[1]Kristopher G. Klein [2]Harlan Spence

1]Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721,
USA2]Space Science Center, University of New Hampshire, Durham, NH 03824,
USA3]Observatoire de Paris, LESIA, Meudon, 92190, France4]Institute for
Advanced Study, Princeton, NJ, 08540, USA5]Department of Astronomy &
Astrophysics, University of Chicago Chicago, IL, 60637, USA6]BWX Technologies,
Inc. Washington DC, 20001, USA7]Heliophysics Science Division, NASA Goddard
Space Flight Center, Greenbelt, MD, 20771, USA8]Blackett Laboratory, Imperial
College London London, SW7 2AZ, UK9]Mullard Space Science Laboratory,
University College London Dorking, RH5 6NT, UK10]Institut de Recherche en en
Astrophysique et Planetologie, 31028 Toulouse, France11]Department of Physics
and Astronomy, University of Iowa, Iowa City, IA 52242, USA12]Ames Research
Center, NASA, Mountain View, CA, 94043, USA13]Laboratoire de Physique et
Chimie de l’Environnement et de l’Espace, CNRS and Université d’Orléans, 45071
Orléans, France14]Department of Astrophysical Sciences, Princeton University
Princeton, NJ, 08540 USA15]Laboratoire d’astrophysique de Bordeaux, Univ.
Bordeaux, CNRS, Pessac, France16]Laboratoire de Physique des Plasmas,
CNRS/Sorbonne Université/Université Paris-Saclay/Observatoire de Paris/Ecole
Polytechnique Institut Polytechnique de Paris, Paris, 75005 France17]Space
Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720,
USA18]Department of Physics & Astronomy, University of Delaware, Newark, DE
19716, USA19]Institute of Astronomy, University of Cambridge, Cambridge, CB3
OHA, UK20]Space Research Institute, Austrian Academy of Sciences, Graz,
Austria21]Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
Oxford, OX1 3PU, UK22]Los Alamos National Laboratory, Las Alamos, NM 87545,
USA23]Center for Astrophysics, Harvard & Smithsonian Cambridge, MA 02138,
USA24]Department of Space Science, University of Alabama in Huntsville
Huntsville, AL 35899, USA25]Department of Astronomy, University of Wisconsin-
Madison Madison, WI 53706, USA

# HelioSwarm: A Multipoint, Multiscale Mission to Characterize Turbulence

<EMAIL_ADDRESS><EMAIL_ADDRESS>Olga Alexandrova  Matthew Argall
Lev Arzamasskiy  Jay Bookbinder  Theodore Broeren  Damiano Caprioli  Anthony
Case  Benjamin Chandran  Li-Jen Chen  Ivan Dors  Jonathan Eastwood  Colin
Forsyth  Antoinette Galvin  Vincent Genot  Jasper Halekas${}^{\textrm{11}}$
Michael Hesse  Butler Hine  Tim Horbury  Lan Jian  Justin Kasper  Matthieu
Kretzschmar  Matthew Kunz  Benoit Lavraud  Olivier Le Contel  Alfred Mallet
Bennett Maruca  William Matthaeus  Jonathan Niehof  Helen O’Brian  Christopher
Owen  Alessandro Retino  Christopher Reynolds  Owen Roberts  Alexander
Schekochihin  Ruth Skoug${}^{\textrm{22}}$ Charles Smith  Sonya Smith  John
Steinberg${}^{\textrm{22}}$ Michael Stevens  Adam Szabo  Jason TenBarge  Roy
Torbert  Bernard Vasquez  Daniel Verscharen  Phyllis Whittlesey  Brittany
Wickizer  Gary Zank  Ellen Zweibel  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [  [  [  [

###### Abstract

HelioSwarm (HS) is a NASA Medium-Class Explorer mission of the Heliophysics
Division designed to explore the dynamic three-dimensional mechanisms
controlling the physics of plasma turbulence, a ubiquitous process occurring
in the heliosphere and in plasmas throughout the universe. This will be
accomplished by making simultaneous measurements at nine spacecraft with
separations spanning magnetohydrodynamic and sub-ion spatial scales in a
variety of near-Earth plasmas. In this paper, we describe the scientific
background for the HS investigation, the mission goals and objectives, the
observatory reference trajectory and instrumentation implementation before the
start of Phase B. Through multipoint, multiscale measurements, HS promises to
reveal how energy is transferred across scales and boundaries in plasmas
throughout the universe.

###### keywords:

Turbulence, Space Plasma, Heliophysics, NASA Mission, HelioSwarm

##  1 Introduction

Turbulence is multiscale disorder. It is the process by which energy that has
been injected into a system is transported between fluctuating magnetic fields
and plasma motion with larger and smaller spatial scales. Once this cascade of
energy reaches sufficiently small scales, dissipation mechanisms can act
efficiently to remove energy from the fluctuations, leading to heating of the
constituent particles. Observations from single spacecraft provide only
information along a single path through a turbulent system; leveraging such
measurements to understand turbulence relies on assumptions about the
underlying spatial and temporal structure. Clusters of four spacecraft provide
more information about spatial structure, but are sensitive to only a single
scale for a given configuration. Understanding fundamental processes such as
turbulence requires characterizing the underlying fluctuations and their
dynamic evolution across many characteristic scales simultaneously. HelioSwarm
(HS) is a Heliophysics Division NASA Medium-Class Explorer mission designed to
make such multiscale observations.

HS, currently in Phase B-prep, is the first mission that will make the
required measurements to transform the current understanding of space plasma
turbulence, using a first-ever swarm of nine spacecraft (SC), composed of one
Hub and eight Nodes. The nine spacecraft, comprising the HelioSwarm
Observatory, co-orbit in a lunar resonant Earth orbit, with a 2-week period, a
mean-apogee radius of $\sim 60R_{E}$ and a mean-perigee radius of $\sim
11R_{E}$, where $R_{E}=6.371\times 10^{6}$ m is the Earth’s radius. This
orbit, illustrated in Figure 1, enables measurements of a variety of near-
Earth plasma environments, including the pristine solar wind (SW), the
magnetically connected foreshock, the magnetosheath, and the magnetosphere.
Carefully designed trajectories produce separations between the spacecraft
spanning magnetohydrodynamic (MHD) and sub-ion (e.g., ion gyroradius) spatial
scales, allowing us to address a broad set of questions about the three-
dimensional dynamics of magnetized turbulence. Answering these open questions
was identified as a science priority in the 2013 Heliophysics Decadal
Surveyheliodecadal:2013  and is deeply rooted in decades-earlier
recommendations by the space science community in the 1980 report by the
Plasma Turbulence Explorer Study GroupMontgomery:1980 . As the first
multipoint, multiscale mission, HS gives an unprecedented view into the nature
of space plasma turbulence.

Figure 1: HelioSwarm Observatory Configuration drawn from the Phase B Design
Reference Mission (DRM) trajectory. In the right row, the Observatory location
(black pentagon) and orbits relative to Earth over the 12-month science phase
are shown in both the X-Y and X-Z GSE planes. Lunar position is indicated by
an open circle for reference, with different regions of near-Earth plasmas
indicated with color. The connected foreshock is here defined based on a
typical Parker spiral orientation. The remaining panels characterize two-
dimensional projections of the relative configurations of the eight Nodes
(black) with respect to the central Hub (red) in Geocentric Solar Ecliptic
(GSE) coördinates. A video of the HS DRM Configuration throughout the Science
Phase is available in Online Resource 1.

In Sec. 2, we describe the scientific motivation for HS and in Sec. 3 we
enumerate the specific mission goals and objectives. Secs. 4 and 5 describe
the requirements on what the missions will measure and the observatory
trajectories and instrumentation. Sec. 6 illustrates the application of
analysis methods to synthetic data modeling future observations drawn from
numerical simulations of turbulence. Conclusions and ongoing work towards the
scheduled launch date at the end of this decade are discussed in Sec. 7.

##  2 Scientific Background and Motivation

Turbulent systems consist of fluctuations spanning a wide range of spatial and
temporal scales. Fluctuations interact nonlinearly, typically with a net
transfer of energy from larger to smaller spatial scales. This process, the
energy cascade, couples the injection range of scales, through a lossless
inertial range, into a dissipation range where heating occurs. Turbulence in
plasmas is significantly more complex than in hydrodynamics: plasma motion
couples to dynamically significant electromagnetic fields, the system
possesses many characteristic spatial and temporal scales, supports many
different waves and fluctuations, and in weakly collisional systems many
mechanisms other than viscosity can act to dissipate the cascade. Additional
details on the current state of plasma turbulence research can be found in
recent reviews, e.g. Bruno:2013 ; Kiyani:2015 ; Verscharen:2019 . The most
energetic SW fluctuations are non-compressive Alexandrova:2008  with
properties resembling Alfvén waves Belcher:1971 ; Matthaeus:1999 ; Howes:2015
. Different types of fluctuations nonlinearly interact in different ways
Zank:1996 ; Matthaeus:1999 ; Schekochihin:2009 ; Kunz:2015 ; Kunz:2018
resulting in dramatically different outcomes. The ubiquity of turbulence in
space and astrophysical plasmas makes it a leading candidate for the process
governing the thermodynamics of a wide range of systems. For instance,
turbulence is conjectured to enable angular momentum transport in accretion
disks (Balbus:1998, ), amplify galactic magnetic fields (Kulsrud:2008, ),
affect transport processes Kunz:2022  and establish high temperatures
(Zhuravleva:2018, ) in the intracluster medium of galaxy clusters, determine
the dispersal and mixing of elements in the Interstellar Medium (ISM)
(Scalo:2004, ), and play a key role in star formation (McKee:2007, ).

SW turbulence at the injection scales is driven by large-scale
structuresSmith:1995 ; Matthaeus:2007 . Recent observations from Parker Solar
Probe (PSP) indicate that this regime is formed due to SW processing in the
near-Sun environment Huang:2023 ; Davis:2023 . Measurements of the scale-to-
scale rate of energy transfer, the cascade rate, near the end of the injection
range generally agree with rates near the start of the inertial range, as has
been explicitly demonstrated with Magnetospheric Multiscale (MMS) observations
Bandyopadhyay:2018 .

Inertial-range observations Matthaeus:1982c  exhibit scale-invariant energy
transfer consistent with Kolmogorov theory Kolmogorov:1941 ; Kolmogorov:1962 :
turbulent structures splitting into ever-smaller fluctuations while conserving
energy. The inertial range plasma behaves like a MHD fluid Davidson:2001 ,
with MHD turbulence theory describing relevant phenomena in space physics and
astrophysics Moffatt:2019 ; Parker:1979 ; Kulsrud:2005  and predicting some SW
features Matthaeus:2011 ; Horbury:2012 ; Verscharen:2019 . For example, Figure
2 shows a composite interplanetary magnetic field (IMF) power spectrum from
three magnetometers at 1 AU measured over different time intervals from tens
of days to an hour as the SW rapidly sweeps past the spacecraft.

