ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-481-2017Effect of water vapour absorption on hydroxyl temperatures measured from SvalbardChadneyJoshua M.https://orcid.org/0000-0002-5174-2114WhiterDaniel K.https://orcid.org/0000-0001-7130-232XLanchesterBetty S.Department of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UKJ. M. Chadney (j.m.chadney@soton.ac.uk)24March201735348149112December201628February20171March2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/35/481/2017/angeo-35-481-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/481/2017/angeo-35-481-2017.pdf
We model absorption by atmospheric water vapour of hydroxyl
airglow emission using the HIgh-resolution TRANsmission molecular absorption
database (HITRAN2012). Transmission coefficients are provided as a function
of water vapour column density for the strongest OH Meinel emission lines in
the (8–3), (5–1), (9–4), (8–4), and (6–2) vibrational bands. These
coefficients are used to determine precise OH(8–3) rotational temperatures
from spectra measured by the High Throughput Imaging Echelle Spectrograph
(HiTIES), installed at the Kjell Henriksen Observatory (KHO), Svalbard. The
method described in this paper also allows us to estimate atmospheric water
vapour content using the HiTIES instrument.
Atmospheric composition
and structure (airglow and aurora; middle atmosphere – composition and
chemistry; pressuredensityand temperature)Introduction
A layer of excited hydroxyl molecules near the mesopause is responsible for
strong airglow emission that can be detected over a wide wavelength range.
First identified by , this emission is produced
by molecular vibration–rotation transitions. It is possible to derive
rotational OH temperatures by determining ratios between emission lines. If
one considers that the lower rotational and vibrational states have undergone
enough collisions to be in local thermodynamic equilibrium (LTE), then the
neutral atmospheric temperature can be assumed to match the rotational
hydroxyl temperature. This method has been used by many authors to calculate
mesospheric temperatures
e.g.. In this study,
we use the same nomenclature as to describe hydroxyl
emissions, as detailed in Appendix A of that paper.
Atmospheric profiles used in the water vapour absorption model,
measured by radiosonde over Ny-Ålesund, Svalbard, from
. Panel (a) shows the pressure as a function of
altitude, panel (b) shows the temperature, and
panel (c) shows the number density of H2O. The coloured lines
each represent a daily observation during the month of December 2003; the
mean profiles from this month are plotted in solid black lines with 1
standard deviation from the mean shown in dashed black
lines.
OH temperatures have been derived in previous works using different
vibrational bands, depending on the wavelength range of the spectroscope in
use. and measured the OH(6–2) band,
compared OH(6–2) band observations with measurements
from OH(8–3), and made use of the OH(8–4) band. In this
study, we use HiTIES, the High Throughput Imaging Echelle Spectrograph
, part of the Spectrographic Imaging Facility (SIF),
located at the Kjell Henriksen Observatory (KHO), Svalbard
(78.148∘ N, 16.043∘ E), to record hydroxyl emission line
intensities. We use the HiTIES filter panel with a bandpass from 728 to
740 nm, allowing measurement of the OH(8–3) band. The emission peak
altitude is dependent on vibrational level , the mean
thickness of the OH layer being about 8 km with higher
vibrational levels located at the top of the layer and vice versa. Due to the
spread in altitude of the different vibrational excited states, there is
expected to be a difference in the rotational temperatures derived from
emission from different vibrational bands. The temperature difference between
bands is thought to vary from a few kelvin to up to 20 K between bands with
the largest difference in upper vibrational state ν′.
A difficulty in determining accurate temperatures from OH lines is the
absorption of this emission by other atmospheric species, in particular water
vapour. found that water vapour absorption decreases the
intensity of the OH(8–3) P1(4)
line by 2.3 %, affecting the temperature derived using a ratio between
this line and other P-branch lines in the (8–3) band by up to 17 K. This
calculation was carried out only for average winter conditions at Davis
Station, Antarctica; however, this absorption is dependent on the
concentration of water vapour in the atmosphere, which is highly variable,
and will also be on average higher at lower latitudes.
