FIP fractionations for K, Ar, Cl, S, Si and Al have been given by Sylwester et
al. (2008), with more detailed results given in subsequent papers (Sylwester
et al., 2010a, b, 2011, 2012, 2013, 2014), and given in more detail in Table 3
for comparison with model results. Warren (2014) finds almost no FIP
fractionation in 21 flares observed with the EUV variability Experiment (EVE)
on the Solar Dynamics Observatory (SDO), while Del Zanna and Mason (2014) find
a typical enhancement in the abundance of Fe compared to O and Ne of 3.2 in 9
flares observed by the Flat Crystal Spectrometer (FCS) on the Solar Maximum
Mission (SMM). Brooks and Warren (2012) and Widing and Feldman (2008) find
similar enhancements.

Coronal mass ejections (CMEs) observed in situ often have strong FIP
fractionation as shown in a survey (Reisenfeld et al., 2007) conducted with
data the Advanced Composition Explorer (ACE Gloeckler et al., 1998) and the
Genesis mission (Burnett et al., 2003). Zurbuchen et al. (2004) and Smith et
al. (2001) show that frequently these large FIP fractionations are associated
with the CME flux rope, possibly implying that the flux rope forms in the
corona and does not emerge preformed from the photosphere (in which case it
might be expected to exhibit photospheric abundances). CMEs can also exhibit
mass dependent fractionation (Wurz et al., 2000), most likely indicating the
role of processes other than FIP fractionation in modifying the elemental
composition of the solar upper atmosphere.

We have already mentioned the difference in FIP effect between fast and slow
speed streams observed by Ulysses (Zurbuchen et al., 1999; von Steiger et al.,
2000). Lepri et al. (2013) study the evolution of abundances of fast and slow
solar wind observed in the ecliptic by the Advanced Composition Explorer
(ACE). While other solar wind parameters such as charge states, magnetic
fields and freeze-in temperatures show marked variation, the degree of FIP
fractionation (i.e. element abundances relative to O) does not. Absolute
abundances, measured relative to H, do show some variation, with the lowest
abundances of He, C, O, Si and Fe being observed in the slowest wind at solar
minimum. Interestingly, their absolute abundances for O support the revised
photospheric abundance of O (Asplund et al., 2009), with the highest mean O/H
abundance ratio in the slow speed wind at solar maximum being 8.68 (in
logarithmic notation), with upper and lower limits of 8.42 and 8.94
respectively, from their Table 1. Previous studies of this sort (Bochsler,
2007b; von Steiger et al., 2010) have favoured the “older” O abundance (8.83
Grevesse and Sauval, 1998). von Steiger et al. (2010) assume that the high
speed solar wind gives the most faithful representation of photospheric
abundances (their O/H value in slow speed solar wind agrees with Asplund et
al., 2009). Bochsler (2007b) plots O/H against He/H in his Figure 3. and shows
a striking correlation between them, arguing that inefficient Coulomb drag is
the cause of both variations. This might appear to be problematic, in that O
is not always depleted relative to H (unless one takes oldest O/H value from
Anders and Grevesse, 1989, as the correct value), but He is, with the problem
becoming worse for the more recent O/H values.

Figure 1: Full disk solar emission measure distribution, with low FIP elements
depicted by solid symbols, high FIP elements by open symbols. The transition
from photospheric abundances at $\log T<6.0$ to coronal abundances at higher
temperatures, indicating that different types of coronal structures are
emitting above and below this temperature. Figure from Laming et al. (1995).
Reproduced by permission of the AAS.

Figure 2: _Left:_ Helium abundance relative to hydrogen in the slow speed
solar wind over 1.5 solar cycles. The curve colors denoted the wind speed at
which at abundance is measured. He is more variable in the slowest slow speed
solar wind. The black curve gives the monthly smoothed sunspot number. Figure
from Kasper et al. (2012). Reproduced by permission of the AAS. _Right:_
Helium abundance relative to oxygen in the slow speed solar wind, measured by
ACE/SWICS. The same trend of depletion with wind speed as for He/H is seen,
but the solar cycle dependence is less pronounced. Figure from Rakowski and
Laming (2012). Reproduced by permission of the AAS.

Figure 3: Maps of intensity, non-thermal velocity, photospheric magnetic field
and coronal FIP bias derived for a sigmoidal anemone active region observed by
EIS/Hinode. The overlaid black ellipses show the footpoints of loops,
coincident with strong photospheric magnetic field concentrations, while the
dotted black lines show the loop connections between them. The strongest FIP
effect can be seen at the loop footpoints, also coincident with strong
nonthermal velocities. Figure from Baker et al. (2013). Reproduced by
permission of the AAS.

The solar slow speed wind helium abundance has long been known to be depleted
from the photospheric value, and this depletion is now established to vary
with solar slow wind speed and the phase of the solar activity cycle (Aellig
et al., 2001; Kasper et al., 2007) in a similar manner to that mentioned above
(Lepri et al., 2013). Figure 2 (left panel, from Kasper et al., 2012) shows
the He/H abundance ratio measured by the Wind spacecraft over 1.5 solar
cycles, in bins of different solar wind speed. Greater variability is seen in
the slowest wind speed bins, and in the slowest bin, the variability matches
the smoothed sunspot number. The right panel of Figure 2 show a similar study
with ACE by Rakowski and Laming (2012), where the abundance ratio He/O is
plotted. Broadly similar behavior with solar wind speed is seen, but the solar
cycle dependence is less pronounced, but still apparent. The fast solar wind
helium abundance is also depleted, but to a lesser degree than the slow wind,
and is also less variable. Measurements of the helium abundance in coronal
holes (Laming and Feldman, 2003) and in the quiet solar corona (Laming and
Feldman, 2001) generally show similar values to those measured in the solar
wind, indicating that the helium depletion occurs lower down in the solar
atmosphere than the corona. A spectroscopic measurement in a solar flare
(Feldman et al., 2005) showed a much higher abundance of helium, in
qualitative agreement with the by now routine observation in situ of enhanced
(i.e., less depleted) helium abundance in CMEs (e.g., Wimmer-Schweingruber et
al., 2006).

Widing and Feldman (2001) studied the variation of the FIP effect with time in
newly emerged active regions. They found that the new loops emerged with
photospheric abundances, and gradually changed to coronal abundances, i.e.,
developed a FIP effect, over the course of a few days. Consequently a new
active region should have a weak FIP effect, while an older one would show
strong fractionation. Recent observations by Baker et al. (2013) corroborate
this view. Figure 3 shows maps of intensity, non-thermal velocity,
photospheric magnetic field and coronal FIP bias derived for a sigmoidal
anemone active region observed by EIS/Hinode. The overlaid black ellipses show
the footpoints of loops, coincident with strong photospheric magnetic field
concentrations, while the dotted black lines show the loop connections between
them. The strongest FIP effect can be seen at the loop footpoints, also
coincident with strong nonthermal velocities. Weaker FIP effect is seen along
the loop connections, suggesting that the FIP effect originates in the
chromosphere, and is communicated to the coronal loop by transport processes.
The active region seen here would then be considered relatively new, since
only weak FIP effect is seen in its coronal connections.

##  4 Stellar FIP and Inverse FIP Effects

The measurement of element abundances in stellar coronae became possible for
the first time with the 1992 June 7 launch of the Extreme Ultraviolet Explorer
(EUVE) satellite. The use of grazing incidence gratings allowed strong lines
in the EUV spectra of stars to be resolved, allowing the acquisition of data
on stellar corona of similar quality to the early solar spectra analyzed by
e.g., Pottasch (1963). The Advanced Satellite for Cosmology and Astrophysics
(ASCA) launched on 1993 February 20 provided X-ray spectroscopy of stellar
coronae with CCD-level spectral resolution. Individual lines could not be
resolved, but the He- and H-like line complexes of different elements could.
Other missions from which a few results emerged were Ginga (launched on 1987
February 5), ROSAT (launched 1990 June 1) and BeppoSAX (launched 1996 April
30) which carried proportional counters for spectroscopy (gas scintillation
proportional counters in the cases of Ginga and BeppoSAX) which allowed
measurements of the He-like Fe complex of lines with respect to the
surrounding continuum. More recently Suzaku (launch 2005 July 10) has also
provided CCD X-ray spectra, although the higher resolution calorimeter failed
soon after launch.

The review of Feldman and Laming (2000) covered the status of this new field
at a time when the results from EUVE and ASCA were available, but before the
launches of Chandra and XMM-Newton (launched 1999 July 23 and 1999 December
10, respectively). These satellites both carry grating instruments as well as
CCD cameras, and so high spectral resolution X-ray spectra of stellar coronae
are acquired with significantly higher throughput than was the case with EUVE
or ASCA. Testa (2010) provided a more updated view of the field, encompassing
a wider variety of stellar targets. Here we will recap the discussion of
Feldman and Laming (2000), and the later reviews of Favata and Micela (2003),
Drake (2003) and Testa (2010), concentrating on stars of varying spectral type
but in other respects similar to the Sun, with a view to the theoretical
discussions to follow in Sections 6 and 7.

Table 2: List of FIP Bias Assessments for Main Sequence Stars (updated from Wood et al., 2012). Star | Alternate | Spectral | Radius | $\log L_{X}$ | $\log F_{X}$ | Ne/Fe | $F_{\mathrm{bias}}$ | Ref.
---|---|---|---|---|---|---|---|---
| Name | Type |  ($R_{\odot}$) |  (ergs s-1) |  ergs cm-2 s-1 |  |  |
$\pi^{3}$ Ori | … | F6V | 2.05 | 28.96 | 5.55 | 2.00 | $-0.55$ | Wood and Laming (2013)
$\beta$ Com | HD 114710 | G0 V | 1.08 | 28.21 | 5.36 | 0.73 | $-0.668$ | Telleschi et al. (2005)
$\pi^{1}$ UMa | HD 72905 | G1 V | 0.91 | 28.97 | 6.27 | 1.60 | $-0.645$ | Telleschi et al. (2005)
$\chi^{1}$ Ori | HD 39587 | G1 V | 0.98 | 28.99 | 6.22 | 1.91 | $-0.555$ | Telleschi et al. (2005)
Sun | … | G2 V | 1.00 | 27.35 | 4.57 | 1.02 | $-0.600$ | Feldman and Laming (2000)
$\alpha$ Cen A | HD 128620 | G2 V | 1.22 | 27.00 | 4.04 | 1.41 | $-0.410$ | Raassen et al. (2003b)
$\kappa$ Ceti | HD 20630 | G5 V | 0.98 | 28.79 | 6.02 | 2.51 | $-0.462$ | Telleschi et al. (2005)
$\xi$ Boo A | HD 131156A | G8 V | 0.83 | 28.86 | 6.24 | 3.80 | $-0.344$ | Wood and Linsky (2010)
70 Oph A | HD 165341A | K0 V | 0.85 | 28.27 | 5.63 | 2.40 | $-0.403$ | Wood and Linsky (2006)
36 Oph A | HD 155886 | K1 V | 0.69 | 28.10 | 5.64 | 4.90 | $-0.250$ | Wood and Linsky (2006)
36 Oph B | HD 155885 | K1 V | 0.59 | 27.96 | 5.63 | 3.80 | $-0.328$ | Wood and Linsky (2006)
$\alpha$ Cen B | HD 128621 | K1 V | 0.86 | 27.60 | 4.95 | 1.44 | $-0.478$ | Raassen et al. (2003b)
$\epsilon$ Eri | HD 22049 | K2 V | 0.78 | 28.32 | 5.75 | 4.68 | $-0.050$ | Wood and Linsky (2006)
$\xi$ Boo B | HD 131156B | K4 V | 0.61 | 27.97 | 5.62 | 7.41 | $-0.185$ | Wood and Linsky (2010)
70 Oph B | HD 165341B | K5 V | 0.66 | 28.09 | 5.67 | 8.71 | $0.138$ | Wood and Linsky (2006)
GJ 338AB | … | M0 V+M0 V | 0.59+0.58 | 27.92 | 5.30 | … | $0.305$ | Wood et al. (2012)
EQ Peg A | GJ 896A | M3.5 V | 0.35 | 28.71 | 6.84 | 16.12 | $0.450$ | Liefke et al. (2008)
EV Lac | GJ 873 | M3.5 V | 0.30 | 28.99 | 7.25 | 13.92 | $0.474$ | Liefke et al. (2008)
EQ Peg B | GJ 896B | M4.5 V | 0.25 | 27.89 | 6.31 | 14.67 | $0.417$ | Liefke et al. (2008)
AD Leo | GJ 388 | M4.5 V | 0.38 | 28.80 | 6.86 | 18.25 | $0.536$ | Liefke et al. (2008)
Proxima Cen | GJ 551 | M5.5 V | 0.15 | 27.22 | 6.08 | 11.98 | $0.471$ | Liefke et al. (2008)
Very Active Star Sample ($\log L_{X}>29$)
EK Dra | HD 129333 | G1.5 V | 0.89 | 29.93 | 7.25 | 3.71 | $-0.277$ | Telleschi et al. (2005)
AB Dor | HD 36705 | K0 V | 0.79 | 30.06 | 7.48 | 17.11 | $0.488$ | Güdel et al. (2001a)
AU Mic | HD 197481 | M1 V | 0.61 | 29.62 | 7.27 | 24.30 | $0.695$ | Liefke et al. (2008)

Wood and Linsky (2010) conducted a survey of stellar FIP effects restricted to
stars with X-ray luminosities less than $10^{29}$ erg s-1, with results shown
in Figure 4 (adapted from Figure 9 of their paper, also including points for
$\alpha$Cen B, $\pi^{3}$ Ori and GJ338). By excluding the most active stars, a
clear trend of decreasing stellar FIP effect with later spectral type is
uncovered. The Sun is at the bottom left of Figure 4, with a logarithmic FIP
bias $-0.6$ (expressed as
$\log\left(X/H\right)_{\mathrm{phot}}-\log\left(X/H\right)_{\mathrm{cor}}$, so
that this is a coronal low FIP enhancement of $10^{0.6}\simeq 4$), along with
other stars of similar spectral type ($\pi^{1}$UMa (G1V) and $\chi^{1}$Ori
(G0V); Güdel et al., 2002). The degree of fractionation diminishes as one
moves to later spectral type, becoming zero at about K5, and an inverse FIP
effect is observed in the M stars. The magnetic fields of a sunspot umbra and
penumbra are likely to be similar to those found more ubiquitously in the
atmospheres of later type (i.e. M) stars (see Donati and Landstreet, 2009), so
the results of Feldman and Widing (1990) and Phillips et al. (1994) appear
consistent with these stellar results. At the extreme left hand side, the
result for $\pi^{3}$ Ori suggests that the FIP effect saturates at a value
similar to that found in the Sun, a low FIP enhancement of about 4, and does
not continue increasing at spectral types earlier than this. The properties of
the various stars are summarized in Table 2. Wood et al. (2012) show that the
sample of T Tauri stars from Güdel et al. (2007) show a similar trend of Ne/Fe
as in Figure 4, but generally at higher FIP bias (i.e., stronger Inverse FIP
effect). Other more active stars (X-ray luminosities larger than $10^{29}$ erg
s-1) also show generally larger inverse FIP than those in Figure 4. We do not
consider these further because of complications to the stellar physics
introduced by fast rotation and tidal interactions in close binaries that are
typical of these types of stars.

