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 Geophysical Monograph 283

Helicities in Geophysics, Astrophysics,

and Beyond
Kirill Kuzanyan
Nobumitsu Yokoi
Manolis K. Georgoulis
Rodion Stepanov
Editors

This Work is a co-publication of the American Geophysical Union and John Wiley and Sons, Inc.
This edition first published 2024
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Title: Helicities in geophysics, astrophysics, and beyond / editors Kirill
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CONTENTS

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Part I Helicity Essentials: Basic and Fundamental Concepts 1

1 Introduction to Field Line Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Anthony R. Yeates and Mitchell A. Berger

2 Magnetic Winding: Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

David MacTaggart

3 Transport in Helical Fluid Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Nobumitsu Yokoi

Part II Helicity Manifestations in Nature and Their Observations 51

4 Observations of Magnetic Helicity Proxies in the Solar Photosphere: Helicity With Solar Cycles . . . . . . . . . 53
Hongqi Zhang, Shangbin Yang, Haiqing Xu, Xiao Yang, Jie Chen, and Jihong Liu

5 Chirality of Solar Filaments and the Supporting Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Peng-Fei Chen

6 Solar Flares and Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Shin Toriumi and Sung-Hong Park

7 Magnetic Helicity Measurements in the Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Yasuhito Narita

8 Magnetic Helicity in Rotating Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Maxim Dvornikov

9 Writhing From Biophysics to Solar Physics and Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Christopher Prior and Arron N. Bale

10 Kinetic Helicity in the Earth’s Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Otto Chkhetiani and Michael Kurgansky

Part III Theoretical and Numerical Helicity Modeling 167

11 Effects of P-Noninvariance, Particles, and Dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Victor B. Semikoz and Dmitry D. Sokoloff

v
vi CONTENTS

12 Stability of Plasmas Through Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Simon Candelaresi and Fabio Del Sordo

13 Helicity-Conserving Relaxation in Unstable and Merging Twisted Magnetic Flux Ropes . . . . . . . . . . . . . . . . . . 189
Philippa K. Browning, Mykola Gordovskyy, and Alan W. Hood

14 Emergence of Magnetic Structure in Supersonic Isothermal Magnetohydrodynamic Turbulence. . . . . . . . . . 203

Jean-Mathieu Teissier and Wolf-Christian Müller

15 Nonlinear Mean-Field Dynamos With Magnetic Helicity Transport and Solar Activity: Sunspot Number
and Tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Nathan Kleeorin, Kirill Kuzanyan, Igor Rogachevskii, and Nikolai Safiullin

16 The Spatial Segregation of Kinetic Helicity in Geodynamo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Avishek Ranjan and Peter Davidson

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
LIST OF CONTRIBUTORS

Arron N. Bale Mykola Gordovskyy

Department of Mathematical Sciences Department of Physics and Astronomy
Durham University University of Manchester
Durham, UK Manchester, UK

Alan W. Hood
Mitchell A. Berger School of Mathematics and Statistics
Department of Mathematics University of St Andrews
University of Exeter St Andrews, UK
Exeter, UK
Nathan Kleeorin
Philippa K. Browning Department of Mechanical Engineering
Department of Physics and Astronomy Ben-Gurion University of the Negev
University of Manchester Beer-Sheva, Israel
Manchester, UK
and
Simon Candelaresi Institute of Continuous Media Mechanics
School of Mathematics and Statistics Russian Academy of Sciences
University of Glasgow Perm, Russia
Glasgow, UK
Michael Kurgansky
Jie Chen A. M. Obukhov Institute of Atmospheric Physics
National Astronomical Observatories Russian Academy of Sciences
Chinese Academy of Sciences Moscow, Russia
Beijing, China
Kirill Kuzanyan
Peng-Fei Chen Pushkov Institute of Terrestrial Magnetism, Ionosphere and
Key Lab of Modern Astronomy & Astrophysics Radiowave Propagation (IZMIRAN)
School of Astronomy and Space Science Russian Academy of Sciences
Nanjing University Moscow, Russia
Nanjing, China
and
Otto Chkhetiani Institute of Continuous Media Mechanics
A. M. Obukhov Institute of Atmospheric Physics Russian Academy of Sciences
Russian Academy of Sciences Perm, Russia
Moscow, Russia
Jihong Liu
Peter Davidson National Astronomical Observatories
Department of Engineering Chinese Academy of Sciences
University of Cambridge Beijing, China
Cambridge, UK
and
Fabio Del Sordo School of Science
Institute of Space Sciences Shijiazhuang University
Barcelona, Spain Shijiazhuang, China
and
David MacTaggart
Catania Astrophysical Observatory School of Mathematics and Statistics
National Institute for Astrophysics University of Glasgow
Catania, Italy Glasgow, UK
Maxim Dvornikov
Pushkov Institute of Terrestrial Magnetism, Ionosphere and
Radiowave Propagation (IZMIRAN)
Russian Academy of Sciences
Moscow, Russia

vii
viii LIST OF CONTRIBUTORS

Wolf-Christian Müller Dmitry D. Sokoloff

Center for Astronomy and Astrophysics Pushkov Institute of Terrestrial Magnetism, Ionosphere and
Technical University of Berlin Radiowave Propagation (IZMIRAN)
Berlin, Germany Russian Academy of Sciences
and Moscow, Russia

Max-Planck/Princeton Center for Plasma Physics and

Princeton, NJ, USA Department of Physics
Moscow State University
Yasuhito Narita Moscow, Russia
Institute for Theoretical Physics
Technical University of Braunschweig Jean-Mathieu Teissier
Braunschweig, Germany Center for Astronomy and Astrophysics
Technical University of Berlin
Berlin, Germany
Sung-Hong Park
Korea Astronomy and Space Science Institute Shin Toriumi
Daejeon, Republic of Korea Institute of Space and Astronautical Science
and Japan Aerospace Exploration Agency
Sagamihara, Japan
Institute for Space-Earth Environmental Research
Nagoya University Haiqing Xu
Nagoya, Japan National Astronomical Observatories
and Chinese Academy of Sciences
Beijing, China
W. W. Hansen Experimental Physics Laboratory
Stanford University Shangbin Yang
Stanford, CA, USA National Astronomical Observatories
Chinese Academy of Sciences
Christopher Prior
Beijing, China
Department of Mathematical Sciences
Durham University and
Durham, UK School of Astronomy and Space Science
University of Chinese Academy of Sciences
Avishek Ranjan
Beijing, China
Department of Mechanical Engineering
Indian Institute of Technology Bombay Xiao Yang
Mumbai, India National Astronomical Observatories
Chinese Academy of Sciences
Igor Rogachevskii
Beijing, China
Department of Mechanical Engineering
Ben-Gurion University of the Negev Anthony R. Yeates
Beer-Sheva, Israel Department of Mathematical Sciences
and Durham University
Durham, UK
Nordic Institute for Theoretical Physics
KTH Royal Institute of Technology and Stockholm University Nobumitsu Yokoi
Stockholm, Sweden Institute of Industrial Science
University of Tokyo
Nikolai Safiullin
Tokyo, Japan
Department of Information Security
Ural Federal University Hongqi Zhang
Ekaterinburg, Russia National Astronomical Observatories
and Chinese Academy of Sciences
Beijing, China
Institute of Continuous Media Mechanics
Russian Academy of Sciences
Perm, Russia

Victor B. Semikoz
Pushkov Institute of Terrestrial Magnetism, Ionosphere and
Radiowave Propagation (IZMIRAN)
Russian Academy of Sciences
Moscow, Russia
PREFACE

Helicities, defined by the volume integral of the inner all the participants for their energy and for directly and
product of a vector field and its curled counterpart in a indirectly contributing to the contents of this book.
generalized sense (including cross helicity and general- This book has two primary aims. The first is to provide
ized helicity in Hall magnetohydrodynamics [MHD]), are a perspective on helicities relevant to geophysics, astro-
known to play essential roles in several geophysical, astro- physics, physical and space plasma sciences (including
physical, and space plasma phenomena, from dynamo biological and quantum fluids). The second is to propose
actions in geo/planetary magnetism and astrophysical future directions for helicity studies in these fields using
objects to solar and stellar magnetic field evolution, mass cohesive theoretical, observational, experimental, and
condensation in star-forming regions, accreting jet forma- numerical strategies for constructing models applicable to
tion near compact massive objects, amplification of the real-world phenomena. Compact, readable introductions
magnetic field in the Universe, magnetic confinement in in each chapter acquaint the reader with advanced topics
fusion plasmas, etc. Aligning with these important roles and aspects of helicity studies.
in various research fields, helicities have been studied We could not cover all topics in the natural sciences
using various methods from different viewpoints. that involve helicities, but we have tried to give a reader
This book is a synopsis of recent developments and the broadest possible representation of these fields. Cer-
achievements in helicity studies by leading scientists. tain important topics are lacking, such as the possible
It grew from a series of gatherings entitled “Online detection of magnetic helicity proxies in observable
Advanced Study Program on Helicities in Astrophysics fast-rotating stars, a perspective on kinetic and magnetic
and Beyond” (https://helicity2020.izmiran.ru) in Fall helicities in the Earth’s core, and laboratory fluid and
2020 and Spring 2021, which were held online as a plasma experiments. We hope such topics will be covered
substitute for in-person interactions within the research in future publications and books.
community during the Covid-19 pandemic. The book consists of 16 chapters divided into three
These gatherings built on years of scientific studies and parts. Part I discusses helicity basics and fundamental
previous events. Magnetic helicity has been intensively concepts. In Chapter 1, Anthony Yeates and Mitchell
studied in recent decades from observational, theoret- Berger propose that field-line helicity provides a finer
ical, and modeling viewpoints in fields such as general local topological description of magnetic flux than the
astrophysics, solar physics, plasma and fluid dynamics, usual global magnetic helicity integral, with invariant
and pure mathematics. A series of focused events on properties preserved. They present a way to appropriately
helicity have also taken place, such as the 1998 AGU define field-line helicity in different volumes. They also
Chapman Conference in Boulder, Colorado, USA; the discuss the time evolution of field-line helicity under
2009 and 2013 Helicity Thinkshops in China (https:// both boundary motions and magnetic reconnection. In
sun10.bao.ac.cn/old/meetings/HT2009/ and https:// Chapter 2, David MacTaggart introduces the notion of
sun10.bao.ac.cn/old/meetings/HT2013/); the 2017 Helic- magnetic winding from a theoretical perspective. Mag-
ity Thinkshop in Japan (http://www.iis.u-tokyo.ac.jp/ netic winding is a renormalization of magnetic helicity,
~nobyokoi/thinkshop/), part of the SEIKEN Symposium directly measuring field-line topology. This new notion
(https://science-media.org/conference/23); and the Pro- is complemented by an application to observations of
gram on Solar Helicities in Theory and Observations solar active regions. In Chapter 3, Nobumitsu Yokoi
at NORDITA in 2019 (https://indico.fysik.su.se/event/ constructs a turbulent transport model including helicity.
6548/), among others. The model reveals that the helical contribution may
The Online Advanced Study Program in 2020 and suppress momentum transport with a remarkable feature
2021 was hosted by IZMIRAN, the Russian Academy on the induction of a large-scale flow caused by an inho-
of Sciences, in Moscow. It featured seminars delivered mogeneous coupling between helicity and rotation. This
by international experts in astrophysics, geophysics, and flow appears to have numerous applications to astro- and
many fields of natural sciences involving observational geophysical phenomena.
and theoretical studies of magnetic, kinetic, and other Part II contains several reviews of manifestations of
helicities. We deliberately took an interdisciplinary and helicities in various natural phenomena and their obser-
multidisciplinary approach to attract broader audiences. vations. Chapter 4, by Hongqi Zhang et al., reviews
Some events exceeded 200 online attendees, and we thank longstanding diagnostic tools available for inferring

