# A solar source of Alfvénic magnetic field switchbacks:
in situ remnants of magnetic funnels on supergranulation scales

S. D. Bale Physics Department, University of California, Berkeley, CA
94720-7300, USA  Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA  T. S. Horbury The Blackett Laboratory, Imperial
College London, London, SW7 2AZ, UK  M. Velli Earth Planetary and Space
Sciences, UCLA, CA 90095, USA  M. I. Desai Southwest Research Institute, 6220
Culebra Rd, San Antonio, TX 78238, USA  3University of Texas at San Antonio,
San Antonio, TX 78249, USA  J. S. Halekas Department of Physics and Astronomy,
University of Iowa, Iowa City, IA 52242, USA  M. D. McManus Physics
Department, University of California, Berkeley, CA 94720-7300, USA  Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
O. Panasenco Advanced Heliophysics, Pasadena, CA 91106, USA  S. T. Badman
Physics Department, University of California, Berkeley, CA 94720-7300, USA
Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450,
USA  T. A. Bowen Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA  B. D. G. Chandran Department of Physics &
Astronomy, University of New Hampshire, Durham, NH 03824, USA  Space Science
Center, University of New Hampshire, Durham, NH 03824, USA  J. F. Drake
Department of Physics, University of Maryland, College Park, MD 20742, USA
Institute for Physical Science and Technology, University of Maryland, College
Park, MD 20742, USA  Joint Space Science Institute, University of Maryland,
College Park, MD 20742, USA  J. C. Kasper BWX Technologies, Inc Washington DC
20002  Climate and Space Sciences and Engineering, University of Michigan, Ann
Arbor, MI 48109, USA  R. Laker The Blackett Laboratory, Imperial College
London, London, SW7 2AZ, UK  A. Mallet Space Sciences Laboratory, University
of California, Berkeley, CA 94720-7450, USA  L. Matteini The Blackett
Laboratory, Imperial College London, London, SW7 2AZ, UK  T. D. Phan Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
N. E. Raouafi The Johns Hopkins Applied Physics Laboratory, Laurel, MD 20723,
USA  J. Squire Physics Department, University of Otago, Dunedin 9010, New
Zealand  L. D. Woodham The Blackett Laboratory, Imperial College London,
London, SW7 2AZ, UK  T. Woolley The Blackett Laboratory, Imperial College
London, London, SW7 2AZ, UK

###### Abstract

One of the striking observations from the Parker Solar Probe (PSP) spacecraft
is the prevalence in the inner heliosphere of large amplitude, Alfvénic
magnetic field reversals termed ’switchbacks’. These $\delta
B_{R}/B\sim\mathcal{O}(1$) fluctuations occur on a range of timescales and in
patches separated by intervals of quiet, radial magnetic field. We use
measurements from PSP to demonstrate that patches of switchbacks are localized
within the extensions of plasma structures originating at the base of the
corona. These structures are characterized by an increase in alpha particle
abundance, Mach number, plasma $\beta$ and pressure, and by depletions in the
magnetic field magnitude and electron temperature. These intervals are in
pressure-balance, implying stationary spatial structure, and the field
depressions are consistent with overexpanded flux tubes. The structures are
asymmetric in Carrington longitude with a steeper leading edge and a small
($\sim$1∘) edge of hotter plasma and enhanced magnetic field fluctuations.
Some structures contain suprathermal ions to $\sim$85 keV that we argue are
the energetic tail of the solar wind alpha population. The structures are
separated in longitude by angular scales associated with supergranulation.
This suggests that these switchbacks originate near the leading edge of the
diverging magnetic field funnels associated with the network magnetic field -
the primary wind sources. We propose an origin of the magnetic field
switchbacks, hot plasma and suprathermals, alpha particles in interchange
reconnection events just above the solar transition region and our
measurements represent the extended regions of a turbulent outflow exhaust.

solar physics — solar wind — solar corona

††journal: Astrophys. J.

##  1 Introduction

The NASA Parker Solar Probe (PSP) mission (Fox et al., 2016) was executed to
make direct in situ measurements of the source regions of the solar wind and
to understand the heating and acceleration mechanisms using those
measurements. A foundational observation from PSP has been the prevalence of
large $\delta B_{R}/B\sim\mathcal{O}(1)$ Alfvénic field reversals that had
earlier been termed ’switchbacks’ (hereafter ’SBs’), which are interspersed
among intervals of ’quiet’ primarily radial magnetic field (Bale et al., 2019;
Kasper et al., 2019; Horbury et al., 2020; Dudok de Wit et al., 2020). While
the SBs had been identified previously using spacecraft measurements at 0.3 AU
(Horbury et al., 2018), at 1 AU (Gosling et al., 2009, 2011) and at high solar
latitudes (Balogh et al., 1999; Matteini et al., 2014), the PSP measurements
are notable for the abundance of events and the apparent patterns of SBs and
quiet wind. Several models of SB generation and evolution have been developed
recently with sources in velocity shear in the corona (Landi et al., 2005;
Ruffolo et al., 2020; Schwadron & McComas, 2021), magnetic reconnection (Fisk
& Kasper, 2020; Drake et al., 2021; Zank et al., 2020), impulsive energy
injection within magnetic funnels (Magyar et al., 2021a, b), and the nonlinear
radial evolution of large amplitude Alfvénic fluctuations (Landi et al., 2006;
Squire et al., 2020; Mallet et al., 2021; Shoda et al., 2021). A key question
is whether the SBs are generated at the source of the solar wind, and serve as
a fundamental diagnostic of the energization mechanism, or if rather they are
a product of radial evolution.

Because of their ubiquity, and apparent nonlinearity, the possibility that
switchbacks provide a direct, in situ diagnostic of solar wind heating or
energization is tantalizing. A variety of heating mechanisms have been
proposed: nanoflares, wave-dissipation, footpoint shearing, magnetic
reconnection, and turbulence, to name a few. Most of these theories find some
support in remote-sensing or in situ data and many of them rely in impulsive
magnetic activity at the coronal base. There are many good review papers on
the topic, recently by Cranmer & Winebarger (2019).