Figure 4: Survey of FIP fractionation
($\log\left(X/H\right)_{\mathrm{phot}}-\log\left(X/H\right)_{\mathrm{cor}}$)
observed in a sample of dwarf stars, updated from Wood and Linsky (2010),
where values below zero indicate a solar-like FIP effect and values above zero
indicate an inverse FIP effect. Points for GJ338 and $\pi^{3}$ Ori have been
added from Wood et al. (2012) and Wood and Laming (2013), respectively, and
$\alpha$Cen B from Drake et al. (1997). Diamonds indicate measurements from
Telleschi et al. (2005), triangles from Liefke et al. (2008), with a solar
value from Feldman and Laming (2000). For all GK stars the FIP bias
calculations include corrections for stellar photospheric abundances from
Allende Prieto et al. (2004), but for the M stars there are in general no
stellar photospheric measurements available so we have to simply assume solar
photospheric abundance apply. Among these, EV Lac is the strongest case, due
to chromospheric evaporation of photospheric abundance material observed
during flares (Laming and Hwang, 2009, and references therein). For the
purposes of this figure, we avoid extremes of stellar activity, confining our
attention to stars with $\log L_{X}\leq 29$. Reproduced by permission of the
AAS.

Anticipating a connection between the coronal abundance anomalies, and the
chromospheric wave field, arising either from waves produced in the corona and
propagating down, or as a result of helioseismic or asteroseismic p-modes, we
make the following comments about some of the important stars in Figure 4.

$\alpha$ Cen AB: This inactive G2 V+K1 V binary is not in the original sample
of Wood and Linsky (2010), but it is of prime interest since its fundamental
parameters (photospheric abundances, $p$-mode frequencies, mass, radius,
surface gravity, and effective temperatures) are very well known for both
stars (e.g., Bedding et al., 2004; Butler et al., 2004; Kjeldsen et al., 2005;
Porto de Mello et al., 2008; Bruntt et al., 2010; Karoff et al., 2007; Chaplin
et al., 2009). Coronal abundances were first measured by Drake et al. (1997)
from EUVE data, yielding a FIP effect of about a factor of two in the
unresolved binary. Subsequent observations determined that $\alpha$ Cen B is
the dominant coronal source, which places it in its spot in Figure 1. There
are many observations by both XMM and Chandra (e.g., Raassen et al., 2003b;
Liefke and Schmitt, 2006), and model chromospheres of both stars have been
developed (Ayres et al., 1976; Jordan et al., 1987; Vieytes et al., 2009). Fe
XII 1242 Å linewidths are available in Ayres et al. (2003).

$\epsilon$ Eri: Laming et al. (1996) first measured coronal abundances for
this moderately active K2 V star with EUVE. More recent spectra from XMM and
Chandra have been extensively analyzed as well (Sanz-Forcada et al., 2004;
Wood and Linsky, 2006; Ness and Jordan, 2008). Drake and Smith (1993) give
fundamental parameters (see also Table 2 in Vieytes et al., 2009). Only
theoretical estimates of $\epsilon$ Eri’s $p$-mode spectra are available (Gai
et al., 2008). Several sources provide model chromospheres (Jordan et al.,
1987; Sim and Jordan, 2005; Vieytes et al., 2009). Ayres et al. (2003) give
the Fe XII 1242 Å width from which coronal turbulence may be deduced.

$\xi$ Boo A: This is a G8 V dwarf with coronal abundances first measured with
EUVE by Laming and Drake (1999) and Drake and Kashyap (2001), and later with
Chandra (Wood and Linsky, 2010). Model chromospheres are available from Kelch
et al. (1979) and Jordan et al. (1987).

70 Oph A: This K0 V dwarf with an X-ray spectrum was analyzed by Wood and
Linsky (2010). Its fundamental parameters are given by Bruntt et al. (2010),
including $p$-mode oscillation frequencies (Carrier and Eggenberger, 2006;
Eggenberger et al., 2008). No empirical model chromosphere is available for
this star, but there is a theoretical one from Schmitz and Ulmschneider
(1980). Further guidance may come from the K dwarf model chromospheres of
Vieytes et al. (2009). It is also included in the survey of forbidden line
observations by Ayres et al. (2003).

M Dwarfs: The cluster of M dwarfs at top right in Figure 1 are studied by
Liefke et al. (2008). A model chromosphere for AD Leo is given by Fuhrmeister
et al. (2005), and a grid of model chromospheres for M1 dwarfs is given by
Houdebine and Stempels (1997). Ayres et al. (2003) give linewidths for the Fe
XII 1242 Å and Fe XXI 1354 Å forbidden lines. EV Lac is most recently studied
by Laming and Hwang (2009) using Suzaku observations before and during a
flare. It exhibits abundance change during flares; the inverse FIP quiescent
corona giving way to a more “normal” (i.e., solar photospheric) abundance
pattern. This is interpreted as evidence of chromospheric evaporation, where
unfractionated plasma is evaporated up into a flaring loop, lending more
confidence to the inverse FIP interpretation.

Favata and Micela (2003) and Sanz-Forcada et al. (2004) caution that for
several stars with apparent metal depletion in their coronae, determined from
the Fe/H ratio, the coronal abundances merely reflect metal poor photospheres.
In Figure 4, AD Leo maybe such a case (Jones et al., 1996), although it
clearly shows a nonsolar Ne/Fe abundance ratio. However metal depleted
coronae, or inverse FIP effect, do appear to clearly exist in some cases,
e.g., II Peg (K2 IV plus an unseen companion; Huenemoerder et al., 2001), AR
Lac (G and K subgiants in a 1.98 day orbit; Huenemoerder et al., 2003) and AB
Dor (K2 IV–V with a 0.515 day spin period; Sanz-Forcada et al., 2003).
Further, the variation of element abundance during stellar flares, in which
initially metal depleted plasma evolves towards the standard composition,
interpreted in terms of the chromospheric evaporation of unfractionated plasma
to the coronal flare site, seems to require the existence of such abundance
anomalies. Such phenomena are observed in HR 1099 (K1 IV and G5 IV; Audard et
al., 2001), Algol (B8 V and K2 IV; Favata and Schmitt, 1999), and UX Ari (G5 V
and K0 IV; Güdel et al., 1999) where the later type subgiant is taken to be
the main source of coronal emission, as well as AB Dor (Güdel et al., 2001b),
YY Gem (dMe and dMe with 0.814 day orbit; Güdel et al., 2001a), II Peg (Mewe
et al., 1997), AT Mic (dM4.5 and dM4.5; Raassen et al., 2003a), and EV Lac
(dMe3.5e; Laming and Hwang, 2009). Many of these more active stars show
stronger Inverse FIP effects than shown in Figure 4.

Not included in Figure 4, but also of interest for having neither a solar-like
FIP effect nor an inverse FIP effect is Procyon. Coronal abundances measured
with EUVE, Chandra and XMM (Drake et al., 1995b, a; Raassen et al., 2002;
Sanz-Forcada et al., 2004). At spectral type F5 does not follow the trend of
Figure 1 (it is, however, a subgiant, not a solar-like dwarf star). Extensive
asteroseismological observations are available for Procyon (e.g., Mosser et
al., 2008; Leccia et al., 2007). In fact, its $p$-mode lifetimes are known to
be significantly shorter than those of the Sun (Bedding et al., 2010).
Chromospheric models are available for us to use for this well-studied star
(Ayres et al., 1974; Evans et al., 1975; Brown and Jordan, 1981).

Another interesting case is $\tau$ Bootis (Maggio et al., 2011). It is a F7 V
dwarf, with a M2 V companion. Following the assumption of Maggio et al. (2011)
that the F7 V star is responsible for the X-ray emission, it is also
discrepant from the trend in Figure 4, with $F_{\mathrm{bias}}=-0.17$ at
spectral type F7. $\tau$ Bootis A has already attracted interest because it
hosts a close-in giant planet ($\tau$ Boo b: $P_{\mathrm{rot}}=3.31$ days,
$M\sin i=3.9M_{J}$) Butler et al., 1997). A star-planet interaction has been
noted, in that the planet appears to be inducing an active region on the star,
that leads the planet by $\sim 70^{\circ}$ in longitude (Shkolnik et al.,
2008; Walker et al., 2008), and this opens the possibility of the planet also
affecting the coronal abundances, due to waves induced in the stellar
atmosphere. Another possibility of course is that the XMM observation is
contaminated by emission from the M2 companion. Another curiosity is that the
photospheric abundances of $\tau$ Boo A do not agree with the trend reported
by Lopes and Silk (2013), being of order 0.2 dex higher than solar values
(Maggio et al., 2011). $\epsilon$ Eridani also hosts a Jupiter mass planet,
but shows no coronal abundance discrepancy from the trend in Figure 4.

##  5 Early Theoretical Models: Overview

###  5.1 Diffusion Models and Variations

The earliest attempts to explain that FIP effect invoked various processes
like thermal and ambipolar diffusion (in a stationary atmosphere) or
inefficient Coulomb drag (in a moving case). Thermal diffusion arises in a
temperature gradient, with minor ions diffusing towards higher temperatures.
This happens because collision cross sections decrease with increasing
collision energy, and so the force from the hotter particles from the
direction in which the temperature is rising is lower than than from the
colder direction. This obviously will select ions over neutrals, and
accelerate them towards the corona. However such a process is inherently slow
(Hénoux, 1995, 1998), and requires static conditions for periods ranging from
tens of hours (Hansteen et al., 1994) to days or weeks (Killie and Lie-
Svendsen, 2007). This appears increasingly at odds with the modern view of the
solar chromosphere as a dynamic environment, continually being perturbed by
the passage of shocks (Carlsson and Stein, 2002) or reconnection at
chromospheric layers (Isobe et al., 2008). It also conflicts with the argument
that in the absence of mass supply from below, the solar wind would empty the
corona within 1-2 days. Clearly, any fractionation mechanism that takes longer
than this to change coronal abundances cannot be right. Ambipolar diffusion
refers to the diffusion of neutrals along an ionization balance gradient, and
must also be considered in such models, though by itself does not appear
capable of causing FIP fractionation. Hansteen et al. (1997) also demonstrate
that chromospheric mixing, in their case between hydrogen and helium, is also
necessary to prevent gravitational settling of helium in the chromosphere and
to give a realistic abundance of helium in the solar wind.

In an attempt to speed up the fractionation process by thermal means, various
authors have modeled the separation of elements entrained in a flow of neutral
hydrogen and protons. von Steiger and Geiss (1989) considered such a flow
driven across magnetic field lines, either a horizontal flow across vertical
field lines, where gravity plays no role in the fractionation, or a vertical
flow driven across horizontal field lines, where it does. FIP fractionations
matching observations reasonably well are achieved, but the models themselves
must be considered highly idealized and unlikely to represent the real sun. No
mechanism is suggested to produce such a cross-field flow. In a treatment of
fractionation in a (more plausible) vertical flow along vertical field lines,
Marsch et al. (1995) solve diffusion equations for ions and neutrals
separately in the background flow. These equations only include
photoionization of neutrals. No recombination of ions and electrons is
accounted for (their equations 6 and 9). At the lower boundary the gas is
assumed completely neutral, with density $n\left(s=0\right)=n_{0}$. The
neutral density gradient is specified at the upper boundary, taken to be where
the gas is completely ionized, and is accelerated into the solar wind. Here
$dn/ds\left(s=S\right)=0$ and the ion density also $n^{+}\left(s=S\right)=0$.
Finally the ion density gradient at the lower boundary
$dn^{+}/ds\left(s=0\right)=0$. The boundary conditions on $n$ and $n^{+}$ are
justifiable, and do not greatly affect the solutions. Different choices can
lead to the same diffusive fluxes. The choices of boundary conditions for the
density gradients are less clear, and these are crucial to obtaining FIP
fractionation in the model. In fact it is precisely the choice of
$dn^{+}/ds\left(s=0\right)=0$ that specifies the ion flux at the upper
boundary, with value $n_{0}\sqrt{D/\tau}$ where $D=v_{j}^{2}/\nu_{jH}$, the
diffusion coefficient for neutrals of element $j$ in background of neutral H,
in terms of its thermal speed and collision frequency with H, and $\tau$ is
photoionization time for neutral $j$ in the chromospheric radiation field. The
fact that the ion flux at the top of the chromosphere depends ultimately on
the characteristics of neutrals at the bottom through the diffusion equation,
and especially so for the FIP fractionated low FIP elements that in reality
have very small neutral fractions throughout the region of interest, is
indicative of problems. McKenzie et al. (1998) and McKenzie (2000) point out
that in a gas where collisions play the dominant role in coupling species
together, FIP fractionation cannot occur in a one dimensional steady state
model. If all elements enter at the lower boundary with the speed of H, and
leave at the upper boundary with at the proton speed, continuity demands that
no fractionation occur. They also conclude that the boundary condition on the
density gradient at the lower boundary is the cause of the fractionation in
this model, and that such fractionation must have taken place below the lower
boundary. Indeed, equation 71 of Marsch et al. (1995) giving the upper
boundary condition on the exiting ion fluxes suggests that the FIP
fractionation is already embodied in the inputs to the model, rendering the
model explanation spurious.