ix
x PREFACE

proxies of both the magnetic and current helicity in the balance and fluxes are used to analyze atmospheric vor-
solar atmosphere. They analyze solar atmospheric helicity tices such as tropical cyclones, tornadoes, dust devils, and
at short and long timescales, in the latter showing analogs Ekman boundary layer dynamics. The helical properties
of the butterfly diagram for sunspots using the mean cur- of turbulence within the atmospheric boundary layer
rent helicity and twist, that have important implications have been probed by direct pioneering measurements of
for the solar dynamo. Chapter 5, by Peng-Fei Chen, casts turbulent helicity in natural atmospheric conditions.
doubt on how one can observationally infer the chirality Part III on theoretical and numerical modeling of
of solar filaments by observing the skewness or bearing helicities opens with Chapter 11 by Victor Semikoz and
of the filament barbs. While this suggestion is gaining Dmitry Sokoloff, who review cosmological dynamos and
traction, it is far from unanimously accepted in the solar explore the P-noninvariance of elementary particles. This
physics community but is eloquently presented to spur presents a possibility for nonclassical dynamo genera-
discussion and debate. Chapter 6, by Shin Toriumi and tion in the early Universe based on the intrinsic mirror
Sung-Hong Park, discusses various diagnostic tools for asymmetry of elementary particles. In Chapter 12, Simon
magnetic helicity in local (i.e., active-region) solar scales, Candelaresi and Fabio Del Sordo point out that magnetic
aiming to understand and ultimately predict solar flares. helicity constrains the dynamics of plasmas. They discuss
Describing immense magnetic complexity, they make how magnetic helicity stabilizes the plasma and prevents
a twofold effort to distinguish populations of flaring its disruption, with reference to observations, numerical
active regions from the majority pool of non-flaring experiments, and analytical results. Several illustrative
regions in order to understand the separator(s) of these examples in the solar corona are presented, as well as
populations and then use this knowledge for prediction fusion devices, galactic and extragalactic medium, and
purposes. Chapter 7, by Yasuhito Narita, reviews meth- extragalactic bubbles. Chapter 13, by Philippa Browning
ods of evaluating magnetic helicity in the solar wind et al., addresses the contentious topic of magnetic relax-
in terms of both single- and multipoint measurements.
ation in the solar corona and its implications for magnetic
Methodologies invariably aim to resolve the transport
helicity. They arrive logically at the concept of magnetic
problems of helicity, magnetic flux, and energy of the Sun
relaxation, likely in terms of the Taylor hypothesis, and
into the heliosphere. On short (compared to magnetohy-
show how ideal instabilities and merging magnetic flux
drodynamic turbulence) spatial scales in the ion-kinetic
ropes in the corona can lead to relaxation. Implications
domain, the observations reveal nonzero helicity as a
for laboratory plasmas and the elusive coronal heat-
signature of linear-mode wave excitation, such as the
ing mechanism(s) are also discussed. Chapter 14, by
kinetic Alfven waves and whistler waves. This differs in
Jean-Mathieu Teissier and Wolf-Christian Müller, deals
each solar hemisphere and varies with the solar cycle. In
with the formation and sustainment of magnetic struc-
Chapter 8, Maxim Dvornikov reviews the role of magnetic
helicity evolution in rotating neutron stars. He utilizes tures in supersonic isothermal magnetohydrodynamic
the conservation law for the sum of the chiral imbalance turbulence. They review the first results obtained through
of charged particle densities and the density of magnetic direct numerical simulations of isothermal compressible
helicity and explores the possibility of X-ray or gamma MHD at Mach numbers ranging from subsonic to about
bursts observed in magnetars due to this mechanism. 10, finding contributions of the local and nonlocal, direct,
He argues that the quantum contribution dominates the and inverse transfer of magnetic helicity in supersonic
classical contribution in the surface terms in standard MHD regimes. Chapter 15, by Nathan Kleeorin et al.,
MHD but only for neutron stars with rigid rotation. He discusses nonlinear mean-field dynamos, with special
shows that the characteristic time of the helicity change reference to various mechanisms for sunspot formation
is in accord with the magnetic cycle period of certain and the prediction of solar activity. Based on nonlinear
pulsars. Chapter 9, by Christopher Prior and Arron Bale, dynamo equations, including the model equation for
deals with writhing and its prospects for wider interdisci- magnetic helicity, they explain existing observations of
plinary applications and interaction between biophysics, magnetic helicity in the Sun and dynamical solar activity.
solar physics, and other disciplines. Writhing quantifies a The contributions of magnetic helicity, large-scale mag-
structure’s global self-entanglement (knotting) and plays netic fields, and differential rotation to the mean tilt angle
a fundamental role in DNA compactification (super- of sunspot bipolar regions are also discussed. Finally,
coiling). Earlier results on magnetic helicity in solar Chapter 16, by Avishek Ranjan and Peter Davidson,
physics can be used for biophysical applications such as is dedicated to the origin of the spatial segregation of
understanding protein structures through their writhing kinetic helicity in dynamo simulations. They discuss
measures. Chapter 10, by Otto Chkhetiani and Michael various sources of kinetic helicity, including helical waves
Kurgansky, explores kinetic helicity in the Earth’s atmo- such as inertial waves. Strong spatial correlations of
sphere and its role in atmospheric turbulence. The helicity the segregation pattern of helicity and the source term
 PREFACE xi

due to buoyancy exhibited in numerical simulations are Our thanks go to the AGU Books Editorial Board
interpreted using helical wave propagation. and the reviewers of our proposal, who encouraged us
Together, these chapters present different aspects of and helped improve this book. We acknowledge the
helicities in diverse scientific contexts, from basic concepts dedication of each of the chapter contributors for the
and fundamental properties to manifestations in several time they spent preparing their manuscripts. We greatly
natural phenomena, as well as theoretical and numerical appreciate the patience and professionalism of those who
models. The book also presents the possibility of tackling reviewed chapters to ensure that this book was of the
real-world problems via the many forms and flavors of highest standards as well as to increase its readability
helicity. We hope the breadth of information and evi- and appeal to the community. We also acknowledge
dence presented in this book will spur discussions and support from AGU Publications and Wiley, includ-
debates that will lead to an enhanced, broader scientific ing technical assistance and advice from their staff.
understanding of complexity in various natural systems Finally, we thank our colleagues, friends, and families
and networks. We trust that this volume contributes to for their patience and inspiration while we worked on
further developments in this fascinating, challenging, and this project.
ever-evolving subject.
We are grateful to the organizations that have hosted Kirill Kuzanyan
various in-person and online gatherings relating to Pushkov Institute of Terrestrial Magnetism, Ionosphere,
helicity studies, such as the National Astronomical and Radio Wave Propagation
Observatories of China, University of Tokyo in Japan, Russian Academy of Sciences, Moscow, Russia
NORDITA in Sweden, and IZMIRAN in Russia. We Nobumitsu Yokoi
thank all the participants of these meetings for their con- Institute of Industrial Science
tributions and discussions, especially the speakers who University of Tokyo, Tokyo, Japan
gave talks, posters, and online presentations. We would Manolis K. Georgoulis
also like to acknowledge the support and hospitality of Research Center for Astronomy and Applied Mathematics
the Isaac Newton Institute for Mathematical Sciences in Academy of Athens, Athens, Greece
Cambridge, UK, during the Dynamo Theory Programme Rodion Stepanov
(DYT2) held in Fall 2022, where the contents of this book Institute of Continuous Media Mechanics
were finalized. Russian Academy of Sciences, Perm, Russia
 Part I
Helicity Essentials: Basic and
Fundamental Concepts

1
 1
Introduction to Field Line Helicity
Anthony R. Yeates 1 and Mitchell A. Berger 2

ABSTRACT
Field line helicity measures the net linking of magnetic flux with a single magnetic field line. It offers a finer
topological description than the usual global magnetic helicity integral while still being invariant in an ideal
evolution unless there is a flux of helicity through the domain boundary. In this chapter, we explore how to
appropriately define field line helicity in different volumes in a way that preserves a meaningful topological inter-
pretation. We also review the time evolution of field line helicity under both boundary motions and magnetic
reconnection.

1.1. DEFINITIONS OF FIELD LINE HELICITY do not cross the boundary of  (or within an infinite
space). Historically, Gauss (1809) discovered a double
We briefly review topological measures of magnetic field line integral, which measures the linking of two closed
structure. Some of these refer to the structure of the total curves L1 and L2 . Let positions on the curves be given by
field within a volume (the magnetic helicity), and others x⃗ (𝜎) and y⃗ (𝜏). Then
to the relationship between individual pairs of field lines
(linking and winding). Field line helicity is intermediate 1 d⃗x ⃗r d⃗y
12 = − ⋅ × d𝜎 d𝜏, (1.1)
between these ideas, as it measures the net linking or wind- 4𝜋 ∮L2 ∮L1 d𝜎 r3 d𝜏
ing of one field line with the total field. Helicity integrals
depend on both the tangling of field lines with each other where ⃗r = x⃗ − y⃗ . This may be calculated by counting
and the topology and geometry of the volume in which the signed crossings (see Fig. 1.1).
field lines reside. A strong rationale for considering topo- For a magnetic field consisting of a finite collection of
logical measures is that field line topology is conserved in closed magnetic flux tubes (a very special case), we can
any ideal evolution of the field (Moffatt, 1969). define an overall invariant

∑ ∑
N N
1.1.1. Definitions of Field Line Helicity for Closed HN = ij Φi Φj , (1.2)
Volumes i=1 j=1

The simplest situation occurs with two closed field where Φi represents the magnetic flux of each tube and ij
lines residing within a volume  where the field lines is the linking number between the tubes. This invariant is
called the magnetic helicity. For a more general magnetic
1
field consisting entirely of closed field lines (still a special
Department of Mathematical Sciences, Durham University, case), we can take N → ∞ in equation (1.2) so that the
Durham, UK
2 sums become integrals. Accounting for the magnetic flux,
Department of Mathematics, University of Exeter, Exeter,
UK
the tangent vectors in equation (1.1) become magnetic

Helicities in Geophysics, Astrophysics, and Beyond, Geophysical Monograph 283, First Edition.
Edited by Kirill Kuzanyan, Nobumitsu Yokoi, Manolis K. Georgoulis, and Rodion Stepanov.
© 2024 American Geophysical Union. Published 2024 by John Wiley & Sons, Inc.
DOI:10.1002/9781119841715.ch01

3
4 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

the equivalent expression for the field line helicity will

simply be
(L) = ⃗ ⋅ dl.
A (1.7)
∮L

This is just the net magnetic flux encircled by L, and

since L is a closed curve, it is manifestly gauge invariant.
We remark that Yahalom (2013) interprets (L) as a
magnetohydrodynamic analog of the Aharonov-Bohm
effect from quantum mechanics: if B ⃗ = ∇𝜒 × ∇𝜂, then
⃗
A = 𝜒∇𝜂 + ∇𝜁 , so that a nonzero H requires jumps in 𝜁
around closed field lines, with such jumps giving (L).
Figure 1.1 For this link (the Whitehead link), the Gauss linking Suppose our volume  is simply connected (e.g., a
number, 12 , is zero. In general, the linking number equals half sphere, not a torus). Note that because L is closed, we
the difference between the number of positive crossings and the can gauge transform A ⃗ →A ⃗ + ∇𝜙 for an arbitrary gauge
number of negative crossings, as seen in any plane projection. function 𝜙 without affecting (L). If we have a multiply
connected volume, a difficulty arises: the wrong choice of
field vectors, and the magnetic helicity may be written A⃗ may imply the existence of magnetic flux in the external
region that threads through a hole, which makes ∮L A ⃗ ⋅ dl
H=−
1 ⃗ x) ⋅ ⃗r × B(⃗y) d3 x d3 y.
B(⃗ (1.3) dependent on the unknown external field. To remedy this,
4𝜋 ∫ ∫ r3 one can restrict the gauge of A ⃗ so that ∮ A⃗ ⋅ dl = 0 for
L
Here the integral is over all space or over a volume any closed curve on the boundary encircling a hole the
containing all field lines. In fact, in this equation, we long way around (see also MacTaggart and Valli, 2019).
do not need to assume that the field lines close upon
themselves – some may ergodically fill a subvolume or
1.1.2. Definitions of Field Line Helicity for Open
twist around a toroidal surface with an irrational winding
Volumes
number. Such fields can be constructed as the limit as
N → ∞ of a sequence of fields consisting of N thin closed We now go to volumes with flux crossing the bound-
flux tubes (Arnol’d and Khesin, 1998). Thus, regardless ary. Suppose a field line forms a loop, with both ends
of whether the field lines close upon themselves, we can on the same boundary (as, for example, in Fig. 1.2b).
approximate any magnetic field as a collection of such An early approach (Antiochos, 1987) involves drawing
tubes. a geodesic between the two endpoints, thus forming a
Returning to the finite set of N flux tubes with fluxes Φi , closed curve for which ∮ A ⃗ ⋅ dl can be measured. This
notice that equation (1.2) can be written as scheme does not integrate to the full helicity, however.
∑
N
∑
N We wish to define field line helicity to be consistent with
HN = i Φi ; i = ij Φj . (1.4) full helicity but also so that gauge transformations of
i=1 j=1 A⃗ do not change our results. There are two principal
approaches to this problem. One consists of defining
The limit as N → ∞ of i is another topological mea-
topological quantities such as winding numbers and
sure, this time defined for every individual magnetic field
then summing to find the helicities. If the boundary
line – this is what we call the field line helicity.
flux does not move, ideal motions of the field lines will
Since the field lines do not cross the boundary of the
not change the windings, so field line helicity will be
volume, we can now use the Biot-Savart formula
conserved. If the boundary flux does move, the topo-
⃗ x) ≡ − 1
⃗ x) = curl−1 B(⃗ ⃗r ⃗ y) d3 y logical structure can cross the boundary, leading to
A(⃗ × B(⃗ (1.5)
∫
4𝜋  r3 changes in helicity measures. This will be discussed in
section 1.2.
to find The second approach is to measure helicity relative to
H= ⃗ ⋅B
A ⃗ d3 x. (1.6) a potential field: i.e., a field with zero current in the vol-
∫
ume. As the potential field is the minimum energy state for
Historically, invariance of the expression in equation (1.6) given boundary conditions, its structure depends only on
was known before the interpretation in terms of Gauss the shape of the boundary and the distribution of flux on
linking number was identified (Moffatt, 1969; Moffatt the boundary (and for multiply connected volumes, spec-
and Ricca, 1992). For a closed line L within the field, ified new fluxes around each hole). For highly symmetric
 INTRODUCTION TO FIELD LINE HELICITY 5

θa line helicity of Li will be

(Li ) = 𝑤(̃xi , x̃ j )Bz (̃xj ) d2 x̃ j . (1.9)