(a)  (b)  (c)

Figure 2: Single spacecraft missions only provide statistical properties of SW
turbulence averaged over both long times and different kinds of turbulence.
This approach relies on Taylor’s hypothesis to map observed time series to
advected structures, measuring only a single 1D slice of the turbulence (red
line in inset) and thus only provides a crude measure of turbulent properties.
Previous multipoint missions, e.g. MMS and Cluster, are only able to
characterize spatial structure at a single-scale. The HS observatory will
encompass MHD and ion scales scales simultaneously, enabling the
characterization of multiscale structure and dynamics of turbulence in near-
Earth plasmas. Adapted from Verscharen:2019  and Arzamasskiy:2019 .

MHD theory adequately describes the inertial range spectral slope, but
provides no guidance in the critical higher-frequency transition connecting
inertial and dissipation ranges, which begins near the proton gyrofrequency
$f_{p}\equiv\Omega_{p}/2\pi$. At observed frequencies of
$f_{\textrm{break}}\sim 0.33$ Hz in the SW at 1 AU—approximately equivalent to
advected length scales of
$L_{\textrm{break}}=v_{\textrm{SW}}/f_{\textrm{break}}\sim 1200$ km, the
inertial range scale-invariance ends. This break arises before sub-ion scales
(e.g., ion gyroradius, $\rho_{p}$ ), typically at apparent frequencies of
$v_{\textrm{SW}}/\rho_{p}\sim 3$ Hz (length scales $\sim 100$ km) Klein:2019 .
In Figure 2 this breakdown is seen as a change in spectral slope at the
transition between inertial and dissipation ranges. The spectral break
suggests a change in the dominant physical processes and a loss of cascaded
energy. The energy removed from the cascade is partitioned between the ions
and electrons, with its dissipation leading to the heating of these charged
particles.

This process of turbulent dissipation is why SW plasma is much hotter than
simple theories of adiabatic expansion would predict Marsch:2012 . If the SW
were an adiabatically expanding ideal gas, the protons at Earth would be much
cooler than observed Smith:1995  and protons at Jupiter orbit ($\sim 5$ AU)
would be $8$ times cooler than at 1 AU, in contrast to Voyager observations
Richardson:1995 . Non-adiabatic heating via turbulence dominates plasma
thermodynamics throughout much of the solar system, and is a leading candidate
for accelerating the SW Cranmer:2012 ; Verdini:2010 .

The exact heating mechanisms leading to this heating are a matter of
substantial debate. Determining the nature of these mechanisms requires
observing 3D distributions of the turbulent fluctuations. Plasma turbulence is
inherently anisotropic due to preferred directions associated with the IMF
Boldyrev:2006 ; Schekochihin:2009 , radial expansionWoodham:2021b  and large-
scale gradients Volk:1972 ; Grappin:1993 ; Greco:2012b . If turbulent
fluctuations vary primarily parallel to the IMF (slab-like) Ghosh:1998a , then
non-compressive, Alfvénic fluctuations would, at small scales, ultimately
dissipate energy via ion-cyclotron wave (ICW) heating Kasper:2013 . However,
for fluctuations that vary mostly perpendicular to the IMF (quasi-2D
Matthaeus:1990  or “critically balanced” Mallet:2015 ), ICW heating is
exceedingly weak. In this regime, dissipation instead occurs via other
mechanisms such as Landau damping TenBarge:2013a  or stochastic heating
Chandran:2010a . Recent work on imbalanced cascades Meyrand:2021 ; Squire:2022
complicates these models by providing a pathway for low-frequency, anisotropic
turbulence fluctuations to develop small scale structure parallel to the
magnetic field, enabling dissipation via ICWs. If turbulent structures are
highly anisotropic sheets, they may undergo magnetic reconnection, which
causes heating and particle acceleration Matthaeus:1986a ; Mallet:2017b . To
distinguish among these requires an accurate determination of the 3D power
distribution. Previous determinations using single spacecraft use long time
series for sufficient statistics, Horbury:2012 ; Chen:2011a ; Chen:2012b ;
Chen:2016 , combining together intervals of turbulence with very different
properties. The regulation of energetic particle transport, in both SW
Jokipii:1972  and astrophysical plasmas Zweibel:2013 , is also sensitive to
the turbulence spectrum and its anisotropies.

At a fundamental level, the nature of turbulent fluctuations in magnetized
plasmas remains unknown: is it an MHD extension of hydrodynamic eddies
Matthaeus:2011 , a quasi-2D system Zank:1992 , critically balanced wave-like
fluctuations Schekochihin:2009 ; Mallet:2015 , or a dynamically evolving
mixture? The complexity of plasma turbulence precludes simple, analytic
solutions. Numerical simulations are invaluable but limited by incomplete
physics and small system sizeParashar:2015 . Confined laboratory plasmas
Brown:2015 ; Forest:2015 ; Gekelman:2016  have similarly limited scale
separations and access to SW-like physical parameters.

The SW is a natural laboratory where we can finally answer these questions by
concurrently observing turbulent energy transfer and ion heating over a
targeted range of scales. However, single-spacecraft observations of SW
turbulence are fundamentally limited. Multi-spacecraft missions enabled
advances by creating geometric configurations to sample single-scale plasma
structure without relying on Taylor’s hypothesis (futher discussed in Section
3.1.1). While four- (ClusterEscoubet:2001 , MMSBurch:2016 ) and five-
spacecraft (Time History of Events and Macroscale Interactions during
Substorms (THEMIS)Angelopoulos:2008 ) missions produce configurations that
allow for single-scale measurements Chen:2019:WP ; Escoubet:2021 , they still
cannot explore multiple scales simultaneously in three dimensions. Even with
advanced analysis techniques, scales sampled using four spacecraft cover at
most a factor of $\sim 10$, as demonstrated for instance with the wave-
telescope technique Sahraoui:2010a ; Sahraoui:2010b , nowhere near the $>2$
orders of magnitude necessary to simultaneously measure across inertial and
dissipation ranges. HS’s configurations created by 9 spacecraft provide the
first simultaneous multiscale view of plasma turbulence, targeting key scales
from MHD to sub-ion scales. By measuring plasmas at multiple scales
simultaneously, the HS Observatory promises transformative impacts in our
understanding of turbulence, which will be a boon for heliophysics,
astrophysics, and plasma physics Armstrong:1981 ; Elmegreen:2004 ; MacLow:2000
.

##  3 HelioSwarm Goals and Objectives

HS advances Goal 4 of the 2013 National Academy of Sciences (NAS) Heliophysics
Decadal Survey heliodecadal:2013  (DS) which calls on the community to
”[d]iscover and characterize fundamental processes that occur both within the
heliosphere and throughout the universe.” Magnetized plasma turbulence is the
primary mechanism responsible for transforming energy injected at largest
scales into small-scale motions, eventually dissipating as plasma heat. Plasma
turbulence is universal, responsible for energy transfer in such diverse
systems as the solar corona, SW, pulsar wind nebulae, accretion discs,
interstellar medium, planet formation regions, and laboratory fusion devices.
Only the SW is both of sufficient size for multiscale observations and
accessible for in situ measurements. Turbulence is identified as one of eight
DS Goals for SW/Magnetosphere Interactions (SWMI): “Understand the origins and
effects of turbulence and wave particle interactions.” Because of that
importance, turbulence is also identified as a SWMI Decadal Imperative:
“Implement…a multi-spacecraft mission to address cross-scale plasma physics.”
Likewise, the NASA Heliophysics Roadmap Roadmap:2014  highlights
“Understand[ing] the role of turbulence and waves in the transport of mass,
momentum, and energy” as one of its key Research Focus Areas of high priority.
Long standing heliophysics mysteries — as how the solar coronal temperature
increases by orders of magnitude and how the SW is accelerated and heated —
remain unanswered after decades of research because we lack detailed
understanding of how energy in turbulent plasmas heat particles. HS advances
these NAS and NASA science priorities, and will specifically resolve six
science objectives (O) associated with two overarching science goals (G).

  * •

(G1) Reveal the 3D spatial structure and dynamics of turbulence in a weakly
collisional plasma.

    * –

G1O1 Reveal how turbulence energy transfers in the typical SW plasma as a
function of scale and time.

    * –

G1O2 Reveal how the turbulent cascade of energy varies with background
parameters in different SW environments.

    * –

G1O3 Quantify the transfer of turbulent energy between fields, flows, and
proton heat.

    * –

G1O4 Identify the thermodynamic impacts of intermittent structures on protons.

  * •

(G2) Ascertain the mutual impacts of turbulence, variability, and boundaries
near large scale structures.

    * –

G2O1 Determine how SW turbulence affects and is affected by large-scale
structures such as Coronal Mass Ejections (CMEs) and Corotating Interaction
Regions (CIRs).

    * –

G2O2 Determine how driven turbulence differes from that in undisturbed SW.

The HS goals and objectives in turn define the observatory and instrument
requirements, detailed in Secs. 4 and 5.

###  3.1 G1: Reveal the 3D spatial structure and dynamics of turbulence in a
weakly collisional plasma

Most of our limited present understanding of turbulence is based on single
point observations. Clusters of four spacecraft provide improvements by
exploring processes occurring at a single size scale at a single time. As any
three points define a plane, extraction of non-coplanar 3D information (such
as curls or gradients) requires four points and appropriate analysis methods
Paschmann:1998 ; Paschmann:2008 . However, turbulence is fundamentally
multiscale; HS for the first time simultaneously explores the dynamics of
processes at multiple size scales.

####  3.1.1 G1O1: Reveal how turbulent energy transfers in the typical SW
plasma as a function of scale and time

Using the undisturbed SW as a natural laboratory, with typical plasma
parameters, HS measures fluctuations in the plasma velocity and density
($\delta\mathbf{v}$ and $\delta n$) and magnetic field ($\delta\mathbf{B}$) at
MHD to sub-ion scales simultaneously using the instrument suite described in
Sec. 5. These data reveal how turbulent energy is distributed and transferred
as a function of space and time. Turbulent fluctuations are affected by local
magnetic fields Iroshnikov:1963 ; Kraichnan:1965 ; Schekochihin:2009 ;
Matthaeus:1990 , so we must characterize SW turbulence relative to the local
IMF direction. Such studies have been performed with data from single
spacecraft; c.f. the review in Chen (2016) Chen:2016 , and necessarily rely
upon the assumption of essentially frozen turbulence structures, an
approximation known as the Taylor hypothesis Taylor:1938 ; Fredricks:1976 ;
Osman:2007 ; Klein:2014b  that neglects temporal variations and can infer only
1D variation along the SW flow direction. These studies also frequently assume
that the turbulence is insensitive to the angle between the SW velocity and
the magnetic field, using variations in $\theta_{vB}$ to study the functional
dependence of the turbulence on the angle between the wavevector and magnetic
field $\theta_{kB}$. Recent work Woodham:2021b  suggests that this assumption
may not be valid; verifying this claim will require sampling the turbulent
structures both along and transverse to the magnetic field direction
simultaneously, a measurement that HS is designed to produce.

