2021

[1]Ariel A. Valladares

1]Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de
México, Apartado Postal 70-360, Ciudad Universitaria, 04510, CDMX, México

2]Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado
Postal 70-542, Ciudad Universitaria, 04510, CDMX, México

# Nanomagnetism in porous amorphous palladium, a sequel. A possible light-
weight magnet.

Isaías Rodríguez<EMAIL_ADDRESS>Renela M. Valladares
<EMAIL_ADDRESS>David Hinojosa-Romero<EMAIL_ADDRESS>Alexander Valladares<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [

###### Abstract

Magnetism is a very relevant subject that permeates our everyday lives.
However, magnetism keeps taking us from surprise to surprise which seems to
indicate that it is a phenomenon not well understood. For example, we found
that bulk amorphous palladium becomes magnetic; so, naturally one should ask,
could defective palladium develop magnetism? In particular, would amorphous
porous palladium become magnetic? Here we show that the answer to that
question is affirmative, this defective topology of Pd is magnetic, with a
magnetism that depends on the amount of sample porosity and on the topology of
their structures. Clearly, if magnetism exists in porous amorphous palladium,
this indicates the possibility of developing light-weight magnets, useful when
a maximized magnetism/weight ratio is demanded, well suited for space and
aeronautical applications.

###### keywords:

Amorphous Materials, Porous Materials, Magnetic Materials, First-Principles
Simulations

##  1 Introduction

Porous science is in the making. Porous materials have existed before
humanity, but a systematic and fundamental approach to its understanding is
recent. This explains the limited number of papers that deal with the basics
of the subject. We pretend to contribute to this understanding by performing
computational simulations of the topology, the electronic structure and the
magnetic properties of some porous and amorphous metallic systems, pure and
alloyed. The present study follows on the footsteps of our prediction that
amorphous palladium, $a$-Pd, displays magnetism in the bulk
(rodriguez_emergence_2019, ). Our reasoning is that since a defective
structure of palladium, like the amorphous phase, displays magnetism then an
even more defective one, like the porous-amorphous topology, $pa$-Pd, should
display magnetism as well. If so, this could become an important asset in the
technological world since it would be possible to have light weight magnets.

Palladium behaves in a non-expected manner when expanded or contracted
(moruzzi_magnetism_1989, ), or confined to a restricted dimensionality
(sampedro_ferromagnetism_2003, ) and certainly when amorphous in the bulk
(rodriguez_emergence_2019, ), as opposed to the behavior of the lone atom. It
has also been found that Pd develops magnetism when alloyed with H
(rodriguez_superconductivity_2020, ; mott_theory_1958, ), or silicon, both
liquid (muller_magnetic_1978, ) or solid (rodriguez_ab_2020, ). Moruzzi and
Marcus (moruzzi_magnetism_1989, ) studied the electronic properties of
crystalline Pd and Rh from below the equilibrium volume to an expanded state
to simulate free atoms, and they discovered that palladium, like most
condensed transition metals, must also exhibit magnetic behavior at
sufficiently large volumes, although the free lone atom of Pd is not a
magnetic entity but acquires the $4d^{10}5s^{0}$ structure. The expectation
that bulk crystalline Pd could be magnetic even at equilibrium volumes has led
to a sustained interest in its electronic structure and, consequently, its
magnetic properties.

Porous palladium, $p$-Pd, was first reported by Kabius, Kaiser and Kaesche
(KKK) (kabius_micromorphological_1986, ) in 1986. KKK produced $p$-Pd by
selective chemical dissolution of copper in a CuPd alloy. However, their
accomplishment remained archived until 2009 when Balk and coworkers resumed
its study and produced nanoporous Pd thin films by a selective dissolution,
dealloying, of PdNi alloy films in dilute sulfuric acid (li_preparation_2009,
). Their objective was to investigate the hydrogen absorption/desorption
behavior in this hydride, but, in the process they were able to generate
structures that were sponge-like with ligaments/pores $\sim$10 nm. depending
on the experimental conditions. Dealloying nickel was an incomplete process
since the final result was a 46 at. % nickel remnant, although the XRD scans
did not detect the PdNi alloy peak (li_achieving_2010, ). They studied the
topology of films that were 90 nm thick, with an initial concentration of 18
at. % Pd and 82 at. % Ni, and they found finer pores and ligaments in the
final sample. These pores and the ligaments are larger than the ones reported
in this work as shall be shown later. Selective dissolution, or dealloying, is
a process used by KKK to create porous palladium by dissolving copper in a
PdCu alloy, and it generates large stresses in the material and the structure
created is in a metastable state that would evolve into a different, stable,
structure when equilibrium is reached. Nevertheless, lacking an experimental
method to expand (negative pressures) solid matter, this dealloying approach
may become an interesting route to produce porous materials.

Other works on the experimental generation of porous palladium are found in
references (hakamada_preparation_2009, ; yu_nanoporous_2008, ;
guo_synthesis_2008, ), all since 2008. However, to the best of our knowledge,
no reference to magnetic properties of porous-amorphous palladium has been
made; therefore, we decided to extend the investigation of the magnetism found
in $a$-Pd (rodriguez_emergence_2019, ) to enquire into a possible magnetism of
a defective structure like $pa$-Pd. In this work we report findings that
indicate the presence of nanomagnetism in samples with different porosities
(generated using the expanded- lattice approach cultivated in our group
(santiago-cortes_computational_2012, )), and with different structures; some
resulting from a molecular dynamics (MD) procedure applied to the expanded-
lattice structures and some others in local equilibrium, consequence of a
geometry optimization (GO) procedure applied to the outcome of the MD process.
In both cases magnetism appears and the intensity depends on the structure for
a given porosity. Due to the size of the supercells we used, we speak of the
magnetism found, as nanomagnetism, or perhaps more appropriately, sub-
nanomagnetism, because the dimensions of the “domains” found are in the sub-
nano range. The relative orientations determine the final magnetic moment of
our supercells.

##  2 Methodological considerations

When dealing with a material as interesting as palladium the implications of a
given result should be fully appreciated and extended to study related
situations that may occur. Since the magnetism discovered for amorphous bulk
Pd may be due to defects present in the samples (rodriguez_emergence_2019, )
the natural thing to do was to study other defective structures and ponder
their possible magnetism. Amorphous alloys could be considered as defective
matter and that is why we undertook the study of the solid amorphous palladium
“ides”, (Pd1-x(H,D,T)x), and found that they developed magnetism as a function
of the “ides” concentration (rodriguez_superconductivity_2020, ). Also, the
study of amorphous solid PdSi alloys showed magnetism as well
(rodriguez_ab_2020, ). However, the natural extension of our work on amorphous
palladium is to generate porous amorphous samples using two approaches that we
have developed in our workgroup: the undermelt-quench approach
(valladares_new_2008, ) used to generate amorphous materials and the expanded-
lattice approach (santiago-cortes_computational_2012, ) instrumented to
generate porous samples. The undermelt-quench approach has been applied to Pd
and to many other materials to generate their amorphous structures (see Refs.
rodriguez_emergence_2019  and valladares_new_2008 ). The expanded-lattice
approach has also been applied to lattice sizes manageable with first-
principles methods, since these methods are very computer-intensive and demand
vast amounts of computational resources. This limitation restricts us to using
supercells with a number of atoms of the order of a few hundred, and therefore
the apparent existence of magnetic regions, found in this work, smaller than
the well-known magnetic domains, (normally of the order of nanometers), should
be studied more extensively and intensively. We pursued our goal,
nevertheless, and generated supercells that contain 216 atoms with lattice
parameters given in Table 1.