∫
2– 1–
β
x y
1+
α
Let us build up (Li ) one plane at a time – in other
2+
words, find expressions for d(Li )∕dz. For now, we make
θa
the simplifying assumption that Bz > 0 everywhere. There
are two contributions to the z derivative: first (I), hori-
(a) (b) zontal flux in the plane can wrap around the field line Li .
Second (II), Li can move horizontally about other field
lines. Now 𝜃̂ ij = ẑ × r̂ ij is the angular direction at xj about
xi , with ⃗rij = xj − xi . Also, a field line traveling from (r, 𝜃, z)
to (r + dr, 𝜃 + d𝜃, z + dz) satisfies the equation
dr rd𝜃 dz
= = . (1.10)
Br B𝜃 Bz
⃗ j = B(xj ),
Thus for a field line at xj = (x, y, z) (writing B
(c) (d) etc.),
( ) ⃗ j ⋅ ẑ × ⃗rij ⃗j
Figure 1.2 Examples of a pair of magnetic field lines in different d𝜃ij B𝜃j B ⃗rij × B
= = = ẑ ⋅ . (1.11)
domains, including (a) the volume between two parallel planes; dz I rij Bzj r2ij Bzj r2ij Bzj
(b) the volume above a plane; (c) a spherical shell (the volume
between two concentric spheres); and (d) the volume between Next, the field line Li can move horizontally around other
two more general surfaces. field lines. In this case,
( ) ⃗i ⃗ i ⃗rij × Bzj ẑ
d𝜃ij ⃗rij × B B
volumes such as those bounded by planes or spheres, the = −̂z ⋅ = ⋅ 2 . (1.12)
two approaches give identical results. dz II r2ij Bzi Bzi rij Bzj
For more complicated volumes, the shape of the bound-
ary or boundaries can add extra terms that distinguish the Summing the last two equations, the total change in wind-
two approaches, as discussed in section 1.1.2. A definition ing between lines Li and Lj is
of field line helicity can also be based on relative helicity
d𝜃ij B ⃗j
⃗ i ⃗rij × B
(Moraitis et al., 2019), to be discussed in section 1.1.3. = ⋅ 2 . (1.13)
dz Bzi rij Bzj
Volumes between Parallel Planes
Suppose the volume of interest  is the space between Let Sz be the horizontal plane at height z. Suppose we
parallel planes, say z = 0 and z = h, as illustrated in add up contributions from all over the plane: we multiply
Fig. 1.2a. We will first assume that all field lines start at the previous expression by Bzj d2 xj and integrate. Dividing
z = 0 and end at z = h. Then no two field lines link in the by 2𝜋 to make complete turns into integers,
Gauss sense; however, they will twist about each other ⃗j
d(Li ) ⃗i
1 B ⃗rij × B
through some angle 𝛿𝜃ij . The winding number makes this = ⋅ d2 xj . (1.14)
an integer for complete twists: dz 2𝜋 Bzi ∫Sz rij 2

1 This expression for the field line helicity can be

𝑤ij = 𝛿𝜃 . (1.8)
2𝜋 ij expressed very simply using a special vector potential –
For the example in Fig. 1.2a, we have 𝑤ij = (𝜃b − the winding gauge (Prior and Yeates, 2014): for a point on
𝜃a )∕(2𝜋) + 1. field line Li at (x, y, z),
We can now adopt formulae for the helicity and field ⃗j
⃗rij × B
⃗ (x, y, z) = 1
W
line helicity similar to the closed field case, with the A d2 xj . (1.15)
winding number replacing the linking number. We denote 2𝜋 ∫Sz rij 2

horizontal positions by x = (x, y). Also, we distinguish

positions in the bottom x − y plane by x̃ = (x, y, 0). Let We have
Li be the field line with foot point at x̃ i . Also let 𝑤(̃
xi , x̃j ) W W dl
(Li ) = ⃗
A ⋅ dl = ⃗
A ⋅ dz, (1.16)
be the winding number between Li and Lj . Then the field ∫ ∫ dz
6 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

so We assume T, P, Bz , and Jz all vanish at infinity. Then the

⃗ (x, y, z) surface Laplacians have unique solutions,
d(Li ) W B
⃗ ⋅ dl = A
W
=A ⃗ ⋅ i . (1.17) 1 ′ ′
dz dz Bzi P(x, y) = − B (⃗x ) ln |⃗x − x⃗ | d 2 x′ ; (1.21)
2𝜋 ∫Sz z
Volume above a Plane 1 ′ ′
T(x, y) = − J (⃗x ) ln |⃗x − x⃗ | d 2 x′ . (1.22)
In the previous section, we assumed that Bz > 0 every- 2𝜋 ∫Sz z
where. Thus the field lines were braided about each other,
and simple winding numbers sufficed to measure pair- We can now create a vector potential
wise entanglements. If we remove the restriction on Bz , ⃗ = ∇ × P̂z + T ẑ .
A (1.23)
field lines can go up and down. We will also remove the
restriction of an upper plane in the following discussion, Note that the horizontal components of A ⃗ only involve the
although that is not strictly necessary. Such a domain is poloidal function P. Furthermore, the horizontal diver-
illustrated in Fig. 1.2b. gence of A ⃗ vanishes: ∇∥ ⋅ A ⃗ ∥ = 0. One can then employ
If field lines can form loops that go up and down, we equations (1.21) and (1.22) to show that this vector poten-
W
can measure topological structure in several ways. One tial is identical to the winding gauge, A ⃗ = ∇ × P̂z + T ẑ ,
method is to cut the field lines at any local minima or max- provided that the volume considered is infinite in the x and
ima where Bz = 0 and then sum the winding numbers as y directions. We can again employ this vector potential to
before (Berger and Prior, 2006). The winding gauge will calculate field line helicities. The total helicity can also be
still work in this situation, as the formulae for calculating regarded as the net linking of toroidal and poloidal fields
W
A⃗ do not require Bz > 0. (Berger and Hornig, 2018),
A second method, when considering the winding
between two loops, consists of examining the angles of H=2 ∇ × (T ẑ ) ⋅ ∇ × (P̂z) d3 x. (1.24)
∫
the quadrilateral formed by the four foot points of the
loops (Berger, 1986; Demoulin et al., 2006). Consider the
Volumes Bounded by a Sphere
upper half space {z > 0}, shown in Fig. 1.2b. The foot
Spherical boundaries – such as in Fig. 1.2c – can, for
points of loops 1 and 2 are labeled 1+ , 1− , 2+ , and 2− , the most part, be treated the same way as planar bound-
where Bz > 0 at 1+ and 2+ . If the loops cross as seen from aries, with the radial unit vector r̂ replacing the vertical
above, we assume that loop 1 is the upper loop. Consider vector ẑ . In essence, winding angles become azimuthal
the quadrilateral 1+ 2+ 1− 2− . Let 𝛼 and 𝛽 be the angles at angles. Suppose we consider three points A, B, and C on a
vertices 1+ and 1− , respectively. Then (from considering sphere. Rotate the spherical coordinates so that B is at the
the net change of helicity found when bringing the two North pole and A is at the azimuthal coordinate 𝜙 = 0.
loops in from infinity) Berger (1984) showed that Then the angle ∠ABC equals the azimuthal coordinate
1 of C.
𝑤12 = (𝛼 + 𝛽). (1.18) The poloidal and toroidal flux functions are (Kimura
2𝜋
and Okamoto, 1987)
Poloidal-Toroidal Representation 1 ′
P(𝜃, 𝜙) = B (⃗x ) ln(1 − cos 𝜉) d 2 x′ ; (1.25)
The winding gauge employed in the previous sections is 4𝜋 ∫S r
equivalent to the vector potential found when employing
1 ′
a poloidal-toroidal representation for the field (Moffatt, T(𝜃, 𝜙) = Jr (⃗x ) ln(1 − cos 𝜉) d 2 x′ , (1.26)
∫
4𝜋 S
1978; Berger and Hornig, 2018). Because the mag-
netic field B ⃗ is a three-vector subject to one condition where
(∇ ⋅ B ⃗ = 0), we can often express it in terms of two scalar
cos 𝜉 = cos 𝜃 cos 𝜃 ′ + sin 𝜃 sin 𝜃 ′ cos(𝜙 − 𝜙′ ) (1.27)
functions. Here we let P and T be the poloidal and toroidal
functions. Let the horizontal surface Laplacian be Δ∥ = is the spherical distance between (𝜃, 𝜙) and (𝜃 ′ , 𝜙′ ). With
𝜕 2 ∕𝜕x2 + 𝜕 2 ∕𝜕y2 . In any plane z = const., P is determined these functions,
by the vertical flux,
⃗ = ∇P × r̂ + T r̂ ;
A ⃗ = ∇ × (∇P × r̂ ) + ∇T × r̂ .
B
Δ∥ P = −Bz , (1.19) (1.28)
Some care must be taken in spherical geometries. One
while T is determined by the vertical current, must ensure that no magnetic monopoles are hiding inside
the sphere – i.e., the net flux through the sphere must van-
⃗ z.
Δ∥ T = −Jz = −(∇ × B) (1.20) ish (similarly for the net current).
 INTRODUCTION TO FIELD LINE HELICITY 7

Also, in the special case of a spherical shell geometry where ′

where flux enters at the inner sphere and exits at the outer ⃗ P ⋅ n̂ .
∇ × ∇ × As ⋅ n̂ = −∇ × ∇ × A (1.34)
sphere (as in Fig. 1.2c), the separation of helicity into self
helicity and mutual helicity can be ambiguous (Campbell Putting it all together, the generalized winding gauge
and Berger, 2014). Self helicity measures the twist and becomes
W
writhe of individual tubes, while mutual helicity measures A⃗ = −1 Bn + (T + TS )∇𝑤. (1.35)
linking and intertwining between tubes. This will not
affect the field line helicity, however. 1.1.3. Relative Field Line Helicity
More General Volumes In the previous sections, helicity and field line helicity
For volumes bounded by planes or spheres, there are have been defined in terms of geometrical quantities such
natural definitions of angle and winding angle depending as winding and linking. The helicity of open fields in a
only on the Euclidean metric. The winding gauge, or the volume  can also be defined in terms of how much the
poloidal-toroidal vector potential, captures this natural magnetic structure within  contributes to the helicity of
definition. We could also consider a volume with bound- all space (Berger and Field, 1984). Let the field inside 
aries consisting of a bottom plane and side boundaries, be called B ⃗  . Also consider a potential (zero current) field
where flux only crosses the bottom plane. Such a volume ⃗ ⃗  ⋅ n̂ |S =
P inside  with the same boundary conditions B
could represent a closed active region. We could choose
⃗
P ⋅ n̂ |S . A potential field minimizes the magnetic energy
winding angles without regard to the side boundaries,
using the same winding gauge. An alternative will be in the volume subject to the constraint of the given bound-
described later when using relative measures of helicity. ary flux distribution. Hence potential fields can be said to
Next, consider a volume bounded by one or two simply have the minimum structure (more accurately, their struc-
connected surfaces that lack the symmetry of planes ture only depends on the boundary conditions, so they
or spheres. An example is shown in Fig. 1.2d. Here, add zero additional structure to the field).
definitions of angle are less obvious. However, we can Here we compare the total helicity of all space with
the helicity that would be obtained if, inside , B ⃗  were
still employ a generalization of the poloidal-toroidal
representation (Berger and Hornig, 2018) to define the replaced by P ⃗  . All integrals are gauge invariant, so the
winding gauge. difference between the two helicities will also be gauge
Let our volume  be sliced into nested surfaces. Employ invariant. Let space external to  be  ′ . Then the relative
coordinates (u, 𝑣, 𝑤), where 𝑤 = const. labels one of the helicity HR () is
nested surfaces. The toroidal field (which is responsible for
the current perpendicular to surfaces of constant 𝑤) can HR () = H(B , B ′ ) − H(P , B ′ ). (1.36)
be written in terms of a toroidal function T as before: It is important to note that the calculation of HR () does
B ⃗T;
⃗T = ∇ × A ⃗ T = T∇𝑤.
A (1.29) not require knowledge of the external field B ′ (Berger and
Field, 1984; Finn and Antonsen, 1985).
However, the poloidal field is more complicated. Let the For symmetric volumes (i.e., boundaries consisting
normal component of the curl be given by the operator , of a plane or parallel planes or one or two concentric
⃗ = n̂ ⋅ ∇ × V
⃗. spheres), the topological and relative definitions give
V (1.30)
the same answers (Berger and Field, 1984; Berger and
This operator has an inverse −1 . To make the inverse Hornig, 2018). For more complicated geometries, they
unique, we require that the inverse normal curl gives a may differ. For example, suppose the volume  consists
divergence-free vector field parallel to the surface. Thus of a helical tube rising between parallel planes. Suppose
we place a potential field both inside and outside . Then
−1 f ⋅ n̂ = 0; ∇ ⋅ −1 f = 0. (1.31)
the helicity of all space will be nonzero because of the
Let Bn be the normal magnetic field. Ordinarily, we helical structure. Now add some axial current inside 
would write the poloidal field as so the field has an extra twist. Then the relative helicity
′ ′ inside  will only measure this extra twist because the
⃗ P = −1 Bn ;
A ⃗P = ∇ × A
B ⃗ P. (1.32) helicity due to the axis writhe is subtracted with the
However, without spherical or planar symmetry, this mag- potential term.
netic field may have nonzero normal current Jn – but the The relative helicity of a potential field is, by defini-
toroidal field has already taken care of Jn . Thus (Berger tion, zero. Suppose again that the boundary is planar
and Hornig, 2018) we must add a shape term or spherical. Then the topological helicity (summing
winding numbers as in previous sections) will also be
⃗ S = TS ∇𝑤,
A (1.33) zero: the potential field is purely poloidal, so there is
8 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