To quantify the effects of atmospheric absorption on OH emission,
modelled atmospheric absorption of a number of species (such
as H2O, CO2, and O3), as well as OH line profiles to provide
transmission coefficients for OH Meinel rotational lines in a number of
different vibrational bands. These coefficients are provided for four
different atmospheric conditions: mid- and high-latitude summer, and mid- and
high-latitude winter. Unfortunately, their table giving transmission
coefficients for the OH(8–3) (Table 6) is a repeat of the OH(6–2) table
(Table 5) and is thus erroneous.
In this work, we model water vapour absorption in the atmosphere and provide
transmission of various OH Meinel band lines as a function of the
concentration of atmospheric H2O. Whereas provided
transmission values for four discrete concentrations of water vapour, here we
provide the transmission as a continuous function of water vapour
concentration. This paper is organised as follows: Sect.
describes the instruments, models, and methods used in this study; Sect. provides the transmission coefficients for OH
lines; and Sect. shows the effect of water vapour absorption on
temperatures derived from OH lines in the (8–3) vibrational band measured by
HiTIES.
MethodsModelling water vapour absorption
The transmission of emission at a given wavelength λ as it passes
through the atmosphere can be determined using the Beer–Lambert law:
τ(λ)=∫0zOHσH2Oabs(λ,z)nH2O(z)dz,
where τ is the optical depth,
σH2Oabs is the absorption
cross section of H2O, nH2O is the number density
of H2O, and z is the altitude. This expression gives the optical depth
over a column of atmosphere between z=zOH, the altitude at
which the hydroxyl emission occurs, and z=0, where the instruments
measuring the emission are located. Equation () is valid for a
vertical beam, where there is only extinction in the beam (i.e. no emission
within the beam or scattering into the beam). The transmission coefficient
can then be obtained from
Tλ=exp(-τ(λ)).
To determine the absorption cross section, we use the latest version of the
HIgh-resolution TRANsmission molecular absorption database, HITRAN2012
. This gives the H2O absorption cross section
σH2Oabs for given temperature and
pressure conditions, used to determine line broadening. Therefore to
calculate transmission coefficients, we need atmospheric profiles of pressure
p(z), temperature T(z), and water vapour number density
nH2O(z). These are obtained from radiosonde
measurements over Ny-Ålesund, Svalbard by . Daily
observations have been made since 1993.
Since in Sect. we take a week of OH line intensity
measurements from December 2003 as an example to determine temperatures, we have used
atmospheric profiles from this month from the Ny-Ålesund dataset. These
are plotted in Fig. , where the thin coloured lines
each represent a daily observation from December 2003 (all days of this month
are included); the thick black lines are the mean of all the daily values and
so represent a monthly average during winter.
Example clear-sky HiTIES spectrum, showing labelled OH emission
lines. Taken on 23 December 2003 at 22:30 UT, with an integration time of
120 s.
Water vapour is principally located in the troposphere, at altitudes less
than about 10 km. Therefore, we need only the atmospheric profiles below
this altitude. To take into account varying water vapour column densities,
the monthly mean profile from Fig. c is scaled such
that the integrated column density matches the required value. We have tested
using a selected daily profile instead of the monthly mean and obtain little
difference in the results.
Atmospheric water vapour content is often provided as precipitable water
vapour (PWV), normally expressed in millimetres, which is the height of a column of
liquid water that would result from the condensation of all the water in a
column of atmosphere. A PWV of 1 mm corresponds to a column density of
1 kg m-2. consider PWVs of 29, 21, 8.5, and 4.2 mm,
corresponding respectively to low-latitude summer, winter, high-latitude
summer, and winter.
Determining hydroxyl temperatures
If we assume a Boltzmann distribution for the population of rotational
levels, the intensity I of a Meinel band line is I=Nν′A22J′+1Qν′(T)exp-hcFkT,
where Nν′ is the total number of molecules in the ν′ vibrational
level, A is the transition probability, J′ is the upper state total
angular momentum quantum number, Qν′ is the partition function, F is
the energy level of the initial rotational level, T is the rotational
temperature, h is Planck's constant, c is the speed of light, and k is
Boltzmann's constant.
One method to obtain the temperature is to take the ratio of intensities of
OH lines with different upper rotational states . Thus,
from Eq. (), we obtain
T=hc(Fb-Fa)klnIaAb(2Jb′+1)IbAa(2Ja′+1)-1,
where the indices a and b represent the two emission lines from different
upper states.