Fractionation in such a class of models is not completely ruled out. If the
chromosphere can respond to the enhanced coupling of ions to protons in the
upper chromosphere (compared to that of neutrals) by allowing diffusion to
supply ions from below at a greater rate, i.e. $dn/ds$ has to vary with
species. But this reliance on diffusion places a limit on the speed with which
abundance modification may take place. There are some newer models (Pucci et
al., 2010) investigating the changes in abundances with hydrogen flux through
a chromospheric magnetic funnel, but this requires significant fine tuning for
each element (Pucci et al., 2010, consider O, Ne, and Fe), and appears
unlikely to be able to get every element right at the same time, even if the
atmosphere was sufficiently quiescent (see also Byhring et al., 2011; Byhring,
2011). Bøet al. (2013) make the point that above the chromospheric temperature
minimum, the atmosphere is convectively stable, but ignore the fact that waves
from the convectively unstable regions may continue to propagate upwards to
perturb higher altitudes, as in e.g. Heggland et al. (2011). They consider the
effect of gravitational settling within the chromosphere as a means of
fractionation among O, Ne, S, and Fe. The mechanism only appears capable of
producing a depletion of high FIP elements (no enhancement of low FIPs as
observed), and then only with a static atmosphere. An episodic upflow to
supply the corona, punctuated by periods of stasis to allow the fractionation
to occur. Both Pucci et al. (2010) and Bøet al. (2013) use somewhat
unrealistic chromospheric models, in that the degree of hydrogen ionization
for a given density is lower than that in Avrett and Loeser (2008), at least
for the models which come closest to matching observed abundance anomalies.
Although the Avrett and Loeser (2008) model is quasi-static designed to match
spectroscopic observations, it matches quite well with average chromospheric
structures modeled in Heggland et al. (2011), as does its antecedent (Vernazza
et al., 1981, VALC) compared with Carlsson and Stein (2002), at least in terms
of electron density.

The helium depletion explained by such processes (Byhring, 2011) is also at
odds with the observation that it (and the other minor ions) flow faster than
hydrogen in the solar wind at 1 AU (Neugebauer et al., 1996; Kasper et al.,
2008; Bourouaine et al., 2011). A further stage of solar wind acceleration
must set in at a level above that where the He depletion sets in (Wang, 2008).
Noci et al. (1997), following Geiss et al. (1970), make the point that if such
processes were the origin of the solar wind He abundance depletion, other
heavy ions should also be depleted, which is now known not to be the case.
Bochsler (2007b) revisits this, comparing He, O, and Ne for which He appears
most affected by gravitational settling, or inefficient Coulomb drag, in the
corona. In this case the He abundance should correlate with the proton flux,
which is not observed (Wang, 2008). However all these elements appear also to
vary with respect to H, leaving open the possibility that quasi-thermal or
diffusive processes play some role in establishing the absolute coronal
abundances.

###  5.2 Thermoelectric Driving

Perhaps the best early model was that of Antiochos (1994). Cross B diffusion
of chromospheric ions into a flux tube by a thermoelectric force associated
with downward heat conducting electrons from a closed loop enhances the loop
footpoint abundances of ions, but not neutrals. Given $\nabla\times{\bf E}=0$
in steady state conditions, the transverse gradient of the longitudinal
electric field (due to the plasma resistivity) must be balanced by a
longitudinal gradient of the transverse electric field. This transverse
electric field points into the flux tube, thus concentrating ions therein. The
predicted fractionation pattern is proportional to ion mass1/2 (it is
independent of mass for the ponderomotive force). In a sense, this is
conceptually very similar to the ponderomotive force model, in that it is the
loop’s response to coronal heating that causes the fractionation, except here
it is mediated by heat conduction rather than Alfvén waves. The main problem
is that heat must be conducted down to regions of the chromosphere without
transverse magnetic structuring (there must be plasma surrounding the flux
tube), and this appears unlikely. It is unclear where in the chromosphere
horizontal structuring sets in, but a natural place to expect it is likely to
be the equipartition layer, where the Alfvén speed and sound speed are equal,
roughly where the plasma $\beta\simeq 1$. The stopping distance of 1 keV
electrons is of order 1 km at a density of $10^{12}$ cm-3, so downwards heat
conduction is unlikely to reach this far, and other authors have recently
considered energy transport by Alfvén waves (Fletcher and Hudson, 2008;
Haerendel, 2009). Alfvén waves may still fractionate material in the upper
magnetically structured layers of the chromosphere, because ions move
vertically along the flux tube rather than horizontally across it. Antiochos
(1994) also does not allow for an “Inverse FIP Effect”. Laming and Hwang
(2009) discuss how the ponderomotive force may also inhibit downwards heat
conduction, but large Alfvén wave amplitudes are required.

###  5.3 Chromospheric Reconnection

Arge and Mullan (1998) consider chromospheric reconnection giving ions a
larger density scale height than neutrals. They model this using the Zeus
code, iterating between its hydrodynamic and magnetohydrodynamic
implementations to treat the neutrals and ions respectively, with these two
fluids being coupled by collisions. They only calculate Si and Ne as examples,
so it is not possible to evaluate the full fractionation pattern such a
process would produce. The ion-neutral coupling rate they use appears to be
that due to charge exchange collisions between protons and neutral hydrogen
(their equation 6), and this appears to be applied to all elements, with
possible modifications to the collision velocities. This is not correct, and
the coupling should more properly be described by elastic scattering (cf.
Malyshkin and Zweibel, 2011). In fact, looking at these ion-neutral collision
rates a mass dependent fractionation appears likely, not the FIP effect, which
might be relevant to the observations of Wurz et al. (2000).111 This is
similar to the case in Drake et al. (2009), where pick-up ion behavior leads
to mass dependent fractionation. By contrast Laming (2004a) and Laming (2012)
devoted considerable effort to the correct description of these atomic
processes. Arge and Mullan (1998) also predict FIP effect everywhere, with no
difference between coronal holes and closed loops, unless extra assumptions
are made about the presence or absence of chromospheric reconnection in open
and closed field, and cannot reproduce an “Inverse FIP Effect”.

###  5.4 Ion Cyclotron Wave Heating

Schwadron et al. (1999) offered the first mechanism of fractionation by wave-
particle interactions. Coronal ion cyclotron (IC) waves propagate down and
heat chromospheric ions, not neutrals. The FIP effect arises from the
preferential heating combined with the coupling of the chromospheric minor
ions to a background flow of H atoms and protons. The formalism for this
coupling is elegant, and forms the basis for models invoking the ponderomotive
force to be discussed at greater length elsewhere in this paper.

The fractionation pattern depends on the assumed spectrum of IC waves, since a
resonant wave-particle interaction is assumed. This is not specified in their
paper, except through the wave interaction rate given as
$\nu_{sw}=N_{w}\Omega_{s}$ where $\Omega_{s}$ is the gyrofrequency of ions of
element $s$, and $N_{w}$ is a constant of order 70. Equating this to the pitch
diffusion coefficient, $D=\left(\pi/4\right)\Omega_{s}\left(\delta
B^{2}/B^{2}\right)$, where $\delta B$ is the magnetic field perturbation due
to the wave, and $B$ is the ambient magnetic field, we find $\delta B/B\sim
10$.222Bohm diffusion corresponding to $N_{w}\sim 1$ would give $\delta
B/B\sim 1$. With $B\sim 10$G and a density of order $10^{10}$ cm-3, the wave
velocity amplitude is 2000 km s-1. Of course these estimates should be treated
with caution, because they are derived by applying quasi-linear theory well
beyond its regime of validity, but the essential point that the wave energy
requirements of this model are implausible remains. If the coronal ion
cyclotron wave derived form a turbulent cascade, then even higher wave
amplitudes should be found at lower frequencies, assuming for example a
Kolmogorov spectrum. Loop resonant frequencies are lower than ion cyclotron
frequencies by several orders of magnitude (typically 6-8), leading to a huge
increase in the intensity of these waves if a -5/3 spectrum is assumed.
Further, waves in the ion cyclotron frequency range rapidly damp by charge
exchange reactions (e.g. Kulsrud and Pearce, 1969), with rate 1 - 100 s-1,
depending on the neutral fraction. Thus waves traveling at the chromospheric
Alfvén speed are damped after traveling a few km, increasing the energy
requirements still further.

###  5.5 Summary

To summarize, while some of the models described above might come close to
describing the solar FIP effect, they do so for rather contrived magnetic
field geometries or for various other extreme assumptions. In all cases except
those of Antiochos (1994) and Schwadron et al. (1999), whose mechanisms both
derive from the coronal response to energy deposition, no natural account of
the difference between closed and open field is given. Further, none of the
models appear capable of explaining the Inverse FIP Effect.

Given the relative success of Antiochos (1994) and Schwadron et al. (1999), it
seems reasonable to pursue the byproducts of coronal heating as the key to the
fractionation. Given the high frequency waves do not work due their energy
requirements, and that lower frequency Alfvén waves carry far more of the wave
energy, the next question to ask is how such waves could interact with
chromospheric ions. Clearly a resonant interaction is not possible, but such
waves can interact nonresonantly through the ponderomotive force. This is
discussed more fully in the following sections.

##  6 The Ponderomotive Force Model

###  6.1 Overview

We now turn to a description of our model of the FIP effect, which invokes the
ponderomotive force due to magneto-hydrodynamic waves as the agent that
separates ions from neutrals. Ponderomotive forces in magnetospheric and space
plasmas are reviewed by Lundin and Guglielmi (2006). According to them,
“Ponderomotive forces are time-averaged nonlinear forces acting on media in
the presence of oscillating electromagnetic fields. The word ponderomotive
comes from the Latin words pondus (ponderis) meaning “heaviness” and motor.”
The complex dynamics of a system acting under Lorentz forces may be
considerably simplified by averaging over the period of oscillations and
described instead by ponderomotive forces. They can be seen to arise from the
effects of wave refraction in an inhomogeneous plasma. In a nonmagnetic
plasma, the refractive index, $\sqrt{\epsilon}$, is given by
$\epsilon=1-\omega_{p}^{2}/\omega^{2}$ where $\omega_{p}$ is the plasma
frequency. Waves are refracted to high refractive index, which means low
plasma density. The increased wave pressure can then expel even more plasma
from the low density region, leading to ducting instabilities. In magnetic
plasma, $\epsilon=1-\omega_{p}^{2}/\left(\omega^{2}-\Omega^{2}\right)$ for
linearly polarized parallel propagating transverse waves, where $\Omega$ is
the ion cyclotron frequency. Thus waves with $\omega<<\Omega$ refract to high
density regions, and plasma is attracted to regions of high wave energy
density, especially when the energy and momentum of the waves are large. A
simple expression for the ponderomotive force on an ion is derived below.

The refraction of waves to high density regions, and the corresponding
attraction of ions to locations of high wave energy density when the wave
pressure dominates over the thermal pressure of the ionized component of the
plasma, means that the FIP effect depends crucially on details of wave
propagation through the chromosphere. A high wave energy density in the corona
will lead to a strong FIP effect. Weak coronal waves and strong waves lower
down may lead to an inverse FIP effect, as chromospheric ions are attracted
downwards. We will explore precise conditions under which these various
phenomena may occur in more detail below.

We concentrate solely on the fractionation, referring the reader to the
reviews of Reale (2014) for the properties of plasma in closed coronal loops,
and to Marsch (2006), Cranmer (2009) and Ofman (2010) for the physics of the
acceleration of the fractionated gas into the fast and slow speed solar wind.
In the subsections that follow, we discuss in detail the various components of
the model. Figure 5 illustrates the basic scenario. We consider a coronal loop
meeting the chromosphere at both footpoints. For the purposes of calculating
the Alfvén wave propagation, we ignore the loop curvature. A variety of wave
processes can occur at each footpoint, some illustrated at one and some
illustrated at the other in the figure for clarity. A steady evaporative flow
with speed much less than the local Alfvén speed is assumed to take material
from the chromosphere to the corona. Such a flow arises from the chromospheric
response to coronal heating, as heat is conducted downwards increasing the
chromospheric temperature (e.g Bray et al., 1991). Imada and Zweibel (2012)
calculate evaporative upflows for a variety of coronal heating scenarios, and
find upflows of order one to a few km s-1 in the chromosphere, similar to
previous work (Warren et al., 2002; Bray et al., 1991). We ignore transient
effects, although in a magnetic filament heated episodically, these could play
an important role. For instance, if Alfvén waves released by a coronal heating
event reach the chromosphere before the heat conduction front, the
chromospheric flow velocity under which the fractionation occurs will be much
smaller than these estimates. In the opposite case, of heat conducting down
before the Alfvén waves arrive, no fractionation could occur. More exotic
cases where the Alfvén waves themselves carry the heat that causes the
evaporation, for example in solar flares (Haerendel, 2009) are not considered
here.