Here we use PSP measurements to demonstrate that an interval of periodically-
modulated SBs observed below 25 solar radii ($R_{S}$) of PSP Encounter 06 is
related directly to underlying photospheric magnetic field concentrations.
While the bulk radial proton (’solar wind’) speed here ranges from 200-400
km/s, these measurements are akin to the ’microstream’ structures measured
previously in the fast solar wind (Neugebauer et al., 1995). However these
intervals are clearly pressure-balanced, hence spatially stable, and show
enhanced alpha particle abundance, hot proton beams, energetic particles, and
depressed electron temperature - all suggestive of fast wind-like sources. The
alpha particles are heated to $T_{\alpha}\geq$ 8$T_{p}$ and the temperature
anisotropies of the alpha particles and beam protons suggest a common origin
and/or evolution. The ballistically-mapped longitudinal source structure and
the presence of suprathermal ions suggests a source at the edge of magnetic
funnels associated with interchange reconnection. A narrow ($\sim$1∘) leading
edge shows enhanced proton parallel heating, similar to measurements from
Helios at the leading edge of high speed streams (Marsch et al., 1982).

We present a schematic scenario of the source suggesting that PSP is
transiting through the coronal extension of magnetic funnels (Kopp & Kuperus,
1968; Gabriel, 1976; Dowdy et al., 1986; Tu et al., 2005) and possibly plumes
(Wilhelm et al., 2011; Poletto, 2015) associated with the photospheric network
magnetic field, where the Alfvénic switchbacks are generated. The underlying
magnetic configuration should be favorable to interchange magnetic
reconnection between adjacent funnels and/or closed loop structure above the
photosphere, as developed in the furnace solar wind model of Axford et al.
(1999). Reconnection would also explain the energetic ions and the inherent
intermittent nature of the switchbacks. Although we highlight the impact of
processes and structure in the low solar atmosphere, some properties of SBs
may develop or be further amplified by in situ evolution in the solar wind,
which increases $\delta B_{R}/B_{0}$ and can lead to abrupt rotations of B
(e.g. Squire et al., 2020; Shoda et al., 2021; Mallet et al., 2021).

##  2 Parker Solar Probe Measurements and context

We use magnetic field and electron density measurements from the FIELDS
instrument (Bale et al., 2016), plasma ion and electron measurements from the
Solar Wind Electrons Alphas and Protons (SWEAP) instrument suite (Kasper et
al., 2016), and energetic ion measurements from the Integrated Science
Investigation of the Sun (ISOIS) suite (McComas et al., 2016) on the NASA
Parker Solar Probe (PSP) spacecraft (Fox et al., 2016). Measurements are made
near perihelion of PSP Encounter 06 from September 27-28, 2020 when the
spacecraft was between 20.4-22.7 solar radii ($R_{S}$) from the Sun center and
232∘-270∘ heliographic (HG) longitude; the spacecraft trajectory dips south of
the ecliptic near perihelion and was below -3∘ HG latitude during this
interval and stayed below the heliospheric current sheet (HCS). Perihelion at
20.39 $R_{S}$ occurred at 09:16 UT on September 27, 2020. The spacecraft
tangential (Keplerian) speed in HG coordinates ranges from 87 km/s at the
start of the interval to 68 km/s at the end, passing through the maximum 89
km/s at perihelion. The spacecraft radial speed ranges from -11 km/s at the
start of the interval to 39 km/s at the end, passing through the 0 km/s at
perihelion. The measurements we analyze here are primarily in the outbound leg
of the orbit.

Figure 1: Magnetic field connectivity during PSP Encounter 06. The perihelion
loop is seen between 210∘ and 300∘ Carrington (heliographic) longitude. A PFSS
model (described in the text) maps PSP magnetic connection (in 12 hour
intervals) from the source surface at 2.2 Rs to the height 1.02 Rs above the
photosphere. A solid black line shows the model neutral line on the source
surface. Black contours indicate magnetic field pressure at 1.02 Rs ($\sim$ 14
Mm) showing the spatial scales associated with network magnetic field and
supergranulation cell boundaries. During the interval studied here, the source
surface was connected to the southern coronal hole boundary near -60∘
Carrington latitude. The ballistic projection of the PSP trajectory (blue
diamonds) on the source surface (blue crosses) and down to the solar wind
source regions (blue circles) calculated for the height R = 1.02 $R_{S}$ and
measured in situ solar wind speed $\pm$ 80 km/s. Open magnetic field regions
are shown in blue (negative) and pink (positive). The $B^{2}$ map is
calculated for 27 Sep 2020, different PSP positions along the trajectory shown
for selected days are displayed under the blue diamonds with a time cadence of
12 hours during Sep 26-29, 2020.

Figure 1 shows the magnetic connectivity of the PSP spacecraft footpoints from
the source surface at $R_{SS}$ = 2.2 $R_{S}$ mapped down the solar surface,
near the low latitude boundary of a coronal hole (CH) feature at -60∘ HG
latitude. The mapping uses a Potential Field Source Surface (PFSS) model
(Schatten et al., 1969; Altschuler & Newkirk, 1969; Hoeksema, 1984) as
implemented by Schrijver & De Rosa (2003). PFSS has proved to be remarkably
robust during early PSP (solar minimum) orbits (Bale et al., 2019; Badman et
al., 2020; Panasenco et al., 2020). As a lower boundary condition, the PFSS
model incorporates magnetic field maps produced by an evolving surface-flux
transport model based on magnetic fields observed by the Helioseismic and
Magnetic Imager (HMI) (Scherrer et al., 2012; Schou et al., 2012) on the Solar
Dynamics Observatory (SDO). The model evolves these fields in accordance with
empirical prescriptions for differential rotation, meridional flows, and
convective dispersal processes. PFSS models are parametrized by a radial
distance at which all magnetic field lines become open and radial: the source
surface (SS) height. This is typically set at $R_{SS}$ = 2.5 $R_{S}$, but is a
free parameter of the model and we choose $R_{SS}$ = 2.2 $R_{S}$ to be
compatible with in situ magnetic field measurements by PSP and Solar Orbiter
for this time interval (Telloni et al., 2021).