Boltzmann plot determined from the OH(8–3) P-branch lines shown in
the HiTIES spectrum from Fig. , taken on 23 December 2003 at
22:30 UT, with an integration time of 120 s. Blue symbols represent P1
lines, and red symbols are P2 lines. Crosses indicate intensities as
measured by the HiTIES instrument, whereas open circles have been corrected
for water vapour absorption. The solid black line is a linear fit to the
water-vapour-corrected P1 lines. The slope of this fit gives a rotational
temperature of T=188.7 K.
Values for the energy levels are taken from . There is a
discrepancy in the literature over transition probabilities that can affect
the temperature values obtained from Eq. () by up to ∼10-20 K . The most common sources are
, , and . It is
therefore important to be consistent with the set of transition probabilities
used whenever comparing temperature results. In the results shown in this
paper, we use the values from .
Water vapour transmission at wavelengths of OH(8–3) lines. Vacuum
wavelengths are from , with conversion to air
wavelengths undertaken as specified by . At each
wavelength, water vapour transmission is given by a function of precipitable
water vapour (PWV): Tλ=exp(-A×PWV), with PWV in
mm.
Water vapour transmission at wavelengths of OH(5–1) lines. Vacuum
wavelengths are from , with conversion to air
wavelengths undertaken as specified by . At each
wavelength, water vapour transmission is given by a function of PWV: Tλ=exp(-A×PWV), with PWV in
mm.
Figure shows a clear-sky spectrum taken with the HiTIES
instrument. Only OH airglow emission has been recorded; there are no auroral
emissions which, when present, can contaminate the OH lines, rendering the
measurement of OH line intensities more difficult. We have developed a
fitting routine with which we can determine the intensities of OH lines,
even when there is auroral contamination. A least squares fit is used,
involving four different components present in the measured spectrum: a constant
background, six OH emission lines, two auroral O+ doublets (at wavelengths
λ=7320.121 Å and 7319.044 Å and λ=7330.755 Å and 7329.675 Å; from ), and auroral
N2 1P(5–3) band emission, which can be modelled .
Water vapour transmission at wavelengths of OH(9–4) lines. Vacuum
wavelengths are from , with conversion to air
wavelengths undertaken as specified by . At each
wavelength, water vapour transmission is given by a function of PWV: Tλ=exp(-A×PWV), with PWV in
mm.
Water vapour transmission at wavelengths of OH(6–2) lines. Vacuum
wavelengths are from , with conversion to air
wavelengths undertaken as specified by . At each
wavelength, water vapour transmission is given by a function of PWV: Tλ=exp(-A×PWV), with PWV in
mm.
For the spectrum in Fig. , the OH line intensities can be
easily determined as there is no auroral contamination. We determine
temperatures using the three strongest lines: P1(2), P1(3), and
P1(4). Since there are three possible line ratios using these lines, we
obtain three temperature values. Applying Eq. (), we obtain
T(P1(2)/P1(3))=189.1 K,
T(P1(3)/P1(4))=183.7 K, and
T(P1(3)/P1(4))=180.0 K. The difference between these
three values is due to water vapour absorption that affects the P1(4)
lines, whereas P1(2) and P1(3) are much less affected. Correcting for
water vapour absorption using the modelling described in
Sect. , we obtain very similar values from all three line
ratios: T(P1(2)/P1(3))=188.7 K,
T(P1(3)/P1(4))=188.6 K, and
T(P1(3)/P1(4))=188.5 K.
In order to ensure that the airglow layer is in LTE, we produce a Boltzmann plot e.g.
for each temperature retrieval performed. This consists of plotting
lnI/A(2J+1) as a function of (hc/k)F
for each of the OH(8–3) P-branch lines measured by HiTIES. If
Eq. () is valid, i.e. the airglow layer is in LTE, it can be seen
that the plotted function should be linear and the slope is the rotational
temperature T. As an example, the Boltzmann plot for the HiTIES spectrum in
Fig. (measured on 23 December 2003 at 22:30 UT) is shown
in Fig. . The linear fit (solid black line) is determined
using the three P1 lines (blue symbols), since these have the highest
intensities (and thus lower associated errors). The points corresponding to the
P2 lines (represented by red symbols in Fig. ) follow
the same linear fit, indicating that the airglow layer is indeed in LTE at
this time. To assert that there is LTE, we use the same criteria as
, namely a variance between the fit and the P1 values
of less than 0.05 and between the fit and the P2 values of 0.3.