In the absence of the ponderomotive force, we assume that the chromosphere is
unfractionated, i.e. that sufficient turbulence exists to inhibit any
gravitational settling of other forms of diffusion that might occur. A similar
approach was taken by Hansteen et al. (1997). We suggest that the reflection
and refraction of acoustic waves at discontinuities associated with
chromospheric shock waves set up the conditions necessary for a turbulence.
Interactions between oppositely directed waves lead to fluctuations of smaller
and smaller size scales, right down to microscopic dimensions where mixing can
occur. Following on from the discussion above, we neglect diffusion processes.
Although the thermal force is expected to develop over a similar region to the
ponderomotive force (the region of strong density and temperature gradients in
the upper chromosphere), the acceleration associated with the thermal force is
of order 1% - 10% of the ponderomotive acceleration, and is negligible. The
thermal acceleration is also dependent on $1/A$, where $A$ is the element
atomic mass, unlike the ponderomotive acceleration which is mass independent,
and can never produce an inverse fractionation as seen for example in the
coronae of M dwarfs.

Figure 5: Schematic diagram of model loop and wave processes, adapted from
(Laming, 2012), which follows Hollweg (1984). All footpoint wave processes may
happen at both footpoints, but are here split between the two for clarity.
Alfvén waves shown as thick solid lines are assumed to be generated inside the
coronal portion of the loop, and to bounce back and forth from the loop
footpoints, with a probability of leaking out and being transmitted deeper
into the chromosphere at each bounce (shown on left hand side). Approximately
equal amplitudes of waves propagating in each direction result. Reflecting
Alfvén waves can also generate slow mode (here labelled as “p-mode”) waves by
a parametric process (shown on the right hand side as thin dashed lines).
Other acoustic waves (“p-modes” in helioseismology parlance) propagating
within the solar envelope can mode convert to fast mode waves upon reaching
the chromospheric layer where the sound speed and Alfvén speed are equal
(approximately where the plasma $\beta=1$). The upgoing fast mode waves (shown
as thin solid lines) are refracted back downwards in the chromospheric region
where the Alfvén speed increases with height. Fast mode wave may also mode
convert to Alfvén waves and then propagate up to the loop to be reflected or
transmitted, depending on the match between their frequency and the loop
resonance. This provides an alternative means of seeding the loop with
propagating Alfvén waves. Our non-WKB wave propagation calculations in fact
follow an Alfvén wave being injected at one loop footpoint, being reflected or
transmitted into the corona, and the successive bounces it undergoes. For
waves at the loop resonance, this behaviour (without growth or damping
included) is indistinguishable from waves generated within the loop itself,
necessarily at the loop resonance or its harmonics. Reproduced by permission
of the AAS.

###  6.2 The ponderomotive force

An expression for the ponderomotive force is derived as follows, updated from
Appendix A in Laming (2009). Consider the Lagrangian for a system of $n$
particles with mass $m$ and electromagnetic waves in a box of unit volume

$L=\sum_{i}{1\over 2}m_{i}\left({\bf v_{th,i}}+\delta{\bf
v_{i}}\right)^{2}+\sum_{i}{q_{i}\over c}\left({\bf v}_{th,i}+\delta{\bf
v}_{i}\right)\cdot\delta{\bf A}+{\epsilon\delta{\bf E}^{2}-\delta{\bf
B}^{2}\over 8\pi}$ (1)

where ${\bf v_{th,i}}$ is the thermal speed and $\delta{\bf v_{i}}$ is the
oscillatory speed induced by the wave of particle $i$, with mass $m_{i}$, and
charge $q_{i}$. Wave electric and magnetic fields are given by $\delta{\bf E}$
and $\delta{\bf B}$, respectively, $\delta{\bf A}$ is the wave vector
potential, and $c$ is the speed of light. We have chosen the radiation gauge
where the electrostatic potential $\phi=0$. In any case, this is constant in
an electrically neutral plasma (in the absence of electrostatic waves).

For Alfvén waves, energy is partitioned according to $\delta
B^{2}/8\pi=\sum_{i}mv_{\mathrm{osc},i}^{2}/2+\delta E^{2}/8\pi$. Also
$\delta{\bf B}\cdot{\bf B}=0$ where ${\bf B}$ is the ambient magnetic field,
and $\delta{\bf v}_{i}\cdot\delta{\bf A}=0$, not just on average but for all
time. We also take the time average of ${\bf v}_{th,i}\cdot\delta{\bf
v}_{i}=0$ and ${\bf v}_{th,i}\cdot\delta{\bf A}=0$ to find the Lagrangian

$L=\sum_{i}{1\over 2}mv_{th,i}^{2}+{\left(\epsilon-1\right)\delta E^{2}\over
8\pi}\rightarrow\sum_{i}{1\over 2}mv_{th,i}^{2}+\sum_{i}{q_{i}^{2}\delta
E\left(z_{i}\right)^{2}\over 2m_{i}\left(\Omega_{i}^{2}-\omega^{2}\right)},$
(2)

where in the second line we have written
$\epsilon-1=\omega_{p}^{2}/\left(\Omega^{2}-\omega^{2}\right)=4\pi
nq^{2}/m\left(\Omega^{2}-\omega^{2}\right)$ and $\left(\epsilon-1\right)\delta
E^{2}\rightarrow\Sigma_{i}4\pi q_{i}^{2}\delta
E\left(z_{i}\right)^{2}/m_{i}\left(\Omega_{i}^{2}-\omega^{2}\right)$ for
linearly polarized parallel propagating transverse waves. The “$z$”
Euler–Lagrange equation for particle “i” gives

${d\over dt}\left(mv_{th,iz}\right)={q_{i}^{2}\over 2m_{i}}{d\over dz}{\delta
E^{2}\over\left(\Omega_{i}^{2}-\omega^{2}\right)},$ (3)

evaluating for the component of $v_{th,i}$ orthogonal to $\delta{\bf A}$ and
$\delta{\bf v}_{i}$. In uniform magnetic field, this is the same as the
expression derived by Landau et al. (1984), and agrees with earlier work (e.g
Lee and Parks, 1983; Li and Temerin, 1993) if $\delta E^{2}=\delta
E_{p}^{2}/2$, where $\delta E_{p}$ is the peak electric field in the wave,
giving a ponderomotive force

$F_{i}={q_{i}^{2}\over 4m_{i}\left(\Omega_{i}^{2}-\omega^{2}\right)}{d\delta
E_{p}\left(z_{i}\right)^{2}\over dz}.$ (4)

When $\omega\ll\Omega_{i}$, the ponderomotive acceleration is thus independent
of ion mass, which is one crucial property relevant to obtaining an almost
mass independent fractionation as observed. It is also independent of ion
charge, so long as the ion is charged (and not neutral). Litwin and Rosner
(1998) give a similar expression derived from the ${\bf j}\times{\bf B}$ and
other second order terms in the MHD momentum equation. Away from the low
frequency limit, circular polarized waves may give slightly different forces,
with left and right circular polarization being different from each other as
well as linear polarization (e.g., Nekrasov and Feygin, 2013). In nonuniform
${\bf B}$ in the low frequency limit, the ponderomotive force is given by

$F_{i}={m_{i}c^{2}\over 4}{d\over dz}\left[{\delta
E_{p}\left(z_{i}\right)^{2}\over B\left(z_{i}\right)^{2}}\right],$ (5)

which agrees with the first two terms in Equation 2.6 of Lundin and Guglielmi
(2006).

In fast mode waves, energy is now partitioned according to $\delta
B^{2}/8\pi+\delta E_{th}=\sum_{i}mv_{\mathrm{osc},i}^{2}/2+\delta E^{2}/8\pi$,
where $\delta E_{th}$ is the wave thermal energy arising from compression and
rarefaction. For the fast mode wave $\delta{\bf B}\cdot{\bf B}=0$ only on
average, but not instantaneously, though $\delta{\bf v}_{i}\cdot\delta{\bf
A}=0$ remains unchanged. The thermal energy also time averages to zero,
leaving the Lagrangian and ponderomotive force described by the same
expressions as above. However the extra longitudinal pressure associated with
obliquely propagating fast mode waves will reduce the eventual fractionation.
See Sections 6.5 and 6.6 below.

With slow mode waves, energy partitioning is similar to that for fast mode
waves. Whereas Alfvén and fast mode waves always have $\delta{\bf v}\perp{\bf
B}$, slow modes have $\delta{\bf v}\|{\bf k}$. In parallel propagation the
electric field, and hence a ponderomotive force capable of ion-neutral
separation, are absent. Transverse electric field may arise with a slow mode
wave in parallel propagation along an isolated flux tube (a “sausage” mode)
(e.g., Mikhalyaev and Solov’ev, 2005). We do not consider such possibilities
further.

###  6.3 Chromospheric model

The ponderomotive force depends on the gradient of the wave transverse
electric energy, $\partial\delta E_{\perp}^{2}/\partial z$. Since $\delta
E_{\perp}=\delta v_{\perp}B/c$, such a gradient will develop where $\delta
v_{\perp}$ varies in response to varying chromospheric density. Previous
(mainly analytic; e.g. Hollweg, 1984) works on wave propagation through the
chromosphere have approximated the vertical density structure as exponential,
with a scale height of order 200 km. While adequate for studying wave
transmission into the corona, more detail is required for the FIP effect. The
chromosphere is heated, presumably by means similar to those thought to heat
the corona, and cooled in the region where most FIP fractionation occurs
principally by radiation in H Lyman $\alpha$. In this region where H starts to
become ionized, its radiative cooling becomes increasingly inhibited and the
temperature must rise. Correspondingly, the density drops, and the density
gradient so produced is steeper than the typical hydrostatic scale height
elsewhere. Consequently, the region of the chromosphere where the FIP
fractionation has long been assumed to occur, i.e. the region where H becomes
ionized, is highly likely to be a region with a strong density gradient due to
the physics of radiative cooling, and hence a location where the ponderomotive
force will be strongest. Figure 6 shows this structure in the left panel,
taken from the empirical model (C7) of Avrett and Loeser (2008), an update to
the older VALC model (Vernazza et al., 1981).

Figure 6: _Left:_ Empirical model of the solar chromosphere plotted from data
given in Avrett and Loeser (2008). The density is the solid black line, to be
read on the left axis, while the temperature is the dashed gray line, to be
read on the right axis. The ordinate (“distance along the loop”) is the
altitude above the photosphere. The steep density gradient at 2150 km altitude
is where strong ponderomotive force can develop. _Right:_ Force-free magnetic
field structure calculated in Rakowski and Laming (2012) following Athay
(1981). Altitude is given by the $y$-axis. The $x$-axis give lateral
expansion. The dashed lines show contours of the Alfvén speed. In this plot
the plasma $\beta=1$ layer is at an altitude of approximately 650 km in the
Sun. Reproduced by permission of the AAS.

The right panel of Figure 6 shows the model magnetic field. Altitude here is
given by the $y$-axis. The $x$-axis gives lateral expansion on the same scale.
The dashed lines show contours of the Alfvén speed. The magnetic field emerges
from the photosphere in tight fibrils. At the atmospheric layer, typically in
the low chromosphere, where gas pressure and magnetic pressure are equal, the
field begins to expand and fill the whole volume. In Figure 6 (right panel)
this corresponds to $y=650$ km approximately. In the real Sun, this occurs at
altitudes typically between 400 and 800 km, depending on the magnetic field
strength. The total magnetic field expansion from $y=600$ km upwards in this
model is a factor of 5, similar to that suggested by Gary (2001).

Observations of hard X-ray bremstrahlung during flares with RHESSI (Lin et
al., 2004) corroborate many of these features (Kontar et al., 2008; Saint-
Hilaire et al., 2010). An average scale density scale height of 130 – 140 km
is derived. Kontar et al. (2008) infer chromospheric flux tube expanding in
radius by a factor of 3 at an altitude between 900 km and 1200 km. This would
imply magnetic field decreasing by a factor of 9, albeit at a higher height
than suggested above.

Avrett and Loeser (2008) also give an empirical electron density coming from
the ionization balance for H, from which we calculate the ionization balance
for all other elements of interest. Collisional processes (ionization,
radiative and dielectronic recombination) are all taken from the tabulations
provided by Mazzotta et al. (1998). Subsequent refinements to the ionization
and dielectronic recombination rates as implemented by Bryans et al. (2009)
are not crucial here, since our concern is only with neutrals and singly
ionized species, but the suppression of dielectronic recombination by finite
density effects (Nikolić et al., 2013) is important and is now included as an
update to previous calculations (Laming, 2009, 2012). We also include charge
exchange ionization using rates given by Kingdon and Ferland (1996), updated
and amended as detailed in Laming (2012). This treats the capture by a free
proton of an electron from an initially neutral atom. It is most important for
O, and helps lock the O ionization balance to that of H, but can be of
significance for other low FIP elements as well. In the Avrett and Loeser
(2008) model, H begins to become ionized at heights between 1000 – 1500 km,
reaching about 50% ionization at 2000 km. For comparison, Saint-Hilaire et al.
(2010) infer a unit step change in ionization level at a height of $1.3\pm
0.2$ Mm.