Black contours in Figure 1 indicate magnetic field pressure at 1.02 $R_{S}$
($\sim$14 Mm altitude above the photosphere) and reveal a network of stronger
magnetic field concentrations, which can be inferred up to $\sim$1.04 $R_{S}$,
or 28-30 Mm and then quickly dissipate higher in the corona due to magnetic
field line expansion. This low altitude, about 10 times the height of the
chromosphere, is found to be critical in filament channel formation and
dynamics (Panasenco & Velli, 2009) and is comparable to a typical
supergranulation cell diameter. The boundary of the southern coronal hole
along which PSP footpoints were moving has a sequence of these magnetic
concentrations (nodes) with much weaker field between; we estimate from PFSS
that PSP crossed 6-8 strong field concentrations during the Sep 26-29 interval
studied here. Note that this PSP perihelion occurred behind the limb, so that
there were no current magnetogram data available for the longitudes below the
spacecraft. Hence there is no expectation of a one-to-one correspondence
between individual network field concentrations seen in the PFSS, which evolve
on timescales of several hours (Rieutord & Rincon, 2010), and our in situ
measurements.

Proton beam and core parameters are obtained by fitting drifting bi-
Maxwellians to the SWEAP/SPAN-Ion proton spectra. The proton beam is
constrained to lie along the magnetic field direction relative to the core
velocity. For the alpha parameters, 4 successive spectra were first summed
together to obtain better statistics. The SPAN-Ion alpha channel contains a
small ($\sim 2\%$) proton contamination. This is compensated for by taking the
previously fitted proton parameters and scaling the total density to represent
the spurious protons in the alpha channel. This scaled down function, as well
as a single bi-Maxwellian to represent the alpha particles, are then fitted to
the alpha channel counts spectra and the alpha core parameters obtained. The
exact scaling factor from proton channel to alpha channel is a free parameter
in the fit, and it was confirmed that there was no energy or angle dependence
to the proton contamination, so that an overall simple scaling was sufficient.
Distribution functions are fitted only when the solar wind distribution is
within the SPANi field-of-view (e.g. Woodham et al., 2021). For a more
detailed discussion of SPAN-Ion fitting procedures, see (Finley et al., 2021).
SWEAP/SPAN-E measurements (Kasper et al., 2016; Whittlesey et al., 2020) are
used to determine electron core and strahl parameters by a combination of
fitting and partial moment calculations, as described in Halekas et al.
(2020). The electron core component is fitted to a drifting bi-Maxwellian
function and the halo component, where present at measurable levels, to an
isotropic non-drifting Maxwellian function. Partial moments of the strahl are
computed by integrating over the residual of the measured distribution with
respect to the core (and halo, if present), for the portion of the
distribution in the strahl direction, within 45∘ of the magnetic field, and
for velocities greater than two thermal speeds.

Figure 2: Measurements of the suprathermal (314 eV) electron PAD (panel a),
magnetic field magnitude (b), plasma density (c), radial magnetic field
$B_{R}$ (d), and radial proton core velocity $V_{R}$ (e) during the PSP
Encounter 06 perihelion. Red horizontal arrows highlight the magnetic field
compressions/depletions, velocity modulations and the accompanying modulations
in the switchback occurrence and amplitude. The red and green bars in panel
(d) are the 10-minute minimum and maximum values of $B_{R}$. The
ballistically-mapped Carrington longitude is shown at the bottom: these
modulations occur on angular scales of a few degrees. The outward electron
strahl flux implies that the field geometry is uniform outward.

Magnetic field and plasma measurements from PSP Encounter 06 are shown in
Figure 2. A normalized (to 90∘) pitch angle distribution (PAD) of 314 eV
suprathermal (strahl) electrons after $\sim$ 01:00 on September 27, 2020 show
primarily outward flux along $-B_{R}$ indicating a unipolar magnetic field
geometry below and consistent with the PSP trajectory being below the HCS as
implied in Figure 1. Periodic modulations of the magnetic field magnitude
$|B|$ (panel b) and radial proton core velocity $V_{R}$ (panel e) are
emphasized using red arrows. Panel (d) shows the radial component of the
magnetic field $B_{R}$ and the minimum (green) and maximum (red) value
envelopes in 10 minute intervals. The $B_{R}$ dynamics are dominated by
magnetic field ’switchbacks’ (Bale et al., 2019; Horbury et al., 2020) that
here are clearly modulated in occurrence and amplitude by the structure in
$|B|$ and $V_{R}$. Small plasma density enhancements are also measured for
some of these intervals (panel c). It has been reported previously that the
Alfvenic magnetic field switchbacks are interspersed in regions of ’quiet
radial flow’ (Bale et al., 2019; Horbury et al., 2020); here at 20 $R_{S}$
they are clearly modulated periodically and in correspondence with field
amplitude and proton radial flow. Along the bottom of Figure 2 the spacecraft
position in Carrington (HG) longitude is shown, as mapped ballistically using
the measured proton speed (e.g. Nolte & Roelof, 1973; Badman et al., 2020).
The time interval of the modulated features (red arrows) corresponds to
Carrington longitudes of $\sim$ 250∘ \- 270∘ in Figure 1. The modulations of
switchbacks and plasma parameters occur on angular scales of a few degrees,
similar to the network magnetic field and supergranulation scales on the Sun
as in Figure 1.

Figure 3 highlights the interval of modulated switchbacks between
2020-09-27/01:00:00 and 2020-09-28/08:00. The top panel shows ion flux
measurements from ISOIS in 3 energy channels from the EPI-Lo time-of-flight
(TOF) system between 30-84 keV. An examination of the triple coincidence data
shows that these TOF-only measurements are likely to be dominated by a mix of
He and/or O ions and are field-aligned (anti-sunward). Note that these
suprathermal ions appear most clearly within the last two modulations (in
yellow bars), which also contain higher speed flow (panel (e)), suggesting
that these ions may be the suprathermal tail of the modulated solar wind
distribution. Panel (b) is the measured alpha particle abundance
$A_{He}=n_{\alpha}/n_{total}$ showing strong modulations to relatively large
values ($\sim 4\%$) associated with the switchback patches and magnetic field
amplitude modulations in panel (c). The dashed line in Panel (c) indicates the
nominal heliospheric field of 2.5 nT AU2 (Badman et al., 2021) and the
depressed intervals are colored blue. Light yellow vertical panels indicate
intervals with enhanced $A_{He}$ and depressed $|B|$. Alpha particle
temperature $T_{\alpha}$ (green) and proton core temperature $T_{p}$ (red) are
enhanced within these intervals and electron core (lower blue) and strahl
(upper blue) temperatures are depressed; note that $T_{\alpha}/T_{p}\approx
4-15$ as will be discussed below. Plasma radial velocities (panel e) are
enhanced within the structures with the core proton speed (blue) approaching
the Alfvén speed (black) on the edges and Alfvén Mach $M_{A}\approx$ 2 within.
The alpha (green) and proton beam (red) speeds are a large fraction of an
Alfvén speed. The inverse correlation of electron temperature to wind speed
$T_{e}\propto 1/v_{sw}$ is well known and observed quite universally during
PSP solar encounters (e.g. Maksimovic et al., 2020).