With HS, the Taylor hypothesis can be directly evaluated. Spectral information
is also available from proven analysis techniques (Sec. 6) such as 2-point
correlations, structure functions, space-time correlations, and cascade rate
analysis Matthaeus:1982c ; Horbury:2012 ; Matthaeus:1990 ; Hamilton:2008 ;
Chen:2011a ; Horbury:2008 ; Mallet:2016 , from which it is possible to extract
information about 3D spectral structure Osman:2011b ; Hamilton:2008 ;
Bieber:1996  and its intrinsic, scale-dependent decorrelation times. These
techniques frequently use measurements of the velocity and magnetic fields
directly, or the Elsasser variables
$(\mathbf{z}^{\pm}=\delta\mathbf{v}\pm\delta\mathbf{b})$ Elsasser:1950  in
which the magnetic field is expressed in Alfvén(velocity) units
$(\delta\mathbf{b}=\delta\mathbf{B}/\sqrt{\mu_{0}n_{p}m_{p}})$ and $\delta$
indicates the use of a fluctuating quantity.

A prominent example of the use of Elsasser variables is the MHD
$3^{\textrm{rd}}$-order law

$\nabla\cdot\left<\Delta\mathbf{z}^{\mp}|\Delta\mathbf{z}^{\pm}|^{2}\right>=-4\epsilon^{\pm},$
(1)

an analytic result involving spatial increments $\Delta\mathbf{x}$ of the
Elsasser fields
$\Delta\mathbf{z}^{\pm}=\mathbf{z}^{\pm}(\mathbf{x}+\Delta\mathbf{x})-\mathbf{z}^{\pm}(\mathbf{x})$,
and $\left<...\right>$ denotes ensemble average. This relation can be used to
determine the energy cascade rate associated with the forward and backward
Elsasser fields $\epsilon^{\pm}$ Osman:2011b . Formally, this requires
knowledge of 3D anisotropies. Previous studies have usually made assumptions
about isotropy MacBride:2005 ; MacBride:2008 ; Stawarz:2009 ; Coburn:2012 ;
Hadid:2017  or only measured $\epsilon$ over limited range of scales
Bandyopadhyay:2018 . HS can implement the isotropic form at all nine
spacecraft, but also can integrate the 3D form of the $3^{\textrm{rd}}$-order
law at several scales simultaneously, making use of all 36 spacecraft pairs to
compute the 2-point spatial increments. HS provides simultaneous 3D multipoint
knowledge needed to infer spatial gradients contained in the
$3^{\textrm{rd}}$-order equation, quantifying directly those key terms for the
first time, bypassing simplifying assumptions about isotropy, to measure
cross-scale energy transfer rates definitively.

No comprehensive observational evidence exists to distinguish between proposed
theories of turbulent energy transfer. A review of such theories can be found
in NAS 2020 Plasma Decadal Panel white papers Klein:2019:WP ;
Matthaeus:2019:WP ; TenBarge:2019:WP  and other reviews Schekochihin:2009 ;
Oughton:2017 . Candidate energy transfer processes are related to relevant
dynamical timescales that include wave propagation, random and coherent
sweeping of small structures by larger structures, and nonlinear wave
distortion Orszag:1972 ; Tennekes:1975 ; Nelkin:1990 ; Sanda:1992 ;
Servidio:2011 . Numerical simulations provide insights regarding which of
these are important but results remain inconclusive due to fundamental
limitations associated with the necessary trade offs between the volume of
space simulated and the physical processes included in the equations evolved.
HS provides observations to distinguish and refine our understanding of the
relevance of these processes.

####  3.1.2 G1O2: Reveal how turbulent cascade of energy varies with
background parameters in different SW environments

Turbulence and plasma conditions in fast and slow SW differ systematically in
terms of density, temperature anisotropy, and collisional age Belcher:1971 ;
Dasso:2005 ; MacBride:2005 ; MacBride:2008 ; Borovsky:2019 ; Kasper:2008 .
Slow SW turbulence is more highly variable in nature than the fast SW
Dasso:2005  and due to its longer transit from the Sun, has more time to
evolve toward a fully developed state. These differences have been assessed in
limited fashion with single-point (Breech:2008 ; Vech:2017 , e.g., Wind,
Voyager) and single-scale Bandyopadhyay:2018  (e.g., MMS) measurements. The
varying SW speed is also associated with variations in proton number density,
temperature, alpha particle density, and IMF strength. Plasma $\beta=8\pi
nk_{b}T/B^{2}$, the ratio of thermal to magnetic pressure, a particularly
important regulator of plasma processes Chen:2014 , and power imbalances (such
as cross helicity $\sigma_{C}$ and residual energy $\sigma_{R}$, Wicks:2013 ),
are also highly variable in the SW. These parameters influence the underlying
energy cascade from MHD to sub-ion scales. HS targets to study the impact of
this variability on the dynamics of the turbulence.

####  3.1.3 G1O3: Quantify transfer of turbulent energy between fields, flows,
and proton heat

Dissipation of turbulence is one of the most important factors influencing
heating and particle energization in the universe. Consequently, our goal of
investigating energy transfer must include how the cascade heats protons.
Protons are of primary importance as they are the dominant species in terms of
both mass and momentum. How and how much energy is delivered to protons via
dissipation processes determines the overall partitioning of energy across all
species. Primary candidate mechanisms include: ICWs and cyclotron resonances
Hollweg:2002 ; Landau damping TenBarge:2013a ; stochastic heating by large
amplitude turbulent fluctuations Chandran:2010a ; and energization through
intermittent structures, including magnetic reconnection Dmitruk:2004  and
trapping in secondary magnetic islands Ambrosiano:1988 .

Current observations do not provide clarity. For example, intense ICWs are
commonly observed during times when plasma instabilities are present Gary:2016
in extended “storms” during quiet SW and radial IMF Jian:2014 . Because ICWs
are capable of substantial heating of SW ions, it is important to understand
exactly how often they occur. ICWs may be omnipresent but can only be detected
by a single spacecraft when the SW flow is aligned with the local IMF (i.e.,
radial field configurations). Applying methods such as the wave telescope
technique to HS observations, Sec. 6, will identify ICWs when the IMF is not
radial, thus establishing definitively whether ICWs are always present or not.

All aforementioned mechanisms occur at ion time and length scales and create
characteristic signatures in underlying proton velocity distribution functions
(VDFs); each mechanism deposits differing fractions of energy to the protons
Chandran:2010a ; He:2015 ; Matthaeus:2016a . The absence or presence of these
signatures reveals which dissipation pathways operate; their relative
strengths quantify their relative importance. Measurements of proton
temperature at ion heating time scales allows HS to quantify proton heating
directly. One analysis method, colloquially referred to as ’PiD’, makes use of
the measured pressure tensor $\Pi_{ij}$ and flow gradients
$S_{ij}=\nabla_{i}\mathbf{u}_{j}$ to compute the full pressure-strain
interactions $\Pi:S$ which is the rate of production of proton internal energy
Yang:2019 . These methods are enabled in HS by simultaneous measurement of
proton distribution functions and 3D multiscale turbulence, a combined
capability lacking in all previous missions. HS will allow us to directly
quantify relationships between the distribution of turbulence fluctuations and
transformation into proton heat.

####  3.1.4 G1O4: Identify thermodynamic impacts of intermittent structures on
protons

Intermittency is a universal property of turbulence in which dissipation
processes concentrate into small fractions of available volumes Horbury:1997 ;
Matthaeus:2015 , giving rise to current sheet structures. Such structures have
been studied in numerical simulations Karimabadi:2013  and in situ
observations Osman:2011 . Intermittency dramatically impacts how turbulence
heats plasma Mallet:2019 . Cluster and MMS pioneered the ability to resolve
thin structures with 4-point curlometer and gradient techniques Dunlop:2002a ;
Dunlop:2002b . While revolutionary, such techniques probe only a single scale
at a time. HS provides combinatorically more spacecraft groupings and
simultaneous access to multiple scales, tremendously expanding 3D anisotropic
measures of intermittency with well-developed analysis tools, as described in
Sec. 6.

By measuring intermittency of turbulent fluctuations at inertial and ion
scales simultaneously, HS differentiates between models of nonlinear coupling,
that predict enhanced amplitudes of Elsasser fluctuations $\delta z$ at small
scales compared to a scale-independent normal distribution of amplitudes. HS
also resolves the intermediate scales to provide further differentiation.

###  3.2 G2: Ascertain the mutual impacts of turbulence, variability, and
boundaries near large scale structures

While undisturbed SW is a pristine environment, disturbed SW occurs from
impacts of either large scale structures of solar or heliospheric origin or
Earth’s magnetosphere and provides different environments to explore. Impacts
are mutual: turbulence can impact large-scale structures and boundaries and
those same structures can in turn change the nature of turbulence. Goal Two
focuses on these mutual interactions.

####  3.2.1 G2O1: Determine how SW turbulence affects and is affected by
large-scale structures such as CMEs and CIRs

Passage of interplanetary coronal mass ejections (CMEs)Jian:2018  or
corotating interaction regions (CIRs)Goldstein:1984 ; Jian:2019  disturb the
SW from its pristine state. These large-scale features re-inject energy and
thus modify SW turbulence. While turbulence levels are reduced within CME
structures, HS enables 3D characterization of this (possibly weak) low-plasma
$\beta$ turbulence contained within a large scale force-free structure. Near
CMEs, driven turbulence departs significantly from that of the pristine SW; HS
will diagnose 3D turbulence modifications associated with diffusive shock
acceleration LeRoux:2015  near fast CMEs and waves driven by the CME’s
propagationZhao:2021a . Passage of both CMEs and CIRs through pristine SW
turbulence allows us to explore differences in these environments, enabling us
to determine when and how specific energy transfer and heating processes
become important.

####  3.2.2 G2O2: Determine how driven turbulence differs from that in
undisturbed SW

The terrestrial bow shock, foreshock, and magnetosheath are permeated with
magnetic and plasma fluctuations, strongly driving and modifying the turbulent
spectrum across inertial and dissipation scales both in amplitude and
shapeChen:2019:WP . These regions represent parameter regimes not accessible
in pristine SW. The dynamics in these locations are significantly different;
for example, ions reflected off the bow shock can lead to the self-generation
of turbulence, which takes the form of non-linear wave penetrating into the
inner magnetosphere Takahashi:2016 , while at the shock, turbulence generates
high-speed jets that regularly impact the magnetopause, resulting in dayside
reconnection Hietala:2018 . Turbulence is also seen to drive magnetic
reconnection in these regions Retino:2007 . Finally, magnetospheric regions
can be turbulent Chasapis:2017 ; Chasapis:2020 ; Bandyopadhyay:2020a ;
Bandyopadhyay:2020b , but of a different nature (e.g. magnetically dominated)
Maruca:2018 . Objective Two explores this variety of accessible systems to
compare how driven environments differ from pristine SW.