We evaluate photoionization rates using incident spectra based on Vernazza and
Reeves (1978) with the extensions and modifications outlined in Laming
(2004a). In most cases, the “quiet region” spectrum is used. An additional
contribution from trapped Lyman $\alpha$ in regions where this line is
optically thick is also included, with approximate flux $\int
n_{\mathrm{H}}\,dlAC_{ex}/\left(A+\tau C_{de-ex}\right)$ where $\int
n_{\mathrm{H}}\,dl$ is the column density of H atoms, $\tau=\sigma\int
n_{\mathrm{H}}\,dl$ is the opacity in terms of the absorption cross section
$\sigma$ at line centre, $A$ is the upper level decay rate, and $C_{ex}$ and
$C_{de-ex}$ are the collisional excitation and de-excitation rates for the
upper level. We take photoionization cross sections from the compilation of
Verner et al. (1996). Our ionization balance so computed is necessarily a
steady state model, but is based on the empirical electron density found by
Avrett and Loeser (2008), which is higher than that which we would compute
based on their inferred chromospheric temperature. Carlsson and Stein (2002)
study the effects of dynamic ionization of H, and find that the passage of
shock waves through the chromosphere does indeed increase the electron density
over that coming from simple photoionization-recombination equilibrium for H.
Individual photoionization and recombination rates however are too slow for
the ionization balance to respond to individual shock waves. The elevated
electron density instead represents a time averaged response to the dynamics,
and is itself quasi-steady state. Wedemeyer-Böhm and Carlsson (2011) consider
the Ca+ to Ca2+ ionization balance. It is more variable than that between H
and H+, but is of less concern to us, since the ponderomotive force
experienced by an ion is independent of ion charge, so long as the wave
frequency is much lower than the ion gyrofrequency.

In isolation, the action of the ponderomotive force would produce a local
abundance variation close to the region where it is strongest, around
chromospheric altitudes of 2150 km. A region where low FIPs are depleted by
the upwards acceleration would sit below this altitude, and above it a region
of FIP enhancement. A coronal abundance anomaly would not result without
either a flow through the chromosphere, or diffusion (in a static case). The
typical timescales associated with such processes would be $H_{D}/v\sim
2\times 10^{7}/10^{3}\sim 2\times 10^{4}$ s where $H_{D}$ is the density scale
height and $v$ the flow velocity lower down in the chromosphere, or
$H_{D}^{2}/D\sim 4\times 10^{14}/6\times 10^{9}\sim 6\times 10^{4}$ s where
$D$ is the diffusion coefficient for low FIP ions in a neutral H atmosphere.
Both timescales evaluate to several hours to days, similar to that observed
(Widing and Feldman, 2001), as do similar timescales evaluated for the coronal
loop. The chromospheric response may be altered by the mixing induced by
turbulence associated with shocks in the lower chromosphere (e.g., Reardon et
al., 2008), and so the coronal diffusion is likely to be the controlling
timescale. We do not consider such processes further, restricting ourselves to
a steady state chromosphere and upward flow speed, and emphasizing again that
collisional coupling in the chromosphere is crucial to the coronal abundance
anomaly, since it is the source for the extra low FIP ions. Once collisionless
(in the solar wind) no further bulk fractionation can occur. A local FIP
enhancement must be accompanied by another local depletion, since the
collisional coupling to the lower solar atmosphere has been lost.

###  6.4 The Alfvén wave transport equations

Alfvén wave propagation is often treated in the Wentzel-Kramers-Brillouin
(WKB) approximation, where it is assumed that the wavelength is much shorter
than the typical length scale of inhomogeneities, and hence that the effects
of reflection and refraction can be neglected. This is emphatically not the
case in the solar chromosphere. Since the ponderomotive force depends on the
spatial variation of the Alfvén wave electric field in the inhomogeneous
plasma of the chromosphere, a direct integration of the transport equations is
required. We evaluate this spatial variation paraphrasing and updating the
treatment in Appendix B of Laming (2009). We start from the linearized MHD
force and induction equations,

$\rho{\partial\delta{\bf v}\over\partial t}+\rho\left({\bf
u}\cdot\nabla\right)\delta{\bf v}+\left(\delta{\bf
v}\cdot\nabla\right)\rho{\bf u}={\left(\nabla\times\delta{\bf
B}\right)\times{\bf B}\over 4\pi}={\left({\bf B}\cdot\nabla\right)\delta{\bf
B}-\left(\nabla\delta{\bf B}\right)\cdot{\bf B}\over 4\pi},$ (6)

and

$\displaystyle{\partial\delta{\bf B}\over\partial t}$ $\displaystyle=$
$\displaystyle\nabla\times\left(\delta{\bf v}\times{\bf
B}\right)+\nabla\times\left({\bf u}\times\delta{\bf B}\right)$ (7)
$\displaystyle=$ $\displaystyle-{\bf B}\nabla\cdot\delta{\bf
v}-\left(\delta{\bf v}\cdot\nabla\right){\bf B}+\left({\bf
B}\cdot\nabla\right)\delta{\bf v}-\delta{\bf B}\nabla\cdot{\bf u}-\left({\bf
u}\cdot\nabla\right)\delta{\bf B}+\left(\delta{\bf B}\cdot\nabla\right){\bf
u}\,,$

where ${\bf u}$ and ${\bf B}$ are the unperturbed velocity and magnetic field,
$\delta{\bf v}$ and $\delta{\bf B}$ are the perturbations, and $\rho$ is the
density. Equation (6) is rewritten using $\left(\nabla\delta{\bf
B}\right)\cdot{\bf B}=\nabla\left({\bf B}\cdot\delta{\bf
B}\right)-\left(\nabla{\bf B}\right)\cdot\delta{\bf B}$ to yield

${\partial\delta{\bf v}\over\partial t}+\left({\bf
u}\cdot\nabla\right)\delta{\bf v}={\bf V}_{A}\cdot\nabla\left(\delta{\bf
B}\over\sqrt{4\pi\rho}\right)+{\delta{\bf B}\over\sqrt{4\pi\rho}}{{\bf
V}_{A}\cdot\nabla\rho\over 2\rho}+{\left(\nabla{\bf B}\right)\cdot\delta{\bf
B}\over 4\pi\rho}-{\left(\delta{\bf v}\cdot\nabla\right)\left(\rho{\bf
u}\right)\over\rho}\,,$ (8)

where ${\bf V}_{A}={\bf B}/\sqrt{4\pi\rho}$ is the Alfvén velocity, and we
have assumed ${\bf B}\cdot\delta{\bf B}=0$ for Alfvén waves and parallel
propagating fast modes.

Specializing to plane Alfvén waves, we write $\left(\nabla{\bf
B}\right)\cdot\delta{\bf B}=\left(\partial B_{x}/\partial x\right)\delta{\bf
B}=-\left(\partial B_{z}/\partial z\right)\delta{\bf B}/2$ since
$\nabla\cdot{\bf B}=0$ (assuming $\partial B_{x}/\partial x=\partial
B_{y}/\partial y$ in cylindrical symmetry, with $\delta{\bf B}=\delta
B{\bf\hat{x}}$ where ${\bf\hat{x}}$ is a unit vector along the x-axis).
Similarly, $\left(\delta{\bf v}\cdot\nabla\right)\left(\rho{\bf
u}\right)=\delta v_{x}\partial\left(\rho u_{x}\right)/\partial x=-\delta
v_{x}\partial\left(\rho u_{z}\right)/\partial z/2$ since
$\nabla\cdot\left(\rho{\bf u}\right)=0$, and using $\partial\left(\rho
u_{z}/B_{z}\right)/\partial z=0$ gives

${\partial\delta{\bf v}\over\partial t}+\left({\bf
u}\cdot\nabla\right)\delta{\bf v}={\bf V}_{A}\cdot\nabla\left(\delta{\bf
B}\over\sqrt{4\pi\rho}\right)+{\delta{\bf B}\over\sqrt{4\pi\rho}}{V_{A}\over
2H_{D}}-{\delta{\bf B}\over\sqrt{4\pi\rho}}{V_{A}\over 2H_{B}}+\delta{\bf
v}{u\over 2H_{B}}\,.$ (9)

Here $1/H_{B}=\partial\ln B_{z}/\partial z$, $1/H_{D}=\partial\ln\rho/\partial
z$, and below $1/H_{A}=\partial\ln V_{A}/\partial z$.

Similar manipulations using $\nabla\cdot{\bf u}=-{\bf
u}\cdot\nabla\rho/\rho=-u/H_{D}$ and assuming $\nabla\cdot\delta{\bf v}=0$
give the induction equation in the form

${\partial\over\partial t}\left(\delta{\bf
B}\over\sqrt{4\pi\rho}\right)+\left({\bf u}\cdot\nabla\right){\delta{\bf
B}\over\sqrt{4\pi\rho}}=\left({\bf V}_{A}\cdot\nabla\right)\delta{\bf
v}+{\delta{\bf B}\over\sqrt{4\pi\rho}}{u\over 2H_{D}}+\delta{\bf v}{V_{A}\over
2H_{B}}-{\delta{\bf B}\over\sqrt{4\pi\rho}}{u\over 2H_{B}}\,.$ (10)

Taking Equation (9) plus or minus Equation (10) and rearranging gives the
final result,

${\partial I_{\pm}\over\partial t}+\left(u\mp V_{A}\right){\partial
I_{\pm}\over\partial z}=\left(u\pm V_{A}\right)\left({I_{\pm}\over
4H_{D}}+{I_{\mp}\over 2H_{A}}\right),$ (11)

where $I_{\pm}=\delta{\bf v}\pm\delta{\bf B}/\sqrt{4\pi\rho}$, representing
waves propagating in the $\mp$ z-directions. We have focussed on plane
polarized Alfvén waves. Clearly circularly polarized Alfvén waves will obey
the same transport equations, as do torsional waves in cylindrical
coordinates. These different wave polarizations may still produce subtly
different fractionations, due to their different couplings to other wave
modes. The case of parametric generation of slow mode waves is discussed
below.

Such transport equations have been presented in several different forms by
various authors. Cranmer and van Ballegooijen (2005) review these in their
Appendix B and demonstrate their equivalence. We have assumed a cylindrically
symmetrical magnetic field above. One could easily generalize this treatment
to incorporate Alfvén wave transport on arbitrary (i.e., observed or
extrapolated) magnetic field lines for a more realistic description of
observations. We have also restricted treatment to field aligned wave
propagation, rendering the waves incompressible. For plane and circularly
polarized waves this amounts to assuming that the wave follows the field line
on which it propagates. For torsional waves on a flux tube, it requires that
the flux tube magnetic field have no $B_{\phi}$ component. In the case that
this is not so, the torsional wave is inevitably of mixed Alfvén/fast mode
polarization and has ${\bf B}\cdot\delta{\bf B}\neq 0$. In this case Equations
15 are augmented by extra terms $-\nabla\left({\bf B}\cdot\delta{\bf
B}\right)/4\pi\rho\mp{\bf V_{A}}\nabla\cdot\delta{\bf v}$ on the right hand
side.

In integrating Equations (11), we put $\partial I_{\pm}/\partial t=i\omega
I_{\pm}$ to derive four coupled equations for the real and imaginary parts of
$I_{\pm}$. Equations (11) are integrated from a starting point in the left
hand side chromosphere where Alfvén waves leak down into the chromosphere,
back through the corona to the right hand side where waves are fed up from
below. In this way the reflection and transmission of Alfvén waves at the loop
footpoints and elsewhere is naturally reconstructed. The velocity and magnetic
field perturbations are calculated from

$\displaystyle\delta v$ $\displaystyle=$ $\displaystyle{I_{+}+I_{-}\over 2}$
(12) $\displaystyle{\delta B\over\sqrt{4\pi\rho}}$ $\displaystyle=$
$\displaystyle{I_{+}-I_{-}\over 2}\,.$ (13)

The wave energy density and positive and negative going energy fluxes are

$\displaystyle U$ $\displaystyle=$ $\displaystyle{\rho\delta v^{2}\over
2}+{\delta B^{2}\over 8\pi}={\rho\over 4}\left(I_{+}^{2}+I_{-}^{2}\right)$
(14) $\displaystyle F_{+}$ $\displaystyle=$ $\displaystyle{\rho\over
4}I_{+}^{2}V_{A}$ (15) $\displaystyle F_{-}$ $\displaystyle=$
$\displaystyle{\rho\over 4}I_{-}^{2}V_{A}\,,$ (16)

and the wave peak electric field appearing in equation 4 is

$\delta E_{p}^{2}={B^{2}\over 2c^{2}}\left(I_{+}^{2}+I_{-}^{2}\right).$ (17)

###  6.5 Fractionation

We recap and expand the treatment given in Laming (2004a). Our model starts
with a fully mixed chromosphere, upon which pondermotive forces due to Alfvén
wave reflection and transmission act to provide the fractionation. The low
solar chromosphere is of much higher density than the upper layers where the
FIP fractionation will occur in our models. Consequently the lower boundary
condition of completely mixed photospheric composition material gives an
essentially infinite particle “reservoir” to supply the extra fractionated
elements. To calculate the fractionation, we follow in part the approach and
notation of Schwadron et al. (1999). Consider first the motion of ions and
neutrals of element $s$ in a background flow of protons and hydrogen with
speed $u$. We neglect the ambipolar force, which is generally much less than
gravity. We also neglect an inertial term $\partial/\partial
z\left(\rho_{s}u_{s}^{2}/2\right)$ since the flow speed will turn out to be
much lower than the particle thermal speeds. Then the momentum equations for
ions and neutrals are

$\displaystyle{\partial P_{si}\over\partial z}$ $\displaystyle=$
$\displaystyle-\rho_{si}g-\rho_{si}\nu_{si}\left(u_{si}-u\right)$ (18)
$\displaystyle{\partial P_{sn}\over\partial z}$ $\displaystyle=$
$\displaystyle-\rho_{sn}g-\rho_{Sn}\nu_{sn}\left(u_{sn}-u\right),$ (19)

where $P_{si}$ and $P_{sn}$ are the partial pressures of ions and neutrals of
element $s$, $\rho_{si}$ and $\rho_{sn}$ are the corresponding densities,
$\nu_{si}$ and $\nu_{sn}$ the collision rates with ambient gas (assumed
hydrogen and protons), $u_{si}$ and $u_{sn}$ the flow speeds, and $u$ the
hydrogen flow speed imposed on the loop. Detailed expressions for $\nu_{si}$
and $\nu_{sn}$ are given in Laming (2004a). The partial pressures are given by
$P_{s,i,n}=\left(k_{\mathrm{B}}T/m_{s}+v_{\mathrm{turb}}^{2}+v_{\mathrm{wave}}^{2}\right)\rho_{s,i,n}/2$,
where the various terms on the right hand side represent the particle thermal
speed, the microturbulent velocity coming from the chromospheric model, and
the particle motion in longitudinal (i.e., acoustic) waves, either induced by
the Alfvén waves themselves, or propagating up from the photosphere. These are
discussed in more detail in the next subsection. In true collisionless plasma,
neutrals would not respond to the wave. However the solar chromosphere is
sufficiently collisional that neutrals move with the ions in the wave motion
(e.g., Vranjes et al., 2008) for wave frequencies well below the charge
exchange rate that couples neutrals and ions, and so neutrals require the same
form for their partial pressure as the ions. We also neglected inertial terms
above. Inclusion of such terms would lead to

$P_{si,n}=\left(k_{\mathrm{B}}T/m_{s}+v_{\mathrm{turb}}^{2}+v_{\mathrm{wave}}^{2}+u_{s}^{2}\right)\rho_{si,n}/2=v_{s}^{2}\rho_{si,n}/2$
(20)

assuming $u_{si}-u_{sn}\ll u_{s}\sim u$.