Figure 3: Measurements of the suprathermal ion number flux (panel a), alpha
particle abundance $A_{He}$ (b), scaled magnetic field magitude $|B|R^{2}$,
electron core and strahl temperatures (blue), core proton temperature (red),
and alpha temperature (green) in panel (d), radial velocities (core proton =
blue, alpha = green, proton beam = red) and Alfven speed (black) in panel (e).
Number flux of core protons (blue), alphas (green) and beam protons (red) is
in panel (f) and plasma pressures (core proton = red, alpha = green, electron
= blue) and magnetic field energy density (black) in panel (g). Panel (h) is
the total pressure, plasma and magnetic, scaled by its fitted radial scaling
$p\propto r^{-3.17}$. The bottom panel shows plasma beta (red = core proton,
green = alpha, blue = core electron). The intervals of depressed field (panel
b), which correspond to enhanced switchbacks, have enhancements in alpha
abundance $A_{He}$ consistent with fast solar wind values. Likewise, the
intervals of enhanced switchbacks have higher speed radial wind speeds (panel
d) with (core) Mach $\sim$ 1 near the edges but higher Mach flows in the
center. Ion temperatures are higher within these structures and electron
temperatures are slightly lower. Alpha abundance, speed and temperature
measurements suggest that the switchback intervals are more like ’fast wind’.
Panel (g) demonstrates that the set of structures is in local pressure-balance
consistent with spatially stable structure on the spacecraft transit
timescale.

Panel (f) in Figure 3 show the number flux of core protons (blue), beam
protons (red) and alpha particles (green). The core proton flux is relatively
steady, while the proton beam and alpha particle flux are more modulated by
the plasma structure; the proton beam abundance (not shown) is not so clearly
modulated, rather the proton beam speed is enhanced within the high alpha
abundance structures. Magnetic field pressure (black) and plasma pressure are
shown on Panel (g), with proton (red), core electron (blue) and alpha (green)
all showing anti-correlation with the magnetic field pressure. This is best
seen in Panel (h) where the total pressure (magnetic plus plasma) is shown,
normalized to a slowly varying factor to remove the large-scale radial trend -
these structures are in local pressure balance. Pressure-balanced structure
(PBS) is seen throughout the solar wind in the inner heliosphere (Thieme et
al., 1990), at 1 AU (Borovsky, 2016), and on Ulysses (McComas et al., 1995,
1996) and Reisenfeld et al. (1999) reported an association between PBS and
enhanced alpha particle abundance. Here we take this as evidence that these
structures are spatial and stable over the spacecraft transit time, at least.
The lower panel shows plasma $\beta$ for proton (red), alphas (green), and
electrons (blue) demonstrating the ion $\beta$ is enhanced inside the
structures. In addition to the measurements shown in Figure 3, these intervals
show modulations of the proton core temperature anisotropy
$T_{\perp}/T_{\parallel}$ that often exceed the anisotropy-driven ion
cyclotron instability threshold (e.g. Hellinger et al., 2006), relatively
large proton beam speeds and alpha-proton drifts (comparable to $v_{A}$) and
depressed $T_{e}/T_{p}$ and the interval is richly populated with ion
cyclotron-frequency waves (Bowen et al., 2020; Verniero et al., 2020).
Enhanced proton-alpha drifts within PBSs have been reported previously by
Yamauchi et al. (2004). The kinetic features measured here are also associated
with enhanced electrostatic plasma waves and will be investigated more fully
in a future study.

In summary, Figure 3 demonstrates that PSP is transiting over spatial plasma
structures that are a few degrees of HG longitude across mapped to the Sun.
The structures contain hotter and faster ions (protons and alpha particles), a
markedly enhanced alpha particle abundance $A_{He}$, suprathermal ions,
depressed magnetic field $|B|$ and electron temperature $T_{e}$ and a clear
increase in the amplitude and occurrence of magnetic field switchbacks. Since
the alpha abundance, and arguably the electron temperature, are frozen-in from
the solar wind source in the transition region, we argue that the suprathermal
ions have an origin at these altitudes and that the physics of these solar
wind sources organizes the spatial and temporal distribution of the
switchbacks.

Histograms of ion and electron temperatures are shown in Figure 4, normalized
to unity maximum and characteristic values are collected in Table 1. The
cross-hatched histograms are accumulated within the high-alpha structures and
the beige histograms are from outside the structures. The proton and alpha
populations are generally hotter within the magnetic structures while both
core and strahl electrons are slightly cooler, reminiscent of fast solar wind.
Variations from structure-to-structure mask some of the trends in the
unnormalized temperature that become more apparent in the dimensionless ratios
in Figure 5.

Figure 4: Distribution of temperatures (normalized with the peak to one) for alpha particles, beam protons, core protons, and core and strahl electrons, both perpendicular to the magnetic field (upper row) and parallel (lower row). The hatched histograms are within the funnel structures and the beige histograms are outside of the funnels. Temperatures are given in electron volts (lower axis) and MKelvin (upper axis). Protons and alpha particles are all hotter within the funnel structures, as seen in Figure 3 as well. Electron $T_{\perp}$ is similar inside and out, with electron $T_{\parallel}$ slightly enhanced within the funnels. Characteristic values are given in Table 1 |  | $\tt{inside/outside}$ |  |  | $\tt{inside/outside}$ |
---|---|---|---|---|---|---
Species |  $\langle$ T${}_{\perp}\rangle$ |  $Mo($T${}_{\perp})$ |  $\sigma($T⟂) |  $\langle$ T${}_{\parallel}\rangle$ |  $Mo($T${}_{\parallel})$ |  $\sigma($T∥)
He++ | 501/335 | 360/305 | 171/79 | 570/400 | 464/340 | 247/241
Beam H+ | 114/60 | 75/47 | 96/25 | 117/81 | 105/74 | 54/40
Core H+ | 78/47 | 58/40 | 37/13 | 30/21 | 17/21 | 23/8
Core e- | 39/40 | 36/39 | 6.2/6.3 | 45/43 | 39/41 | 6.7/6.6
Strahl e- | n/a | n/a | n/a | 217/215 | 216/220 | 15/15
Table 1: Average, mode, and standard deviation of the distribution of T⟂ and
T∥ in units of eV, inside and outside of the funnel structures
(inside/outside). Figure 5: Distribution of temperatures (upper panel) and
temperature anisotropy (low panel) for alpha particles (black), beam protons
(red), core protons (green) and core electrons (blue). The alpha particle
total temperature shows clear evidence of heating above mass ratio
$T_{\alpha}/T_{p,c}>m_{\alpha}/m_{p}=4$ and peaks near
$T_{\alpha}/T_{p,c}\approx$ 8\. The distribution of proton beam temperatures
peaks at near $T_{p,b}/T_{p,c}\approx$ 2 and the core electron population is
slightly cooler than the protons $T_{e,c}/T_{p,c}<$ 1\. While the core protons
are strongly anisotropic $T_{\perp}/T_{\parallel}\geq$ 2, the alpha particles,
beam protons, and electrons all show slightly enhanced parallel temperatures
$T_{\perp}/T_{\parallel}\lesssim$ 1\. Note that the distributions of alpha and
beam proton temperature anisotropies are very similar suggesting a common
heating and/or evolution mechanism.