no linkage between toroidal and poloidal components. 1.2. IDEAL EVOLUTION

However, for a potential field, the field line helicities of
individual field lines may not be zero, even if they sum Just like the total magnetic helicity, the field line helic-
to zero (Yeates, 2020). These potential field line helicities ities within a given domain  may change in two ways:
reflect irregularities and lack of mirror symmetry in the by (i) boundary motions or (ii) non-ideal processes in
distribution of boundary flux (Bourdin and Brandenburg, the interior. In this section, we show how to derive an
2018). Thus the field line helicity of the potential field can evolution equation for case (i), while case (ii) is deferred to
help characterize magnetic distributions, for example, in section 1.3. We will give some examples where these have
the solar photosphere. been applied, although much remains to be explored.
This raises a question, however: if there are currents As we have seen,  is determined entirely by the choice
inside a volume, how do they change the field line helici- of tangential components A ⃗ × n̂ on the boundary 𝜕,
ties from their minimal (potential field) values? We need along with an extra condition when the boundary has
to subtract the potential field line helicities to remove two disconnected components. Typically, we think of
the contributions from the boundary flux distribution. setting A ⃗ × n̂ = A
⃗ P × n̂ for some reference vector potential
Thus, we can adopt the relative helicity viewpoint for A⃗ P . Thus when B ⃗ ⋅ n̂ on 𝜕 is evolving, we cannot keep
field line helicity as well (Yeates and Page, 2018; Moraitis a fixed reference but must evolve A ⃗ P in some suitable
et al., 2019; Moraitis et al., 2021). First, to avoid gauge way over time that preserves the physical interpretation.
ambiguity, we employ the same gauge conditions as apply For example, Yeates (2020) chooses the so-called minimal
in the poloidal-toroidal formulation. Let the potential gauge.
field be P,⃗ with vector potential A ⃗ P . In particular, for If B⃗ evolves ideally in some volume  so that
the components of the vector potentials parallel to the
⃗ ( )
boundary, we have 𝜕B ⃗ ,
= ∇ × u⃗ × B (1.42)
𝜕t
⃗∥ = ∇ ⋅ A
∇⋅A ⃗ P∥ = 0; (1.37)
then it is well known that magnetic field lines retain their
⃗∥ = A
A ⃗ P∥ . (1.38) identity: if two fluid elements lie on the same magnetic
field line at some initial time, they will continue to do so at
It can be shown (Yeates and Page, 2018) that this choice all later times. Moreover, by Alfvén’s theorem, field lines
of gauge for A⃗ P minimizes the boundary integral ∮ |A ⃗P × cannot change their mutual linkage. It follows immedi-
S
n̂ | d x. Next, let 𝛼+ be the positive endpoint of a field line.
2 2
ately that (L) is preserved for every closed field line L in
The field lines through 𝛼+ for B ⃗ and P ⃗ will generally have . Moreover, any functional of  will also be preserved,
different trajectories and connect to different exit points including not only the total helicity
𝛼− and 𝛼P− . We can now take the difference in field line
helicities H= ⃗ ⋅B
A ⃗ d3 x = (L)dΦ (1.43)
∫ ∫L
𝛼− 𝛼P−
+ (𝛼+ ) = ⃗ ⋅ dl −
A ⃗ ⋅ dl.
A (1.39) but also the helicity of any subregion composed of closed
∫𝛼+ ∫𝛼 +
field lines, as well as the unsigned helicity defined by
We could also do this for the negative endpoints. Starting
at the same negative endpoint 𝛼− , we could track the field H= |(L)|dΦ. (1.44)
∫L
lines backward to either 𝛼+ or 𝛼P+ . In this case, we could
write To inject or remove field line helicity by continuous
𝛼− 𝛼P− ideal evolution therefore requires open field lines where
− (𝛼− ) = ⃗ ⋅ dl −
A ⃗ ⋅ dl.
A (1.40) Bn ≡ B ⃗ ⋅ n̂ || ≠ 0 along with u⃗ ≠ 0⃗ on the boundary S, as
∫𝛼+ ∫𝛼P+ |S
illustrated in Fig. 1.3(b). In this case, the fundamental
Finally, we can take the average of the two (for the end- issue of gauge choice is important. A physically mean-
⃗ to find
points 𝛼+ and 𝛼− of the true field line B) ingful approach is to fix something like the winding or
poloidal-toroidal gauge for A ⃗ at all times. Consider an
1 + open field line Lt , where the subscript t denotes that this
0 = ( (𝛼+ ) + − (𝛼− )). (1.41)
2 line moves with the fluid. To derive the evolution equation
for (Lt ), note first that uncurling equation (1.42) gives
This is the measure of relative field line helicity adopted
by Moraitis et al. (2019), although those authors use a ⃗
𝜕A
slightly different gauge in their computations. ⃗ + ∇𝜙
= u⃗ × B (1.45)
𝜕t
 INTRODUCTION TO FIELD LINE HELICITY 9

x1 (t)

Lt

x0 (t)

Figure 1.3 Examples of closed (left) and open (right) field lines in a domain V with connected boundary 𝜕. Foot
⃗ ⋅ n̂ || < 0, and x (t), where B
points of the open field line Lt are denoted x0 (t), where B ⃗ ⋅ n̂ || > 0.
|S 1 |S

for some scalar function 𝜙(⃗x, t) that depends on the cho- during an ideal evolution, their field line helicities (Lt )
sen gauge of A ⃗ and will generally be nonzero for the depend on the chosen gauge of A ⃗ on the boundary over
poloidal-toroidal gauge. ⃗
time, since this changes both A ⋅ u⃗ and 𝜙, the latter being
Now let t,𝜖 be an infinitesimal flux tube surrounding Lt defined for a given choice of A ⃗ through equation (1.45).
and moving with the fluid. Let Φ(t,𝜖 ) and h(t,𝜖 ) = ∫ A ⃗⋅ For meaningful results, this gauge should be chosen
t,𝜖

⃗ d x be the magnetic flux and helicity of this flux tube.

3 objectively – such as the poloidal-toroidal gauge – rather
B
than for computational convenience. Indeed, it is always
Then the rate of change of field line helicity is
possible to “cancel” the effect of an ideal evolution by
d 𝜕 h(t,𝜖 ) choosing 𝜙 = −A ⃗ ⋅ u⃗ in equation (1.45) so that  remains
(Lt ) = lim (1.46)
dt 𝜕t 𝜖→0 Φ( t,𝜖 )
invariant for all field lines, but this is a highly artificial
choice; in effect, it corresponds to measuring wind-
1 dh(t,𝜖 ) ing with respect to a frame moving with the boundary
= lim . (1.47)
𝜖→0 Φ(t,𝜖 ) dt motions. Of course, if there are no boundary motions,
By the Reynolds transport theorem, since t,𝜖 is a material  will be invariant in time for all field lines provided
volume, we have that 𝜙 on the boundary is not varied. In particular, if
the boundary is a connected surface, it is sufficient to fix
d 𝜕 ⃗ ⃗ 3 ⃗ ⋅ B)⃗
⃗ u ⋅ n̂ d 2 x, A⃗ × n̂ . In cases like the volume between two spheres, one
h(t,𝜖 ) = (A ⋅ B)d x + (A
dt ∫t,𝜖 𝜕t ∮St,𝜖 must also fix the integral of A ⃗ along a line between the two
(1.48) boundaries.
When there are boundary motions, the winding gauge
Which, using equation (1.45), reduces to
makes it possible to express the evolution of  in terms of
( ) the mean angular motions of field line endpoints around
d ⃗ ⋅ u⃗ B⃗ ⋅ n̂ d 2 x.
h(t,𝜖 ) = 𝜙+A (1.49)
dt ∮St,𝜖 one another (Berger, 1988; MacTaggart and Prior, 2021).
In the restricted case where there are boundary motions
This is just the standard expression for the evolution of but they preserve the boundary distribution B ⋅ n̂ |S and,
magnetic helicity in a material volume (e.g., Moffatt and in addition, the magnetic field has a “simple topology”
Dormy (2019) p. 61). Since t,𝜖 is a magnetic flux tube, B ⃗⋅ (meaning the mapping from positive to negative endpoints
n̂ is nonzero only on the two portions of the tube bound- is continuous), Aly (2018) derived the alternative formula
ary St,𝜖 that coincide with the boundary of  – in other [ ]x1 (t)
words, the ends of the tube. So substituting into equation d 𝜕𝜁
(Lt ) = U −𝜁 . (1.51)
(1.47) and taking the limit, noting that B ⃗ ⋅ n̂ → Φ(t,𝜖 ) at dt 𝜕U x0 (t)
both ends of the tube, gives
[ ]x1 (t) Here the boundary motions have been written as u⃗ = (̂n ×
d
(Lt ) = 𝜙 + A ⃗ ⋅ u⃗ , (1.50) ∇𝜁 )∕B⃗ ⋅ n̂ , and the gauge of A
⃗ has been chosen in a partic-
dt x0 (t)
ular way. Specifically, the parallel components of A ⃗ on the
where x0 (t) and x1 (t) denote the endpoints of Lt on the boundary are set by equation (1.38), but with A ⃗ P = U∇V ,
boundary of . where U and V are Euler potentials for the reference field
The expression in equation (1.50) shows that even ⃗ = ∇U × ∇V . Formula (1.51) may be shown to be equiv-
P
though field lines Lt can be uniquely tracked in time alent to equation (1.50) for this gauge choice.
10 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

1.2.1. Simple Examples field line helicity:

The simplest examples are obtained by considering d 2

(r) = (r2 + 1)e−r . (1.54)
a magnetic field between two planes z = 0 and z = 1 dt
on which B ⃗ ⋅ n̂ || = Bz = 1 and applying axisymmetric This is shown by the solid lines in the right-hand panels
|S
boundary motions on the upper plane. In this situation, of Fig. 1.4. One obtains the same  pattern in the topo-
the potential reference field is P ⃗ = ẑ at all times, with logically equivalent situation of a toroidal flux ring local-
poloidal-toroidal vector potential A ⃗ P = (r∕2)𝜃.
̂ We fix the ized in z, as shown by Yeates and Hornig (2014). Indeed,
⃗ ⃗
gauge of  by setting A × ẑ = AP × ẑ on z = 0 and z = 1 the initial condition in Fig. 1.8 (later) is produced by the
and, by the additional condition (because the boundary superposition of six such flux rings, offset so the field lines
has two disconnected components z = 0 and z = 1), follow a tangled pattern.
that ∫L A⃗ ⋅ dl = ∫ A ⃗ P ⋅ dl = 0 for the vertical line L at A final analytical example demonstrates how field line
L
r = 0. This gives us  in the winding, or equivalently helicity may be injected into a magnetic field defined on
poloidal-toroidal gauge (the two are equivalent in this the half-space z > 0 through shearing motions on the
volume). lower boundary z = 0 instead of twisting. This models
Suppose first that we apply a rigid rotation of the upper the shearing of an arcade of coronal magnetic loops in the
boundary with respect to the lower one, given by u⃗ = rz𝜃. ̂ Sun’s atmosphere. We suppose that the arcade is initially
⃗ = u𝜃 Bz r̂ = rẑr, so maintaining the gauge current-free and (for simplicity) invariant in x, with B ⃗=
In this case, u⃗ × B
⃗ = ẑy − ŷz, the solar surface being the plane z = 0 and
P
condition A ⃗ × ẑ = A ⃗ P × ẑ requires, according to equation
the volume V with z > 0 representing the corona. A suit-
(1.45), that
able poloidal-toroidal reference vector potential is A ⃗P =
𝜕𝜙 r2 z
2 2
(y ∕2 + z ∕2)x.̂ Suppose we apply the constant shearing
= −rz ⟹ 𝜙=− . (1.52) velocity u⃗ = ye−y x,
2
̂ localized near the polarity inversion
𝜕r 2
line y = 0. Here u⃗ × B ⃗ = −ux Bz ŷ + ux By ẑ , so requiring
Thus for a field line at radius r, equation (1.50) gives ⃗ = n̂ × A
that n̂ × A ⃗ P on the boundary z = 0 implies that
[ 2 ( ) ]z=1
d rz r 𝜕𝜙 2
(r) = − + rz = 0. (1.53) = ux Bz = −y2 e−y ⟹
dt 2 2 𝜕y
z=0 (√ )
1 2

This shows that a rigid rotation generates no field line 𝜙=− 𝜋 erf(y) − 2ye−y . (1.55)
4
helicity (as noted by Yeates et al. (2021)). In effect, all the
Substituting into equation (1.50) and evaluating on the
field lines rotate together so that none of them acquires
field line rooted at y > 0 then gives
any twist with respect to the others. This example is
shown by the dashed lines in the right-hand panels of √
d 3 −y2
𝜋
Fig. 1.4. (y) = (y + y )e − erf(y). (1.56)
For an example with nontrivial injection of field line dt 2
helicity, we can apply a localized rotation u⃗ = 2re−r 𝜙̂ for
2
This profile is shown in Fig. 1.5. The first term vanishes
which a similar computation leads to a localized patch of on the unsheared field lines at large y. But the second

3
→
u z=1
2
uθ

1
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0

1.0
dA / dt

0.5
z=0
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
r

Figure 1.4 Simple example where the boundary has two parallel components, one of which is rotated with respect
to the other. The graphs show the azimuthal velocity (top) and rate of generation of field line helicity (bottom) for
the two examples in the text (dashed and solid lines).
 INTRODUCTION TO FIELD LINE HELICITY 11

2
0.25

0.00

ux
1 –0.25
0.4
0.3
0.2 z −4 −2 0 2 4
0

y
0.1
1
0.0
2
1 –1

dA / dt
0 0
−0.4
–1 y
−0.2
0.0
0.2 –2
x 0.4 –2
–1
−0.4 −0.2 0.0 0.2 0.4 −4 −2 0 2 4
x y