The momentum equations are combined by forming $\nu_{sn}\partial
P_{si}/\partial z+\nu_{si}\partial P_{sn}/\partial z$ to yield

${\partial P_{s}\over\partial
z}=-\rho_{s}g-\nu_{\mathrm{eff}}\rho_{s}\left(u_{s}-u\right)+{\partial\xi_{s}\over\partial
z}{\rho_{s}v_{s}^{2}\over
2}{\nu_{si}-\nu_{sn}\over\left(1-\xi_{s}\right)\nu_{si}+\xi_{s}\nu_{sn}}\,,$
(21)

with
$\nu_{\mathrm{eff}}=\nu_{si}\nu_{sn}/\left(\xi_{s}\nu_{sn}+\left(1-\xi_{s}\right)\nu_{si}\right)$
and $\xi_{s}$ being the ionization fraction of element $s$. For

$u_{s}=u-g/\nu_{\mathrm{eff}}\left(1-\mu/m_{s}\right)+\partial\xi_{s}/\partial
z\left(v_{s}^{2}/2\right)\left(\nu_{si}-\nu_{sn}\right)/\nu_{si}/\nu_{sn}\,,$
(22)

where $\mu$ is the mean molecular weight, all elements are lifted by the
background flow to the same scale height given by $k_{\mathrm{B}}T/\mu g$.
This obviously requires $u\nu_{\mathrm{eff}}>g$, assuming the term in
$\partial\xi_{s}/\partial z$ negligible. Generally
$u-g/\nu_{\mathrm{eff}}\left(1-\mu/m_{s}\right)\sim 10^{3}$ cm s-1 or more,
while ${\partial\xi_{s}\over\partial z}\sim 10^{-8}$ cm-1 and
${\rho_{s}v_{s}^{2}\over
2}\left(\nu_{si}-\nu_{sn}\right)/\nu_{si}/\nu_{sn}\sim 10^{9}$ cm2s-1. For
$u\nu_{s}\ll g$, we get gravitationally stratified solutions with
$\rho_{s}\propto\exp\left(-m_{s}gz/k_{\mathrm{B}}T\right)$. With $g=2.74\times
10^{4}$ cm s-2, and $\nu_{s}\sim 10^{2}-10^{3}$ the former case is valid for
the Sun for flow speeds in the chromosphere greater than a relatively modest
10 – 100 cm s-1. Laming (2012) considers the upward flow speed due to
chromospheric evaporation, and finds speeds of at least $10^{3}$ cm s-1.

A gravitational settling timescale may be estimated from equations 18 and 19.
We write $\partial P_{s}/\partial z\simeq c_{S}^{2}\partial\rho_{s}/\partial
z\simeq\left(c_{S}^{2}/u_{sett}\right)\partial\rho_{s}/\partial t$ where the
sound speed is $c_{S}$ and the settling speed $u_{sett}\sim g/\nu_{eff}$.
Integrating with respect to $t$ gives $t\sim c_{S}^{2}\nu_{eff}/g^{2}\sim
10^{4}\left(n_{H}/10^{10}\right)/A$ s, where $A$ is the atomic mass of element
$s$, assuming that $\nu_{eff}$ is dominated by collisions with neutral H. The
settling time increases with the density as one moves deeper into the
chromosphere, and with the presence of turbulence with amplitude added in
quadrature to $c_{S}$, and can easily be on the order of days or weeks or
more. Higher in the chromosphere, $\nu_{eff}$ will be dominated by collisions
with protons, and a longer timescale would result as the chromospheric density
approaches $10^{10}$ cm-3.

The solar chromosphere is doubtless a more dynamic environment than
represented by Equations (19) – (22). For our purposes the net result of this
dynamic behavior is assumed to completely mix up the plasma to give uniform
elemental composition with height, which is obtained in our model with the
above choice for $u_{s}$. Other choices may be possible which would provide
chemical fractionation in the unperturbed chromosphere, and one could choose
$u_{s}$ to provide the required FIP effect. However the physics behind such a
specification for $u_{s}$ in most cases remains obscure, and is probably
unrealistic, leading to an unsatisfactory explanation for the FIP effect.

We now include a ponderomotive force, $\rho_{si}a$ on the ions in the momentum
equations;

$\displaystyle{\partial P_{si}\over\partial z}$ $\displaystyle=$
$\displaystyle-\rho_{si}g-\rho_{si}\nu_{si}\left(u_{si}-u\right)+\rho_{si}a$
(23) $\displaystyle{\partial P_{sn}\over\partial z}$ $\displaystyle=$
$\displaystyle-\rho_{sn}g-\rho_{sn}\nu_{sn}\left(u_{sn}-u\right).$ (24)

Again writing $P_{s}=\rho_{s}c_{S}^{2}$, taking $u_{s}$ as specified above and
integrating with respect to $z$ we find

${\rho_{s}\left(z_{u}\right)\over\rho_{s}\left(z_{l}\right)}=\exp\left\\{2\int_{z_{l}}^{z_{u}}{\xi_{s}a\nu_{\mathrm{eff}}\over\nu_{si}v_{s}^{2}}\,dz\right\\},$
(25)

where the constant of integration has been chosen to keep
$\rho_{s}\left(z_{u}\right)=\rho_{s}\left(z_{l}\right)$ when $a=0$. A
quantitative assessment of coronal element abundances anomalies requires an
evaluation of Equation (25) with a realistic model chromosphere in the region
of Alfvén wave reflection, with $a$ coming from Equation (5).

As will be seen below, most the the FIP fractionation develops over a small
range in $z$, where the ponderomotive acceleration $a$ is strong. The
timescale to establish this local abundance anomaly is short, basically
$\left(z_{u}-z_{l}\right)/u\sim\left(10^{6}-10^{7}\right)/10^{4}\sim 10^{3}$
s. By itself, the ponderomotive force would only produce a deficit of low FIP
ions just below this region, and a surplus just above. To produce the
fractionation seen in coronal loops and the solar wind, thermal transport or
diffusion processes must continually supply further low FIP ions from below to
erase the deficit, and communicate the resulting fractionation above to still
higher levels of the solar atmosphere. Thus while the fractionation can be
produced locally rather quickly, and is therefore immune from processes that
would otherwise disrupt fractionation produced solely by diffusive processes,
the timescale to change the composition of a coronal loop is still on the
order of days, as observed (Widing and Feldman, 2001) due to the necessity of
thermal transport of the fractionation.

###  6.6 Compressional chromospheric waves

####  6.6.1 Introduction

We have introduced an extra longitudinal pressure associated with the Alfvén
waves proportional to $v_{\mathrm{wave}}^{2}$ (see Equation (20)), which has
the effect of causing some saturation of the FIP fractionation. Here we give
some physical motivations for this extra term. It has three possible sources.
The first is the inevitable generation of slow mode waves by the Alfvén
driver. Physically, the periodic variation in magnetic pressure of the Alfvén
wave drives longitudinal compressional waves. These generated acoustic waves
can act back on the Alfvén driver, as the compressional wave introduces a
periodic variation in the Alfvén speed, which generates new Alfvén waves. The
second will be obliquely propagating fast mode waves, which are necessarily
compressive. Torsional Alfvén wave on a twisted magnetic flux tube are
inevitably mixed with the kink mode, and hence are compressive. The third
relates to acoustic waves propagating up to the chromosphere from the
convection zone.

####  6.6.2 Parametric generation by Alfvén waves

Following Laming (2012) we illustrate the generation of slow mode or acoustic
waves by the ponderomotive force associated with plane Alfvén waves with a
simple 1D calculation. The linearized momentum equation keeping terms to all
orders in perturbed quantities is (all symbols have their usual meanings as
defined above),

$\left(\rho+\delta\rho\right){\partial\delta v_{z}\over\partial
t}+\left(\rho+\delta\rho\right)\delta v_{z}{\partial\delta v_{z}\over\partial
z}=\left(\rho+\delta\rho\right){\partial\over\partial z}{\delta B^{2}\over
8\pi\left(\rho+\delta\rho\right)}-{\partial\delta P\over\partial
z}-g\delta\rho\,,$ (26)

where

$\displaystyle\delta\rho$ $\displaystyle=-\rho\nabla\cdot{\bf\delta{\bf
r}}-\delta r_{z}{\partial\rho\over\partial z}=-\rho\delta{\bf
r}\left(ik_{S}+{1\over H_{D}}\right)$ (27) $\displaystyle\delta P$
$\displaystyle=-\gamma P\nabla\cdot{\bf\delta{\bf r}}-\delta r_{z}{\partial
P\over\partial z}=-P\delta{\bf r}\left(ik_{S}\gamma+{1\over H_{P}}\right),$
(28)

for Eulerian displacement ${\bf\delta r}\propto\exp
i\left(\omega_{S}t+k_{S}z\right)$, where $H_{P}=P/\left(\partial P/\partial
z\right)$ and $H_{D}=\rho/\left(\partial\rho/\partial z\right)$ (signed)
pressure and density scale heights, respectively. The first term on the right
hand side of Equation (26) represents the instantaneous ponderomotive force.
In cases where the WKB approximation applies, $\delta B\propto\rho^{1/4}$, and
this expression is equivalent to the more usual form $-\partial\left(\delta
B^{2}/8\pi\right)/\partial z$. Substituting for $\delta\rho$ and $\delta P$
from Equations (28), keeping terms as high as second order in small
quantities, Equation (26) becomes

$-i{\omega_{S}\over H_{D}}\delta
v_{z}^{2}+\left(-\omega_{S}^{2}+k_{S}^{2}c_{S}^{2}-i{k_{S}c_{S}^{2}\over
H_{P}}-{c_{S}^{2}\over\gamma H_{P}^{2}}-i{k_{S}c_{S}^{2}\over\gamma
H_{P}}-ik_{S}g-{g\over H_{D}}\right)\delta
v_{z}-i\omega_{S}{\partial\over\partial z}{\delta B^{2}\over 8\pi\rho}=0.$
(29)

This is considerably simplified in isothermal conditions, $\gamma=1$,
$H_{P}=H_{D}=-c_{S}^{2}/g$, so that

$-i{\omega_{S}\over H_{D}}\delta
v_{z}^{2}+\left(-\omega_{S}^{2}+k_{S}^{2}c_{S}^{2}+ik_{S}g\right)\delta
v_{z}-i\omega_{S}{\partial\over\partial z}{\delta B^{2}\over 8\pi\rho}=0.$
(30)

We put $\Im k_{S}=-g/2c_{S}^{2}$, and $\sqrt{\left(\Re
k_{S}\right)^{2}+g^{2}/4c_{S}^{4}}=2\Re k_{A}/n$, $\omega=2\omega_{A}/n$,
where $k_{A}$ and $\omega_{A}$ are the wavevector and angular frequency of the
Alfvén wave with $n=1,2,3,..$ (anticipating the result below). We find

$\delta v_{z}^{2}-\delta v_{z}iH_{D}\omega_{S}\left(1-{c_{S}^{2}\over
V_{A}^{2}}\right)+H_{D}{\partial\over\partial z}{\delta B^{2}\over
8\pi\rho}=0$ (31)

with solution

$\delta v_{z}={-i\over 2}\left[\sqrt{\delta
v_{A}^{2}+H_{D}^{2}\omega_{S}^{2}\left(1-{c_{S}^{2}\over
V_{A}^{2}}\right)^{2}}-H_{D}\omega_{S}\left(1-{c_{S}^{2}\over
V_{A}^{2}}\right)\right]$ (32)

where we have put ${\partial\over\partial z}\left(\delta
B^{2}/8\pi\rho\right)=\left(\delta B^{2}/4\pi\rho\right)/4H_{D}=\delta
v_{A}^{2}/4H_{D}$ using the WKB result for Alfvén waves in a density gradient,
and assuming $1/H_{D}\gg\Re k_{A}$. When $c_{S}^{2}\sim V_{A}^{2}$ or
$H_{D}\rightarrow 0$, $\delta v_{z}\simeq-i\delta v_{A}/2$. In the opposite
limit $\delta v_{z}\simeq-i\delta
v_{A}^{2}/4H_{D}\omega_{S}\left(1-c_{S}^{2}/V_{A}^{2}\right)$. In these two
cases $\omega_{S}=\omega_{A}$ or $\omega_{S}=2\omega_{A}$, respectively. In
fact, acoustic waves can be generated with $\omega_{S}=2\omega_{A}/n$, with
higher $n$ becoming more intense as the nonlinearity increases (Landau and
Lifshitz, 1976). This is shown explicitly by Laming (2012), who extends the
treatment above to include higher powers of $\delta v_{z}$. Vasheghani
Farahani et al. (2011) treat the case of slow mode wave generation by a
torsional Alfvén wave. This is subtly different to the case of a plane Alfvén
wave considered here, and the FIP fractionation resulting from such a model
will be investigated in future work.