Stansby et al. (2019b) compared Helios measurements of alpha and proton
temperatures down to 0.3 AU to expectations from double-adiabatic theory (Chew
et al., 1956) (or CGL). They found that $T_{\alpha,\parallel}$ decreases
faster with radial distance than predicted by CGL, while $T_{\alpha,\perp}$
decreases more slowly, consistent with isotropization associated with
temperature anisotropy instabilities (Hellinger et al., 2006; Bale et al.,
2009; Maruca et al., 2012). Our average values within the funnel intervals at
$\sim$ 20$R_{S}$ ($\sim$ 0.1 AU) of $\langle
T_{\alpha,\parallel}\rangle\approx$ 570 eV (6.6 MK), $\langle
T_{\alpha,\perp}\rangle\approx$ 501 eV (5.8 MK), $\langle
T_{p,\parallel}\rangle\approx$ 30 eV (0.35 MK), and $\langle
T_{p,\perp}\rangle\approx$ 78 eV (0.9 MK) are consistent with the trends in
Figure 1 of Stansby et al. (2019b) and $\langle T_{\alpha,\perp}\rangle$ seems
to trend with their CGL curve. In Figure 5 we show the distribution of
relative (to proton core) temperatures (top panel) and the temperature
anisotropy $T_{\perp}/T_{\parallel}$ for each population, over the interval.
Notably, the alpha particles are heated substantially relative to the core
protons with $T_{\alpha}$ peaking near $T_{\alpha}\sim$ 8 $T_{p,c}$, well
above, even twice, the mass-proportional heating $T_{\alpha}\sim$ 4 $T_{p,c}$
rate. More-than-mass proportional heating has been measured in the inner
heliosphere previously (Marsch et al., 1982; Gershman et al., 2012), and at 1
au (Kasper et al., 2017) and here we identify it with discrete solar wind
sources and report, we believe, one of the largest ever heating fractions
$T_{\alpha}\gtrsim$ 10 $T_{p,c}$ for thermal alpha particles in the solar
wind. Notably, there is no thermalized population $T_{\alpha}\sim T_{p}$.
Indeed, Maruca et al. (2013) used measurements of $T_{\alpha}/T_{p}$ at 1 AU
and a model of collisional evolution to infer large temperature ratios in the
inner heliosphere; their model removed the 1 AU isotropic population
$T_{\alpha}\sim T_{p}$ and the distribution was predicted to peak at
$T_{\alpha}\sim$ 5.4 $T_{p}$ at 0.1 AU ($\sim$ 21 $R_{S}$). If robust and
general, our results imply source of isotropization in addition to collisions,
presumably wave-particle effects (e.g. Klein et al., 2018). Note that alpha-
to-proton temperature ratios of $\sim$10 were predicted at 15 $R_{S}$ by
Chandran (2010) based on stochastic ion heating by low frequency MHD/KAW
turbulence.

The proton beam population is heated to $T_{p,b}\sim 2~{}T_{p,c}$, while the
core electrons are cooler than the core protons $T_{e,c}\lesssim T_{p,c}$ as
observed for faster solar wind sources (Marsch et al., 1989; David et al.,
1998; Maksimovic et al., 2020). The lower panel of Figure 5 shows the
temperature anisotropy $T_{\perp}/T_{\parallel}$ of the core electron (blue),
core proton (green), beam proton (red) and alpha particle (black) populations.
The core proton population is notably more anistropic; the proton beam and
alpha particle populations show similar distributions of temperature
anisotropy. This may be a result of a common heating or energization
mechanism, or rather may indicate similar physics during radial evolution.
Since the beam protons and alpha particle stream ahead of the core protons,
these populations will interact differently with Alfvénic fluctuations
propagating in the wave frame with respect to the bulk (core proton) solar
wind.

Figure 6: Energy spectra in units of differential number flux during the
interval within the last two yellow-highlighted funnel structures of Figure 7
(labeled N4 and N5 there). A dotted line with spectral index $\sim-11$ extends
between the SPANi alpha and ISOIS measurements; we suggest that the ISOIS ions
are the suprathermal tail of the solar wind alpha spectrum.

Figure 6 shows energy spectra of protons and alpha particles measured by SPANi
compared with the ISOIS EPI-Lo spectrum; these spectra are in units of
differential number flux $j$ and averaged over the intervals
2020-09-27/21:28:00 - 2020-09-28/00:10:00 (left) and 2020-09-28/02:44:30 -
2020-09-28/04:55:30 (right), corresponding to the final two yellow boxes in
Figure 3 and labeled N4 and N5 in Figure 7. The ISOIS suprathermal spectrum is
shown assuming a spectrum of He++ (as described above); if the ISOIS spectrum
is oxygen the fluxes will be a factor of 4 higher. While suprathermal He++ is
observed routinely in the ecliptic, slow solar wind (e.g. Collier et al.,
1996) most usually associated with shocks or CIRS (or pickup ions), Ulysses
measurements show a negligible ambient suprathermal He++ population in the
fast solar wind (Gloeckler & Geiss, 1998). Our measurements here, with the
striking correlation in Figure 3 and the plausible continuity of the spectrum
in Figure 6, suggest that the EPI-Lo ion measurements are the tail of a very
steep alpha particle spectrum out to 85 keV. A dotted line in Figure 6
connects the SPANi and EPI-Lo spectra and has a spectral slope $j\sim
E^{-11}$. There is no notable radio emission nor flaring activity at this
time.