Table 1: Porosities, lattice parameters, and densities of the supercells studied in this work. Name | Porosity | Lattice Parameter | Density
---|---|---|---
| (%) | [a=b=c] (Å) | (g cm-3)
$a$-Pd | 0 | 14.7058 | 12.00
$pa$-Pd25% | 25 | 16.1858 | 9.00
$pa$-Pd50% | 50 | 18.5281 | 6.00
$pa$-Pd62.5% | 62.5 | 20.3929 | 4.50
$pa$-Pd75% | 75 | 23.3440 | 3.00
|  |  |

We started by expanding the lattices to generate the porosities mentioned in
Table 1 and of course the 0% porosity sample reported in Ref
rodriguez_emergence_2019 . The computer simulations were conducted in CASTEP
(clark_first_2005, ) an ab-initio software included in the Materials Studio
suite (dassault_systemes_biovia_materials_2016, ). The parameters used in the
simulations are as follows. For both the MD and GO, the PBE exchange-
correlation functional was used (perdew_generalized_1996, ), with a cut-off
energy of 300 eV for the plane-wave basis. The MD process consists of 225
steps with a time-step of 5 fs with an initial increasing linear run of 100
steps starting at a temperature of 300 K and reaching a maximum temperature of
1500 K with a slope of 12 K per step. Then a quenching phase of 125 steps was
executed also with a slope of 12 K per step going from 1500 K to a minimum of
12 K, according to the undermelt-quench procedure (valladares_new_2008, ).
Care was exercised to keep the atomic spins free to orient according to the
potentials (forces) in the quantum mechanical ab-initio computing process. The
thermostat used in both thermal processes was Nóse-Hoover within an NVT
ensemble. After the MD runs were accomplished we proceeded to optimize the
resulting structures to find a (local) minimum of energy by performing a GO.
For the energy minimization, the density mixing was used with Pulay Scheme,
with a charge mixing of 0.5, and a spin mixing of 2.0. Thermal smearing in the
occupancy of 0.1 eV was utilized, and an energy convergence threshold of
5x10-7 eV for the SCF cycles. For this process, the BFGS minimizer was
selected with a geometry energy tolerance of 2x10-5 eV, a force tolerance of
5x10-2 eV Å-1, a stress tolerance of 5x10-2 eV Å-3, and a max displacement of
2x10-3 Å.

We analyzed the topology of the resulting structures using Correlation, a
software developed by our workgroup (rodriguez_correlation_2021, ), and
obtained their electronic and magnetic properties after both the MD and the GO
runs.

##  3 Results

The generated atomic topologies were characterized by obtaining the Pair
Distribution Functions (PDFs), $g(r)$, and the reduced Pair Distribution
Functions (rPDFs), $G(r)$, at the end of each MD and GO run
(musgraves_springer_2019, ; waseda_structure_1980, ); PDFs and rPDFs are shown
in Figures 1 and 2. In a previous work (santiago-cortes_computational_2012, ),
we tried to relate the extent of the region where the corresponding PDF has a
value below 1, beyond the middle-range order, to the size of the pore and the
conclusions were not completely satisfactory. However, this depletion appears
again in this work and by reductio ad absurdum there must be some relation
between this region and the existence of pores, phenomenon to be elucidated.
To emphasize this feature, we decided to plot the rPDFs in the corresponding
Figures 1 and 2 by increasing the number of points used in the Fast Fourier
Transform (FFT) to smooth the functions and, in fact, the over-smoothed red-
dotted curves in the rPDFs indicate the presence of the pores. The regions
where the PDFs are less than 1, and where the rPDFs are less than 0, beyond
the middle-range order, signal the existence of pores, and a comparison
between these values and those obtained by a geometrical estimation, using
Connolly surfaces, is given in Table 2. At the end of the MD processes the
topology of the structures is more dendritic-like, whereas when the GO is
performed large through pores (voids) appear, (See Figures 3 and 4 where the
50% porosity structures are shown).

Table 2: Comparison of the size of the pores obtained by a geometrical estimation and by evaluating the depletion of the $G(r)$. Name | Geometrical estimation | Depletion222Calculated from the depletion of the average $G(r)$ in Figure 2. (Å)
---|---|---
| Pore diameter111Average diameter of all the samples for every porosity. (Å) |  |
| MD | GO | MD | GO
$a$-Pd | - | - | - | -
$pa$-Pd25% | 6.52 | 11.32 | 3.16 | 7.78
$pa$-Pd50% | 10.53 | 12.41 | 7.79 | 7.94
$pa$-Pd62.5% | 16.34 | 17.14 | 8.86 | 9.60
$pa$-Pd75% | 16.67 | 18.83 | 10.30 | 12.73
|  |  |  |
Figure 1: a) Pair Distribution Functions and b) reduced Pair Distribution
Functions, after the MD processes. Average of the cells of every porosity
shown. Color online. Figure 2: a) Pair Distribution Functions and b) reduced
Pair Distribution Functions, after the GO processes. Average of the cells of
every porosity shown. Color online. Figure 3: Typical topology of a porous
structure after the MD run. The porosity displayed corresponds to 50% and the
view is (a) perpendicular to the xy plane, (b) perpendicular to the yz plane,
and (c) perpendicular to the zx plane. Color online. Figure 4: Typical
topology of a porous structure after the GO run. The porosity displayed
corresponds to 50% and the view is (a) perpendicular to the xy plane, (b)
perpendicular to the yz plane, and (c) perpendicular to the zx plane. The
difference with Figure 3 is to be noted. Color online.

Once the MD and GO processes terminated, we investigated the electronic
structure and the magnetism of the samples after each process. After the MD
runs, the integrated spin density displays a systematic increase with
porosity, with a maximum at about 40% and then diminishes to practically zero
for porosities larger than 80%, as shown in Figure 5 a). Figure 5 b) shows the
behavior of the integrated spin density as a function of porosity after the GO
jobs terminated. For porosities greater than 50% large voids were observed and
a magnetic behavior different from the one obtained after the MD runs was
observed.

These results merit a more careful analysis. Since the number of atoms in
every cell is the same, 216, Figure 5 a) indicates that the orientation of the
magnetic moments becomes more aligned as the 40% porosity region is
approached, whereas before and after 40% this orientation does not reinforce
the magnetism and a decrease is observed when departing from this region. This
indicates an increasing randomness in the magnetic moments before and after
due to the MD processes that the supercells underwent, and as the middle of
the porosity range is approached the tendency for the magnetic moments is to
become more parallel. This phenomenon is observed with clarity when we look at
the results after the GO processes. Figure 5 b) shows that when the geometry
optimization is conducted for each sample, the way to minimize the energy is
by aligning all the magnetic moments in each atom, and since the number of
atoms is constant the net magnetism is practically the same for each cell due
to the fact that each atom contributes the same amount to the total magnetic
moment. In our expanded-lattice method, the total number of atoms in a sample
does not change and that explains why the magnetism is practically constant
after the GO for each cell no matter what the porosity value is. If we could
generate a supercell with a 100% porosity (infinite volume) then the number of
atoms would still be 216 and the net magnetic moment would remain the same for
the cluster that agglutinates all 216 atoms.

Figure 5: Behavior of the integrated spin density as a function of the
porosity. a) after the MD runs. b) after the GO runs. Color online.

If we consider the magnetism per unit volume this behavior can be clearly
seen, as in Figures 6 a) and b). As the volume tends to infinity with the
atoms in the supercells remaining constant, then the magnetism per unit volume
would tend to zero, as shown. The technological implications should be
considered: If the total number of atoms in the solid diminishes due to the
creation of porosity, the magnetism would also diminish and in the 30 to 40%
region the non-equilibrium MD process would generate the maximum of magnetism
congruent with the number of atoms in the porous sample. After the GO process
the total magnetic moment would be constant but if the number of atoms is
scarce then the total magnetism would be also small and not very practical.
The best thing to do is to create a geometry-optimized porous sample with a
large number of atoms to take advantage of both aspects. A compromise is
needed between the total number of atoms, porosity and equilibrated
structures.

Figure 6: Behavior of the integrated spin density per cell volume as a
function of the porosity. a) after the MD runs. b) after the GO runs. Color
online.

##  4 Possible applications: A light-weight magnet.

The possibility of constructing magnets with medium to strong magnetism by
developing amorphous porous palladium exists. The structures that are
generated after the molecular dynamics processes indicate that the non-
optimized samples behave in a parabolic manner and their magnetism is similar
to that obtained after the geometry optimization procedure. This magnetism
goes to zero for the highly porous cells. Evidently, the structures after the
GO runs, both magnetic and topological, are different from the corresponding
structures after the MD processes. This is due to the search for a local
stable topology during the GO processes whereas no local equilibrium can be
invoked after the MD jobs. The porosity affects seriously these properties as
opposed to the non-porous samples.

If a process like chemical dissolution (dealloying) is invoked to generate
porous palladium it would lead to a stressed atomic topology that would relax
to a more stable structure in the course of time. Therefore, as these
structures evolve, the magnetism would change and a mechanism to make them
stable is required to avoid the changes in the magnetic value. The MD
arrangements are the ones that would evolve more readily since they are not at
(local) equilibrium, whereas the ones generated after the GO processes would
have acquired a (local) stability. How long would the arrangements after MD
remain static? It is hard to know with the computer simulation techniques
applied in this work. We can only hope that the MD structures would remain
static long enough to be useful.