##  4 HelioSwarm Observatory Design

The specific design of the HS Observatory is driven by decades of measurements
from near-Earth plasmas of characteristic length and time scales as well as
derived dimensionless parameters that are predicted to govern the behavior of
magnetized turbulence.

###  4.1 Quantities to be Measured

As discussed in Sec. 3.1.1, the primitive variables that describe magnetized
turbulence at MHD scales are the Elsasser variables Elsasser:1950  composed of
magnetic fields and particle densities and velocities. G1O1 requires
measurements of the IMF, SW proton density, and SW velocity. It must do so in
undisturbed, most-probable SW for which the range of proton densities is $1.6$
to $20$ cm-3 and magnetic field can be as large as 25 nT , but typically
larger than 2.6 nT (at the 90% occurrence rate) Wilson:2018 ; Klein:2019 . To
resolve at the lowest typical field strength, we require 10% resolution (0.26
nT), corresponding to 0.15 nT per axis. Such measurements allow construction
of Elsasser variables, needed for magnetized turbulence analysis at each
measurement location.

Measurements of the SW proton density, velocity and IMF must be made at
multiple points in 3D encompassing the turbulent cascade during average SW
conditions, within large scale structure analysis intervals—equivalent to
approximately one hour long continuous observations— at cadences, time
knowledge, and sensitivities required to resolve and align SW and IMF
variations down to sub-ion scales.

###  4.2 Spatial Resolution

To measure the multiscale nature of turbulence, HS’s baseline separations
between the nine spacecraft are designed to simultaneously span MHD scales and
ion kinetic scales, enabling the simultaneous resolution of MHD and sub-ion
processes and the transition between these scales, exemplified by the observed
spectral break Goldstein:1994 ; Leamon:1999 ; Hamilton:2008 ; Chen:2014 ;
Vech:2017 ; Vech:2018 ; Woodham:2018  (see also Figure 2). Values for these
physical dimensions are empirically known from decades of SW
observationsBorovsky:2019 ; Wilson:2018 ; Woodham:2018 ; Klein:2019 . Figure 3
shows the joint probability distribution function (PDF) of the proton
gyroscale $\rho_{p}$ and spectral break scale
$L_{\textrm{break}}=v_{\textrm{sw}}/f_{\textrm{break}}$. These observations
define three ranges: MHD scales at $>1200$ km; transition scales between 100
and 1200 km; and sub-ion structures $<100$ km. HS’s baseline requirements are
established to resolve these characteristic scales simultaneously in 85% of
the pristine SW, enabling the Observatory to “encompass the turbulent
cascade.”

(a)  (b)

Figure 3: (a) Joint PDF of proton gyroradius $\rho_{p}$ and spectral break
scale $L_{\textrm{break}}$ as measured by the Wind spacecraft at Earth’s L1
point Wilson:2018 ; Klein:2019 . HS’s baseline separations between spacecraft
will cover from 3000 km to 50 km (blue box), allowing the observatory to
simultaneously measure MHD, transition, and sub-ion physical processes in 85%
of the pristine SW. (b) PDF of SW velocity drawn from the same database,
compared to FC instrumental requirements and project performance, illustrating
that HS will capture both typical and extreme proton velocities. (c) PDF of
the advected SW ion timescale $\rho_{p}/v_{\textrm{sw}}$, compared to HS
instrument cadences, demonstrating that HS will resolve the IMF past ion-
scales in nearly all the SW, and resolve both the ion-scale plasma processes
in typical SW conditions. In all panels, the red numbers indicate the
percentile of the cumulative distribution below the given value.

###  4.3 Temporal Resolution

The Observatory measurement cadence and timing knowledge provide the temporal
resolution necessary to resolve advected SW structures. This analysis requires
measuring at time cadences from MHD scales down to the sub-ion scales.

Given the observed distribution of SW velocities, see Figure 3b, we can
calculate the ratio of the proton gyroscale $\rho_{p}$ to $v_{\textrm{SW}}$ to
construct an advected proton timescale, Figure 3c, which plots the observed
distribution against instrumental measurement rates. The fluxgate magnetometer
(MAG) measures at 16 samples per second (Sps) overlapping with the searchcoil
magnetometer (SCM, at 32 Sps) providing continuous coverage of larger and/or
more slowly advecting structures, while also resolving ion-scale structures
traveling at the fastest $v_{\textrm{SW}}(\sim 800$ km/s); The proton density
($n$) and velocity ($v$) are measured by Faraday cups (FCs) at a rate of 8
Sps, resolving ion scale structures ($\sim 100$ km) traveling at typical
speeds (400 km/s); Measurements of the proton temperature by the ion
electrostatic analyzer (iESA) provide the necessary context for the kind of
turbulence HS is embedded in, with sufficient temporal resolution to resolve
changes in proton velocity distributions to help determine the energy transfer
processes associated with ion scale structure.

In order to resolve characteristic SW wave propagation directions across
multiple points, HS requires post-facto, relative pairwise separation
knowledge of 10% the separation distances. Timing requirements are driven by
applying analysis methods described in Section 6 to synthetic data combined
with models for temporal uncertainty.

###  4.4 Observatory Stability

Simultaneous statistical analysis of turbulence (e.g. Sec. 6.1) requires not
only separations spanning the previous described spatial scales but also
samples taken over long enough periods of time to capture the nonlinear
reshaping of the underlying structures. One can calculate the correlation time
scale $\tau$ by determining the time lag necessary to reduce an
autocorrelation of some measured quantity $F$ by $1/e$ from it’s zero-lag
value

$A\left[F(t),F(t+\tau)\right]=\left<F(t)F(t+\tau)\right>=\left<F(t)F(t)\right>/e,$
(2)

where $<...>$ denotes an appropriate ensemble average. Analysis performed on
intervals measured within a correlation time are effectively sampling the same
population of turbulent fluctuations, and thus can be combined to study the
statistical properties of that plasma. Observations of the correlation time
scale in the SWIsaacs:2015 ; Smith:2018 , illustrated in Figure 4, typically
find it ranges from tens of minutes to approximately an hour. This duration of
SW data provides robust turbulence analysis yet is short enough to effectively
sample the same parcel of SW. These observations drive the timescales over
which the observatory spacecraft separations need to be constant, a
requirement the HS Design Reference Mission (DRM) satisfies, enabling the
accrual of usable intervals for the application of analysis approaches
outlined in Sec. 6; the average relative change in the vector baselines
increases slowly in time (red line), reaching 0.7% at 60 minutes and 1.5% at
120 minutes.

Figure 4: PDF of correlation time measured by ACE (left panel) Isaacs:2015
compared to the average change in HS DRM baseline separation magnitudes as a
function of time (red line). At right, the $x_{\textrm{GSE}}-y_{\textrm{GSE}}$
projection of the evolving Node positions relative to the Hub, with color
indicating time since apogee; the significant overlap in positions illustrate
the relative stability of the observatory configuration.

###  4.5 Spatial Configurations

# Spacecraft | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---
Baselines | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55
Tetrahedra | 1 | 5 | 15 | 35 | 70 | 126 | 210 | 330
Polyhedra | 1 | 6 | 22 | 64 | 163 | 382 | 848 | 1816
Table 1: Number of baselines, tetrahedral, and polyhedral (with at least four
vertices) configurations that can be constructed from $N$ Spacecraft.

In conjunction with spatial separation requirements, the application of the
analysis approaches in Sec. 6 require specific spatial configurations. Given
$N$ spacecraft, there are $N(N-1)/2$ distinct pair-wise baseline separations.
Similarly, for $N$ spacecraft, one can construct ${N\choose
4}=\frac{N!}{4!(N-4)!}$ unique tetrahedral configurations, or
$\sum_{i=4}^{N}{N\choose i}$ polyhedral configurations with at least four
vertices. The number of baselines, tetrahedra, and polyhedra are tabulated as
a function of the number of spacecraft in Table 1. The orientation and
geometries of these configurations have been carefully tailored so that they
span the appropriate size-scales and directions to address the mission
objectives, as discussed in the following subsections and illustrated in Fig.
5. Determining when the HS Observatory satisfies these configurational
requirements is characterized in 1-hour units, during which baseline
separations are effectively constant, see Fig. 4. The number of hours
satisfying these requirements are laid out in Table 2.

Figure 5: Summary plot of HelioSwarm Observatory Phase-B DRM positions and
separations. Top Row Relative positions between the Hub (red) and eight Nodes
(black) projected into the GSE coördinate system at hour 2472 from the DRM.
Bottom Left Projected vector components of the 36 inter-spacecraft baseline
separations (black dots) demonstrate coverage of MHD and ion-kinetic scales,
as well as the transition region in-between. The lunar resonant orbit of the
observatory (black dot) in the GSE coördinate system is shown as colored lines
in the upper-right inset, with the moon’s location (open circle) included to
illustrate scale. Times with orthogonal coverage over all three scales,
highlighted in color, arise in the pristine SW (red lines), the magnetically
connected SW (green) and the magnetosphere/magnetosheath (blue). Bottom Right
The size and geometric configurations of the polyhedra constructed by
spacecraft subsets of the HelioSwarm observatory. The number of spacecraft is
indicated by color, while the size of the polyhedra $L$ and its regularity
(the RMS of the elongation $E$ and planarity $P$) are indicated on the
ordinate and abscissa respectively. The times when there are at least two
regular polyhedra with characteristic sizes more than a factor of three
different are indicted in the upper inset, using the same color scheme as the
3D Configuration inset. As quantified in Table 2, due to the high eccentricity
of the orbit, the Observatory samples these regions near apogee for a
substantial fraction of the orbit period. A video of the HS DRM Geometries
throughout the Science Phase is available in Online Resource 2.

####  4.5.1 3D Configurations

To calculate cascade rates, correlation scales, and structure functions to
characterize the multiscale and 3D nature of turbulence, the 36 unique
baselines between HS’s nine spacecraft have vector components spanning three
orthogonal directions along, transverse, and normal to the Earth-Sun line
(Radial, Tangential, Normal (RTN) coördinates) with amplitudes covering MHD,
transition, and sub-ion scales, while simultaneously the magnitudes of the
baseline vectors also span these three ranges of scales. These 3D
configurations, illustrated in Figure 5, resolve variations along and across
the local magnetic field and flow directions, necessary for verifying theories
of anisotropic turbulent transfer and distribution of energy.

####  4.5.2 Polyhedral Configurations

The polyhedral configuration is satisfied when at least two of HS’s
constituent polyhedra, each satisfying $\sqrt{E^{2}+P^{2}}\leq 0.6$, have at
least a factor of three difference in L, simultaneously measuring the spatial
structure of turbulence at multiple scales.