The effect of taking into account water vapour absorption in the OH line
intensities is also shown in the Boltzmann plot of Fig. .
Circles are used to represent water vapour-corrected OH intensities, whereas
crosses represent uncorrected intensities. From this figure it can be seen
that the lines most absorbed by water vapour are P1(4) and P2(2).
Without correcting the hydroxyl emission line intensities for this effect, we
would obtain a bad linear fit and a significant error on the rotational
temperature.
In the following section, Sect. , we detail the results of
water vapour absorption modelling, providing coefficients for scaling OH line
intensities to take this into account. More details on the resulting
temperatures determined from HiTIES measurements are shown in
Sect. .
Water vapour transmission at wavelengths of OH(8–4) lines. Vacuum
wavelengths are from , with conversion to air
wavelengths undertaken as specified by . At each
wavelength, water vapour transmission is given by a function of PWV: Tλ=exp(-A×PWV), with PWV in
mm.
We have determined water vapour transmission for hydroxyl emissions for the
strongest transitions in the (8–3) band
(Table ), the (5–1) band
(Table ), the (9–4) band
(Table ), the (6–2) band
(Table ), and the (8–4) band
(Table ).
Water vapour absorption lines originating in the troposphere are mainly
subject to pressure-broadening. Using the HITRAN model, we find that for a
temperature of about 300 K and a pressure of 1 bar (most H2O is
concentrated near the Earth's surface), the full width at half maximum of an
H2O line at a wavelength close to the OH(8–3)P1(4) line is about
0.1 Å. On the other hand, since OH emission is produced near the
mesopause, the principal broadening mechanism is Doppler broadening. The full
width at half maximum of such a Gaussian line profile is given by
Δλ=2λ02ln(2)kTmc2,
where λ0 is the centre wavelength of the line. Applying
Eq. () using the value of temperature determined in
Sect. from the HiTIES spectrum in Fig. ,
T=188.7 K, gives Δλ/λ0=2.38×10-6. Therefore, at the
wavelength of the most absorbed OH line in the HiTIES spectrum, P1(4), we
obtain a full width at half maximum of 0.018 Å. We consider that this is
sufficiently small compared to the width of the water vapour absorption lines
that we do not calculate the precise shape of the OH lines to determine the
H2O transmission values in
Tables –, instead only calculating the values at the line centre wavelength.
In addition, we do not take into account any Doppler shifts that may occur to
the centre wavelengths, or effects of non-vertical lines of sight. Thus, the
values given are only valid when observing in the zenith, assuming that any
vertical flows are small.
Water vapour transmission of the OH(8–3) lines, measured by HiTIES,
as a function of precipitable water vapour (PWV). The left panel shows the
Q-branch lines and the right panel, the P branch. Solid lines represent the
e component of the lambda-doubled
line, and the dashed lines represent the f
component.
Water vapour transmission of the OH(5–1) lines as a function of
precipitable water vapour (PWV). The left panel shows the Q-branch lines and
the right panel, the P branch. Solid lines represent the e component of the
lambda-doubled line, and the dashed lines represent the f component.
Water vapour transmission of the OH(9–4) P-branch lines as a
function of precipitable water vapour (PWV). Solid lines represent the e
component of the lambda-doubled line, and the dashed lines represent the f
component. The Q-branch transmissions are not represented for this band since
they are all equal to 1 within 0.01 % over the plotted range of
PWV.
The water vapour transmissions of OH(8–3) band emission lines, some of which
are observed by the HiTIES instrument (see Fig. ), are
plotted in Fig. . The two lambda-doubled components of
each line are shown, even though HiTIES does not have sufficiently high
wavelength resolution to be able to distinguish these. Of the lines observed
by HiTIES, it is mainly P1(4)e that is affected. For the case of the
HiTIES observation from 23 December 2003 at 22:30 UT, described in
Sect. , a value of PWV of 2.7 mm was
taken in order for the three hydroxyl temperatures (obtained from ratios
P1(2) / P1(3), P1(2) / P1(4), and
P1(3) / P1(4)) to match. This represents an absorption of 6.7 %
of the OH(8–3)P1(4)e line.