To infer the spatial structure at the origin, we project the PSP spacecraft
location in HG longitude down to the solar surface using a ballistic mapping
(Nolte & Roelof, 1973; Badman et al., 2020) in Figure 7. The $T,N$-plane
magnetic field (upper panel) and proton velocity (lower panel) vectors are
computed as 10-minute statistical mode values, in an attempt to remove the
dominant and rapidly varying switchback fields. A time series plot (not shown)
suggests that this is largely successful and the vectors shown here represent
larger scale deviations of the underlying field and flow. Where no vector is
apparent, the flow/field is primarily radial. The vectors are centered on the
spacecraft coordinates - note that the spacecraft latitude is not corrected
for ballistics or footpoint location and as shown in Figure 1 the local
magnetic field footpoint ultimately maps to southern HG latitudes of around
-60∘. The time series measurements of the plasma and magnetic field are
overplotted in their ballistically-propagated HG longitude coordinates (and in
arbitrary units here). The result of the ballistic backprojection shows that
the sources are highly-structured and steepened at the leading edge; the
smooth profile at PSP altitudes is a result of time-of-flight. The intervals
with enhanced $A_{He}$ are colored yellow and labels N1-N5. The switchback
intervals (red bars $B_{R,Max}$ and $V_{R,Max}$) cluster near the leading edge
or just within. Magnetic field intensity $|B|$ and electron temperature
$T_{e}$ are just inside this boundary; we will suggest below that this minimum
in magnetic field and electron temperature correspond to the central region of
an asymmetric magnetic structure.

Figure 7: Magnetic field (upper panel) and proton velocity (lower panel)
vectors in the T-N plane and centered on the spacecraft position in latitude
and ballistically-mapped Carrington longitude. Field magnitude $|B|~{}R^{2}$,
electron core temperature $T_{e}$ and radial field $B_{R}$ (upper panel) and
proton $T_{\perp}$ and $T_{\parallel}$, $A_{He}$, and radial core proton speed
$V_{R}$ (lower panel) are overplotted in arbitrary units. Maximum, minimum,
and mode values are shown for $B_{R}$ and $V_{R}$ and maximum and mode for the
proton temperatures. Data are accumulated in bins of 0.2∘ longitude. Yellow
vertical bars show intervals (labeled N1-N5) of enhanced $A_{He}$ within the
PBS and blue vertical bars indicate subintervals with enhanced temperature and
radial flow. The T-N plane field and velocity vectors show non-radial
structure near the boundaries of the $A_{He}$ enhancements. $B_{R,Max}$ (red
bars, upper panel) indicates the distribution of switchbacks; occurrence and
amplitude is organized by the PBS and peaks near, but somewhat within, the
leading edge at the source on the Sun. Intervals of enhanced proton
$T_{\parallel}$, radial speed, and electron temperature $T_{e}$ occur within
$\sim 1^{\circ}$ of the (spacecraft-frame) leading edge of structures N3-N5.

It is interesting to note that for the last 3 events here (N3-N5), there
appears to be a small ($\sim$ 1∘) leading edge (in the spacecraft frame)
interval for which the maximum radial field and speed ($B_{R,Max}$ and
$V_{R,Max}$) leads the step in the minimum values $B_{R,Min}$ and $V_{R,Min}$.
These intervals (colored light blue in Figure 7) also correspond to decreasing
trends in $|B|$ and $T_{e}$ and enhanced proton core temperatures
$T_{\parallel}$ and $T_{\perp}$. Similar structure was measured at stream
interface boundaries with instruments on the Helios spacecraft in the inner
heliosphere (Marsch et al., 1982).

Figure 8 shows some data from Figure 7 represented in polar coordinates (i.e.
in the plane of the heliographic ecliptic). While the information here is
equivalent to Figure 7, the polar representation emphasizes the angular extent
of these structures on the Sun and the potential relationship to the
underlying photospheric structure.

##  3 Interpretation and Discussion

Figure 8: Polar representation of the data from Figure 7 with the noted
parameters in arbitrary units. This shows clearly the longitudinal structure
and angular scale of the sources. The solar rotation is counter-clockwise. The
structures show steep leading edge features in magnetic field intensity,
radial proton speed, and proton $T_{\perp}$; proton $T_{\parallel}$ is peaked
sharply near the leading edge of events N2-N5.

To summarize the observations in the previous section, our measurements show
the following:

  1. (i)

PSP passed through a $\sim$34 hour interval of modulated magnetic field
switchbacks and ion and electron flux. This interval corresponds to $\sim$25∘
of heliographic longitude at the Sun during which PSP was magnetically
connected to a coronal hole boundary near -60∘ heliographic latitude. The
spacecraft was below 25 $R_{S}$.

  2. (ii)

The modulations correspond to angular scales of 3∘-5∘ in heliographic
longitude at the Sun - similar to the supergranulation and network magnetic
field structure. PFSS-mapped footpoints show magnetic field concentrations on
similar angular scales.

  3. (iii)

The modulated intervals are in total pressure-balance implying that the
structures are spatial on the spacecraft transit timescale ($\sim$6 hours).

  4. (iv)

The thermal alpha particle abundance $A_{He}$ is enhanced to typical fast-wind
values and the core and strahl electron temperatures are depressed within the
modulations, implying solar wind sources on open field lines at the base of
the corona similar to fast coronal hole wind.

  5. (v)

Proton and alpha particle radial speeds and temperatures and ion plasma
$\beta$ are enhanced within the structures. Alpha particles and beam protons
are streaming ahead of the core protons and their number flux is strongly
modulated by the structures. The alpha particles are $\sim$ 5-15 $\times$
hotter than the core protons. The alpha particles are fairly isotropic
$T_{\perp}/T_{\parallel}\lesssim 1$, while the core protons have
$T_{\perp}/T_{\parallel}>1$ and are often unstable to an anisotropy-driven ion
cyclotron instability.

  6. (vi)

Suprathermal ions with energies up to $\sim$ 85 keV are measured within a
subset of the structures with higher bulk velocities, suggesting a
suprathermal tail. There is no notable radio emission nor time-of-flight
dispersion that suggests a classical flare origin of the suprathermal ions. We
suggest that PSP is transiting field lines populated by these ions.