HS configurations are also designed for multi-point analysis techniques which
determine spatial gradients and distributions of power (e.g., wave-telescope,
curlometer, and related gradient analysis techniques Paschmann:1998 ;
Paschmann:2008 . Spatial gradient methods require the SC be arranged in a
quasi-regular fashion, occupying vertices of pseudo-spherical polyhedra. One
can characterize the geometry of these polyhedra by calculating the
eigenvectors of the volumetric tensor

$\underline{\underline{R}}=\frac{1}{N}\sum_{\alpha=1}^{N}\left(\mathbf{r}_{\alpha}-\mathbf{r}_{b}\right)\left(\mathbf{r}_{\alpha}-\mathbf{r}_{b}\right)^{T}$
(3)

where $\mathbf{r}_{b}=\frac{1}{N}\sum_{\alpha=1}^{N}\mathbf{r}_{\alpha}$ is
the mesocenter of the configuration, and $\mathbf{r}_{\alpha}$ represents the
positions of the individual SC. The square roots of the three eigenvalues of
$\underline{\underline{R}}$ represent the major, middle, and minor semiaxes of
the configuration, $a$, $b$, and $c$. These values can be interpreted directly
by defining a characteristic size $L=2a$, as well as the elongation
$E=1-\frac{b}{a}$ and planarity $P=1-\frac{c}{b}$. Figure 6 illustrates the
distribution of polyhedra from a single hour of the HS Observatory
configuration. Polyhedra with small elongation E and planarity P,
$\sqrt{E^{2}+P^{2}}\leq 0.6$, can be used to accurately measure structure of
sizes on the order of the characteristic size $L$ Sahraoui:2010b ;
Roberts:2015b . HS’s 9 spacecraft produce 382 polyhedra with at least 4
vertices, the minimum needed for 3D analysis techniques, and at many different
scales. Additionally, HS has configurations where at least two pseudo-
spherical polyhedra exist with at least a 3:1 ratio in L. These formations,
referred to as polyhedral configurations, simultaneously measure spatial
structure of turbulence at multiple scales.

Figure 6: The distribution of planarity (P), elongation (E), and
characteristic size (L, with blue and red representing the smallest and
largest scales respectively) of all 382 polyhedra with at least 4 vertices for
the HS DRM at an arbitrarily selected hour 2474. The different regions in E-P
space are labeled to characterize the geometries of these polyhedra. The
Observatory trajectories are designed to have multiple pseudo-spherical
polyhedra with significantly different sizes to enable measurements of spatial
structures at MHD- and ion-scales simultaneously.

###  4.6 Observatory Orbits

The HS Observatory accesses the near-Earth regions of interest with a 2-week,
lunar-resonant, high Earth orbit (HEO) Plice:2019 ; Levinson-Muth:2021a ;
Levinson-Muth:2021b ; Levinson-Muth:2022 . The HS Observatory design and
onboard propulsion produce inter-spacecraft separations both along and across
the Sun-Earth line. The Nodes perform routine trim maneuvers to maintain
customized configurations that satisfy the 3D and Polyhedral requirements over
the mission lifetime. Since the science orbit is nearly inertially fixed (with
a low rate of apsidal precession), the apogee rotates through the SW,
foreshock, and magnetosphere-dominated regions as the Earth completes a single
orbit of the Sun. This progression allows the Observatory to sample the
pristine SW and regions of strongly driven turbulence during the 12-month
Science Phase, addressing both G1 and G2.

Given an empirical model for the extent of the magnetosphere Formisano:1979
and the average orientation of the IMF combined with the phase B DRM
trajectories, the HS Observatory spends thousands of hours in the required
near-Earth regions of interest, with hundreds of hours in both of the required
spatial configurations in each region; see Table 2 for summary of hours and
Figure 5 for an illustration of the residence time in the regions.
Measurements from all instruments are recorded throughout the orbits outside
of thruster operations, eclipses, and calibration activities and transmitted
regardless of the Observatory configuration.

Phase B DRM; LRD 2028 | Fig. 5 | Total | 3D | Polyhedral
---|---|---|---|---
Solar Wind | Red | 2881 | 777 | 1068
Foreshock | Green | 2470 | 977 | 852
Magnetosphere/Magnetosheath | Blue | 3149 | 650 | 639
Science Phase |  | 8850 | 2404 | 2559
| Objective | Total | 3D | Polyhedral
Pristine SW | G1O1 | 2015 | 544 | 747
Extreme SW | G1O2 | 58 | 16 | 21
SW w/ Large Scale Structure | G2O1 | 546 | 147 | 202
Strongly Driven Turbulence | G2O2 | 5619 | 1627 | 1491
Table 2: HS measures thousands of hours in targeted near-Earth regions of
space, with hundreds of hours in optimal polyhedral (Sec. 4.5.2) and 3D
configurations (Sec. 4.5.1) for the application of analysis approaches
outlined in Sec. 6, providing measurements to advance the understanding of
turbulence in typical (G1O1,G2O2) uncommon (G2O1) and extreme (G1O2) plasma
conditions.

###  4.7 Mission Duration

As discussed in Sec. 3.1.2, SW parameters drive the behavior of turbulence,
and more extreme values of these parameters are useful for distinguishing
competing theories. To establish the minimum number of hours needed for HS
science data sufficiency, we note that $\sim 10$ hours in extremely high
($\geq 10$) and low ($\leq 0.1$) plasma $\beta$ enabled strong
characterization of the spectral break, differentiating between predicted
dissipation mechanisms Chen:2014 . Using this assessment, the required number
of hours of observation for the Baseline Mission was developed by analyzing
two decades of Wind Wilson:2018 ; Klein:2019  data to ensure that we would
adequately sample the full range of SW variability. Our methodology was to
generate PDFs of parameters controlling turbulent behavior, e.g. SW speed
($v_{\textrm{SW}}$), plasma beta ($\beta$), proton temperature ($T_{p}$),
balance of power between Sunward and anti-Sunward propagating fluctuations
(cross helicity, $\sigma_{C}$ Parashar:2018 ), difference between kinetic and
magnetic energy (residual energy $\sigma_{R}$ Wicks:2013 ), SW collisionality
(Coulomb number $N_{C}$ Neugebauer:1976 ; Kasper:2017 ), and Alfvén Mach
number ($v_{\textrm{SW}}/v_{A}$). Obtaining $\sim 10$ hours of measuring
turbulence at relatively large and small values of these parameters determines
the overall requirements for the number of hours in polyhedral and 3-D
configurations; from the widths of the parameter PDFs, we determined that
measuring 500 (100) hours in the 3D (polyhedral) configuration in the pristine
SW, which then result in HS measuring 10 (2) hours of turbulence with extreme
parameters both higher than the 98th percentile and lower than 2nd percentile
of those values (corresponding to $\beta\sim 0.1$ and $\beta\sim 10$ Chen:2014
), a sufficient number of intervals at the very most extreme parameters to
accomplish Mission science. Magnetosheath plasmas that will also be measured
by HS typically have even higher values of $\beta_{\parallel,p}$Maruca:2018 .
Measurements of these extreme intervals allow for the identification of
different turbulent processes that are preferred in different parameter
regimes, and will also be useful for providing accessible analogies to
astrophysical systems where the thermal pressure dominates, e.g., the ISM
($\beta_{\parallel,p}\gtrsim 10$) or accretion disks
($\beta_{\parallel,p}\gtrsim 1$).

Large scale structures (LSS) generated by the Sun– e.g., CMEs, or produced as
the SW propagates, e.g., CIRs can drive different kinds of turbulence compared
to SW w/o LSS. By using in situ SW measurements of these structures over the
last two solar cycles Jian:2018 ; Jian:2019 , we calculate the filling
fraction during the 12-month Science Phase of these two kinds of structures
for a 2028 Launch Readiness Date (LRD) based upon equivalent phases from Solar
Cycles 23 and 24; the total anticipated LSS hours for this LRD are tabulated
in Table 2.

The average CME filling fraction is 2.15% (1.9%/2.4% in Solar Cycle 23/24)
while the CIR filling fraction is 16.8% (19.8%/13.8% in Solar Cycle 23/24).
These rates correspond to 62 hours of CME observations, with 16/23 hours in
3D/polyhedral configurations and 484 hours of CIR measurements, with 131/179
hours in 3D/polyhedral configurations. We have repeated this exercise for
other LRDs, and found that regardless of launch date, there will be a
sufficient number of hours of observed CMEs and CIRs to provide data to bring
closure to G2O1.

###  4.8 Resilience, Redundancy, and Robustness of Multi-satellite Observatory

Multi-SC swarm design offers innovations in flexibility and reconfiguration of
the observatory. Orbital mechanics forces create continuously evolving
relative positions among the 9 SC in HS. With known exceptions, the nominal
swarm configuration has redundancy in most of the 3D baselines and tetrahedral
vertices and accrues successful hours of science data collection well above
the requirements.

Robustness above required performance and redundancy in spatial configurations
create resilience in the event of contingencies. For the case of the loss of
any one (or two) Nodes, the required number of hours in both configurations
can be achieved within the duration of the 12-month Science Phase through a
repositioning of the remaining Nodes to construct the configurations for which
sufficient hours have not been achieved.

###  4.9 Place within the Heliospheric System Observatory

HS stands alone, but would also be part of the Heliospheric System Observatory
(HSO) which provides additional opportunities for joint mission studies.
Parker Solar Probe (PSP) Fox:2015  and Solar Orbiter (SolO) Muller:2013  are
making high-cadence plasma and IMF measurements of the innermost heliosphere.
These inner-heliospheric missions provide only single point measurements, but
these inform limits on injection scale structures that cascade into smaller
structures as they propagate to 1 AU. Together with HS, and supplemented by
Polarimeter to UNify the Corona and Heliosphere (PUNCH) imagingDeForest:2018 ,
these observations allow for estimates of 1-AU-scale evolution and radial and
longitudinal gradients. At intermediate scales ($10^{6}$ km), HS observations
in combination with other missions in the HSO positions near the Sun-Earth L1
point (e.g., Advanced Composition Explorer (ACE)Stone:1998 , WindWilson:2021 ,
Interstellar Mapping and Acceleration Probe (IMAP)McComas:2018 , Deep Space
Climate Observatory (DSCOVR)Lotoaniu:2022 ) provide opportunity for long-
baseline correlations and to address the long-open question of local
geometries of interplanetary shocks and flux ropes. These same HSO missions
provide additional SW composition information to augment HS alpha particle
measurements. Conjunctions with MMSBurch:2016  may also prove useful in
extending the range of scales over which energy transfer and dissipation can
be studied. Finally, given that energetic particle propagation is impacted by
SW turbulence, ACE, Wind, Solar Terrestrial Relations Observatory (STEREO),
and IMAP energetic particle measurements can test the effect of turbulence
models and mechanisms HS quantifies. Joint study opportunities will depend on
what HSO assets are operating when HS launches, but the breadth of the
positions and instrumentation of missions within the HSO will enable a variety
of examinations of fundamental processes at play in our Heliosphere.

##  5 HelioSwarm Mission Implementation

HS was selected as a Heliophysics Division Medium Explorer (MIDEX) mission by
NASA Science Mission Directorate in 2022, and is currently in the formulation
phase. MIDEX missions are affordable testbeds for flagship science, from a
cost and risk implementation perspective. The HS hardware and operations
approach are all extremely high heritage to minimize overall project risk.