Experimentalists are ingenious and depending on the magnetism desired they
will be able to develop the necessary processes to design the adequate
structure to produce the correct magnet, by dealing with the porosity and with
the equilibrium characteristics. However, this may not be a trivial task and
researchers would have to be inventive to accomplish the desired results.
Since magnetic materials are used practically everywhere, the quest for the
correct procedure may be worth searching for. For some applications the ratio
(magnetism / weight) needs to be optimized since an excess of weight may
hinder its use due to “fueling” considerations. Therefore, despite the high
cost of palladium in the market, it may be worthwhile to investigate the
option presented here.

##  5 Conclusions

Magnetism pervades our lives. Since primitive man discovered the properties of
lodestone, the search for an understanding of the origin of magnetism in
materials has been pursued. To date, several types of magnetism are reported
and there seems to be more in the future (musgraves_springer_2019, ;
waseda_structure_1980, ). Here we have computationally researched the
properties of palladium with defects, in order to investigate their magnetic
properties. We found that amorphicity generates magnetism in the material and
the porosity in the amorphous also does, fostering the possible development of
light-weight magnets of potential application in our environment. The nature
of magnetism in $pa$-Pd is being investigated and the possible existence of
magnetic domains in the realm of sub-nano dimensions is appealing, since this
would lead to consider the formation of magnetic regions of very small sizes
that contain only a few atoms, an interesting task.

Acknowledgments

I. Rodríguez thanks PAPIIT, DGAPA-UNAM for his postdoctoral fellowship. D.
Hinojosa-Romero acknowledges Consejo Nacional de Ciencia y Tecnología
(CONACyT) for supporting his graduate studies. A. A. Valladares, R. Valladares
and A. Valladares thank DGAPA-UNAM (PAPIIT) for continued financial support to
carry out research projects under Grants No. IN104617 and IN116520. M.T.
Vázquez and O. Jiménez provided the information requested. A. Lopez and A.
Pompa assisted with the technical support and maintenance of the computing
unit at IIM-UNAM. Simulations were partially carried at the Computing Center
of DGTIC-UNAM under project LANCAD-UNAM-DGTIC-131.

## Declarations

  * •

Funding: Financial support was received from DGAPA-UNAM (PAPIIT) under
projects No. IN104617 and IN116520.

  * •

Conflict of interest/Competing interests: The authors have no competing
interests to declare that are relevant to the content of this article.

  * •

Ethics approval: Not applicable.

  * •

Consent to participate: Not applicable.

  * •

Consent for publication: The authors gave their consent for the publication of
this manuscript.

  * •

Availability of data and materials: The datasets analysed during the current
study are available from the corresponding author on reasonable request.

  * •

Code availability: Code Correlation can be found in the GitHub repository:
https://github.com/Isurwars/Correlation (DOI: 10.5281/zenodo.5514113).

  * •

Authors’ contributions: Ariel A. Valladares, Alexander Valladares and Renela
M. Valladares conceived this research and designed it with the participation
of Isaías Rodríguez and David Hinojosa-Romero. All the simulations were done
by Isaías Rodríguez. and David Hinojosa-Romero. All authors discussed and
analyzed the results. Ariel A. Valladares wrote the first draft and the other
authors enriched the manuscript.

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# Conditions for proton temperature anisotropy to drive instabilities in the
solar wind111Draft

Simon Opie Mullard Space Science Laboratory, University College London,
Dorking, RH5 6NT, UK  Daniel Verscharen Mullard Space Science Laboratory,
University College London, Dorking, RH5 6NT, UK  Christopher H. K. Chen
Department of Physics and Astronomy, Queen Mary University of London, London
E1 4NS, UK  Christopher J. Owen  Mullard Space Science Laboratory, University
College London, Dorking, RH5 6NT, UK  Philip A. Isenberg  Space Science
Center, University of New Hampshire, Durham NH 03824, USA

###### Abstract

Using high-resolution data from Solar Orbiter, we investigate the plasma
conditions necessary for the proton temperature anisotropy driven mirror-mode
and oblique firehose instabilities to occur in the solar wind. We find that
the unstable plasma exhibits dependencies on the angle between the direction
of the magnetic field and the bulk solar wind velocity which cannot be
explained by the double-adiabatic expansion of the solar wind alone. The angle
dependencies suggest that perpendicular heating in Alfvénic wind may be
responsible. We quantify the occurrence rate of the two instabilities as a
function of the length of unstable intervals as they are convected over the
spacecraft. This analysis indicates that mirror-mode and oblique firehose
instabilities require a spatial interval of length greater than 2 to 3
unstable wavelengths in order to relax the plasma into a marginally stable
state and thus closer to thermodynamic equilibrium in the solar wind. Our
analysis suggests that the conditions for these instabilities to act
effectively vary locally on scales much shorter than the correlation length of
solar wind turbulence.

Solar wind — Space plasmas — Solar magnetic fields — Plasma physics — Solar
physics

##  1 Introduction

The solar wind is a continuous stream of plasma from the Sun which exhibits
significant measurable variability in its characteristic properties on a range
of spatial and temporal scales (for recent reviews, see Matteini et al., 2012;
Bruno & Carbone, 2013; Chen, 2016; Verscharen et al., 2019). The fundamental
processes that heat and accelerate the solar wind are not at present fully
understood (Tu & Marsch, 1995; Parker, 1965; Cranmer et al., 2015).

A turbulent cascade is generally invoked to explain how energy injected near
the Sun into the solar wind at large scales is transferred to kinetic scales
where it is available to heat and accelerate individual particles as the
plasma travels radially outwards in a practically collisionless environment
(Bruno & Carbone, 2013; Kiyani et al., 2015; Alexandrova et al., 2013;
Chandran et al., 2011). At kinetic scales, a secular energy transfer from
electromagnetic field fluctuations into the particles ultimately increases
entropy (Bale et al., 2009; Chen et al., 2016; Verscharen et al., 2016). In
addition, energy transfer from the particles into the electromagnetic fields
is possible when free energy in the form of temperature anisotropy or other
non-equilibrium particle features is available. This transfer occurs in the
form of instabilities that lead to characteristic wave–particle interactions.
Micro-instabilities act to restore thermodynamic equilibrium in the solar
wind, thereby lowering the driving free energy (Kunz et al., 2014; Verscharen
et al., 2017; Chen et al., 2016). In this way, micro-instabilities play an
important role for the macro-scale energy distribution in the solar wind
(Verscharen et al., 2019).

The solar wind is often studied in a parameter space defined by the
plasma-$\beta$, given by the ratio of plasma pressure to the magnetic
pressure, and the ratio between the temperature $T_{\perp}$ perpendicular to
the magnetic field and the temperature $T_{\parallel}$ parallel to the
magnetic field (Gary et al., 2001; Kasper et al., 2002; Hellinger et al.,
2006; Bale et al., 2009). We refer to plots of the distribution of the data in
this space for any given species in the solar wind as
$T_{\perp}/T_{\parallel}$-$\beta$-plot. When contours of parameter
combinations reflecting marginal stability to individual unstable modes are
added to these $T_{\perp}/T_{\parallel}$-$\beta$-plots, they demonstrate to
what extent the temperature anisotropy is constrained by specific instability
modes (Chen et al., 2016; Verscharen et al., 2016; Klein et al., 2018).

The best-fit constraints to proton temperature anisotropies in
$T_{\perp}/T_{\parallel}$-$\beta$ plots are typically provided by the
thresholds for the oblique firehose and mirror-mode instabilities (Hellinger
et al., 2006; Bale et al., 2009; Gary, 2015), which are non-propagating
unstable modes of the Alfvén-mode and the slow-mode branches of the dispersion
relation (Howes et al., 2006; Schekochihin et al., 2009; Kunz et al., 2015;
Verscharen et al., 2017). Sufficient plasma pressure anisotropy creates the
necessary conditions for the instabilities to act (Chandrasekhar et al., 1958;
Parker, 1958; Hasegawa, 1969; Maruca et al., 2012; Kunz et al., 2014, 2015).
At large scales they are driven by anisotropies in the total pressure
components of all species combined (Chen et al., 2016). In this work, we focus
solely on the proton contribution to the total pressure anisotropy and the
kinetic versions of these instabilities, which create fluctuations on a scale
of order the characteristic proton kinetic scales (Gary, 2015; Howes, 2015).