Figure 1.5 Simple example of an initial potential “arcade” being sheared by a flow on the lower boundary.
Although they are largely unsheared, the overlying field lines acquire negative field line helicity, as may be seen
from their negative crossings with the sheared field lines beneath.

term does not. This demonstrates the nonlocal nature line endpoint is on the outer boundary), injected field line
of helicity: the field lines in the overlying arcade gain helicity is lost at a steady rate through the outer boundary
a field line helicity because the core of the arcade is by relaxation of the field line. By contrast, closed field
sheared. Effectively, a magnetic flux is passing through lines, with both endpoints on the solar surface, can store
them. Indeed, this was precisely the way Antiochos (1987) field line helicity – as in the simple arcade example in
proposed to define field line helicity, as mentioned earlier. Fig. 1.5.
Notice here that the sign of the erf(y) term is negative, This storage of helicity on closed field lines is important
consistent with Stokes’s theorem and the direction of the because eventually, these form twisted flux ropes that
sheared field relative to the orientation of the overlying lose equilibrium and erupt, leading to bursts of helicity
arcade. One could also infer this sign by looking at the flux output and believed to explain the origin of coronal
arcade from above and observing a negative crossing mass ejections. Lowder and Yeates (2017) studied these
(Fig. 1.5). eruptions in the magneto-frictional model and used field
line helicity as a diagnostic tool to define flux ropes in
1.2.2. Application to the Global Solar Corona the first place. The overall magnetic flux and helicity
content of these structures was found to be compara-
To a first approximation, the magnetic field in the Sun’s ble to that estimated in observations of interplanetary
atmosphere evolves ideally in response to the emergence magnetic clouds. Recently, Bhowmik and Yeates (2021)
of new magnetic active regions from inside the Sun, have used field line helicity to show that episodic losses
decay of these strong magnetic fields due to convective of helicity in the magneto-frictional model come not only
shredding, and transport of the resulting magnetic flux by from the eruption of flux ropes formed along polarity
large-scale motions such as the Sun’s differential rotation. inversion lines in the low corona but also from a second
During these processes, magnetic helicity builds up in type of eruption generated in the overlying streamers.
the corona and is ejected into the heliosphere. Field line Similar eruptions are known in magnetohydrodynamic
helicity offers the exciting prospect of a localized measure (MHD) simulations (Linker and Mikic, 1995) and have
for studying where this helicity is located within the been suggested as a possible explanation for so-called
corona. stealth coronal mass ejections (CMEs) that lack an
Yeates and Hornig (2016) used a global magneto- obvious low-coronal source (Lynch et al., 2016). Many
frictional model in a spherical shell to study how field unanswered questions remain, not least the role of active
line helicity evolves in the corona in response to the evo- regions in the global helicity balance or the possibility of
lution of the solar surface magnetic field (Fig. 1.6). In the long-term storage of their helicity in the corona. Field
simulation shown, active region emergence is neglected line helicity will greatly facilitate these investigations.
so that the dominant injection of helicity is shearing by
differential rotation of field line foot points on the solar 1.2.3. Application to Solar Active Regions
surface. The behavior of field line helicity is found to
be different on open and closed magnetic field lines. On Field line helicity also offers the possibility to identify
open field lines (meaning in this context that one field locations of helicity storage on a smaller scale, within
12 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

(a) (b) (c)

Figure 1.6 Evolution of field line helicity in a magneto-frictional model of the solar corona as an initial potential
field (a) is sheared by the Sun’s differential rotation for (b) 25 days and (c) 50 days. Field lines are colored by
their field line helicity in poloidal-toroidal gauge, with red for positive and blue for negative (for details of the
simulation, see (Yeates and Hornig, 2016)).

individual active regions, provided three-dimensional corresponding to resistive MHD. It is well-known that
magnetic field models are available. Even in a potential the total helicity is no longer conserved but can be dissi-
field model, a simple bipolar active region can have pated within the volume when N ⃗ ⋅B⃗ ≠ 0. In this section,
nonzero field line helicity if its field lines are linked with we show how – for magnetic fields of simple topology
the overlying background field (Yeates, 2020). But much without null points – equations (1.50) and (1.51) have been
larger values of field line helicity are expected in more generalized to this non-ideal case. For magnetic fields of
realistic current-carrying models of active regions. This more complex topology – for example, the solar coronal
has been confirmed by Moraitis et al. (2019) both in ide- examples in sections 1.2.2 and 1.2.3 – evolution equations
alized MHD models and in extrapolations. For example, have not yet been derived explicitly. Indeed, differential
Moraitis et al. (2021) have computed field line helicity in equations may not be appropriate since the distribution of
nonlinear force-free extrapolations of a real active region  is discontinuous across magnetic separatrices between
NOAA 11158 (Fig. 1.7). different connectivity domains. Reconnection can trans-
Figure 1.7 shows a highly sheared magnetic field in port  across these separatrices, but its evolution in such
the core of the active region, which has positive field line a situation remains to be studied in detail.
helicity. The authors found that during a flare, the region There is an important caveat to what follows: field
lost 25% of its relative magnetic helicity. Comparing the line helicity can hold physical significance in a non-ideal
field line helicity between the two extrapolations shown evolution only if the magnetic field lines themselves
at 01:11 UT and 01:59 UT reveals that the decrease in retain sufficient identity over time for their topology to
helicity took place within the same region (the green play a physical role. In practice, this means the mag-
box) where emission from a large X-class solar flare was netic Reynolds number (Rm) must be sufficiently large
observed in extreme ultraviolet (EUV). This supports the or, equivalently, the (effective) resistivity must be suffi-
idea that a decrease in helicity was indeed associated with ciently small. There is no precise threshold for this, but it is
the flare. It is worth noting that Moraitis et al. (2021) already accessible for the parameters achievable in numer-
uses relative field line helicity (section 1.1.3). However, ical simulations, as will be illustrated in section 1.3.2.
they show that using the ordinary field line helicity (in
poloidal-toroidal gauge) leads to the same qualitative
1.3.1. Evolution Equation for Non-Null Magnetic
conclusions in this example, albeit with lower values.
Fields

1.3. NON-IDEAL EVOLUTION The trick for generalizing equation (1.50) or equation
(1.51) to non-ideal evolution in the non-null case is
Suppose equation (1.42) is generalized to to decompose N ⃗ into parallel and perpendicular parts
( ) (Yeates and Hornig, 2011), writing
⃗
𝜕B ⃗ − ∇ × N,
= ∇ × u⃗ × B ⃗ (1.57)
𝜕t ⃗ = −𝑣⃗ × B
N ⃗ + ∇𝜓. (1.58)
⃗ =E
where N ⃗ + u⃗ × B
⃗ represents some non-ideal term When the magnetic field has a simple topology, this
in Ohm’s law. A common example would be N ⃗ = 𝜂 J⃗ decomposition exists globally, with 𝑣⃗ and 𝜓 continuous
 INTRODUCTION TO FIELD LINE HELICITY 13

AIA 1600 å
2000.

1000.
120
100

Bz
0.

–1000.
80

y (Mm)
–2000.

60

2.5 e + 21
40

Field line helicity

2e + 21
20
1.5e + 21

1e + 21
0
5e + 20 0 50 100 150 200
1.0e + 20
x (Mm)
(a) (b)
15 Feb 2011, 01:11UT 15 Feb 2011, 01:59UT h (01:59) – h0(01:11)
0

100 100 100

y (Mm)

y (Mm)

y (Mm)
80 80 80

60 60 60

40 40 40

60 80 100 120 140 160 60 80 100 120 140 160 60 80 100 120 140 160
x (Mm) x (Mm) x (Mm)
(c) (d) (e)

Figure 1.7 Field line helicity in a nonlinear force-free model of active region NOAA 11158, from Moraitis et al.
(2021). Panel (a) shows the field lines colored by (relative) field line helicity in the extrapolation at 01:11 UT, while
panel (b) shows an image of EUV emission from SDO/AIA during the X-class flare at 01:47 UT. Panels (c) and
(d) show (relative) field line helicity before and after the flare (blue/red), while panel (e) shows their difference.
Credit: Moraitis et al., 2021/The European Southern Observatory (ESO).

throughout V , although they are not unique (as we shall Similarly, Aly (2018) showed that equation (1.51) becomes
discuss shortly). Substituting this decomposition into [ ]x1 (t)
equation (1.57) shows that in a non-null magnetic field, d 𝜕(𝜁 + 𝜓)
(Lt ) = U − (𝜁 + 𝜓) . (1.62)
we can write dt 𝜕U x0 (t)
⃗ ( )
𝜕B ⃗ , where 𝑤
=∇× 𝑤 ⃗ ×B ⃗ = u⃗ + 𝑣,
⃗ (1.59) x (t)
Notice that [𝜓]x1 (t) = ∫L E ⃗ ⋅ dl, which is precisely the
𝜕t 0 t

which shows that the magnetic field is still frozen, but into parallel electric field used to define the reconnection rate
the field line (transport) velocity 𝑤 ⃗ rather than the plasma in the theory of general magnetic reconnection (Schindler
⃗ the et al., 1988). It represents the change in (Lt ) due to a
velocity u⃗ . Thus, in a non-ideal evolution where 𝑣⃗ ≠ 0,
(nonlocal) change in the magnetic flux linked with Lt .
field lines slip at some velocity 𝑣⃗ through the plasma (New-
The term A ⃗ ⋅𝑤 ⃗ is harder to interpret in general because it
comb, 1958; Priest and Forbes, 1992; Aulanier et al., 2006;
depends on the choices of A ⃗ and 𝑤.
⃗ With the gauge con-
Schindler, 2010). Uncurling, we have
dition n̂ × A⃗ = n̂ × A ⃗ P on the boundary, the [A ⃗ P ⋅ 𝑤] x (t)
⃗ x1 (t)
⃗
𝜕A 0
⃗ + ∇(𝜙 − 𝜓).
⃗ ×B
=𝑤 (1.60) term effectively represents “work done” by motion of the
𝜕t field line Lt with respect to the reference field (Russell
Thus, if we identify the same field line Lt at different times et al., 2015).
by the fact that it is frozen into the flow of 𝑤,⃗ an argument For a given non-ideal evolution of B, ⃗ the field line
similar to section 1.2 shows that velocity 𝑤 ⃗ is not uniquely defined, and consequently, the
[ ]x1 (t) identification of the field line Lt over time is not unique.
d ⃗ ⋅𝑤
(Lt ) = 𝜙 + A ⃗ −𝜓 . (1.61) While the component of 𝑤 ⃗ is arbitrary – as
⃗ parallel to B
dt x0 (t)
14 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

is clear from equation (1.58) – it is the non-uniqueness of achieved by choosing 𝜓 = 0 throughout the region of the
the perpendicular component 𝑤 ⃗ ⟂ that changes the identi- boundary where B ⃗ ⋅ n̂ |S < 0. At the opposite endpoints
x (t)
fication of Lt . To see that this component is non-unique, x1 (t), the values of 𝜓 will then be fixed by [𝜓]x1 (t) , leading
0
note that equation (1.58) implies
to 𝑤⃗ ⟂ (x1 ) ≠ 0⃗ in general so that these endpoints will move
⃗ × (∇𝜓 − E)
B ⃗ in time in a non-ideal evolution. We could equally well fix
⃗ ⟂ = 𝑣⃗⟂ =
𝑤 . (1.63) ⃗ ⋅ n̂ |S > 0, thus defining the field
B 2 𝜓 = 0 on the region with B
On each field line, we can specify an initial value of 𝜓. This lines by fixed x1 positions so that x0 (t) varies over time.
x (t)
⃗ ⟂. If u⃗ is nonzero on the boundary but N ⃗ remains zero
does not change [𝜓]x1 (t) but does change 𝑤
0
One situation where a natural choice of 𝑤 ⃗ ⟂ arises is there (or can be neglected), then it is possible to subtract
the case when the field-line endpoints are line-tied on the the ideal term [A⃗ ⋅ u⃗ ]x1 (t) from equation (1.61) and isolate
x0 (t)
boundary, meaning u⃗ = N ⃗ =E ⃗ = 0⃗ there (Russell et al., the change in field line helicity coming from non-ideal evo-
2015). In that case, we can identify Lt over time by fixing lution. An analogous calculation was implemented for the
one endpoint – say, 𝑤 ⃗ ⟂ (x0 ) = 0⃗ – for all field lines. This is magnetic winding measure by Gekelman et al. (2020) in

t=0 t = 100 t = 400

–24

–12

z 0

12

24
3

y 0

–3
–4 0 4
x
(a) (b) (c)

3 3 3 10
t=0 t = 100 10 t = 400
2 10 2 2
5 5 5
1 1 1
0 0 0 0 0 0
y

–1 –5 –1 –5 –1
–5
–2 –10 –2 –2
–10 –10
–3 –3 –3
–4 –2 0 2 4 –4 –2 0 2 4 –4 –2 0 2 4
x x x
(d) (e) (f)