When the Alfvén wave becomes evanescent in a sufficiently steep density
gradient, $k_{A}\rightarrow i\kappa_{A}$ and

$\delta v_{z}={-i\over 2}\left[\sqrt{\delta
v_{A}^{2}+H_{D}^{2}\omega_{S}^{2}\left(1+{\kappa_{A}^{2}c_{S}^{2}\over\omega_{A}^{2}}\right)^{2}}-H_{D}\omega_{S}\left(1+{\kappa_{A}^{2}c_{S}^{2}\over\omega_{A}^{2}}\right)\right],$
(33)

resulting in reduced $\delta v_{z}$ compared to the case with propagating
Alfvén waves. This will be apparent in results to be presented in Section 7.

####  6.6.3 Fast-mode waves from above

The treatment of Vasheghani Farahani et al. (2011) also considers a second
source of longitudinal pressure. Torsional waves on a flux tube with magnetic
twist are no longer pure Alfvén waves, but have an admixture of fast mode or
kink polarization. Since the magnetic perturbation (in the $\phi$ direction in
cylindrical polar coordinates) now has a component along the background
magnetic field (the twist gives it a $\phi$ component also), the wave becomes
compressive. This is also true more generally of any fast mode wave in oblique
propagation, and in Laming (2009) it was shown that such longitudinal pressure
would lead to a natural saturation of the FIP effect at high wave amplitudes.
In the case where transverse and longitudinal pressures are equal, the FIP
effect saturates at the commonly observed value of about 4. However such
compressive waves are also subject to increased damping, and are therefore
less likely to develop to large amplitudes. The existence or otherwise of
twisted magnetic loops in the corona is also controversial. Accordingly, we do
not consider such possibilities further in this review.

####  6.6.4 P-modes from below

The third contribution to the longitudinal pressure in the chromosphere comes
from the continued propagation upwards of slow modes waves or shocks
developing from turbulence lower down in the atmosphere. We include a
longitudinal pressure with amplitude 6.25 km s–1 motivated by the simulations
in Heggland et al. (2011). The velocity amplitude of these waves would be
expected to grow with altitude, due to WKB effects, although some reflection
back downwards may also occur, and should damp mainly due to the effect of
radiation. The principal coolant in the photosphere is H-, with cooling
timescale $\tau\simeq\left(1.2\times 10^{11}/n_{e}\right)\left(5000\mathrm{\
K}/T\right)^{2}$ minutes (Ayres and Rabin, 1996). Cooling of a similar
magnitude will be present in the low chromosphere, although different
processes may also contribute. Although the plasma density varies, throughout
large sections of the chromospheric model, the electron density and
temperature are roughly constant and of order $10^{11}$ cm-3 and 5000 K,
respectively, leading to a constant cooling time of order 1 minute. Thus for 3
minute or 5 minute waves, the product $\omega\tau\sim 1$, and the wave decay
length would be comparable to the wavelength (Mihalas and Weibel-Mihalas,
1999), and similar to the WKB growth. In fact, the simulations of Heggland et
al. (2011) show approximately constant slow mode wave amplitude with height
for a variety of models, as do the observations of Beck et al. (2013, albeit
at a lower wave amplitude), so we accordingly adopt a constant slow mode wave
amplitude in our models.

While slow mode waves reduce the fractionation by increasing $v_{s}^{2}$ in
the denominator of the integrand in Equation (25), when these steepen into
shock waves, we expect fractionation to cease altogether due to the extra
turbulence. Accordingly we also neglect fractionation at chromospheric
altitudes where the slow mode wave amplitudes added in quadrature come to a
value greater than the local sound speed. The rationale for this is that in
such conditions, a shock will form, and the mixing produced by turbulence
generated in such conditions would preclude further fractionation. Reflection
and refraction of sound waves at the discontinuity will produce oppositely
directed packets of sound waves which can initiate a cascade to smaller, and
ultimately microscopic, spatial scales, assumed to induce microscopic mixing.
This has the effect in most models of limiting FIP fractionation to the upper
regions of the chromosphere.

##  7 Results and Interpretation

###  7.1 Closed loop

As a first illustration of the action of the ponderomotive force in causing
FIP fractionation, we consider the case of a closed magnetic loop. The model
loop is taken to be 100 000 km long, and plasma in the loop has a density
scale height of 125 000 km, extrapolated upwards from the chromospheric model.
The coronal temperature is constant at the maximum temperature of the
chromospheric temperature ($\sim 1.6\times 10^{6}$ K) but this value is not
significant since we are uninterested in the ionization balance in this region
and the Alfvén wave propagation only depends on the density. The coronal
magnetic field is 20 G, increasing by a factor of 5 in the photosphere, as in
Figure 6 (right panel). The chromospheric model is taken from Avrett and
Loeser (2008), and the plasma $\beta=1$ layer is at 615 km altitude above the
photosphere. We simply treat the parallel propagation of undamped shear Alfvén
waves. With reference to Figure 5, the solution of Equations (11) is started
with a downgoing wave at the left-hand chromosphere, and then integrated back
to the chromosphere at the other end of the loop. Choosing a wave angular
frequency of 0.07 s-1 means the wave is on resonance with the coronal loop,
and in our calculation, 98% of the upgoing wave energy from the right-hand
chromosphere is transmitted to the corona. Hence the chromospheric wave
patterns in both chromospheres are very similar, with the roles of upgoing and
downgoing waves interchanged (with 100% transmission, they would be
identical). The ponderomotive force is insensitive to the direction of wave
motion, and so is the same at each footpoint.

Figure 7: Solution for the coronal portion of the loop. The top panel shows
$\delta{\bf v}$ and $\delta{\bf B}/\sqrt{4\pi\rho}$ (Equations (13)), real
components in black, imaginary components in grey. The middle panel shows the
wave energy fluxes, left and right going, and their difference, divided by the
magnetic flux density. Normalized thus, this difference should be constant in
the absence of wave growth or damping, and provide a check on energy
conservation in the integration. The bottom panel shows the ponderomotive
acceleration, $a$.

Figure 7 shows the solution for the coronal portion of the loop. The top panel
shows $\delta{\bf v}$ and $\delta{\bf B}/\sqrt{4\pi\rho}$ (Equations (13)),
real components in black, imaginary components in grey. The middle panel shows
the wave energy fluxes, left and right going, and their difference, divided by
the magnetic flux density. Normalized thus, this difference should be constant
in the absence of wave growth or damping, and provides a check on energy
conservation in the integration. The bottom panel shows the ponderomotive
acceleration, $a$, calculated from Equations (5) and (17). Throughout the
corona (2500 km $<z<$ 97500 km), $a\ll 2.7\times 10^{4}$ cm s-2, the solar
gravitational acceleration. In the chromospheric footpoints, $a$ is much
larger.

Figure 8: The top and bottom panels on the left hand side show the same
variables as the top and bottom panels in Figure 7 (the wave energy fluxes are
not shown). The panels on the right hand side show the chromospheric
ionization balance and FIP fractionation for selected elements. The bottom
panel shows FIP fractionation for the abundance ratios S/O, He/O, Mg/O, Fe/O
and Si/O, with the linestyles indicated as black lines, to be read on the left
axis. One can see Fe/O and Mg/O fractionating to a value close to 4, Si/O to
2.6. S/O to 1.6 and He/O to 0.4. The ionization balance is shown as grey lines
with the same linestyles for Fe, Mg, Si, S, and He. He stays neutral the
longest, and matches calculations in Golding et al. (2014). Also shown are H
and O as solid lines, labelled respectively. The O ionization balance follows
that for H due to the strong coupling by charge exchange. The ionization
balances for Fe, Mg, Si, and S are hard to distinguish on this plot, so the
top right panel shows these in more detail, with the charge fraction scale
running from 0.99 to 1.00.

Figure 8 illustrates important features of the solution in the chromospheric
portion of the loop. The top and bottom panels on the left hand side show the
same variables as the top and bottom panels in Figure 7 (the wave energy
fluxes are not shown). The ponderomotive acceleration shows a large “spike” at
an altitude of 2150 km, coinciding with the steep chromospheric density
gradient shown in Figure 6 (left panel). Also shown on the bottom left panel
is the amplitude of slow mode waves arising from parametric generation by the
Alfvén driver, shown as a dotted line. This is added in quadrature to the
amplitude of slow mode waves propagating upwards from the photospheric,
assumed here to be a constant 6.25 km s–1, following the discussion in Section
6.3. This value is also consistent with the macroscopic velocity field
inferred in the solar chromosphere by Vieytes et al. (2005, see their Figure
3ab). The parametric slow mode amplitude goes through a sharp minimum where
the chromospheric density gradient is steepest at about 2150 km altitude.
Rakowski and Laming (2012) show that this occurs where the Alfvén wave becomes
evanescent. The panels on the right hand side show the chromospheric
ionization balance and FIP fractionation for selected elements. The bottom
panel shows FIP fractionation for the abundance ratios S/O, He/O, Mg/O, Fe/O
and Si/O, with the linestyles indicated as black lines, to be read on the left
axis. One can see Fe/O and Mg/O fractionating to a value close to 4, Si/O to
2.6. S/O to 1.6 and He/O to 0.4. The ionization balance is shown as grey lines
with the same linestyles for Fe, Mg, Si, S, C, and He. Also shown are charge
states fractions for H and O as labelled. The ionization balances for Fe, Mg,
Si, C, and S are hard to distinguish on this plot, so the top right panel
shows these in more detail, with the charge fraction scale running from 0.99
to 1.00.

Fe and Mg have similar charge state fractions for all altitudes of relevance,
and hence fractionate to about the same degree. Si, although low FIP and 99.8%
ionized in the region 1500 km to 2500 km, fractionates noticeably less. Around
1000 km altitude, Si is more highly ionized due to the increased flux in Ly
$\alpha$, which is now trapped by the increased opacity in the line. In the
model presented here, no fractionation occurs at this height. S is also has a
high charge state fraction, at 99.4%, but its ionization potential is higher
than the energy of Ly $\alpha$, and so it is not affected by the trapped
radiation. This dependence on the ionization fraction comes from the form of
$\nu_{eff}=\nu_{si}\nu_{sn}/\left(\xi_{s}\nu_{sn}+\left(1-\xi_{s}\right)\nu_{si}\right)$
in the integrand of equation 25 where $\nu_{si}>>\nu_{sn}$, (Laming, 2004a),
which is the case when the background gas is mainly ionized H. Only when
$\xi_{s}\simeq 1$ to a high degree of precision does the second term in the
denominator become negligible compared to the first, allowing strong
fractionation to occur. Lower down in the chromosphere, where neutral H
dominates, $\nu_{sn}\sim\nu_{si}$, and this precise dependence on element
charge state is lost.

Table 3: FIP Fractionations in Closed Magnetic Field (see text for details). ratio | models | observations
---|---|---
| 75 | 95 | 110 | a | b | c | d | e | f | g
|  ( km s–1) |  |  |  |  |  |  |
He/O | 0.56 | 0.40 | 0.22 | 0.68 – 0.60 | 0.54 – 0.98 |  |  |  |  | 0.29
C/O | 1.14 | 1.15 | 1.03 | 1.36 – 1.41 | 1.23 – 1.55 |  |  |  |  | 0.76
N/O | 0.85 | 0.73 | 0.55 | 0.72 – 1.32 | 0.59 – 0.68 |  |  |  |  | 1.02
Ne/O | 0.71 | 0.55 | 0.36 | 0.38 – 0.75 | 0.79 – 1.15 |  |  |  |  | 0.75
Na/O | 2.57 | 3.95 | 5.83 |  | 2.51 – 3.31 |  1.8${+2\atop-1}$ |  7.8${+13\atop-5}$ |  | $3.43\pm 1.7$ |
Mg/O | 2.37 | 3.52 | 4.95 | 2.58 – 2.61 | 1.95 – 3.55 |  2.7$\pm 0.3$ |  2.8${+2.3\atop-1.3}$ |  |  | 2.71
Al/O | 2.20 | 3.16 | 4.28 |  | 2.29 – 3.02 |  5.6${+3.3\atop-2.1}$ |  3.6${+1.7\atop-1.2}$ |  |  |
Si/O | 1.92 | 2.60 | 3.26 | 2.49 – 3.11 | 2.14 – 3.26 |  |  4.9${+2.9\atop-1.8}$ |  |  2.5$\pm 1.0$ | 2.37
P/O | 1.52 | 1.83 | 2.01 |  |  |  5${+11\atop-3.4}$ |  |  |  |
S/O | 1.37 | 1.56 | 1.61 | 1.62 – 1.92 | 1.20 – 2.09 |  2.1$\pm 0.2$ |  2.2$\pm 0.2$ | $1.7\pm 0.3$ | $1.00{+0.48\atop-0.32}$ | 1.00
Cl/O | 1.04 | 0.97 | 0.84 |  |  |  |  |  |  1.8$\pm 1.8$ |
Ar/O | 0.93 | 0.83 | 0.65 |  |  |  |  | $1.1\pm 0.1$ |  1.00$\pm 0.15$ | 0.73
K/O | 2.71 | 4.29 | 6.58 |  |  |  4.7${+7.0\atop-2.8}$ |  1.8${+0.4\atop-0.6}$ | $3.5\pm 0.9$ |  5.5${+9\atop-3.7}$ |
Ca/O | 2.71 | 4.29 | 6.58 |  | 2.09 – 3.88 |  2.7$\pm 0.25$ |  3.5${+4.3\atop-1.9}$ |  | 3.0 – 9.7 | 2.86
Ti/O | 2.70 | 4.28 | 6.57 |  |  |  |  |  |  |
Cr/O | 2.66 | 4.18 | 6.36 |  | 2.40 – 3.47 |  |  |  |  |
Fe/O | 2.66 | 4.18 | 6.35 | 2.28 – 2.90 | 1.95 – 3.55 |  |  7.0${+1.4\atop-1.2}$ |  |  | 2.27
Ni/O | 2.39 | 3.57 | 5.11 |  |  |  |  |  |  |
Kr/O | 0.99 | 0.89 | 0.73 |  |  |  |  |  |  |