In the case of the oblique firehose instability, excess pressure parallel to
the direction of the magnetic field causes the growth of bending in magnetic
flux tubes. The magnetic tension force is unable to restore this bending if
the pressure anisotropy is sufficiently large. Such a transverse perturbation
does not propagate in the form of Alfvén waves (as it would in the absence of
pressure anisotropy) but grows aperiodically with a polarization similar to
Alfvén waves (Hellinger & Trávníček, 2008; Matteini et al., 2006; Kunz et al.,
2014). For the mirror-mode instability, excess perpendicular pressure leads to
the formation of quasi-periodic mirror structures trapping some of the ions
between mirror points and setting up compressive standing waves with a
wavevector oblique to the direction of the magnetic field (Kivelson &
Southwood, 1996; Kunz et al., 2016; Yoon et al., 2021). Particles accumulate
in the region between the mirror points where the magnetic field strength is
lower, acting to restore perpendicular pressure balance. This results in the
mirror mode being characterised by anti-correlated fluctuations in density and
magnetic field strength when seen by a traversing spacecraft (e.g. Russell et
al., 1999). In both cases, the transfer of kinetic particle energy to the
electromagnetic fluctuations coincides with a reduction in the anisotropy
(Kunz et al., 2016; Yoon, 2016).

Plasma instabilities are usually described in the context of homogeneous and
steady-state plasma conditions (Gary, 1993). However, the solar wind, like
most natural plasmas, is turbulent and thus does not fulfill the assumptions
applied in the standard theoretical treatment of these instabilities (Kivelson
& Southwood, 1996; Howes et al., 2008). Nevertheless, observations clearly
show that instabilities act, at least at some time, in this environment
(Matteini et al., 2012; Maruca et al., 2012; Wicks et al., 2016; Yoon et al.,
2021). Our goal is to quantify the occurrence rates of oblique firehose and
mirror-mode unstable solar wind intervals and their dependence on the
direction of the magnetic field. We also measure the length of unstable
intervals in order to evaluate statistically the spatial homogeneity
requirement for these instabilities to effectively reduce the proton
temperature anisotropy in the solar wind. If the occurrence of unstable
intervals were determined by a scale-independent process, we would anticipate
a smooth and scale-independent statistical distribution of the lengthscales of
unstable solar wind intervals. However, if the effective action of the
associated instabilities is scale-dependent, we anticipate a break in the
statistical distribution of the lengthscales of intervals with unstable plasma
parameters. Even without knowing the underlying hypothetical distribution of
lengthscales, we conjecture that a break in the statistical distribution into
a steeper slope marks the lengthscale above which instabilities are effective.
In this interpretation, the homogeneity assumption of linear theory is only
sufficiently fulfilled in plasma intervals of length greater than the break
scale in the statistical distribution.

##  2 Methods

###  2.1 Dataset

Recent space missions have been launched to study the inner heliosphere in
great detail, with a focus on the processes that heat and accelerate the solar
wind (Fox et al., 2016; Müller et al., 2020; Zouganelis et al., 2020). For
this study, we use data from Solar Orbiter’s Solar Wind Analyser (SWA; Owen et
al., 2020) instrument suite, specifically the Proton Alpha Sensor (PAS), and
the Magnetometer (MAG; Horbury et al., 2020). Solar Orbiter in-situ data are
publicly available at the Solar Orbiter
Archive222https://soar.esac.esa.int/soar/ which is the source for all data in
this study. We use data from the cruise phase of the mission in both 2020 and
2021.

SWA’s PAS measures the 3D velocity distribution function (VDF) of protons and
$\alpha$-particles, whereby the VDF is assembled over an interval of 1 s every
4 s, resulting in a normal-mode cadence of $0.25\,\mathrm{Hz}$ (Owen et al.,
2020). The MAG fluxgate magnetometer provides 8 magnetic-field vectors per
second in its normal mode (Horbury et al., 2020). We use the PAS proton ground
moments data and the MAG normal-mode data in radial, tangential and normal
(RTN) coordinates. We average the corresponding MAG vector data over each 1 s
VDF measurement interval from PAS.

For our statistical analysis, it is convenient to use continuous data
intervals of reasonable length. In compiling the full dataset, we select
intervals of greater than three consecutive days, subject to data
availability. We only include PAS data with a quality factor
$<0.2$333According to its definition, data with higher quality are identified
with lower quality factor values. and solar wind bulk velocity
$>325\,\mathrm{km}\,\mathrm{s}^{-1}$ with initial selection by visual
inspection of the data aided by the SWA-PAS data
log444http://solarorbiter.irap.omp.eu/documents/FEDOROV/. The intervals chosen
are listed in Table 1. The analysed dataset comprises 975 516 points in total.
No attempt was made to eliminate structures such as shocks, interplanetary
coronal mass ejections (ICMEs), or current sheets from the data set.

Table 1: Data selection from the Solar Orbiter Archive with approximate heliocentric distance for each dataset. Interval | Heliocentric Distance ($R_{S}$) | Number of Datapoints
---|---|---
2020 October 07-18 | 205 | 185 923
2021 April 22-28 | 190 | 131 481
2021 May 05-11 | 180 | 131 849
2021 June 10-13 | 200 | 79 641
2021 July 06-11 | 190 | 117 427
2021 July 20-24 | 180 | 88 429
2021 October 09-12 | 150 | 81 362
2021 October 19-26 | 160 | 159 404

We rotate the proton pressure tensor to align with the magnetic field and
create a timeseries for $\beta_{\parallel}\equiv 8\pi
n_{\mathrm{p}}k_{\mathrm{B}}T_{\parallel}/B^{2}$, where $n_{\mathrm{p}}$ is
the proton number density, $k_{\mathrm{B}}$ is the Boltzmann constant, and
$\boldsymbol{B}$ is the magnetic field averaged over the associated 1 s PAS
measurement interval. We then also calculate the ratio
$T_{\perp}/T_{\parallel}$ for each PAS measurement.

###  2.2 Instability thresholds

We base our analysis on the analytical approximation for the instability
thresholds of the anisotropy-driven instabilities in the form

$\frac{T_{\perp}}{T_{\parallel}}=1+\frac{a}{(\beta_{\parallel}-c)^{b}},$ (1)

where $a$, $b$, and $c$ are constants with values given for each instability
by Verscharen et al. (2016). We use a maximum growth rate of
$\gamma_{m}=10^{-2}\Omega_{p}$, where $\Omega_{p}$ is the proton
gyrofrequency. We evaluate these instability thresholds for the oblique
firehose (OF) and for the mirror-mode (M) instabilities. For reference, we
also include the instability thresholds for the Alfvén/ion-cyclotron (A/IC)
and fast-magnetosonic/whistler (FM/W) instabilities in part of our analysis.

###  2.3 Angle analysis

Working in RTN coordinates, we calculate the angle between $\boldsymbol{B}$
and $\boldsymbol{V}$ using the complete 3D vectors as

$\theta_{BV}^{\prime}=\text{arccos}\frac{\boldsymbol{B}\cdot\boldsymbol{V}}{BV},$
(2)

where $\boldsymbol{V}$ is the bulk velocity of the protons. We convert the
angle $\theta_{BV}^{\prime}$ into a full 360∘ distribution in order to capture
the full range of variability in the fluctuations of the magnetic field and to
retain the separation of the sector structure of the solar wind. For this
conversion, we define $b=B_{R}+iB_{T}$ and $v=V_{R}+iV_{T}$. We then calculate
the angle $\phi_{v}=\mathrm{arg}(v)$, where $\mathrm{arg}(\cdot)\in[0,2\pi)$
is the polar angle in the complex plane. After rotating $b$ by $-\phi_{v}$ in
the complex plane, we define the difference angle between $b$ and $v$ as
$\phi_{bv}=180^{\circ}\mathrm{arg}(be^{-i\phi_{v}})/\pi$. If $0<\phi_{bv}\leq
180^{\circ}$, we set $\theta_{BV}=360^{\circ}-\theta_{BV}^{\prime}$.
Otherwise, we set $\theta_{BV}=\theta_{BV}^{\prime}$. This procedure leads to
a representation of the angle $\theta_{BV}$ between $\boldsymbol{B}$ and
$\boldsymbol{V}$ within the range $[0^{\circ},360^{\circ})$.