Figure 1.8 Evolution of field line helicity during line-tied resistive relaxation of a braided magnetic field (see Yeates
et al. (2021) for details of the simulation). Panels a–c show magnetic field lines and isosurfaces of current density
(a) in the initial condition, (b) during the turbulent relaxation, and (c) in the relaxed state. Panels d–f show cross
sections of field line helicity on the z = −24 boundary at the same time. Chen et al. 2021/Cambridge University
Press/Licensed under CC BY 4.0.
 INTRODUCTION TO FIELD LINE HELICITY 15

data from a laboratory experiment of interacting magnetic term 𝑤 ⃗ ⋅ ∇ is effectively a product of two gradients of
flux ropes. field-line integrated quantities, so one might expect this to
be the largest. It is therefore no surprise that the dominant
1.3.2. Application to Turbulent Magnetic Relaxation behavior observed is a rearrangement of the  pattern by
advection.
The non-ideal evolution of  has been explored only in Chen et al. (2021) took this idea and developed a
the context of braided magnetic fields, where all field lines variational model for turbulent magnetic relaxation that
connect between two planar boundaries z = 0 and z = 1 predicts the relaxed state to have the “simplest” pattern
at which u⃗ = N ⃗ The most significant finding to date
⃗ = 0. of field line helicity achievable by pure advection. This
is that when the magnetic field line mapping from z = 0 to already predicts the two oppositely twisted magnetic
z = 1 is complex with sharp gradients, the evolution of  “flux tubes” in Fig. 1.8(f). However, closer inspection of
for high Rm is dominated by redistribution between field the numerical simulations by Yeates et al. (2021) shows
lines, rather than dissipation (Russell et al., 2015; Yeates that the other terms in equation (1.66) play a role in
et al., 2021). establishing the substructure of the final state.
To see this, consider the evolution of  on the field line
traced from a fixed point x0 on the lower boundary z = 0. REFERENCES
Russell et al. (2015) made the natural choice 𝑤 ⃗ = 0⃗ on z =
0, in which case equation (1.61) reduces to Aly, J. J. (2018). New formulae for magnetic relative helicity
and field line helicity. Fluid Dynamics Research, 50(1), 011408.
𝜕(x0 )
⃗ 1 (t)) ⋅ 𝑤(x
= A(x ⃗ 1 (t)) − 𝜓(x1 (t)), (1.64) https://doi.org/10.1088/1873-7005/aa737a
𝜕t Antiochos, S. K. (1987). The topology of force-free magnetic
fields and its implications for coronal activity. The Astrophys-
where Lt is the field line defined by tracing from the
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fixed point x0 at subsequent times. They argued that Arnold, V. I., & Khesin, B. A. (1998). Topological magnetohy-
the A⃗ ⋅𝑤⃗ dominates the 𝜓 term in a magnetic field with drodynamics. Springer.
complex mapping, because |𝑤| ⃗ scales like |∇𝜓|∕B from Aulanier, G., Pariat, E., Démoulin, P., & DeVore, C. R. (2006).
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quantity, and since the field line mapping has sharp Physics, 238(2), 347–376. https://doi.org/10.1007/s11207-006-
gradients, we would expect |A ⃗ ⋅ 𝑤|
⃗ ≫ |𝜓| for high Rm. 0230-2
Berger, M. A. (1984). Rigorous new limits on magnetic helic-
This was demonstrated originally in kinematic examples
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(see also Yeates (2019)) and has recently been confirmed
cal Fluid Dynamics, 30(1–2), 79–104. https://doi.org/10.1080/
by direct calculation of the terms in equation (1.64) in 03091928408210078
resistive MHD simulations (Yeates et al., 2021). One such Berger, M. A. (1986). Topological invariants of field lines rooted
simulation is illustrated in Fig. 1.8. to planes. Journal of Physics A: Mathematical and General, 34,
Watching the full evolution of (x0 ) in the Fig. 1.8 sim- 265–281.
ulation suggests that the  pattern evolves predominantly Berger, M. A. (1988). An energy formula for nonlinear force-free
by rearrangement with some fictitious flow. To see this magnetic fields. Astronomy and Astrophysics, 201, 355–361.
from the evolution equation, we can employ the seemingly Berger, M. A., & Field, G. B. (1984). The topological properties
unhelpful trick of labeling field lines by fixing their end- of magnetic helicity. Journal of Fluid Mechanics, 147, 133–148.
Berger, M. A., & Hornig, G. (2018). A generalized poloidal–
point x1 on z = 1 so that 𝜓 = 0 and 𝑤 ⃗ ⟂ = 0⃗ on z = 1. Then
toroidal decomposition and an absolute measure of helicity.
instead of equation (1.64), equation (1.50) becomes Journal of Physics A: Mathematical and Theoretical, 51(49),
d 495501.
⃗ 0 (t)) ⋅ 𝑤(x
(Lt ) = −A(x ⃗ 0 (t)) + 𝜓(x0 (t)). (1.65) Berger, M. A., & Prior, C. (2006). The writhe of open and closed
dt
curves. Journal of Physics A: Mathematical and General, 39,
To compute the evolution of  at a fixed position x0 , we 8321–8348.
must now account for the fact that the field lines are mov- Bhowmik, P., & Yeates, A. R. (2021). Two classes of erup-
⃗ 0 ).
ing past this point with the local field line velocity 𝑤(x tive events during solar minimum. Solar Physics, 296(7), 109.
Thus the time derivative in equation (1.65) is a Lagrangian https://doi.org/10.1007/s11207-021-01845-x
one, so at the fixed position x0 , we have Bourdin, P. A., & Brandenburg, A. (2018). Magnetic helicity
from multipolar regions on the solar surface. The Astrophysi-
𝜕(x0 ) cal Journal, 869, 3.
= −𝑤 ⃗ 0 ) ⋅ 𝑤(x
⃗ ⋅ ∇(x0 ) − A(x ⃗ 0 ) + 𝜓(x0 ). (1.66) Campbell, J., & Berger, M. A. (2014). Helicity, linking, and
𝜕t
writhe in a spherical geometry. Journal of Physics: Conference
Notice that all the terms here are “local,” although 𝜓 and Series, 544, 012001. https://doi.org/10.1088/1742-6596/544/1/
 are still field-line integrated quantities. The advection 012001
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Chen, L., Yeates, A. R., & Russell, A. J. B. (2021). Optimal Moraitis, K., Patsourakos, S., & Nindos, A. (2021). Relative field
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(2020). Using topology to locate the position where fully .1029/91JA02435
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Kimura, Y., & Okamoto, H. (1987). Vortex motion on a sphere. Russell, A. J.B., Yeates, A. R., Hornig, G., & Wilmot-Smith,
Journal of the Physical Society of Japan, 56, 2024. A. L. (2015). Evolution of field line helicity during magnetic
Linker, J. A., & Mikic, Z. (1995). Disruption of a helmet reconnection. Physics of Plasmas, 22(3), 032106. https://doi
streamer by photospheric shear. The Astrophysical Journal .org/10.1063/1.4913489
Letters, 438, L45. https://doi.org/10.1086/187711 Schindler, K., Hesse, M., & Birn, J. (1988). General magnetic
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doi.org/10.3847/1538-4357/aa86b1 Schindler, K. (2010). Physics of space plasma activity. Cam-
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J. G., Antiochos, S. K., & Fisher, G. H. (2016). A model Yahalom, A. (2013). Aharonov-Bohm effects in magnetohydro-
for stealth coronal mass ejections. Journal of Geophysical dynamics. Physics Letters A, 377(31–33), 1898–1904. https://
Research: Space Physics, 121(11), 10,677–10,697. https://doi doi.org/10.1016/j.physleta.2013.05.037
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MacTaggart, D., & Prior, C. (2021). Helicity and winding fluxes CISM International Centre for Mechanical Sciences. Springer
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physical Fluid Dynamics, 115(1), 85–124. https://doi.org/10 16344-0
.1080/03091929.2020.1740925 Yeates, A. R. (2020). The minimal helicity of solar coronal mag-
MacTaggart, D., & Valli, A. (2019). Magnetic helicity in mul- netic fields. The Astrophysical Journal Letters, 898, L49.
tiply connected domains. Journal of Plasma Physics, 85(5), Yeates, A. R., & Hornig, G. (2011). A generalized flux function
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vortex lines. Journal of Fluid Mechanics, 35, 117–129. Yeates, A. R., & Hornig, G. (2014). A complete topologi-
Moffatt, H. K., & Ricca, R. L. (1992). Helicity and the cal invariant for braided magnetic fields. Journal of Physics
Calugareanu invariant. Proceedings of the Royal Society of Conference Series, 544, 012002. https://doi.org/10.1088/1742-
London Series A: Mathematical, Physical and Engineering Sci- 6596/544/1/012002
ences, 439(1906), 411–429. https://doi.org/10.1098/rspa.1992 Yeates, A. R., & Hornig, G. (2016). The global distribution
.0159 of magnetic helicity in the solar corona. Astronomy and
Moffatt, H. K. (1978). Magnetic field generation in electrically Astrophysics, 594, A98. https://doi.org/10.1051/0004-6361/
conducting fluids. Cambridge University Press. 201629122
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Moraitis, K., Pariat, E., Valori, G., & Dalmasse, K. (2019). Rel- Yeates, A. R., Russell, A. J. B., & Hornig, G. (2021). Evolution
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624, A51. https://doi.org/10.1051/0004-6361/201834668 abs/2108.01346
 2
Magnetic Winding: Theory and Applications
David MacTaggart

ABSTRACT
Magnetic winding is a renormalization of magnetic helicity that provides a direct measure of field line topology.
Despite its close connection to magnetic helicity, magnetic winding can provide different and important infor-
mation about field line topology that is not clear from an analysis of magnetic helicity alone. In this chapter, we
introduce magnetic winding from a theoretical perspective, with a keen eye on its role in understanding magnetic
helicity and field line topology. We also provide practical applications of magnetic winding, highlighting that it
can be used as an important measure for understanding the evolution of active regions in solar observations.

2.1. INTRODUCTION Although equation (2.1) holds for any vector potential
A in domains of arbitrary topological complexity, it is
Magnetic helicity is a conserved quantity of ideal mag- too general to yield useful information about magnetic
netohydrodynamics (MHD). What it conserves relates to field line topology, which, as mentioned at the start, is
the topology of magnetic field lines. To understand this related to what H conserves. If, again with B ⋅ n = 0 on
concept, we will consider magnetic helicity written in a 𝜕Ω, we make the gauge choice A = BS(B), where BS(B) is
useful gauge. In a bounded domain Ω ⊂ ℝ3 with genus g the Biot-Savart operator
that is magnetically closed (B ⋅ n = 0 on 𝜕Ω, for surface
unit normal vector n), the gauge invariant form of mag- 1 x−y 3
BS(B) = B( y) × d y, (2.2)
netic helicity can be written as 4𝜋 ∫Ω |x − y|3
g ( )( ) where x and y are position vectors in ℝ3 , then equation
∑
H = A ⋅ B d3 x − A ⋅ ti dx B ⋅ nΣi d2 x , (2.1) reduces to the classical form of magnetic helicity,
∫Ω i=1
∫𝛾i ∫Σi
(2.1)
H= BS(B) ⋅ B d3 x,
where B is the magnetic field, A is a magnetic vector poten- ∫Ω
tial, 𝛾i are closed paths around “holes” in the domain (e.g., 1 x−y 3 3
= B(x) ⋅ B( y) × d x d y. (2.3)
the hole of a torus) and have unit tangent vectors ti , and 4𝜋 ∫Ω ∫Ω |x − y|3
the Σi are the surfaces of cuts through the domain with
unit normal vectors nΣi . Equation (2.1) is derived in Mac- Equation (2.3) holds the key to understanding how helic-
Taggart and Valli (2019), which describes more details of ity includes field line topology. Indeed, it was this form of
the geometrical setup (see also Faraco et al. (2022) for an helicity that Moffatt considered in his seminal work on the
application to a rigorous proof of Taylor’s conjecture). subject (Moffatt, 1969).
Let us now take a side step and consider a fundamental
topological description of linked loops. Let C1 and C2 be
School of Mathematics and Statistics, University of Glasgow, two closed and distinct loops with assigned directions, as
Glasgow, UK shown in Fig. 2.1.

Helicities in Geophysics, Astrophysics, and Beyond, Geophysical Monograph 283, First Edition.
Edited by Kirill Kuzanyan, Nobumitsu Yokoi, Manolis K. Georgoulis, and Rodion Stepanov.
© 2024 American Geophysical Union. Published 2024 by John Wiley & Sons, Inc.
DOI:10.1002/9781119841715.ch02

17
18 HELICITIES IN GEOPHYSICS, ASTROPHYSICS, AND BEYOND

definition of helicity will be useful for the rest of this

chapter.