  7. (vii)

When mapped ballistically to the solar surface, the structures have a steep
leading edge at lower Carrington/HG longitude. This may be a signature of a
some small differential rotation between the photosphere and the corona
(photosphere moving faster) or maybe be some inherent asymmetry of the source
fields at the coronal hole boundary. The switchbacks cluster near the steep
edge of the structures.

  8. (viii)

A narrow $\sim$1∘ region is found at the (spacecraft-frame) leading edge of
the steepest structures that has enhanced proton and alpha temperatures
$T_{p}$, $T_{\alpha}$, large differences between maximum and minimum radial
field and flow, and a transition of $|B|$ and $T_{e}$.

An enhanced relative abundance of alpha particles $A_{He}=n_{\alpha}/n_{tot}$
is known to be associated with fast solar wind, especially during solar
minimum conditions (e.g. Kasper et al., 2012). Since the alpha abundance
$A_{He}$ is determined in the chromosphere or transition region our
measurements suggest a direct mapping to discrete solar wind sources. The
modulation of wind speed and alpha abundance $A_{He}$ on the angular scales of
solar supergranulation strongly implies that these ’microstreams’ originate
within the network magnetic field that is known to cluster at the boundaries
of the supergranules (viz. Rieutord & Rincon, 2010; Wiegelmann et al., 2014).
Theoretically this was suggested as the source of the fast solar wind in the
“furnace” model of Axford et al. (1999). Open field lines containing hot
plasma pass through the solar transition region and overexpand rapidly, as the
high-order magnetic field falls off radially much more rapidly that the plasma
pressure. The resulting magnetic ’funnel’ structures (Gabriel, 1976, 1977;
Dowdy et al., 1986; Axford et al., 1999; Hassler et al., 1999; Tu et al.,
2005) then expand into pressure-balance to generate solar wind. Our
measurement of magnetic field intensity depressions within the structures is
consistent with the overexpansion of magnetic funnels, as the axial field
should decay as $B_{R}\sim 1/A$ while $B_{T,N}\sim 1/\sqrt{A}$ with $A$ the
flux tube area, so that magnetic field at the center of the funnel (the axial,
more radial component) will be weaker after expansion. The magnetic funnels
expand rapidly at low altitudes and reconnection of impinging small dipoles
and emerging flux are the source of energy generating the solar wind within
the larger coronal hole structure, whose speed is determined by the overall
expansion at higher altitude, found to be anti-correlated with the asymptotic
solar wind speed Wang & Sheeley (1990); a two-step expansion process has been
explored by Suess et al. (1998) and others.

The association of large amplitude, nonlinear Alfvenic fluctuations (i.e. the
switchbacks), enhanced bulk flow, and more than mass-proportional ion heating
with the edge of the funnel structure suggests an asymmetry in the source
region that is associated with the heating. Figure 9 is a schematic of the
magnetic field structure at the source. The field vectors here are a 2D
potential model of a magnetic funnel (Hackenberg et al., 1999) as a set of
unipolar flux concentrations and a dipole field with a shear in the -$\hat{x}$
direction; color intensity in Figure 9 is magnetic field magnitude.

The enhancement of SBs in wind emanating from funnel boundaries clearly
demonstrates the impact of structure and processes near the Sun on the spatial
distribution of SBs in the solar wind. In particular, the enhanced values of
$\delta B_{R}$ in these regions is strong evidence of preferential Poynting-
flux injection near funnel boundaries in the low solar atmosphere, which could
be the result of magnetic reconnection. It is possible that the Poynting flux
injection is not only stronger but also more intermittent near funnel
boundaries, since reconnection is often impulsive. However, as noted
previously, large-amplitude magnetic fluctuations in the solar wind naturally
develop discontinuities as a consequence of spherical polarization (e.g.
Vasquez & Hollweg, 1996; Squire et al., 2020; Shoda et al., 2021), and thus
the abrupt magnetic-field rotations observed by PSP may also originate through
in situ dynamics.

Figure 9: A schematic to illustrate the proposed configuration of magnetic
field lines (white vectors) within overexpanded funnel structure and a small
shear between the photosphere and corona. The color intensity indicates field
magnitude. The bases of the funnels are separated by a few degrees of
heliographic longitude and are generating higher $\beta$ wind with enhanced
alpha particle abundance. Patches of magnetic field switchbacks are localized
within the funnels. Since $B_{R}\sim 1/r^{2}$, but $B_{(T,N)}\sim 1/r$, super-
radial expansion results in a depressed $|B|$ in the center of the funnel,
which has spent more time expanding radially. In this 2D geometry, the ’cusp’
regions (darker blue) where neighboring funnels interact moves to lower
altitudes with increasing shear; note that the ’cusp’ intervals are $\sim$10
Mm-scale, which corresponds to $\sim$1∘. A horizontal band indicates the
transition region. Note that the local funnel geometry can be expanded away by
20 $R_{S}$.

The cores of the funnels are separated by 40 Mm ($\sim$3.3∘) here and already
at 30 Mm altitude the field direction becomes relatively uniform, but is
modulated in intensity as described above due to the superradial expansion
below. The low altitude ’cusp’ regions (shaded dark blue in Figure 9) where
adjacent funnels interactions result in an x point (in 2D) with a spine-fan
intersection and a dome of confined flux, where magnetic reconnection could
leak heated and confined plasma outward. This asymmetry is seen in the hotter,
faster leading edge of the measurements in Figure 7. Indeed, simulations of
reconnection between funnels and emerging flux show this asymmetry (Jiang et
al., 2012; Takasao et al., 2013) and the resulting outflows depend on the
altitude of the reconnection region. Here the cusp regions move to lower
altitude with increasing (applied) magnetic shear ($\hat{x}$ component);
magnetic shear could be due to differential rotation between the photosphere
and corona or a property of the structure of the coronal hole boundary.