Plots of transmission as a function of PWV for lines from the OH bands
(5–1), (9–4), (6–2), and (8–4) are provided in
Figs. –. These plots show
that the (8–4) band is particularly affected by water vapour absorption.
In this study, we focus on the role of water vapour absorption; however, other
species may also participate in the attenuation of hydroxyl line emission. As
a test, we estimate the absorption by CO2 of OH line emission using a
similar method to that used for H2O absorption described in
Sect. . We use a profile of CO2 density with altitude from
, scaled to the high-latitude winter carbon dioxide column
density given by of 7.103×1021 molecules cm-2. The HITRAN database is then used to determine
CO2 absorption cross sections. We find that the transmission coefficients
at the centre wavelengths of all the OH lines included in
Tables to are
equal to 1. Thus, at the hydroxyl line wavelengths included in this study,
carbon dioxide does not appear to be a major absorber.
Water vapour transmission of the OH(6–2) lines as a function of
precipitable water vapour (PWV). The left panel shows the Q-branch lines and
the right panel, the P branch. Solid lines represent the e component of the
lambda-doubled line, and the dashed lines represent the f
component.
Water vapour transmission of the OH(8–4) lines as a function of
precipitable water vapour (PWV). The left panel shows the Q-branch lines and
the right panel, the P branch. Solid lines represent the e component of the
lambda-doubled line, and the dashed lines represent the f
component.
Hydroxyl temperatures
To test the transmission values determined in this paper, we incorporate
these into a calculation of OH(8–3) temperatures from spectral measurements
from the HiTIES instrument. The resulting temperatures are plotted in
Fig. . Twelve days of observations in December 2003 and
January 2004 are used to obtain temperatures, using the method described in
Sect. . The three strongest P-branch lines are used –
P1(2), P1(3), and P1(4) – meaning three temperature estimates can be
obtained for each measured spectrum, since one temperature determination
requires the ratio of two OH line intensities, as per Eq. ().
Without correction for water vapour absorption, the three mean daily
temperatures are different by 10 to 20 K, similar to the discrepancies found
by . This difference in temperature values is due to the
absorption of the P1(4) line by H2O (see Sect. ). The
daily mean temperatures without taking into account water vapour absorption
are shown in dashed coloured lines in Fig. a: in blue
for temperatures obtained by taking the ratio of P1(2) and P1(3), in
red for P1(2) and P1(4), and in yellow for P1(3) and P1(4).
To correct for the absorption of OH emission by water vapour, we use the
expressions and coefficients given in Table
for each of the P-branch lines used to determine temperatures. Since the
three OH lines used are from the same vibrational band, the emission for each
of them is produced at the same altitude. Thus, any difference in temperature
estimation obtained from different ratios of these lines' intensity is
assumed to be due only to differing amounts of water vapour absorption that
their emission undergoes before reaching the HiTIES detector. Therefore, the
column density of water vapour (or PWV) is chosen such that the three
temperature determinations give the same result.
In this fashion, we obtain the temperatures and PWV values plotted in black
in Fig. . Here we show only temperatures obtained
during clear-sky periods, i.e. with no auroral emission present. Each
temperature estimation (represented by a black point in
Fig. a) is calculated with OH(8–3) line intensities
measured from a spectrum composed of 20 min of HiTIES observations. The
corresponding values of PWV for each of these temperature calculations are
shown by the black points in Fig. b. The daily mean of
all of these values is shown by the black solid lines in both panels.
Panel (a): OH(8–3) temperatures obtained using HiTIES.
Each black point is obtained from a spectrum composed of 20 min of
integration time. The solid black line is the daily mean. The coloured dashed
lines represent the daily mean temperatures obtained when water vapour
absorption is not taken into account. Temperatures obtained from the line
ratio P1(2) / P1(3) are shown in blue, those from
P1(2) / P1(4) are in red, and those from P1(3) / P1(4)
are in yellow. Panel (b): precipitable water vapour (PWV) obtained
during each temperature calculation shown in panel (a), represented
by black points. The daily mean of these is plotted in a solid black line.
The red points represent PWV determined from the humidity profiles measured
by the daily Ny-Ålesund radiosonde measurements .