The angle $\theta_{BV}$ is the most appropriate measure for quantifying the
fluctuations of $\boldsymbol{B}$ and $\boldsymbol{V}$ within structures
convected over the spacecraft as a single point of measurement (see also
Woodham et al., 2021, and references therein). This link to the convection
speed $\boldsymbol{V}$ is particularly important when Taylor’s hypothesis is
used to map temporal to spatial data (Taylor, 1938; Treumann et al., 2019). On
average and for large datasets, we expect that
$\theta_{BV}\approx\theta_{BR}$, where $\theta_{BR}$ is the angle between
$\boldsymbol{B}$ and the unit vector $\hat{\boldsymbol{R}}$ in the radial
direction. In the case of $\boldsymbol{V}\propto\hat{\boldsymbol{R}}$,
$\theta_{BV}$ represents the azimuthal angle of $\boldsymbol{B}$ and
statistically approaches the Parker angle (Parker, 1965).

###  2.4 Lengthscale analysis

Figure 1: We identify unstable intervals and calculate their persistence
interval ($\Delta_{j}$) by aggregating the lengthscales ($\delta_{i}$) of each
measurement interval $i$ in appropriate dimensionless units, where
$\delta_{i}=V_{i}\tau/\rho_{\mathrm{p}i}$, $V_{i}$ is the bulk velocity in
interval $i$, $\rho_{\mathrm{p}i}$ is the proton gyroradius in interval $i$,
and $\tau$ is the interval duration ($4\,\mathrm{s}$). We also calculate
$\delta_{i}$ in units of inertial length ($\mathrm{d_{pi}}$) where
$\rho_{\mathrm{p}i}$ is replaced by $\mathrm{d_{pi}}$. We label sets of
consecutive unstable intervals with the number $j$ and calculate their length
$\Delta_{j}$ with Eqs. (3) and (4).

We calculate the lengthscales associated with the persistence of instabilities
in the solar wind using Taylor’s hypothesis (Taylor, 1938). As indicated in
Figure 1, we identify all intervals with parameters above an instability
threshold from Eq. (1) in the complete dataset separately for oblique firehose
and mirror-mode instabilities. Unstable intervals are shown as individual
green boxes in Figure 1. We calculate the lengthscale $l_{i}=V_{i}\tau$ for
each unstable interval $i$, where $V_{i}$ is the proton bulk velocity of
interval $i$ and $\tau=4\,\mathrm{s}$ is the cadence of PAS. Using the proton
gyroradius $\rho_{\mathrm{p}i}$ and the inertial length $d_{\mathrm{p}i}$ for
each individual interval $i$, we then calculate the dimensionless lengthscales
$\delta_{i}^{\rho}=l_{i}/\rho_{\mathrm{p}i}$ and
$\delta_{i}^{d}=l_{i}/d_{\mathrm{p}i}$. The sums of the consecutive
dimensionless lengthscales give the total dimensionless persistance interval
for each occurrence of the respective instability as convected over the
spacecraft. We define them as

$\Delta_{j}^{\rho}=\sum\limits_{i}\delta_{i}^{\rho}$ (3)

and

$\Delta_{j}^{d}=\sum\limits_{i}\delta_{i}^{d},$ (4)

where the index $j$ indicates each set of consecutive unstable intervals, and
the index $i$ sums over all individual intervals that contribute to the
persistence interval $j$.

##  3 Results

###  3.1 Data overview

Figure 2: Data distribution in a
$T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$-plot. We overplot the
instability thresholds according to Eq. (1) for
$\gamma_{m}=10^{-2}\Omega_{p}$. The color-coding indicates the probability
density of datapoints in each bin.

In Figure 2, we show the $T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$-plot of
the probability density function (PDF) of our full dataset. From the total
dataset, 940 598 individual data points are stable to both the mirror-mode and
oblique firehose instabilities, within the regime that we classify as both
mirror-stable and firehose-stable. 4526 individual data points are in the
mirror-mode unstable region ($0.46\%$) and 30 392 individual data points in
the oblique firehose unstable region ($3.12\%$). The instability thresholds
apparently bound the probability distribution, as has been noted by others
(Hellinger et al., 2006; Bale et al., 2009).

The number of data points in the mirror-mode and oblique firehose unstable
regimes is sufficient to allow a separate statistical analysis of these
regions. For this investigation, we define four categories of data: ’all’ data
represents the complete dataset; ’stable’ refers to the data points that are
stable to both the mirror-mode and the oblique firehose instabilities;
’oblique firehose unstable’ and ’mirror-mode unstable’ refers to the data
points in the regions beyond their respective instability thresholds with
$\gamma_{m}>10^{-2}\Omega_{p}$.

###  3.2 Angle analysis

Figure 3: Polar plots of the distributions of the four categories of data
points as functions of $\theta_{BV}$. The three panels show (a) all data
(blue) with stable data (orange) overlaid, (b) oblique firehose unstable data
(green), and (c) mirror-mode unstable data (magenta). Both (b) and (c) are
normalized according to Eq. (5). The grey concentric circles are spaced in
increments of (a) 2500, (b) 0.02, and (c) 0.005, starting from zero.
$0^{\circ}$ is the direction of $\boldsymbol{V}$. The distributions are binned
at $3^{\circ}$ resolution.  Figure 4: PDF of the angular distribution of our
four data categories: all data (blue), stable (orange), oblique firehose
unstable (green), mirror-mode unstable (magenta). The orange histogram is
mostly obscured by the blue. PDFs, calculated according to Eq. (6), are binned
at $5^{\circ}$ resolution.

Figure 3 shows by category of data points (all, stable, oblique firehose
unstable, mirror-mode unstable), the distributions of $\theta_{BV}$,
calculated as described in Section 2.3. We quantify the rate of occurrence of
unstable data for oblique firehose and mirror-mode by normalizing the
distributions by the total number of occurrences of the whole dataset in each
angle bin. This normalization quantifies the statistical significance of the
excess of unstable modes in each angle bin. We calculate the normalized
distribution bin count

$\sigma_{ND}=\frac{\sigma_{R}}{\sigma_{TD}},$ (5)

where $\sigma_{R}$ is the bin count of unstable intervals and $\sigma_{TD}$ is
the total bin count of the whole dataset for a given angle. The resulting
polar plots represent the conformal projection of the 3D angle distribution
onto a 2D (RT) plane.

Panels (b) and (c) of Figure 3 show a clear differentiation in the
distribution of the data points that are oblique firehose and mirror-mode
unstable. We find that oblique firehose unstable data points occur
predominantly when $225^{\circ}\lesssim\theta_{BV}\lesssim 270^{\circ}$. The
mirror-mode unstable data points occur predominantly when $\boldsymbol{B}$ and
$\boldsymbol{V}$ are within $\sim 45^{\circ}$ of alignment or anti-alignment.

We note the uneven distribution of the direction of $\boldsymbol{B}$ between
the sunward and anti-sunward sectors in our dataset. Data in the upper left
quadrants of the plots, where $180^{\circ}\leq\theta_{BV}\leq 270^{\circ}$,
predominates. We attribute this asymmetry to the position of the spacecraft
relative to the current sheet under the quiet solar wind conditions during our
data collection period.

Figure 4 shows by category of data points (all, stable, oblique firehose
unstable, mirror-mode unstable), the probability densities of $\theta_{BV}$,
calculated as described in Section 2.3. For purposes of comparison, we plot
the normalized density bin count

$\sigma_{N}=\frac{\sigma_{R}}{\sigma_{T}W_{b}}$ (6)

for each category, where $\sigma_{R}$ is the bin count, $\sigma_{T}$ is the
total bin count of the plotted dataset across all angle bins, and $W_{b}$ is
the bin width. This distribution is normalized so that
$\sum(\sigma_{N}W_{b})=1$. The maxima of the PDF, shown by the peak values for
the distribution of all data are at $\theta_{BV}\approx 70^{\circ}$ in the
anti-sunward and $\approx 225^{\circ}$ in the sunward direction. The PDF of
mirror-mode unstable points peaks and exceeds the PDF of stable points at
$\theta_{BV}\approx 25^{\circ}$, $155^{\circ}$, $225^{\circ}$, and
$330^{\circ}$, whereas the PDF of oblique firehose unstable points peaks and
exceeds the PDF of stable points at $\theta_{BV}\approx 75^{\circ}$ and
$255^{\circ}$.

Figure 5: Dependence of $T_{\perp}$ and $T_{\parallel}$ on $\theta_{BV}$ in
the solar wind. The upper panels ((a) $T_{\perp}$ and (b) $T_{\parallel}$)
only include mirror-mode unstable points. The lower panels ((c) $T_{\perp}$
and (d) $T_{\parallel}$) only include oblique firehose unstable points. The
color scale indicates the number of data points per bin.