Magnetic reconnection may also naturally explain the energization of alpha
particles (or oxygen) to 85 keV and the altitude of a reconnection site may be
related to alpha particle abundance, if gravitational settling plays a role.
If the full magnetic energy of a reconnection outflow is available to energize
alpha particles (e.g. Phan et al., 2013), we can infer the Alfvén speed at the
source $\frac{1}{2}m_{\alpha}v_{A}^{2}\sim$ 85 keV, which gives $v_{A}\sim$
2020 km/s, not inconsistent with expections at low altitudes (e.g. Axford et
al., 1999; Warmuth & Mann, 2005). Drake et al. (2009) describe simulations of
a ion pickup process in reconnection outflow regions that effectively
energizes alpha particles to $\sim\frac{1}{2}m_{\alpha}v_{A}^{2}$.

The observation that the lower-$|B|$ funnels are replete with switchbacks is
consistent with the theory that switchbacks grow as a consequence of Alfvénic
fluctuations reaching large amplitudes through solar-wind expansion (Squire et
al., 2020; Mallet et al., 2021). Specifically, funnel regions have presumably
undergone more super-radial expansion than neighbouring regions, so even minor
variations in relative fluctuation amplitudes at the source surface could lead
to large differences at the location of PSP (Hollweg, 1974). Fluctuations that
grow to $\delta B/B\gtrsim 1$ then form switchbacks or potentially fluxropes
associated with turbulent reconnection (e.g. Drake et al., 2021). The relation
of this to other solar-wind properties (wind speed, proton temperature, and
alpha fraction) remains an open problem: the funnels described here lead to
solar wind streams dubbed Alfvénic slow streams (Stansby et al., 2019a;
D’Amicis et al., 2021), rather than the fast wind typically associated with
polar coronal hole outflows. The reason is that the speed of the solar wind is
controlled not by the local expansion discussed here occurring within 20 Mm
from the photosphere, but the overall coronal hole expansion. PSP, always
skirting the current sheet in the ecliptic plane as the Sun’s activity picks
up out of the latest minimum, has been mostly observing wind coming from
rapidly expanding open field lines. For this type of wind, the heating and
acceleration is similar to fast solar wind (Chandran, 2021), including plasma
properties, but the global coronal geometry determines the slower acceleration
profile (Wang & Sheeley, 1990; Panasenco et al., 2019).

The stream structure and fluctuations shown here must evolve to become the
familiar solar wind at $1~{}au$ and beyond. Horbury et al. (2021) have used
measurements by Solar Orbiter of the same solar wind stream at $200~{}R_{S}$
as presented here: while the alpha particle modulation was preserved, the
longitudinal speed variations were smoothed out by that distance, leading to
density variations on the same longitudinal scale. The clear variation in
fluctuation power due to switchback modulation that was present at
$25~{}R_{S}$ was replaced by a large scale magnetic field variation on the
same scale, suggesting that the photospheric structure affecting the wind at
PSP still retains a signature in the wind much farther from the Sun.

Coronal magnetic activity is naturally multi-scale. Large-scale activity
(e.g., flares and CMEs) correlate well with the sunspot solar cycle, while
smaller-scale activity (e.g., bright-points, plumes, and jets) is more
ubiquitous regardless of the solar cycle phase. Space and ground-based
observations in the 1970s provided the first evidence for the highly dynamic
nature of the coronal base (Demastus et al., 1973; Bohlin et al., 1975;
Withbroe et al., 1976; Brueckner, 1980). Yohkoh/SXT observations showed most
energetic coronal jets (e.g., Shibata et al., 1992; Strong et al., 1992;
Shimojo et al., 1996, 1998, 2001). These discoveries led to speculations on
the role these transients play in the heating and acceleration of the solar
wind plasma (Brueckner & Bartoe, 1983).

Our in situ measurements imply a connection between impulsive Alfvénic
activity within discrete wind sources and solar wind heating and acceleration
mechanisms. While our measurements appear to be consistent with interchange
reconnection, either through direct heating from the reconnection site itself
or through enhanced Poynting fluxes (e.g., waves) that dissipate at higher
altitudes, there are other viable mechanisms to consider. Axford et al. (1999)
describe a wave-heating process occurring within magnetic funnels: high
frequency Alfvén waves propagating outward enter resonance with ions and heat
perpendicular to the local magnetic field and the resulting mirror force
produces accelerated flow (Czechowski et al., 1998); resonant ion cyclotron
heating is observed at 1 au (Kasper et al., 2013). Martinez-Sykora et al.
(2017) suggest that ambipolar diffusion transports kinked field lines near
network field concentrations into the chromosphere and the stored magnetic
tension is released impulsively to form type II spicules and Alfvénic jets.

Finally, it is interesting to speculate about a connection to coronal plumes -
persistant filamentary structures that disappear into the general solar wind
outflow several tens of solar radii above the Sun’s surface (Wilhelm et al.,
2011; Poletto, 2015; Raouafi et al., 2016; DeForest et al., 1997; Teriaca et
al., 2003; Wang & Muglach, 2008; Yu et al., 2014). Plumes are also known to
emerge from network field concentrations with enhanced heating at the base and
are characterized by higher density (Wang, 1994) and lower electron
temperatures (DeForest et al., 1997) than the surrounding corona. Raouafi et
al. (2008) show that coronal plumes display myriads of small, short-lived jets
at their footpoints (Raouafi & Stenborg, 2014), most probably resulting from
magnetic reconnection due to small emerging bipolar features (see e.g.,
Panesar et al., 2018, 2019). Plumes exhibit oscillations and sub-structures
(Uritsky et al., 2021) and should expand in pressure balance into the solar
wind (Velli et al., 2011), but their potential relationship to the structures
observed by PSP and reported here remains to be explored.

Parker Solar Probe was designed, built, and is now operated by the Johns
Hopkins Applied Physics Laboratory as part of NASA’s Living with a Star (LWS)
program (contract NNN06AA01C). Support from the LWS management and technical
team has played a critical role in the success of the Parker Solar Probe
mission. We would like to acknowledge the efforts of the FIELDS, SWEAP, and
ISOIS instrument and science operations teams and the PSP spacecraft
engineering and operations team at the Johns Hopkins Applied Physics
Laboratory. O. Panasenco and M. Velli were partially supported by the HERMES
DRIVE NASA Science Center grant No. 80NSSC20K0604. B. Chandran was supported
by NASA grants NNX17AI18G and 80NSSC19K0829. J. Squire was supported by the
Royal Society Te Apārangi NZ through Rutherford Discovery Fellowship
RDF-U00180. T. Horbury and L. Woodham were supported by UK STFC grant
ST/S000364/1, T. Woolley by ST/N504336/1 and R. Laker by an Imperial College
President’s scholarship.

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