The HS architecture consists of one central Hub, an ESPA-class (EELV Secondary
Payload Adapter) spacecraft provided by Northrop Grumman, and eight co-
orbiting Nodes, SmallSats provided by Blue Canyon Technologies, both high
heritage, 3-axis stabilized spacecraft. The Hub, Sec. 5.5, carries the eight
Nodes to the science orbit. Pairs of Nodes will then separate from the Hub
over four consecutive 14-day orbits. Each Node, Sec. 5.6, possesses identical
instrument suites (IS) consisting of three high-heritage, high-TRL sensors
optimized for HS: the Faraday Cup (FC, Sec. 5.2), provides high cadence
measurements of the SW density and flow, the Fluxgate Magnetometer (MAG, Sec.
5.1.1), and Search Coil Magnetometer (SCM, Sec. 5.1.2), provide measurements
of the IMF at cadences sufficient to probe fluctuations from MHD to sub-ion
scales. The Hub has the same IS as the Nodes, plus an ion Electrostatic
Analyzer (iESA, Sec. 5.3), another high-heritage, high-TRL instrument that
will provide high cadence measurements of the proton and alpha particles in
order to characterize the local turbulence and to quantify ion heating. An
electron Electrostatic Analyzer, Sec. 5.4), included as a Student
Collaboration Option for installation on the Hub, provides additional context
for the plasma environment sampled by the HS Observatory.

Observable | Requirement | Projected Performance | Instrument
---|---|---|---
Multi-point vector | DC to 2-Sps | DC to 16-Sps | MAG
DC IMF $\mathbf{B}$ | $\pm 100nT$ | $\pm 128nT$ | (all SC)
| 0.15 nT per axis | 0.1 nT per axis |
Multi-point vector | 0.1 to 32-Sps | up to 32-Sps | SCM
AC IMF $\mathbf{B}$ | 15/1.5 pT/$\sqrt{}$Hz at 1/10 Hz | 6/0.6 pT/$\sqrt{}$Hz at 1/10 Hz | (all SC)
Multi-point | 0.15 s | 0.125 s | FC
proton density $n_{p}$ | 0.2 - 20 cm-3 | 0.1 - 50 cm-3 | (all SC)
| $\pm 6\%$ | $\pm 5\%$ |
Multi-point | 0.15 s | 0.125 s | FC
proton velocity $\mathbf{v}_{p}$ | 250 - 800 km/s | 212 - 840 km/s | (all SC)
| $\pm 3\%$ | Accuracy $\pm 1\%$ |
Single-point | 0.3 s | 0.15s | iESA
proton temperature $T_{p}$ |  $10^{4}-5\times 10^{5}$ K |  $10^{4}-10^{6}$ K | (Hub)
| $\pm 5\%$ | $\pm 1.8\%$ |
Single-point temperature | 0.3 s | 0.15s | iESA
anisotropy $\frac{T_{\perp}}{T_{\parallel}}$ | $0.2-5$ | $0.1-10$ | (Hub)
| $\pm 6\%$ | $\pm 3.4\%$ |
Single-point $\alpha$-proton | Hourly Averages | 10 s | iESA
density ratio $\frac{n_{\alpha}}{n_{p}}$ | $0-40\%$ | $0-100\%$ | (Hub)
| $\pm 10\%$ | $\pm 3.4\%$ |
Table 3: Plasma and magnetic field observables measured across the HS
Observatory. Required cadences, ranges, and accuracy for the measurements, as
well as the projected performance and the instrument that will provide the
measurement are organized by column.

The instruments were specifically selected to be both capable of addressing
the science objectives when used as an Observatory and for having high
heritage to ensure the fabrication, integration, and testing approaches for
the required nine copies of flight model instruments, along with their costs
and schedules, would be low risk.

###  5.1 Magnetometers

HS uses a combination of flux gate (MAG) and search coil (SCM) magnetometers
to measure the IMF over the required frequency range indicated by Figure 2 (DC
$\sim 3600$ s to sub-ion $<0.15$ s). Two different magnetometer types are
required owing to sensitivities required, especially at high frequencies (15
pT/ Hz at 1 Hz and 1.5pT/ Hz at 10 Hz – see noise floors on Figure 2); these
same sensitivities impose mission requirements for DC and AC magnetic
cleanliness. The MAG and SCM instruments overlap in frequency allowing for
cross-calibration and the production of a merged data product, as has been
performed for other missions Fischer:2016 ; Bowen:2020c .

####  5.1.1 Flux Gate Magnetometers (MAG)

The MAG is a dual core fluxgate magnetometer designed and built by Imperial
College London (Imperial) which will be carried on every HS SC to measure the
local magnetic field. The MAG design is based on direct heritage from the
successful Solar Orbiter Horbury:2020b  magnetometer (Fig. 7) with
modifications taken from the soon-to-fly JUICE (JUpiter ICy moons Explorer)
instrument. HelioSwarm MAG will carry just one sensor on each spacecraft, at
the end of a dedicated 3 m boom to minimise the effects of spaceraft fields,
connected to the instrument electronics box via a harness. The electronics box
will contain a power supply and Front End Electronics board: the latter will
drive the sensor and digitise the signal, sending it directly to the
spacecraft digital processing unit (DPU) where it will be filtered and
decimated to 16 vectors/s on a common timeline with the SCM.

MAG operations are straightforward, with the instrument operating throughout
the science orbit. The instrument will have a 4 pT resolution in its most
sensitive range of $\pm 128$ nT but can range automatically up to $60,000$ nT
and can therefore operate in a full Earth field before launch.

MAG data will be calibrated at Imperial College, with inter-calibration
between S/C performed to ensure that derived products such as volumetric
currents are reliably estimated. MAG therefore contributes directly to the
multi-point vector magnetic field measurement observable in Table 3, but is
also central to the AC magnetic field measurement as well as some plasma
products such as temperature anisotropies.

Figure 7: MAG is based on the flight-proven Solar Orbiter magnetometer design.

####  5.1.2 Search Coil Magnetometers (SCM)

The SCM is a heritage set of magnetic sensors designed and built by
Laboratoire de Physique des Plasmas (LPP) and Laboratoire de Physique et
Chimie de l’Environnement et de l’Espace (LPC2E) selected to measure the IMF’s
higher frequencies needed to capture advected ion-scale structures. The HS SCM
design is based on the most recent sensor developed for the ESA JUICE mission
by LPP Wahlund:RPWI ; Retino:SCM ; Retino:2020 . LPP and LPC2E will be
responsible for the testing and calibration of the instruments.

The SCM consists of a tri-axial set of 20 cm long magnetic sensors with
associated preamplifier (ASIC) mounted at the tip of a 3 m boom opposite to
the MAG boom. Each sensor axis consists of two windings (a primary and a
secondary) around an internal PEEK mandrel inside which the ferromagnetic core
(mu-metal) used on other flight heritage missions, (e.g. Cluster Cornilleau-
Wehrlin:2003  or THEMIS Roux:2008 ) resides. Windings are connected to the
preamplifier which drives the analog signal down the SCM boom harness to the
IDPU which performs the digitization. SCM ground calibration is performed at
the National Magnetic Observatory of Chambon-la-forêt using a facility
upgraded by LPP for MMS and BepiColombo.

Each primary winding response is modified through a flux-feedback applied via
a secondary winding to produce the frequency response and phase stability
needed for Observatory-level analyses.

SCM has a single science operational mode drawing a steady 0.3 W. The three
differential analog outputs of the SCM preamplifier are anti-alias filtered
and digitized by the IDPU receiving electronics at 128 Sps then filtered to 32
Sps to satisfy HS observational requirements described in Table 3. This
science operational mode is only interrupted during the in-flight calibration
sequence. This sequence, scheduled for one per orbit and following events such
as maneuvers and eclipses, will follow procedures successfully implemented on
MMS, PSP, and SolO. It is performed to assess the stability of the transfer
function through the mission using a calibration signal provided by the IDPU.

Figure 8: JUICE SCM heritage instrument with its ASIC preamplifier.

###  5.2 Faraday Cups (FC)

The Faraday Cup (FC) is a heritage-based design developed at the Smithsonian
Astrophysical Observatory (SAO), in conjunction with University of California,
Berkeley (UCB), and Draper Laboratories. The sensor makes measurements of the
radial VDFs of SW ions along with the flow angle of the incoming beam to
measure proton densities and velocities over the ranges and sensitivities
typical of the pristine SW.

Previous Faraday Cups have been employed on a wide variety of missions
including Voyager Bridge:1977 , WindOgilvie:1995 , DSCOVRLotoaniu:2022 , and
PSPCase:2020 ; Kasper:2015 . Two of the HS FC electronics boards (the
logic/signal analysis board and the low-voltage power supply) are direct
copies of the PSP electronics. A third board (the high-voltage power supply)
is a fully-qualified backup design from the PSP instrument development. The
instrument uses an oscillating electric potential to create an electric field
that accepts or rejects particles based on their energy/charge. Particles with
large enough E/q to successfully transit the electric field deposit their
charge onto collector plates that measure the incoming current of SW
particles.

A Faraday Cup instrument is placed on the sun-facing side of each spacecraft
so that an unimpeded view is maintained in the direction of the Sun. Because
Elsasser analysis involves measurements from both the IMF and SW, cross-sensor
timing, pointing, and alignment requirements between the magnetometers and FC
are levied.

The Faraday Cup operates in a single operational configuration throughout all
phases of the mission. The instrument starts at its lowest voltage
(energy/charge) and steps its way upward through 16 voltage windows while
making measurements of the incoming SW current on each of its four collector
plates in each window. The instrument keeps track of the maximum current
measured in the previous spectrum so that the following spectrum can be
measured with a more focused voltage range with better resolution.

The FC instrument design parameters have been determined by analyzing the
historic distribution of all measurements made by the Wind Faraday Cup
instrument. The aperture sizes, voltage ranges, and field-of-view for the
HelioSwarm instrument are designed to capture more than 98% of the SW
conditions (velocities, densities, and temperatures) that have been previously
observed. The resulting voltage range will allow for measurement of proton
velocities in the range of about 200-850 km/s.

The Faraday Cup instruments provide a measurement of the radial distribution
functions of the SW plasma on each of the nine spacecraft along with plasma
quantities derived from those distributions. By calculating the moments of the
distribution and by fitting an assumed functional form to the distribution,
the vector velocity, density, and radial temperature can be provided. These
data products contain 8 measurements per second and fulfill the multi-point
measurement requirements of the velocity and density of the SW, as shown in
Table 3.

Figure 9: The Faraday Cup mechanical design (as of the concept study report).
The mounting bracket is displayed in a semi-transparent mode. The instrument
consists of two main subassemblies: the sensor (the top cylindrical section)
and the electronics module (the lower rectangular box), which are connected by
flexible coaxial cables.