Table 3 gives model fractionations for three different assumed coronal Alfvén
wave amplitudes, 75, 95, and 110 km s–1, which are initiated with amplitudes
0.6, 0.75, and 0.9 km s–1 low in chromosphere A. Figures 7 and 8 correspond to
the middle entry, 95 km s–1. The model results are compared with observations
taken from the literature; (a) Zurbuchen et al. (2002), given relative to O,
(b) Bochsler (2007a), relative to O, (c) Giammanco et al. (2007, 2008),
relative to H, (d) Bryans et al. (2009), given relative to the mean of O, Ne
and Ar and, (e) Phillips et al. (2003), relative to H, (f) Ar, Sylwester et
al. (2010a); K, Sylwester et al. (2010b); Cl, Sylwester et al. (2011); S,
Sylwester et al. (2012); Si, Sylwester et al. (2013); and Na Phillips et al.
(2010), all relative to H and given relative to the photospheric abundance of
Asplund et al. (2009), and (g) SEP measurements from Reames (2014), given
relative to O. Zurbuchen et al. (2002), Bochsler (2007a) and Giammanco et al.
(2007, 2008) give abundances derived from in situ measurements in the slow
solar wind. They generally agree best with model with coronal Alfvén wave
amplitudes of 75-95 km s–1. Bryans et al. (2009) give abundances derived from
a deep SOHO/SUMER (Wilhelm et al., 1995, 1997) spectrum of the quiet solar
corona, and are more consistent with a higher Alfvén wave amplitude, 110 km
s–1. The remaining references give abundance measured in solar flares with
RESIK (Sylwester et al., 2005). All models assume a constant amplitude with
height for slow mode waves propagating up from the photosphere of 6.25 km s–1.
Fractionations are somewhat sensitive to this, in that increased slow wave
wave amplitude decreases the degree of fractionation. However for the case
considered here, that of a coronal Alfvén wave on resonance with the coronal
loop, most of the fractionation is restricted to a small region close to the
top of the chromosphere where the slow mode waves generated parametrically by
the Alfvén wave driver are also significant. Rakowski and Laming (2012)
extended such considerations to Alfvén waves away from the loop resonance. A
similar FIP effect is produced, but the fractionation occurs over a wider
layer of the chromosphere, as Alfvén waves penetrate deeper. Some subtle
differences occur. S is more strongly fractionated in such a case, and He is
depleted much less, both as a result of the difference in Alfvén wave
behaviour. Slow mode waves are also more strongly generated deeper in the
chromosphere, and in cases such as this, their generation has a stronger
effect on the fractionation. Laming (2004a) also considered the effect of
different chromsopheric models from Vernazza et al. (1981). If VALC may be
considered as a forerunner to the Avrett and Loeser (2008) model used here,
and it does indeed give similar FIP fractionation, the result of Laming
(2004a) in that stronger fractionations are found within cells at dark points
(VALA, VALB) and weaker fractionation at network segments (VALD, VALE).
However these chromospheric models should probably be considered obsolete at
the current time of writing.

###  7.2 Open field

We next consider the FIP effect in open magnetic field. The chromospheric
model is extrapolated upwards assuming a density scale height of 170,000 km,
to give a density profile comparable to that observed and modeled elsewhere
(Laming, 2004b). The temperature is set at a maximum of $10^{6}$K, but is
similarly inconsequential as in the closed field case. Figure 9 shows the same
panels as Figure 7, but extending to an altitude of 500 000 km. Five Alfvén
waves are included, designed to match the spectrum given in Figure 3 of
Cranmer et al. (2007), with amplitude in the transition region from their
Figure 9. Waves of angular frequency 0.010, 0.031, 0.062, 0.093, and 0.124 s-1
are included with amplitudes at 500 000 km of 12.5, 150, 75, 50, and 12.5 km
s–1, respectively. The integration of Equations (11) begins at 500 000 km with
outgoing waves only following Cranmer et al. (2007), and is taken back to the
chromosphere. No account is taken of the motion of the coronal hole plasma.
For the range of altitudes considered, the fast solar wind speed is always
much less than the Alfvén speed, and is probably still insignificant compared
to the uncertainty in the Alfvén speed.

Figure 9: Solution for the coronal hole. The top panel shows $\delta{\bf v}$
and $\delta{\bf B}/\sqrt{4\pi\rho}$ (Equations (13)), real components in
black, imaginary components in grey. The middle panel shows the wave energy
fluxes, left and right going, and their difference, divided by the magnetic
flux density. Normalized thus, this difference should be constant in the
absence of wave growth or damping, and provide a check on energy conservation
in the integration. The bottom panel shows the ponderomotive acceleration,
$a$.

Figure 10: The top and bottom panels on the left hand side and the top panel
on the right hand side show the same variables as in Figure 9. Additionally,
the amplitude of slow mode wave generated parametrically by the Alfvén waves
is also shown in the top right panel. Compared to the closed loop case with a
wave on resonance, the ponderomotive acceleration has a stronger contribution
lower down in the chromosphere. The bottom right hand side panel shows the
chromospheric ionization balance and FIP fractionation for selected elements.
The bottom panel shows FIP fractionation for the abundance ratios He/O, Mg/O,
and Fe/O, with the linestyles indicated as black lines, to be read on the left
axis. The degree of fractionation is reduced from the closed field case. The
ionization balance is shown as grey lines with the same linestyles for Fe, Mg,
Si, S, and He, and is very similar to the previous, except that in the reduced
radiation field of the coronal hole, He stays neutral longer.

Figure 10 shows a similar set of panels to Figure 8 above. Top left shows
$\delta{\bf v}$ and $\delta{\bf B}/\sqrt{4\pi\rho}$ for the five waves in the
chromosphere, with the real and imaginary parts distinguished as before.
Bottom left now shows the chromospheric wave energy fluxes. Although we have
made no assumption about where the waves originate, in the case of the coronal
hole the usual interpretation is that Alfvén waves start out as kink mode
waves on photospheric flux tubes, and propagate up into the coronal hole. This
bottom left panel shows that most of the upgoing wave energy flux is reflected
back down again, at the steep chromospheric density gradient. Of order 10% of
the wave energy makes it out of the chromosphere, and more reflection occurs
higher up (not included in our model). Higher frequency waves suffer less
reflection than the low frequency waves.

The top right panel shows the ponderomotive acceleration. Superficially it is
very similar to the previous case, with a “spike” at the altitude where the
chromospheric density gradient is steepest. However, the strong wave
reflection leads to a stronger component of the ponderomotive acceleration at
lower altitudes, around 1000 km above the photosphere, and a reduced
acceleration associated with the “spike”. In principle, stronger fractionation
can occur low down in the chromosphere because here the hydrogen is mainly
neutral, whereas around the “spike” it is becoming ionized. This difference in
the background H makes a big difference to the value of $\nu_{\mathrm{eff}}$
in Equation (25), and also to the resulting fractionation. The degree to which
this actually happens is determined by the amplitude of slow mode waves
appearing in $v_{s}^{2}$ in the denominator of the integrand in Equation (25).
The model shown in Figure 10 assumes the same slow mode wave amplitude as in
the closed loop case, a constant value of 6.25 km s–1, which when added in
quadrature with the parametrically generated slow mode wave amplitude
eliminates the fractionation low down. Thus the fractionation is restricted to
the top of the chromosphere, and generally agrees well with coronal hole
observations. Treating the Alfvén and acoustic waves in a more complete
fashion (but only considering Fe/O), Cranmer et al. (2007) reach a similar
result. The amplitude of acoustic waves developing low in the chromosphere in
their model is higher than in ours, of order 10 km s–1 from their Figure 9,
though varying with altitude.

Table 4: FIP Fractionations in Open Magnetic Field (see text for details). ratio | models | observations
---|---|---
| 6.15 | 6.25 | 6.5 | a | b | c
|  ( km s–1) |  |  |
He/O | 0.68 | 0.68 | 0.68 | 0.59 – 0.63 | 0.55 – 0.69 | 0.45 – 0.55
C/O | 3.69 | 1.00 | 0.99 | 1.41 – 1.68 | 1.41 – 1.68 | 0.9 – 1.1
N/O | 0.89 | 0.89 | 0.89 | 1.07 – 1.32 | 1.07 – 1.32 |
Ne/O | 0.81 | 0.81 | 0.81 | 0.44 – 0.52 | 0.44 – 0.52 | 0.3 – 0.4
Na/O | 7.12 | 1.56 | 1.43 |  | 1.45 – 1.91 |
Mg/O | 6.74 | 1.48 | 1.37 | 1.61 – 1.85 | 1.29 – 2.82 | 0.95 – 2.45
Al/O | 6.46 | 1.41 | 1.33 |  | 1.51 – 2.00 |
Si/O | 5.97 | 1.31 | 1.26 | 1.86 – 2.26 | 1.29 – 2.34 | 0.9 – 1.8
P/O | 5.22 | 1.18 | 1.15 |  |  |
S/O | 4.80 | 1.13 | 1.11 | 1.46 – 1.60 | 1.17 – 1.86 |
Cl/O | 1.01 | 1.00 | 1.00 |  |  |
Ar/O | 0.95 | 0.95 | 0.95 |  |  |
K/O | 7.76 | 1.62 | 1.48 |  |  |
Ca/O | 7.77 | 1.62 | 1.48 |  | 1.38 – 1.82 |
Ti/O | 7.83 | 1.62 | 1.48 |  |  |
Cr/O | 7.78 | 1.60 | 1.47 |  | 1.81 – 2.63 |
Fe/O | 7.80 | 1.60 | 1.47 | 1.45 – 1.80 | 1.51 – 2.29 | 0.65 – 1.35
Ni/O | 7.25 | 1.50 | 1.39 |  |  |
Kr/O | 0.98 | 0.98 | 0.98 |  |  |

The effect of varying the slow mode wave amplitude is illustrated in Table 4,
where models are given for amplitudes of 6.15, 6.25 and 6.5 km s–1. Above
about 6.25 km s–1, the FIP fractionation slowly decreases with increasing slow
mode waves, assuming the Alfvén waves are kept constant. Below this value, the
FIP fractionation increases dramatically. Observational ratios are taken from,
(a) Zurbuchen et al. (2002), given relative to O, (b) Bochsler (2007a),
relative to O, and (c) Ko et al. (2006), relative to H. FIP fractionations at
the level of those in the first model column in Table 4, or higher, have been
reported (Widing and Feldman, 1992; Young and Mason, 1997), and are always
observed _in open field structures_. Other authors (Doschek and Laming, 2000;
Del Zanna, 2003; Del Zanna et al., 2003) have not found large FIP effects in
polar plumes. This last reference actually challenges the finding of Widing
and Feldman (1992), (but not that of Young and Mason, 1997), arguing that an
isothermal plasma at $\log T=5.9$ would produce the same Ne VI/Mg VI intensity
ratio with photospheric abundances. Hence the reality, and certainly the
ubiquity of strong FIP effects in polar plumes is questionable. Even so, the
possibility of strong FIP effects in these structure is supported by our
model, as Alfvén waves can have considerable amplitudes low in the
chromosphere, and the sensitivity of our open field FIP model to the assumed
slow mode wave amplitude is perhaps realistic.

###  7.3 The helium abundance

The ponderomotive force was originally invoked to explain just the FIP effect
(Laming, 2004a), i.e., the enhancement in coronal abundance of the low FIP
ions. Once more accurate treatments involving the non-WKB analysis of Alfvén
wave transport were implemented (Laming, 2009, 2012), it was noticed that as
well as providing a satisfactory explanation of the FIP effect, the model also
predicted a depletion of the He abundance, (and to a lesser extent Ne also)
with respect to O in the corona and solar wind. As noted in Section 3, in the
slow solar wind especially, the He/H ratio is reduced from its photospheric
value and becomes more variable in slower speed slow solar wind. The He
abundance is also depleted in the fast solar wind, but to a lesser extent, and
is much less variable.

Rakowski and Laming (2012) investigated models for the depletion of He in more
detail. They showed that similar behavior is seen in the ratio He/O measured
by ACE/SWICS (see Figure 11 left panel); a depletion of He to about 0.4 – 0.8
of the solar photospheric value, with greater depletion seen the slower the
solar wind speed, although the solar cycle dependence is not apparent in the
time interval 1998 – 2011. Rakowski and Laming (2012) constructed models for a
variety of loop lengths and corona magnetic fields, all using the Avrett and
Loeser (2008) model chromosphere, and studied the fractionation produced as a
function of the assumed Alfvén wave frequency with respect to the loop
resonance frequency. For each case, the coronal wave amplitude was chosen to
give a fractionation Fe/O by a factor of 4, as observed in the slow speed
solar wind. As we have seen above, for waves that resonate with the coronal