We plot the distributions of $T_{\perp}$ and $T_{\parallel}$ against
$\theta_{BV}$ for datapoints in the mirror-mode and oblique firehose unstable
categories in Figure 5. In both cases, the distributions of $T_{\perp}$ and
$T_{\parallel}$ separately exhibit variability with $\theta_{BV}$ consistent
with the pattern of angular dependence in Figure 4. Similar distributions for
$\beta_{\parallel}$ (not shown here) do not reveal a marked dependence on
$\theta_{BV}$.

We explore the correlation between temperature anisotropy and $\theta_{BV}$ by
investigating the mean values of $T_{\perp}$ and $T_{\parallel}$ as functions
of $\theta_{BV}$ for the complete dataset and for each of the unstable
categories. For this calculation, we first sort all $N_{D}$ data points in
each of the data intervals by $\theta_{BV}$, where $N_{D}$ is the total number
of data points in the interval. We then calculate a running mean over
$\theta_{BV}$ for the separate parameters $T_{\perp}$ and $T_{\parallel}$
using a moving averaging window of length $0.04N_{D}$. The results are plotted
as Figure 6.

Figure 6: Panels (a) to (c) show $T_{\perp}$ (red) and $T_{\parallel}$ (cyan)
dependence on $\theta_{BV}$ in the solar wind for (a) mirror-mode, (b) oblique
firehose unstable, and (c) all data. Panel (d) shows the equivalent dependency
of $T_{R}$ (blue) and $T_{TN}$ (green) for all data. The result shown is a
running mean over $\theta_{BV}$ after sorting the data points by
$\theta_{BV}$.

We show the running mean for each of $T_{\perp}$ and $T_{\parallel}$ for the
mirror-mode instability, the oblique firehose instability, and for all data in
panels (a) through (c). For comparison, we show a similar running mean of
$T_{R}$ and $T_{TN}$ in panel (d), where $T_{R}$ is taken directly as the
radial temperature from the $T_{RTN}$ dataset in the proton ground moments.
$T_{TN}$ is given by $(T_{T}+T_{N})/2$, where $T_{T}$ is the tangential
temperature and $T_{N}$ is the normal temperature.

###  3.3 Lengthscale analysis

Figure 7: PDF plots of persistence intervals in lengthscale for mirror-mode
(magenta) and oblique firehose (green) instabilities in the solar wind. Panels
(a) and (b) show the lengthscale in proton gyroradii ($\Delta^{\rho}$). Panels
(c) and (d) show lengthscale in proton inertial lengths ($\Delta^{d}$). We
indicate breakpoints at $\Delta_{b}^{\rho}$ with cyan vertical lines and
power-law fits in dark blue, giving power indexes of $\approx-2.17$ for both
oblique firehose and mirror-mode instabilities.

In Figure 7, we show the PDFs of instability persistence for both oblique
firehose and mirror-mode instabilities measured in units of the proton
gyroradius and in units of the inertial length. Panels (a) and (b) show a
power-law relationship with a distinct break ($\Delta_{b}^{\rho}$) at $\sim
34\rho_{p}$ for the oblique firehose instability and at $\sim 24\rho_{p}$ for
the mirror-mode instability. At scales smaller than the break, the dependence
of the instability persistence PDF on $\Delta^{\rho}$ exhibits a shallow
gradient. At larger scales, the fitted power-law relationship is appreciably
steeper and shows an exponent of $\approx-2.17$ for both the oblique firehose
and the mirror-mode. This exponent is consistent with those found for
$\Delta^{d}$ in panels (c) and (d), although here breakpoints are not readily
identifiable.

According to previous studies (Gary, 1993; Pokhotelov, 2004), the maximum
growth rate of the oblique firehose and mirror-mode instabilities occurs when
$k_{w}\rho_{\mathrm{p}}\approx 0.5$, where $k_{w}$ is the wavenumber. The
associated instability wavelength at maximum growth is then given by

$\lambda_{w}=\frac{2\pi}{k_{w}}\approx\frac{2\pi\rho_{\mathrm{p}}}{0.5}.$ (7)

Using $\Delta_{b}^{\rho}$ above, Eq. (7) indicates that the space required for
the instabilities to act is $\sim 2.7\lambda_{w}$ for the oblique firehose
instability and $\sim 1.9\lambda_{w}$ for the mirror-mode instability.

##  4 Discussion

###  4.1 Dependence on measurement cadence

By contrast with most previous studies, we find a more extensive distribution
of datapoints in the unstable regions of our
$T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$-plot in Figure 2. This
difference is likely due to the higher sampling cadence of PAS at
$4\,\mathrm{s}$ in normal mode (Owen et al., 2020) than, for example, that of
the WIND SWE instrument. SWE has a cadence for its Faraday cup ion sensor of
$92\,\mathrm{s}$ (Ogilvie et al., 1995; Maruca & Kasper, 2013) and is the data
source for many earlier studies (Bale et al., 2009; Chen et al., 2016;
Hellinger et al., 2006; Maruca et al., 2012; Verscharen et al., 2016). PAS’s
higher cadence enables the instrument to sample characteristic features of the
solar wind without averaging over variations in the distribution at timescales
greater than a few seconds (Verscharen & Marsch, 2011; Nicolaou et al., 2019).

We demonstrate the impact of longer sampling times by averaging our dataset in
Appendix A. In Figure 8, we show the consequent decrease in the proportions of
datapoints in the mirror-mode and oblique firehose unstable regions with
increasing measurement cadence. Although we observe in our data widespread
distributions of data points in the parts of parameter space characterized as
unstable according to Eq. (1), the overall pattern of regulation by non-
propagating instabilities ultimately remains consistent with the earlier work.

###  4.2 Interpretation of our angle analysis

For our analysis, we divide the polar plot in Figure 3 into 8 segments of
$45^{\circ}$ arc. We define the four segments between $315^{\circ}$ and
$45^{\circ}$ and between $135^{\circ}$ and $225^{\circ}$ as ‘quasi-parallel’
with respect to the flow direction of the solar wind. We define the other four
segments as ‘quasi-perpendicular’ with respect to the flow direction of the
solar wind.

The oblique firehose instability is driven by $T_{\parallel}>T_{\perp}$ which,
according to Figure 3 (b), most frequently corresponds to excess pressure in
the direction quasi-perpendicular to the flow velocity. This direction is also
approaching alignment with the Parker spiral angle. The mirror-mode
instability is driven by $T_{\perp}>T_{\parallel}$ which, according to Figure
3 (c), most frequently also corresponds to excess pressure in the direction
quasi-perpendicular to the flow velocity. This finding suggests that the
expansion direction plays a crucial role for the generation of plasma
conditions that drive both oblique firehose and mirror-mode instabilities.
However, the large-scale double-adiabatic expansion alone according to the
Chew–Goldberger–Low (CGL) prediction (Chew et al., 1956; Parker, 1958;
Matteini et al., 2007, 2012) does not produce this observed correlation. In
fact, the observed correlations are opposite to the expectations from CGL
expansion alone, as shown in Appendix B. Instead, we must invoke a non-CGL
expansion of the ions, which is a known observational result (Marsch et al.,
2004; Matteini et al., 2007) and confirmed by the analysis of the two-fluid
thermal energy equation (e.g. Hellinger et al., 2013).

A possible explanation for the angular dependency of the PDF for the oblique
firehose and mirror-mode instabilities lies in the presence and variability of
local, large-scale Alfvénic fluctuations. These fluctuations lead to a time
variation in $\theta_{BV}$ at the location of the spacecraft (D’Amicis et al.,
2019). Greater amplitudes of large-scale Alfvénic fluctuations typically
coincide with increased perpendicular ion heating in the solar wind (Bruno et
al., 2004, 2006). This increased perpendicular heating often generates
temperature anisotropy with $T_{\perp}>T_{\parallel}$ and thus favorable
conditions for the excitation of mirror-mode instabilities (Matteini et al.,
2006; Yoon et al., 2021). As a consequence, we expect a statistically
increased occurrence of mirror-mode unstable data points at times with large-
amplitude Alfvénic fluctuations. These are more likely to be associated with
$\theta_{BV}$ angles away from the direction of the Parker spiral than times
without large-amplitude Alfvénic fluctuations (Bruno et al., 2004).