###  5.3 Ion Electrostatic Analyzer (iESA)

The iESA is a particle sensor designed and built by Institut de Rescherch en
Astrophysique et Planetologie (IRAP, Toulouse, France), Laboratoire
d’Astrophysique de Bordeaux (LAB, Pessac, France), University of New Hampshire
(UNH, USA), and Mullard Space Science Laboratory (MSSL, UK), with IRAP
technical leadership and heritage. The direct heritage instrument is the
Proton and Alpha Sensor Owen:2020  onboard the Solar Orbiter mission
Muller:2020 , with some sub-systems inherited from other past particle
instrumentation led by IRAP (on STEREO, MAVEN, Cluster, etc.). iESA measures
the full proton and alpha particle distribution functions with an
unprecedented combination of high energy, angular and time resolutions (cf.
Table 3).

As illustrated in Figure 10, entrance deflectors allow for the sweeping of
input elevation angles $\pm 24^{\circ}$ from the main detection plane with
$3^{\circ}$ angular binning, which is resolvable with the use of a collimator.
The deflected and collimated ions are then subject to E/q selection through a
classic top-hat electro-static analyzer. The E/q selected ions are focused
onto the main detection plane, which comprises 16 channel electron multipliers
(CEMs). These perform a $10^{7}$ gain in charge collection on anodes with
$3^{\circ}$ resolution in azimuth over an angular range of $\pm 24^{\circ}$ as
well, allowing for a homogeneous $\pm 24^{\circ}$ field-of view with
$3^{\circ}$ angular resolutions, in both elevation and azimuth angles. The
iESA electronics contains (1) a front-end board comprising 16 CEMs with
associated anodes and amplificators, (2) a high-voltage board to supply the
entrance deflectors, analyzer plates, and CEMs with the required (static or
sweeping) high voltages, as well as (3) an FPGA and (4) a low-voltage power
supply board dedicated to instrument control and power.

iESA operations are based on the sequential stepping of the electrostatic
analyzer and entrance deflector high voltages. The instrument implements SW
beam-tracking strategies Owen:2020 , using previous measurements, to
dynamically set the energy and angular bins of the next sample, allowing for
faster measurement cadence. The iESA is highly versatile and the tracking
strategy can implement any combination of energies and angles. Instrument
operation will be adapted to the science target, but a primary operation mode
is expected to be a Proton Tracking mode measuring the 3D VDFs of SW protons
with high energy (8%) and angular ($3^{\circ}$) resolutions at a cadence down
to 150 ms, well into the sub-ion timescales. To characterize alphas, a Proton-
Alpha Tracking mode will be used, with 48 energy bins and a 450 ms cadence,
though longer accumulation times can be used to enhance counting statistics as
needed.

Figure 10: iESA subsystem components (as of the concept study report), include
the deflectors, collimators, and analyzer spheres (top), as well as the 16
channel electron multipliers, front end electronics, low- and high-voltage
power supply, and FPGA board.

###  5.4 Student Electron Electrostatic Spectrograph Student Collaboration

In addition to the magnetic field and ion instruments previously listed in
this section, HS has also proposed to include a Student Electron Electrostatic
Spectrograph (SEE) Student Collaboration project to measure ambient, low
energy electrons. This electron instrument would be mounted on the Hub SC, and
would be used to study the connectivity of the local magnetosphere, solar
wind, and cis-lunar space via measurements of low-energy electron populations.
The project would be co-led by graduate and undergraduate students, with the
prime deliverable from the SEE project a cohort of future scientists educated
in the lifecycle of a NASA mission, including instrument development and
merger of science goals with hardware design. A backup design for SEE has PSP
and ESCAPADE flight heritage Whittlesey:2020 .

###  5.5 The Hub

Northrop Grumman (NG) provides the Hub spacecraft. This ESPA-class spacecraft
serves as the central relay for all Nodes within the Observatory and is based
on the high-heritage ESPAstar line which was designed to carry separable
payloads to orbit. The Hub is 730 kg at launch, including the hydrazine
propellant necessary for carrying it and all the Nodes into the HS science
orbit, illustrated at the top of Fig. 11. The Hub is capable of generating
1165 W of power via its single deployable solar array. As configured for
science, the Hub spans a maximum dimension of 8.4 m.

Figure 11: Transfer (top) and Science (bottom) Configurations for the Hub
(Sec. 5.5) and Node (Sec. 5.6) S/C.

###  5.6 The Nodes

The Node spacecraft are Blue Canyon Technologies (BCT) Venus-class spacecraft
with standard accommodations for hosting the HS payload. As configured for HS,
the SmallSat Nodes are just over 70 kg each and use onboard propulsion to
maintain the proper swarm geometry. The Nodes generate 200 W of power and
provide a single mechanical interface to the HS payload. With booms deployed,
bottom left of Fig. 11, the maximum tip-to-tip dimension of the Nodes is just
over 6m.

###  5.7 Observatory Architecture

The HS flight system will launch with the Hub carrying all 8 Nodes through a
series of phasing loops and a lunar swingby into the science orbit over a
period of approximately 72 days. Once the science orbit (a High Earth Orbit
P/2 lunar resonant orbit with apogee greater than $60R_{E}$ and perigee less
than $12R_{E}$), pairs of Nodes are separated from the Hub spacecraft and
instrument checkouts and calibrations occur during the $\approx 81$-day
commissioning phase. Once the Nodes have been maneuvered into their proper
locations in the observatory, the 12-month science phase begins. HS uses a
Hub-and-spoke communications architecture in which the Hub is the only direct
link to the ground (via S-band, DSN) and each Node receives commands from and
relays science data directly to the Hub via S-band crosslinks.

The Mission Operations Center (MOC) is hosted out of ARC, with engineering
support centers for Hub and Nodes at NG and BCT respectively. The Science
Operations Center (SOC) is hosted out of the PI-institution, UNH, which is
also responsible for delivering L1-L4 data products to NASA SPDF; see Sec.
5.8. The HS missions operations approach strategically uses the inherent
orbital dynamics of the observatory; the swarm naturally “expands” and
“contracts” over each 14-day orbital period. The Flight Dynamics team has
designed the staggering of the Node-Hub closest approaches around perigee to
facilitate periods of high-rate data downlink for each Node each orbit. As the
observatory begins to expand out towards apogee, the polyhedral performance,
Sec. 4.5.2, requirements are satisfied. At apogee and on the contraction back
in towards perigee, the longest-baseline 3D performance requirements are
satisfied, Sec. 4.5.1.

###  5.8 Data Processing and Selection

The Science Data Center (SDC) at the UNH SOC supports HS data processing,
produces timing and other ancillary data that are provided to the instrument
teams, releases and archives L0-L4 data with associated calibrations, and
provides data selection tools to the community.

Automated processes transfer L0 telemetry data from the HS MOC to the SOC
within 24 hours of ground receipt. Upon L0 receipt, the SOC performs packet
format checks (e.g., valid headers, checksums). The SOC prepares a timing
product to correct Node clock differences relative to the Hub timing and
processes IS housekeeping (HK) to calibrated units (L1). L0 data are provided
to the instrument teams for processing, with timing corrections and L1 HK
included as additional inputs. The instrument teams generate and validate L1
(measurements in engineering units), L2 (science data—magnetic field
measurements, particle velocity distributions and the associated moments and
fits— in payload coördinates), and L3 (science data in RTN coördinates) data
products within 30 days of receipt.

L1-L3 data are retrieved by the SOC from the instrument teams within 24 hours
of processing and summary plots generated. L4 data products, e.g. a merger of
the MAG and SCM data or combined magnetic field and proton products from
across the observatory, are produced at the SOC within 5 working days.

Upon completion of IS commissioning, the SOC begins a period of data product
and instrument performance validation. As data for each orbit are downlinked,
they are routinely and automatically processed. During the subsequent orbit,
the SDC lead coördinates instrument team validation of the data. Validation
activities proceed through individual calibration and Observatory-level
calibrations. As calibrations are updated, previous orbits’ data are
reprocessed with the updated calibrations, so the process iterates with
increasing data volumes of increasing refinement. Validated, calibrated L2
through L4 data are provided to NASA SPDF for public access upon completion of
the validation period, anticipated to be no more than 5 months, which may be
shortened if reasonable calibrations are available sooner.

Because HS comprises nine spacecraft with slowly changing relative positions
in an equally dynamic plasma environment, specialized tools to assist
researchers in their data selection will be developed and implemented at the
SOC. Many techniques employed by the researchers require specific Observatory
configurations. To aid the researcher in the selection of processed data, the
SOC is developing interactive queries, but all science data are downlinked and
processed regardless of observatory configuration. As an example, Figure 5
shows a snapshot from a preliminary data selection code using DRM orbit
trajectories.

##  6 Analysis Approaches

A variety of multi-spacecraft Analysis Approaches have been previously
developed for missions such as Cluster and MMS. These methods include
calculations of

  * •

Cascade Rate, measuring of the transfer of turbulent fluctuation energy from
one spatial scale to another (e.g. Hadid:2018, ; Pecora:2023, ),

  * •

2-point correlation, measuring the temporal and spatial scale over which a
spectral element is remade by nonlinear processes (e.g. Horbury:2000, ;
Matthaeus:2016, ),

  * •

Structure Functions, determining the statistics turbulent field increments to
reveal scale-dependent, intermittent turbulence(e.g. Sreenivasan:1997, ;
Chhiber:2021, ),

  * •

Wave telescope, determining the wavevectors of plasma waves and their
associated 3D power distributions(e.g. Pincon:1991, ; Narita:2022, ),

  * •

Pressure-strain interaction, measuring the dilation,
$-(\underline{\underline{P}}\cdot\nabla)\cdot\mathbf{U}$, which describes the
local conversion between flow and thermal energy(e.g. Yang:2017, ;
Cassak:2022, ),

  * •

Curlometer & Gradient Methods, which construct current sheets and other
intermittent structures from spatially distributed measurements(e.g.
Dunlop:2002b, ; Dunlop:2021, ).

Many of these methods and their applications have been documented in ISSI
review articles over the last several decadesPaschmann:1998 ; Paschmann:2008 .
In this section, we apply some analysis approaches to synthetic timeseries
constructed using DRM trajectories corresponding to selected HS configurations
through different numerical simulations of turbulence including two-fluid
Wang:2015 ; Juno:2018  and hybrid-PIC Kunz:2014 ; Arzamasskiy:2019  nonlinear
simulations and quasilinear simulations Klein:2012 ; Klein:2014a . We note
that due to computational limitations, all the numerical codes used make
approximations in terms of the physical processes included and/or the scales
simulated. Therefore, we do not expect the numerical simulations to be
completely representative of actual plasma turbulence at all scales, and
differences between simulations and HS observations will drive improvements of
the modeling of turbulent transport and dissipation.

###  6.1 Multipoint Correlations and Structure Functions

Multipoint spectral analysis, 2nd-order structure functions, and space-time
correlation functions yield distributions of turbulent energy in configuration
space Chen:2011a ; Paschmann:1998 ; Paschmann:2008  and, through time-lagging,
decorrelation times for fluctuations at measured spatial scales Matthaeus:2016
.