These data were recorded in December 2003 and
January 2004.
The daily mean OH rotational temperature shows a variation of around 20 K
over the course of a 2- to 3-day period. This is similar to what has
previously been observed by and in OH
airglow temperature measurements in Antarctica. These studies concluded that
such rapid changes in OH rotational temperatures were due to changes in the
meridional component of the mesospheric wind, caused by gravity waves. It is
possible that such a mechanism is also responsible for the time variation of
the OH temperatures that we measure here. Such changes in OH temperatures
over a few days have also been seen in the Arctic
.
In panel b of Fig. , we compare the values of PWV
obtained during the OH temperature determination (black points) with those
calculated from the daily radiosonde measurements taken from Ny-Ålesund
. These radiosonde measurements match well the daily
mean PWV obtained from HiTIES from 22 to 28 December. Thereafter, the
HiTIES PWV values rise more steeply than those from the radiosonde. However,
there is a corresponding large rise in the radiosonde PWV on
2 January 2004. The difference between the two measurements is not unexpected
given that Ny-Ålesund and the site where HiTIES is located are 124 km
apart, and that the sonde will drift in latitude and longitude as it rises
with altitude and takes its measurements.
We also note the potential presence in the PWV estimations of a diurnal
cycle, such as that seen by over North America. This trend in
atmospheric water vapour content, as well as further study of OH temperatures
obtained through the methods described in this paper, will be the topic of
future studies.
Conclusions
We have modelled the absorption by water vapour of OH airglow emission in a
number of different Meinel bands and tabulated the corresponding transmission
coefficients. Water vapour absorption coefficients from the HITRAN2012
database were used. A number of OH vibrational bands have been chosen to
apply this calculation to (8–3), (5–1), (9–4), (8–4), and (6–2);
however, on request, we can determine water vapour absorption in other bands,
as needed. Knowledge of these transmission coefficients allows the
determination of accurate hydroxyl rotational temperatures.
Validation of the water vapour transmission coefficients in the OH(8–3) band
has been carried out by applying them to 12 days of observations with the
HiTIES instrument, located at the Kjell Henriksen Observatory, Svalbard.
Without taking into account the effect of water vapour absorption on the OH
lines, there was a discrepancy of about 10 to 20 K between temperatures
obtained from different line intensity ratios. Using the transmission
coefficients produced in this study allowed for this discrepancy to be
eliminated.
A by-product of the method described in this paper is the possibility to
obtain water vapour column densities. Using the HiTIES instrument, these can
be determined at much higher cadence than using a radiosonde. Additionally,
we always measure the same column of atmosphere, whereas a radiosonde will
drift horizontally as it rises to take its measurements. Water vapour column
densities determined from HiTIES observations in December 2003 have been
shown to agree with values obtained from nearby radiosonde measurements.
For this study we have been limited to applying the water vapour transmission
coefficients to the (8–3) band emission lines that have previously been
measured with HiTIES. However, we have now developed a new mosaic filter
which in future will allow OH(5–1) and OH(9–4) bands to be observed by the
instrument. Future papers will further explore the trends in temperature and
water vapour concentration that have been observed by HiTIES across different
OH bands.
Recent quick-look keograms from the Spectrographic
Imaging Facility (SIF) are available at http://sif.unis.no. To access any data, please contact the authors.
The authors declare that they have no conflict of
interest.
Acknowledgements
J. M. Chadney, D. K. Whiter, and B. S. Lanchester are funded by the United
Kingdom Natural Environment Research Council (NERC) under grant NE/N004051/1.
The authors acknowledge the support of the Atmospheric Physics Group at
University College London (UCL) in the operation of the Spectrographic
Imaging Facility (SIF) and J. Sullivan, A. Stockton-Chalk, J. Holmes, and
M. Dyrland in running the instruments during the winter campaign in 2003/4.
We would also like to thank the staff at the University Centre in Svalbard
(UNIS) for their support and the use of their facilities. The SIF was funded
by the United Kingdom Particle Physics and Astronomy Research Council (PPARC)
and NERC, and is a joint project between UCL and the University of
Southampton. The topical editor,
C. Jacobi, thanks F. Sigernes and one anonymous referee for help in
evaluating this paper.
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