In this interpretation, we associate oblique firehose unstable intervals with
solar wind parcels with low amplitudes of Alfvénic fluctuations. We expect a
stronger average alignment of the distribution of these oblique firehose
unstable intervals with the average $\theta_{BV}$ of the solar wind. This is
consistent with Figure 3 (a) and (b), which shows that the distribution of
oblique firehose unstable intervals is mostly aligned with the average
direction of $\boldsymbol{B}$, representing the Parker spiral angle (Parker,
1965) – notwithstanding the asymmetry of sampled sector structures in our
dataset.

Figure 6 (a) through (c) demonstrate that the measured solar wind on average
exhibits $T_{\parallel}>T_{\perp}$, so that conditions favorable for the
excitation of the mirror-mode instability are the exception. However, panel
(c) shows that for all data $T_{\parallel}$ and $T_{\perp}$ converge when
$\boldsymbol{B}$ and $\boldsymbol{V}$ approach alignment or anti-alignment,
which is consistent with the statistical distribution of the unstable
datapoints we observe in Figure 4. However, the sector asymmetry of our
dataset makes this convergence stronger in the case of $\theta_{BV}\sim
360^{\circ}$.

The distributions of the unstable data as a function of $\theta_{BV}$ shown in
Figure 3 (b) and (c), together with the required temperature anisotropy to
drive each of the instabilities, suggest that the mirror-mode and oblique
firehose instabilities act predominantly when $T_{TN}>T_{R}$, where for the
mirror-mode $T_{TN}\sim T_{\perp}$ and for the oblique firehose $T_{TN}\sim
T_{\parallel}$. The mean values of the dataset show that, on average,
$T_{R}/T_{TN}$ = 0.985 (all data), 0.447 (oblique firehose unstable data
points), and 0.572 (mirror-mode unstable data points). Figure 6 (d) shows the
$\theta_{BV}$-dependence in the variability of $T_{R}$ and $T_{TN}$ which
derives from the general condition that $T_{\parallel}>T_{\perp}$ on average
as shown in Figure 6 (c). The angular dependency of the observed oblique
firehose unstable data is consistent with this variability. Whereas the
distribution of mirror-mode unstable data peaks at the values of $\theta_{BV}$
where the plots of $T_{R}$ and $T_{TN}$ intersect. We observe that these
points of intersection are the limits to the values of $\theta_{BV}$ where
both $T_{TN}\sim T\perp$ and $T_{TN}\geq T_{R}$.

###  4.3 Interpretation of our lengthscale analysis

In Figure 7, we find that our PDF of $\Delta^{\rho}$ steepens appreciably at
the breakpoint $\Delta^{\rho}_{b}$. The breakpoint is more clearly defined by
lengthscales normalized in units of $\rho_{p}$ than in units of $d_{p}$, which
is expected since these instabilities grow on scales associated with the
gyroscale rather than the inertial length (Howes et al., 2011; Matthaeus et
al., 2014). We interpret the shallower PDF at
$\Delta^{\rho}<\Delta_{b}^{\rho}$ as an indication that, in spatial intervals
shorter than $\Delta_{b}^{\rho}$, the instabilities are less efficient in
reducing the temperature anisotropy to stable values than in intervals longer
than $\Delta_{b}^{\rho}$. The existence of this breakpoint and the transition
into a steeper slope at $\Delta^{\rho}>\Delta_{b}^{\rho}$ is consistent with
our conjecture that the efficiency of oblique firehose and mirror-mode
instabilities is scale-dependent. In this interpretation, $\Delta_{b}^{\rho}$
represents the minimum length of plasma intervals with unstable parameters for
which the instabilities efficiently modify the plasma into a stable state. We
interpret the difference in $\Delta_{b}^{\rho}$ between oblique firehose and
mirror-mode instabilities as an indication that these instabilities set
different requirements on the homogeneity of the unstable plasma volumes. We
observe that the power-law index beyond the breakpoint is consistent for both
categories of unstable data and independent of our lengthscale normalization.
This universality suggests that the power laws themselves are representative
of the underlying distribution of conditions that drive the analyzed
instabilities in the solar wind.

We find that $\Delta_{b}^{\rho}$ corresponds to approximately 2 to 3
wavelengths of the unstable mode at typical maximum growth rates. This result
suggests that the conditions needed for instabilities to act efficiently are
bounded by spatial scales $\mathcal{O}(10^{3}\,\mathrm{km})$ that are very
short relative to the correlation length of solar wind turbulence which has
been measured as $\mathcal{O}(10^{6}\,\mathrm{km})$ (Matthaeus & Goldstein,
1982; Matthaeus et al., 2005). However, the evaluation of the influence of the
turbulent cascade on linear processes requires a scale-dependent comparison of
nonlinear and linear timescales (Matthaeus et al., 2014). While outside the
scope of this study, it would be worthwhile to compare the scale-dependent
eddy turnover times of the plasma turbulence at the scales of the unstable
intervals. Such a comparison would allow the assessment of the timescales that
potentially create and destroy the conditions required for instabilities to
act (Klein et al., 2017, 2018; Qudsi et al., 2020).

In our study, we assume Taylor’s hypothesis to link temporal variations in the
measurements with spatial variations in the solar wind. A single-spacecraft
measurement is unable to disentangle temporal and spatial variations.
Simulations show a temporal latency in the onset of the oblique firehose
instability (López et al., 2022), which complicates the interpretation of the
spatial latency discussed in this work. The average temporal persistence of
datapoints in the unstable regions of our
$T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$-plot is $10.33\,\mathrm{s}$ for
the mirror-mode and $11.49\,\mathrm{s}$ for the oblique firehose instability.
From these numbers, we infer that a sampling cadence of less than $\sim
10\,\mathrm{s}$ is needed to observe the full lengthscale distribution of
unstable regions in the solar wind in our dataset. This cadence is equivalent
to a spatial scale of $\sim 80\rho_{\mathrm{p}}$ convected over the
spacecraft, given the average values for $V$ and $\rho_{\mathrm{p}}$ derived
from our data.

###  4.4 Limitations of our analysis

Our analysis is necessarily limited by the statistics and quality of the
dataset. We use data from the cruise phase of the Solar Orbiter mission
(Zouganelis et al., 2020). Our period of data collection coincides with
relatively quiet solar wind conditions, and the dataset contains mostly slow
solar wind observations. Instrumental effects on our observations have been
carefully evaluated in consultation with the SWA and MAG teams. Our rigorous
application of the available quality filters to the data and our exchanges
with the instrument teams increase the reliability of our analysis. Our result
in Figure 2 depends on the details of the method used to define the
instability thresholds (see also Isenberg et al., 2013). Our method is however
consistent with previous studies concerning the specific instabilities we
consider (Hellinger et al., 2006; Bale et al., 2009). Our Figure 6 differs
from the more straightforward $T_{\perp}$-dependence presented by D’Amicis et
al. (2019) who find a positive correlation between $\theta_{BV}$ and
$T_{\perp}$ in Alfvénic fast solar wind. However, this result does not
contradict our analysis, given that we mostly observe slow solar wind in our
dataset.

##  5 Conclusions

We perform a statistical analysis of a large Solar Orbiter dataset ($\sim
10^{6}$ data points) to investigate the conditions necessary in the solar wind
for the oblique firehose and mirror-mode instabilities to reduce temperature
anisotropies. Our motivation is to use the newly available high-resolution
data from the Solar Orbiter mission to explore energy transfer processes at
small scales.

In our $T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$-plot, we find that, while
the investigated instabilities largely limit the plasma anisotropy, a
significant number of data points ($\sim 3\times 10^{4}$ for the oblique
firehose and $\sim 4\times 10^{3}$ for the mirror-mode) lie in the unstable
regions of parameter space. We interpret these as transient features whose
full extent is revealed by the short measurement time ($1\,\mathrm{s}$) of the
SWA instrument’s PAS (Owen et al., 2020).

We explore the dependency of the distribution of oblique firehose and mirror-
mode unstable solar wind intervals on $\theta_{BV}$ and find that the mirror-
mode instability predominantly occurs when $\theta_{BV}\approx 0^{\circ}\pm
45^{\circ}$ or $\theta_{BV}\approx 180^{\circ}\pm 45^{\circ}$ and hence when
$\boldsymbol{B}$ is close to the radial (or anti-radial) direction. By
contrast, the peak in the PDF of the oblique firehose instability occurs when
$\theta_{BV}\approx 75^{\circ}$ or $\theta_{BV}\approx 255^{\circ}$ and hence
when $\boldsymbol{B}$ is close to a direction perpendicular to
$\boldsymbol{V}$. This result suggests a predominant elevation of the
temperature $T_{TN}$ perpendicular to the radial direction relative to the
temperature $T_{R}$ in the radial direction in the unstable intervals, which
is inconsistent with the predictions from the CGL double-adiabatic expansion
of the solar wind alone. We interpret this dependency of the mirror-mode
(oblique firehose) unstable plasma intervals on $\theta_{BV}$ as due to the
presence (absence) of perpendicular ion heating from local, large-scale
Alfvénic fluctuations.