C1 C2 2.2. RELATIVE HELICITY

So far, we have considered magnetic helicity in a mag-

netically closed domain. However, there are many appli-
Figure 2.1 Two closed and linked loops, C1 and C2 , with spec- cations, particularly in solar physics, for which, at some
ified directions. boundary, nonzero normal components of the magnetic
field must be included. In this situation, equation (2.1)
Any deformation that does not break the loops pre- no longer represents a gauge invariant quantity. Instead,
serves the linkage. Therefore, linkage, in this scenario, is a relative measure of magnetic helicity, which compares
a topological quantity. If position vectors of points on two magnetic fields with the same normal boundary
C1 are denoted by x and those on C2 by y, the Gauss conditions, is required. From this moment onward in this
linking number, which measures the net algebraic linkage, chapter, although not strictly necessary for everything
is given by we introduce, our domain Ω will be simply connected.
This choice not only simplifies many descriptions but also
1 dx dy x−y relates to practical applications – the domains of most
Lk = ⋅ × ds ds , (2.4)
4𝜋 ∫x(s1 ) ∫y(s2 ) ds1 ds2 |x − y|3 1 2 solar simulations and observations are simply connected.
where s1 and s2 are parameterizations of C1 and C2 , Following Berger and Field (1984) and Finn and Anton-
respectively. The linkage Lk is an integer if the loops in sen (1985), relative magnetic helicity has the form
the link are closed. The closure of the links is also neces-
HR (B, B′ ) = (A + A′ ) ⋅ (B − B′ ) d3 x, (2.5)
sary for Lk to be a topological invariant: i.e., indifferent ∫Ω
to deformations that do not break closed loops. For a
Where B′ is a reference magnetic field with B′ ⋅ n = B ⋅ n
historical account of the development of equation (2.4),
on 𝜕Ω and A′ is a reference vector potential. Similar to
the reader is directed to Ricca and Nipoti (2011).
equation (2.1), equation (2.5) is too general to reveal clear
For the example in Fig. 2.1, the two loops are clearly
information about the field line topology. We cannot just
linked once. Due to the directions of the arrows, Lk = 1.
substitute in equation (2.2) as before, since field lines
We can also perform the calculation in equation (2.4) in
are not necessarily closed, so the Gauss linking number
reverse: i.e., assign x to points on C2 and y to points on
is no longer a topological invariant. Instead, we must
C1 and reach the same result. Another way to write this
find another description of field line topology, this time
is Lk12 = Lk21 = 1.
starting from equation (2.5). To achieve this, we will
In Fig. 2.1, we now transform the closed loops into
further restrict our attention to “tubular” domains, as
magnetic flux tubes. That is, each loop becomes the axis
illustrated in Fig. 2.2.
of a thin flux tube with flux Φi , i = 1, 2. All field lines are
By “tubular,” we mean the following: the domain Ω is
parallel to the tube axis: i.e., for the moment, we do not
bounded above and below by horizontal planes. The mag-
consider internal twist. With this construction of the flux
netic field in Ω must be tangent to side boundaries. Field
tubes, we have that
lines can connect to both, one, or neither of the upper
Φi ti dx = B d3 x, and lower boundaries. Although this restricts what kind
which connects flux and line elements to magnetic field
and volume elements. This means by rescaling dx of z=h z
C1 and dy of C2 in equation (2.4) by the magnetic flux
Φi associated with each loop, we find the helicity in Sz
equation (2.3). In other words, the magnetic helicity in
equation (2.3) of closed magnetic fields measures the
(pairwise) Gauss linking number of field lines weighted Ω
by magnetic flux. For the example in Fig. 2.1, the
value is H = Lk12 Φ1 Φ2 + Lk21 Φ2 Φ1 = 2Φ1 Φ2 . Although z=0 y
magnetic helicity can be extended to incorporate more
complex magnetic fields that have internal twist (Moffatt x
and Ricca, 1992) and ergodic field lines (Laurence and
Avellaneda, 1993), the basic picture we have presented Figure 2.2 A “tubular” domain Ω between two horizontal
here holds. That is, magnetic helicity is a measure of field planes at z = 0 and z = h. At each height in Ω, the horizontal
line topology weighted by magnetic flux. This conceptual slice is denoted Sz .
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d’une journée d’orage, dont les brusques sautes de vent devaient
démonter la mer. Le service des transports fait par la Compagnie
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matin.
 J’en ai encore le vertige dans la tête, et derrière les lamelles de
mes persiennes closes qu’enflamme la clarté blanche du dehors, je
crois voir monter et descendre, dans un abominable mouvement de
montagnes russes, les roches de la côte, la houle et le bastingage
du Bocognano, le premier rouleur de la Compagnie avec la Ville-de-
Bastia.
Des fanfares, un bruit de foule m’arrachent du lit où je somnole ;
je me précipite à la fenêtre, j’entr’ouvre les persiennes ; tout un
peuple en fête se presse sur les trottoirs du Cours-Napoléon. A la
terrasse de la caserne, en face, tout un flot d’artilleurs se bousculent,
s’accoudent et cherchent à se faire place, avidement penchés sur la
procession.
La Procession ! une procession comme on n’en voit plus sur le
continent et que M. Combes ne se risquera pas à supprimer encore
ici, car la population, enracinée dans ses coutumes et foncièrement
latine et dévote, tient avant tout à ses manifestations religieuses, et
celle-là a, en effet, un caractère tout particulier et bien local.
Précédé d’une fanfare, un long Christ de grandeur humaine
apparaît et oscille au-dessus de la foule à l’angle du Cours.
Enluminé et peint de plaies saignantes, il s’avance, érigé très haut
par un porteur en froc violet ; des guirlandes de fleurs et des
banderoles violettes l’encadrent. Une confrérie de pénitents violets
l’escorte ; suivent des groupes de femmes en noir, encapuchonnées
à la mode corse, et des hommes en complet de velours ; puis un
autre Christ enguirlandé, lui, de banderoles et de fleurs rouges, la
confrérie qui l’entoure est vêtue de frocs écarlates, et la procession
continue, et un troisième Christ apparaît, tenu très haut par un
porteur et suivi d’une confrérie à ses couleurs, et voici un autre
Christ et un autre Christ encore dans leur faste un peu barbare de
banderoles et de fleurs artificielles. Les cinq corps suppliciés
dominent, tels d’étranges mâts, la marée des têtes nues et des
capuchons ; c’est un défilé de cinq grands Christs planant au-dessus
de confréries et d’une foule recueillie et lente. Un concours de
peuple entoure une statuette de saint portée sur les épaules d’un
groupe de brancardiers, c’est une figurine de moine en robe de bure
qui, une palme à la main, semble bénir, debout sur un amas de
rochers. Une dizaine d’hommes — des gars musclés aux yeux aigus
et noirs dans des faces de hâle — se disputent l’honneur de le
porter, et aux fortes encolures, aux cheveux drus et plantés bas sur
le front, j’en fais des mathurins, des hommes de mer. Des vieux
chenus prêtent aussi leur épaule aux brancards ; mais c’est surtout
une jeunesse ardente qui se dresse autour de la statuette du saint ;
et ces cinq Christs oscillants, cette ferveur odorante autour d’une
figure aux dimensions d’idole imposent à ma mémoire des souvenirs
de pardon de Bretagne en même temps que de processions
espagnoles croisées dans les « calle » de Saragosse et de Valencia.
— « La procession de saint Roch, le plus vénéré de nos saints,
m’est-il répondu par mon hôtelier, il y a quatre ans que la procession
n’avait eu lieu. Nous avons ici deux églises Saint-Roch, et chacune
des paroisses s’entêtait à ne pas céder à l’autre l’honneur de
promener le saint. Elles sont enfin tombées d’accord et la joie de
cette foule lui vient de contempler enfin dans les rues son saint
qu’elle n’y avait pas vu depuis quatre ans. »
Saint Roch ! Sa légende m’en était contée une heure plus tard
sur la place du Diamant par Michel Tavera, un jeune Corse que je
connus il y a quelques années, à Paris, faisant une littérature
savoureuse et colorée comme les montagnes et qui depuis a
abandonné la Capitale, préférant aux odeurs de la rue du Bac
l’atmosphère sauvage et parfumée du maquis.
— « Voyez-vous ces écueils là-bas, de l’autre côté de la baie, en
face, »
Mais le miroitement de la mer, le halo lumineux de la côte
noyaient le point désigné dans une brume de chaleur.
« Regardez bien au bas de ce promontoire, vous les verrez se
dessiner, ils sont sept, ce sont les setto navi, les sept navires. Toute
la légende de saint Roch est là, ces sept rochers affirment sa
puissance. Sept galères barbaresques, chargées d’infidèles atteints
de la peste, étaient entrées dans la baie ; elles menaçaient de
débarquer à Ajaccio, tout le pays était consterné, ces mécréants
allaient y répandre leur mal. Saint Roch, imploré par les populations
accourues des campagnes, s’avança jusqu’au bord de la mer.
S’agenouillant et s’étant mis en prière il adjurait Notre-Seigneur le
Christ d’entraver la marche des navires et de préserver l’île, et sur
un geste du saint, les sept galères s’arrêtaient, devenues pierres
pétrifiées, elles et leurs équipages, changées en sept écueils.
« Ce sont les sept récifs qui s’échelonnent à la file au pied de
Chiavari, Ajaccio célèbre encore aujourd’hui le souvenir de ce
miracle et de sa délivrance. »
Au fond de la baie, les neiges du Monte d’Oro, enflammées par
l’adieu du soleil, étincelaient toutes roses au-dessus des forêts
bleuâtres, les fanfares de la procession éclataient par intervalles
dans le quartier de la Citadelle, les cinq Christs défilaient sur les
anciens remparts.
La maison de Napoléon, c’est le pèlerinage tout indiqué du
lendemain. Je l’ai déjà visitée, il y a trois ans, c’était pendant l’hiver
et la longue enfilade des pièces du premier, le seul étage où soient
admis les visiteurs, en prenait, derrière les persiennes closes, un
lamentable aspect de détresse et d’abandon. Dans la chaleur de
l’été l’impression sera peut-être tout autre.
C’est dans la petite rue étroite et fraîche en août, froide en
janvier, la même maison provinciale à trois étages, façade blanche
et volets verts. Elle se penche un peu en arrière sous sa toiture
comme mal d’aplomb ou redressée d’orgueil. Derrière les volets,
qu’entre-bâille à peine la gardienne, ce sont les mêmes pièces aux
parquets légèrement disjoints, plafonds peints, à la mode italienne,
d’attributs et de fleurs de facture un peu sèche, selon le goût du
temps. Des sièges de l’époque de la Révolution, des cabinets de
Florence incrustés de lapis et de marbre, des bergères Louis XVI,
dont les coussins de velours perdent leur crin, en meublent la
solitude et c’est le salon de famille, et c’est le cabinet de travail du
père de Napoléon, la chambre à coucher de Mme Lætitia, le canapé
sur lequel elle mit au monde le premier Consul ; car, prise pendant la
grande messe des premières douleurs et rapportée en toute hâte de
l’église, on n’eut même pas le temps de la mettre sur son lit, et c’est
sur un canapé que Lætitia Bonaparte accoucha du grand Napoléon,
le 15 août 1769, vers midi, comme finissait l’office de l’Assomption.
Comme il y a trois ans, la gardienne ouvre pieusement deux
petites armoires dissimulées dans le mur, placées l’une au pied du lit
de Mme Lætitia, l’autre à la tête. De la première elle tire avec
précaution une crèche d’ivoire représentant la Sainte Famille dans
l’étable de Bethléem ; le premier Consul la rapporta d’Égypte pour
l’offrir à sa mère, c’est le gage de son culte filial. L’autre cachette
recèle, posée sur un coussin de velours rouge, la couronne de
lauriers du premier Consul. Elle est en or massif et c’est
l’enthousiasme reconnaissant d’Ajaccio qui en a fait les frais par une
souscription récente. L’emblème consulaire repose sous un globe de
verre comme une vulgaire pendule ; les mains de l’Ajaccienne, qui la
montre, n’en tremblent pas moins d’orgueil.
Nous reprenons notre promenade, et c’est au hasard des pièces
fraîches et vides, comme embaumées de silence et de clair-obscur,
la chaise à porteurs dans laquelle Mme Lætitia fut rapportée de
l’église, la chambre de Napoléon Bonaparte avec la fameuse trappe
par laquelle il échappa aux poursuites de Paoli, et enfin ce joli salon
en galerie que j’avais tant aimé à mon premier voyage. Six fenêtres
sur la rue, six autres sur une terrasse intérieure font de la pièce
oblongue une étroite lanterne qu’éclairent encore des petites glaces
à appliques posées entre chaque fenêtre. Un parquet luisant, deux
cheminées à chaque bout de la galerie se faisant face, deux grandes
glaces au-dessus et tous les petits miroirs des appliques donnent à
ce petit salon de fête un faux air de splendeur, et pourtant quel
misérable papier au mur et quelles piteuses peintures au plafond !
Mais il est bien de son époque, ce salon des fêtes de la famille
Bonaparte et prépare déjà les magnificences de Fontainebleau. Il est
raide, élégant et convenu, comme l’Empire lui-même. La vieille
Ajaccienne, qui nous en fait les honneurs, nous fait remarquer la
terrasse carrelée qui borde le salon, Mme Lætitia l’avait fait
aménager pour retenir au logis Nabulione ; c’était le préau où jouait
l’Empereur enfant. Mme Lætitia avait dû prendre le parti de garder
son fils auprès d’elle ; Nabulione, turbulent, batailleur et dominateur,
organisait avec les autres gamins de son âge des guerres et des
embuscades de quartier qui finissaient toujours par des horions, des
bleus et même des effusions de sang. Impérieux et volontaire, il se
mettait à la tête des petits Corses de sa rue, préparait la victoire et
faisait mordre la poussière au parti adverse ; le parti de Nabulione
était toujours vainqueur. Devant les plaintes des voisins et des
mères, Mme Lætitia avait dû se décider à garder l’enfant indiscipliné
auprès d’elle.
Nabulione enfant s’exerçait déjà à conquérir le monde… La
vieille Corse, qui me raconte cette légende faite peut-être à plaisir, la
débite avec une joie évidente, toute sa vieille face crevassée
rayonne, a comme un air de fête. Pour elle, comme pour tout bon
Ajaccien, quand on parle de l’Empereur, c’est toujours le 15 août, la
Saint-Napoléon.
 SOUS LES CHATAIGNIERS

Le châtaignier, cet ancêtre.

Marcagi.

La châtaigne, c’est le blé de la Corse : elle nourrit tout le pays.