Figure 12 illustrates the temporal and spatial decorrelation of signals from
synthetic magnetic field data set constructed by sampling a hybrid-PIC
numerical simulation Arzamasskiy:2019  over trajectories defined by the DRM
described in Section 4.6. The correlation is calculated between all nine
trajectories at each point in the timeseries as

$R(|\mathbf{r}|,\tau)=\left<\mathbf{b}(\mathbf{x},t)\cdot\mathbf{b}(\mathbf{x}+\mathbf{r},t+\tau)\right>.$
(4)

These correlation values, and how quickly they depart from unity,
characterizes the temporal and spatial scales over which fluctuations are
remade by nonlinear terms, and represents a key statistical property of
turbulent distributions Matthaeus:2005  that will be used in G1O1 and G1O2.
For comparison, an example auto-correlation produced using lagged timeseries
from a single trajectory (upper right panel) mixes together spatial and
temporal dependence. HS will disentangle spatial and temporal correlations
which single SC convolve together.

Figure 12:  The spatial-temporal correlation calculated from synthetic HS
measurements extracted from a hybrid-PIC simulation of
turbulenceArzamasskiy:2019 . By using all nine trajectories, we are able to
resolve the spatial and temporal dependences of Eqn. 4 independently, unlike
the auto-correlation from a single trajectory, presented in the upper right-
hand panel, which effectively samples along the red arrow in the lower left
panel, convolving together spatial and temporal variations.

The measurement of intense fluctuations at smaller scales while simultaneously
measuring the distribution of fluctuations at larger scales differentiates
between models of scale-dependent intermittency, testing theoretical
predictionsTennekes:1975 ; Grauer:1994 ; Chandran:2015 ; Mallet:2017a ;
Greco:2009 . Intermittency also affects different types of proton heating
mechanisms in different ways, with the associated coherent structures greatly
affect the efficiency of certain processes. For example, stochastic heating
occurs when fluctuation amplitudes at the scale of a particle’s gyroradius
become large Chandran:2010a ; Dalena:2014 , which are enhanced near coherent
structures. Conversely, dissipation mechanisms such as Landau damping are less
affected by intermittency Mallet:2019 . HS multiscale measurement of higher
order intermittent statistical measures, along with temperature and
temperature anisotropy reveal deep connections between cascade, intermittency
and dissipation.

Figure 13: HS’s multiscale configuration enables the calculation of multiple
orders of structure functions (left panel), that when fit to a power-law
$S_{n}\propto\lambda^{\xi_{n}}$ (center) or recombined into quantities such as
kurtosis $S_{4}/S_{2}^{2}$ (right) can be compared to theoretical predictions
[e.g. Kolmogorov:1941 ; Grauer:1994 ; Chandran:2015 ] to characterize the
scale-dependent intermittency in turbulence (G1O4).

One means of quantifying the intermittency of a system is illustrated in
Figure 13, where we calculate the scale-dependent structure function $S_{n}$
using synthetic times series drawn from the same hybrid numerical simulation
of turbulence Arzamasskiy:2019 . Instead of using increments drawn from a
single timeseries

$S_{n}(\lambda=\tau v_{{SW}})=\left<[\delta z(t+\tau)-\delta z(t)]^{n}\right>$
(5)

Chhiber:2021 , increments are calculated using all nine timeseries combined
with HS’s spatial resolution and configurations and a modified form of
Taylor’s hypothesis Horbury:2000  allowing for orders of magnitude more
samples to be used at a given scale both along and transverse to the magnetic
field and flow directions,

$S_{n}[\mathbf{x}_{i}(t+\tau)-\mathbf{x}_{j}(t)]=\left<\left\\{\delta
z[\mathbf{x}_{i}(t+\tau)]-\delta z[\mathbf{x}_{j}(t)]\right\\}^{n}\right>$ (6)

enabling the calculation of the higher order structure functions spanning ion
kinetic (blue, left panel), transition (purple), and MHD (red) scales; for N
increments measured, structure functions of order $n=\log_{10}[N]-1$ can be
resolved Dudok:2004 ; Dudok:2013 ; HS enables the study of higher order
$S_{n}$ than previous missions, where differences between theoretical
descriptions are more easily distinguishable. From these measurements of
$S_{n}$, theories about intermittency as a function of scale Sreenivasan:1997
, which describe how frequently and how abruptly sharp structures of different
sizes and shapes arise, are tested by fitting $S_{n}\propto\lambda^{\xi_{n}}$
over different scale ranges (center); the trending of the fit parameters with
order n is used to validate, falsify, or improve theories [e.g.
Kolmogorov:1941 ; Grauer:1994 ; Chandran:2015 . Combined with other metrics
such as kurtosis (right panel) as well as the analysis method presented in
Figure 14, HS characterizes turbulent intermittency as a function of scale
addressing G1O4.

Fig. 14 illustrates with the same simulated synthetic DRM timeseries used for
Fig. 12 another way in which HS will provide science closure on G1O4 by using
its 36 baseline separations to simultaneously quantify the distribution of
turbulent fluctuations at large and small scales. Following Mallet et al
2015Mallet:2015 , we define the fluctuation amplitude increment

$\delta
z_{\perp}^{\pm}=|\delta\mathbf{z}_{\perp}^{\pm}|=|\mathbf{z}_{\perp}^{\pm}(\mathbf{r}_{0}+\mathbf{r}_{\perp})-\mathbf{z}_{\perp}^{\pm}(\mathbf{r}_{0})|$
(7)

where $\mathbf{r}_{\perp}$ is the separation in the plane perpendicular to the
mean magnetic field direction $\mathbf{B}_{0}$. Using both the synthetic HS
timeseries drawn from a hybrid-PIC simulation, as well as timeseries
constructed from a collection of randomly phased waves, we calculate $\delta
z_{\perp}^{\pm}$ as a function of scale $\lambda=|\mathbf{r}_{\perp}|$ for an
ensemble of separations. We then calculate the median value of the increment
$\delta\overline{z}_{\perp}^{\pm}$ over a series of bins spaced logarithmicaly
in $\lambda$, and normalize the probability distribution function of the
increments in each bin by the median. For the trivial case of randomly-phased
waves, the increments are normally distributed, and there is no variation as a
function of scale. For the case of simulated turbulent, the intermittency
increases with decreasing scale, seen in the transition from red (large, MHD
scales) to blue (smaller, ion-kinetic scales). As the intermittency of a
turbulent system depends on the nature of the nonlinear interactions that
transport energy through a system, these types of distributions represent a
sensitive test of different models of turbulence.

Figure 14: HS measures intermittency at MHD (red) and sub-ion (blue) scales
simultaneously from different numerical simulations, providing science closure
on G1O4 by differentiating between models of nonlinear coupling that predict
enhanced amplitudes of normalized Elsasser fluctuations $\delta z^{+}_{\perp}$
at small scales (right, from a turbulent hybrid-PIC simulation
Arzamasskiy:2019 ) compared to a scale-independent normal distribution of
amplitudes (left, drawn from ensemble of randomly-phased wave modes.).

###  6.2 Curlometer and Gradient Techniques

Wave telescope and curlometer techniques reveal the nature of fluctuations and
identify structures within turbulence. Gradient estimation is closely related
and is required for determining the pressure-strain tensor and the production
rate of internal energy.

Figure 15: HS’s SC positions (red points) enable a high-fidelity
reconstruction (perfect measurements shown in blue, measurements with included
systematic errors in green) of the magnetic field (arrows) over a larger
volume of space, enabling the simultaneous study of ion-scale structures
(e.g., current sheets, shown as contours at left) with sufficient accuracy to
address G1O4 (regions of reconstruction with less than 5% and 10% error shown
at right, as explained in the text).

As an example of a novel application of gradient techniques enabled by a
multi-spacecraft observatory, estimates for the spatial gradients from
synthetic timeseries drawn from a two-fluid numerical simulation of
turbulenceWang:2015  are used in a first-order reconstruction to reconstruct a
3D synthetic magnetic field, shown in Figure 15 and described in detail in
Broeren:2021 . By using a weighted average of the first-order estimates for
the reconstructed magnetic field drawn from the spatial gradients determined
from the 126 unique tetrahedra that comprise the HS Observatory, we accurately
reproduce the magnetic field over large volumes of the simulation. For a
selected DRM interval in a good polyhedral configuration, we reconstruct the
simulated turbulent magnetic field (black/blue arrows indicate the
simulated/reconstructed field). Averaging over many spatial locations in the
simulation, this analysis method yields $\leq 5$% relative error over a volume
of nearly $1.82\times 10^{9}\textrm{km}^{3}$ (solid blue line, right panel)
and $\leq 10$% error over a volume of $3.208\times 10^{9}$ km3 (dashed blue
line).

To quantify the impact of systematic errors on this analysis technique, we
introduce random offsets replicating estimated systematic errors to each
component of the measured magnetic field at all nine SC (green lines). This
produces reconstructed volumes of $3.66\times 10^{8}(1.77\times 10^{9})$ km3
for the 5% (10%) error thresholds. By leveraging the large number of
tetrahedral configurations sensitive to many length scales simultaneously, HS
enables simultaneous studies of both MHD-scale structure as well as much
smaller current sheets, bringing closure to questions about the transfer of
energy from fields to flows (G1O3) and associated heating near ion-scale
intermittent structures (G1O4).

As discussed at the beginning of this section several other multispacecraft
analysis methods, previously implemented on missions such as MMS and Cluster,
can be immediately applied to HS observations, or extended to incorporate
information from all nine spacecraft in the observatory, e.g. Perri:2017 ;
Zhang:2021 ; Pecora:2023 ; Toepfer:2023 ; Broeren:2023 .

##  7 Conclusions

Turbulence is the process by which energy contained in fluctuating magnetic
fields and plasma motion cascades from larger to smaller spatial scales, and
ultimately into thermal energy of charged particles comprising the plasma. In
addition to being a key process that heats cosmic plasmas, it also creates the
conditions in which all universal plasma processes (e.g., magnetic
reconnection, shocks, particle acceleration) act, both within the heliosphere
and in all astrophysical domains. Due to its fundamentally multiscale nature,
only spatially distributed, simultaneous measurements provide the data needed
to bring closure to outstanding questions about the distribution and transfer
of turbulent energy. HS achieves its mission objectives through an innovative
swarm implementation of high-heritage mission elements, ranging from
instruments, to spacecraft, operations, and analysis tools. HS provides a
paradigm shift in mission design where the many elements of the swarm and the
way they interact form an Observatory that is far more than the sum of its
parts. As the first multipoint, multiscale mission, HS gives an unprecedented
view into the nature of space plasma turbulence.

Acknowledgements We are deeply indebted to the incredible members of the
HelioSwarm science, engineering, and proposal teams whose tireless efforts
enabled this mission. Construction and analysis of the HelioSwarm Observatory
Design Reference Mission trajectories was supported in part by the HelioSwarm
Project funded under NASA’s Prime contract no. 80ARC021C0001.

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