In our analysis of $\theta_{BV}$, we do not confine consideration to Alfvénic
wind intervals (Louarn et al., 2021; D’Amicis et al., 2019; Woodham et al.,
2021) but instead concentrate on the relationship between $\theta_{BV}$ and
the specific proton temperature anisotropy that drives the oblique firehose
and mirror-mode instabilities. This allows us to investigate the conditions
needed for instabilities to act efficiently on the solar wind plasma and
possible explanations for the occurrence of these conditions. However, our
analysis is necessarily constrained by the statistical $\theta_{BV}$
distribution in our dataset, which we expect to become less asymmetrical as
more data are included while the Solar Orbiter mission continues.

We also measure the persistence of unstable solar wind intervals. The oblique
firehose and mirror-mode instabilities require intervals of a size greater
than about $34\rho_{p}$ and $24\rho_{p}$, respectively, in order to regulate
the temperature anisotropy efficiently. These lengthscales are more clearly
defined in units of $\rho_{p}$ rather than in units of $d_{p}$. The minimum
space to develop and regulate anisotropy corresponds to approximately 2 to 3
typical wavelengths of the unstable mode at maximum growth rate.

Our work highlights the intricate connections between expansion effects,
turbulence, and kinetic micro-instabilities in the solar wind. Numerical
simulations show that a combination of plasma expansion and strong 2D
turbulence can drive both oblique firehose and mirror-mode instabilities
(Hellinger et al., 2015, 2017). In addition, the spread of data in the
$T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$ parameter space is increased by
pronounced small-scale intermittency in strong turbulence (Servidio et al.,
2014). The combination of in-situ instruments on the Solar Orbiter mission
allows us to study particle distributions with a very high time resolution,
which helps to gain fresh insight into the underlying processes (Adhikari et
al., 2021; D’Amicis et al., 2021; Louarn et al., 2021; Nicolaou et al., 2021;
Owen et al., 2021). We expect high-cadence in-situ observations in combination
with kinetic simulations of the expanding solar wind (Dong et al., 2014;
Hellinger et al., 2015; Franci et al., 2015) to deliver further insights into
this interplay in the future.

We appreciate very helpful discussions with the MAG and PAS instrument teams.
We thank Thomas Keel, Kris Klein, and Rob Wicks for very useful discussions
and advice, and Jonathan Niehof for very helpful personal correspondence on
the application of SpacePy’s pycdf toolkit. S. O. is supported by the Natural
Environment Research Council (NERC) grant NE/S007229/1. D. V. is supported by
the Science and Technology Facilities Council (STFC) Ernest Rutherford
Fellowship ST/P003826/1. D. V. and C. J. O. are supported by STFC Consolidated
Grants ST/S000240/1 and ST/W001004/1. C. H. K. C. is supported by UKRI Future
Leaders Fellowship MR/W007657/1, STFC Ernest Rutherford Fellowship
ST/N003748/2 and STFC Consolidated Grant ST/T00018X/1. P. A. I. is supported
by NASA grant 80NSSC18K1215 and by National Science Foundation (NSF) grant
AGS2005982. Solar Orbiter is a space mission of international collaboration
between ESA and NASA, operated by ESA. Solar Orbiter Solar Wind Analyser (SWA)
data are derived from scientific sensors which have been designed and created,
and are operated under funding provided in numerous contracts from the UK
Space Agency (UKSA), STFC, the Agenzia Spaziale Italiana (ASI), the Centre
National d’Etudes Spatiales (CNES), the Centre National de la Recherche
Scientifique (CNRS), the Czech contribution to the ESA PRODEX programme and
NASA. Solar Orbiter SWA work at UCL/MSSL is currently funded under STFC grants
ST/T001356/1 and ST/S000240/1. The Solar Orbiter magnetometer was funded by
UKSA grant ST/T001062/1.

##  Appendix A Averaging effect

Figure 8: $T_{\perp}/T_{\parallel}$-$\beta_{\parallel}$-plots for averaged
solar wind parameters with different averaging times. We overplot the
instability thresholds according to Eq. (1) for
$\gamma_{m}=10^{-2}\Omega_{p}$. The color-coding indicates the PDF of
datapoints in each bin. The different panels represent averaging of data over
subsequent sampling steps to simulate instruments with lower measurement
cadences: (a) $4\,\mathrm{s}$ (original dataset), (b) $12\,\mathrm{s}$, (c)
$48\,\mathrm{s}$, and (d) $92\,\mathrm{s}$.

Our analysis in Figure 7 suggests that plasma instruments with low measurement
cadence detect a lower proportion of unstable intervals than actually exist.
In general, all instruments miss unstable intervals with a duration in the
spacecraft frame comparable to the measurement cadence or shorter.

We simulate different measurement cadences by averaging our data over a
successive number of sampling intervals. In Figure 8, we show the results
averaged over 12, 48, and 92 s, compared with the base dataset at 4-s cadence.
The proportional share of datapoints in the regions of parameter space
unstable to oblique firehose and mirror-mode instabilities increases with
increasing cadence. At 4 s cadence, 3.12% of the data are oblique firehose
unstable and 0.46% of the data are mirror-mode unstable. At 92 s, which
corresponds approximately to the sampling cadence of the SWE instrument
onboard WIND, these numbers decrease to 2.45% for the oblique firehose
instability and 0.31% for the mirror-mode instability. Moreover, the number of
data points with extreme $\beta_{\parallel}$-values decreases significantly.

##  Appendix B CGL Analysis

The double-adiabatic expansion according to the CGL theory is often considered
an important contributor to the impact of expansion on the temperature
anisotropy of the solar wind (Matteini et al., 2007; Verscharen et al., 2016).
In this Appendix, we evaluate the consistency of the CGL approach with our
observations.

We start by assuming that the solar-wind response to the expansion is
consistent with the CGL equations (Chew et al., 1956):

$\frac{d}{dt}\left(\frac{T_{\perp}}{B}\right)=0$ (B1)

and

$\frac{d}{dt}\left(\frac{T_{\parallel}B^{2}}{n_{p}^{2}}\right)=0.$ (B2)

When $\boldsymbol{B}$ is purely in the radial direction, $B\propto r^{-2}$ due
to $\nabla\cdot B=0$. Likewise, when $\boldsymbol{B}$ is purely in the
tangential direction, $B\propto r^{-1}$ in a spherically symmetric
configuration (Matteini et al., 2007, 2012; Hellinger et al., 2015). From Eqs.
(B1) and (B2), we obtain

$\frac{T_{\perp}}{T_{\parallel}}\propto\frac{B^{3}}{n_{p}^{2}}.$ (B3)

Hence, under the assumption that $\boldsymbol{B}$ is purely radial, we find

$\frac{T_{\perp}}{T_{\parallel}}\propto r^{-2}.$ (B4)

Under the assumption that $\boldsymbol{B}$ is purely tangential, we find

$\frac{T_{\perp}}{T_{\parallel}}\propto r.$ (B5)

According to Eqs. (B4) and (B5), CGL double-adiabatic expansion in a smooth
magnetic field predicts that $T_{\perp}/T_{\parallel}$ drops more quickly with
$r$ when $\boldsymbol{B}$ is in the radial direction than when
$\boldsymbol{B}$ is in the tangential direction.

Assuming that the solar wind magnetic field obeys Parker’s model, in which the
Parker angle is a monotonously increasing function of $r$ (Parker, 1965), we
expect conditions favorable for the oblique firehose instability especially
when the field is quasi-radial. However, Figure 3 reveals the opposite
behavior and even a higher occurrence of mirror-mode unstable intervals in the
quasi-radial field geometry. This finding suggests that CGL double-adiabatic
expansion alone does not provide a consistent explanation for the angular
dependency result shown in Figure 3. The observed distribution of these
instabilities contradicts the CGL prediction, suggesting that other processes
must dominate the observed occurrence rates of unstable intervals.

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