Sa farine remplace celle du froment ; la frugalité et surtout la paresse
du paysan corse s’en accommodent.
Si le châtaignier met trente ans avant de produire, à partir de cet
âge, il fournit d’année en année une récolte certaine et de plus en
plus abondante. A mesure qu’il pousse ses fortes ramures, la
châtaigne se multiplie hérissée et verte, dans le clair-obscur vernissé
de ses feuilles. Le châtaignier ne demande aucune culture. Pendant
qu’il prolonge à fleur de sol l’enchevêtrement de ses racines
pareilles à des accouplements monstrueux, et, telle une énorme
araignée végétale, étreint de tentacules ligneux le granit du talus et
la pierraille de la sente, les luisantes châtaignes pleuvent des
branches hautes et, couché dans l’ombre, le Corse indolent regarde
tomber les fruits, et c’est le pain d’aujourd’hui, et c’est le pain de
demain, et c’est le pain de l’été, et c’est le pain de l’hiver. Le petit
champ de maïs qu’il cultive à ses moments perdus, derrière la
masure paternelle, ajoute un bien faible apport à l’annuelle récolte.
La châtaigne, c’est la manne de ce désert de cimes et de roches
montagneuses ; que serait la Corse sans ses oasis de
châtaigneraies nourricières !
 Les châtaigneraies de la Corse ! Il faut voir leur moutonnement
de verdure monter du fond des vallées à l’assaut de la montagne !
Elles en ascensionnent les pentes, en escaladent les hauteurs,
cernent la crête, descendent dans le torrent et ne s’arrêtent à la
zone déjà froide où commencent les hêtres, que pour dévaler
précipitamment dans le creux des gorges et des ravins, où leur
rondeur feuillue ondoie comme une mer… Dans leur ombre fraîche
sourdent et jasent des sources ; l’eau froide et bleue, fille des neiges
éternelles, court entre leurs troncs crevassés et chenus. Elles se
rencontrent à mi-flanc de la montagne, attirées l’une vers l’autre, la
source descend des hauteurs, la châtaigneraie monte de la vallée, et
de leur rencontre naît le village Corse… Le village Corse et ses
vieilles maisons grises tout en hauteur et pareilles de loin à quelque
chantier de pierres à l’abandon. Percées d’étroites fenêtres, presque
des meurtrières, elles se dressent à l’ombre des châtaigniers et à
l’ombre de la montagne, déjà assez haut sur les pentes…, dans
quelque repli de ravin dont une route en lacets contourne les hautes
roches. Échelonnés un peu à l’aventure autour d’un clocher isolé,
comme les campaniles d’Italie, les villages corses dominent toujours
la vallée et, contemplatifs en même temps qu’instinctivement
pratiques puisque toujours à portée de l’eau et de l’ombre, ils se
posent invariablement devant un vaste horizon. J’ai déjà dit que la
sobriété et la paresse du paysan corse trouvaient leur compte dans
la farine de châtaigne. D’une incroyable endurance, foncièrement
honnête et probe, frugal, sans besoins même, mais étonnamment
fier et paresseux, le paysan corse, interrogé sur ses moyens
d’existence, a une phrase mélancolique passée maintenant en
proverbe : « Comment je vis ? répond-il au touriste, surpris d’un pays
sans labour presque et sans culture. De pain de bois et de vin de
pierre ! » pane di legno e vino di petra. Le pain de bois, la farine de
châtaigne ; le vin de pierre, l’eau de rocher ; et certains voyageurs se
sont apitoyés sur la tristesse de cette réponse.
Il y a eu là méprise ; la phrase est mélancolique, mais de la
mélancolie du pays même ; elle en a la sauvage fierté. Le paysan
corse aime sa pauvreté, il ne souffre pas de sa condition, il ne
tiendrait qu’à lui de l’améliorer. S’il voulait cultiver la terre et lui faire
rendre ce que l’extraordinaire richesse du sol donnait ici sous la
domination romaine, il serait presque riche ; mais le paysan corse ne
daigne pas. Travailler la terre lui semble indigne de lui, il laisse cette
basse besogne aux Lucquois, et il faut entendre avec quel mépris il
englobe sous le nom de Lucquois, tous les tâcherons italiens
débarqués en Corse par les bateaux de Bastia-Livourne, dont le
labeur est la seule animation du pays. Le paysan corse chasse,
court la montagne, pousse devant lui quelques chèvres à travers les
roches, ou bien le long d’un raidillon un âne chargé de bois. Vêtu de
velours noir et guêtré jusqu’aux cuisses, il chevauche parfois un
mulet ou un petit cheval corse, tandis que sa femme, chargée
d’énormes paquets, une lourde cruche en équilibre sur la tête,
chemine à pied à côté de lui. Plus rarement encore, de quatre et
demie à huit heures, dans la fraîcheur du matin, arrose-t-il le maïs
de son champ ou les quelques légumes de son jardin ; mais la
plupart du temps la pipe à la bouche, il rêve, assis sur le petit
parapet de pierre sèche de la route, ou devise, accoudé à la table
d’un cabaret, avec d’autres hommes vêtus de velours comme lui et,
dans la belle saison, toutes ses journées il les passe dans la
châtaigneraie.
L’Arabe au pied du palmier, le Corse au pied du châtaignier.
O fresco. Au frais, à la fraîcheur ! Dès deux heures, au sortir de
table, le paysan corse, par des sentiers pierreux et brûlés de soleil,
gagne la belle ombre verte. Il retrouve là tous les autres hommes du
village, les jeunes et les vieux. Couchés, vautrés au hasard des
roches et des racines dans la clarté douce qui pleut des hautes
branches, ils forment des groupes pittoresques, jouent à la mora, au
loto ou ressassent entre eux des histoires de bandits. Quelques-uns
font la sieste. Entre les énormes quartiers de granit, une eau
hallucinante tant elle est transparente sanglote ou rit sur le velours
des mousses ; parfois, un des joueurs se lève, va à la source et, se
penchant, boit à même comme un animal. Ceux que le continent a
déjà affinés font pour se coucher des lits de fougère, et la journée se
passe o fresco, parmi le calme et le demi-jour, glauque dans les
cimes feuillues, bleuté près des sources, de la châtaigneraie corse.
Dans le village, assez loin déjà, les femmes peinent et
s’exténuent les unes sur les routes poudreuses, la tête chargée de
pesants fardeaux, les autres aux soins du ménage ; les pourceaux
noirs voguent en liberté par les rues, et autour du forno di campana,
four des cloches, le four à cuire le pain, toujours situé au centre du
village à côté du campanile, d’où son nom four des cloches (le four
en plein air où tout village corse cuit son pain) — des pétrisseuses
de pâte (car le paysan corse laisse aussi les femmes faire le
boulanger) entassent les pains pour la fournée de la nuit.
La châtaigneraie corse et la belle fainéantise de ses paysans. Un
ami Corse, m’en fait aujourd’hui les honneurs. Il m’introduit dans le
cénacle de ces endurcis attardés, philosophes inconscients à la
manière de Lucrèce, puisqu’ils font passer avant toutes choses la
joie de vivre lentement les heures et de les sentir vivre. On m’a
annoncé aux paysans d’Ucciani, et, comme ils ont tous lu, ou plutôt
comme on leur a lu la veille un récent article consacré à Ajaccio et à
la gloire de Napoléon, je suis plus qu’attendu. A ma venue, tous se
lèvent, de fortes mains hâlées se tendent vers moi, on me fait place,
je me trouve assis sur un lit de fougère, je suis environné de sourires
à dents blanches et de larges prunelles étrangement limpides. Il y a
dans les yeux corses, une ardeur et une violence contenues en
même temps qu’une candeur si avide que, dans les premiers temps,
ce regard animal et pourtant très beau me déconcertait et me
troublait.
Nulle part je n’ai rencontré des yeux si sauvagement attentifs.
Si pourtant, en Kabylie, dans les hameaux arabes, en Kabylie
aussi comme dans toute l’Algérie, la femme traitée en bête de
somme est exténuée de maternité et de basses besognes, tandis
que l’homme farniente et, drapé dans son burnous, s’absorbe en de
longues contemplations.
Comme au hameau kabyle, ma venue a dérangé deux conteurs,
deux Uccianais dont l’un de retour de Toulon, où il y a un mois
encore, il servait dans la flotte, et l’autre de Marseille, inscrit
maritime frais débarqué d’un transatlantique, et tous deux auréolés
du prestige des navigateurs. Escales et traversées, villes de mirage
et grèves lointaines, j’ai coupé court aux récits merveilleux ; tous les
yeux, toutes les bouches s’inquiètent fiévreusement de mon
impression sur la Corse : « Quel beau pays, n’est-ce pas, mais
combien méconnu ? Quelles forêts et quelles montagnes ! Et la baie
d’Ajaccio, les calanques de Piana, et les grottes de Bonifacio ! » Tout
Corse a l’orgueil de son île et veut vous en imposer l’admiration ; la
sienne est enthousiaste, délirante, aveugle, et c’est encore le
fanatisme arabe dont s’illuminent leurs yeux cyniques quand ils
vantent leur pays.
— Et Bellacoscia, avez-vous vu Bellacoscia !
Bellacoscia, de son nom Antoine Bonelli, le fameux bandit qui,
durant quarante-sept ans, tint le maquis et abattit, fin tireur
d’hommes, vingt-cinq à trente gendarmes, est une des gloires de la
Corse. Après Napoléon et Sampierro, je ne crois pas que le paysan
des montagnes ait une vénération plus haute. Entre Ajaccio et Corte,
Antoine Bellacoscia est estimé et respecté comme un héros. Ce
tueur d’hommes a une telle légende, il incarne aux yeux du Corse
l’esprit d’indépendance et cet amour de la liberté que lui ont mis au
cœur des siècles de lutte et de guerres perpétuelles contre le
Génois et le Maure. Perpétuellement menacé dans son île par les
galères barbaresques ou les flottes de Gênes, il a eu de tout temps
la montagne pour citadelle et le maquis pour refuge, et le banditisme
à ses yeux d’instinctif prolonge dans les temps modernes la grande
figure de ses héros d’embuscades et d’exil défendant pied à pied la
terre natale contre l’envahisseur. Les deux Bellacoscia, car ils
étaient deux frères, Antoine et Jacques, ont pris le maquis pour ne
pas obéir à la loi militaire. Plutôt que de se laisser enrôler comme
soldats, ils ont gagné la solitude des cimes et, pendant près d’un
demi-siècle, ont vécu en plein air, dormi à la belle étoile, gîté dans
l’antre et bu au torrent, protégés et nourris, d’ailleurs, par toute la
contrée complice, tour à tour sauvés et dénoncés par les paysans en
admiration et en terreur aussi de ces fusils qui ne manquaient pas
leur homme. Bellacoscia ! et, aux éclairs des prunelles ardentes, je
vois combien ces âmes impulsives ont le culte sauvage de leur
bandit légendaire et national. J’ai vu Bellacoscia, l’avant-veille
même, à Bocognano où le vieux proscrit, gracié par le Président
Carnot, passe ses journées assis sur une chaise, au seuil de sa
porte, la pipe à la bouche et vêtu du traditionnel costume de velours
noir.
Les cartes postales ont vulgarisé sa physionomie. Bellacoscia
vieillit là, vénéré de tout le village, entouré de famille et fait même
partie d’une confrérie de pénitents. Il a aujourd’hui soixante-dix-sept
ans, J’ai bu et causé avec lui. C’est un grand vieillard tout blanc, à
barbe de patriarche ; le visage émacié, aux traits fins et creusés, a
les tons jaunis d’un vieil ivoire, les yeux demeurés vifs ont dû être
très beaux. Sous le large feutre noir, que porte ici le paysan, c’est un
peu la tête classique d’un prophète biblique, Job ou Ezéchiel.
Conversation illusoire ! Bellacoscia n’entend pas le français, et
quand la bande de jeunes gens qui, en grande pompe, m’avaient
présenté à lui, lui rappelaient quelques-uns des bons tours de sa vie
d’autrefois — l’histoire des cinq gendarmes abattus l’un après l’autre
comme cinq poupées de tir, l’un atteint au genou, l’autre à l’épaule,
le troisième au front, et tutti quanti, par le bandit tranquillement
couché derrière une roche si fortement inclinée sur le sol qu’on n’y
pouvait soupçonner sa présence — et l’aventure du chat enveloppé
dans une pellone, costume en poil de chèvre, filé et tissé par les
femmes corses, qui fut longtemps le vêtement des montagnards et
qui tend à disparaître de nos jours, et jeté dans l’escalier à toute une
compagnie de gendarmes en train d’envahir sa maison ; les
Pandores croyant avoir affaire à Bellacoscia lui-même, déchargeant
leurs armes sur le pellone et les deux frères tiraient alors à bout
portant sur les assaillants — le vieux bandit se contentait de pencher
de côté un long cou de vautour, et avec un clignement d’yeux à la
fois malicieux et bonhomme. « Eh ! eh ! eh ! » faisait-il de la voix
chantante du paysan corse ; et c’était presque le sourire amusé et
gourmand d’un ancien coureur de femmes, rajeuni au récit d’une de
ses prouesses d’autrefois.
 Ce sont les cent et un bons tours de Bellacoscia aux gendarmes
que me ressasse, sous les châtaigniers, une fervente jeunesse.
Pour tous ces paysans, retour du continent ou toujours demeurés
au village, à Ucciani comme à Bocognano, à Vivario comme à Corte,
Antoine Bellacoscia vaut un roi. Des deux frères, c’était Jacques le
bandit terrible, quelques atrocités lui sont reprochées à lui. Jacques
Bellacoscia est mort au maquis, on ignore… même l’emplacement
de sa tombe, ses enfants seuls connaissent l’endroit où repose son
corps. Antoine aussi le sait, Antoine, le survivant, mais Jacques
Bellacoscia a fait jurer aux siens de ne jamais révéler la place. « Ils
ne m’ont pas eu vivant ils ne m’auront pas mort ! » ont été ses
dernières paroles, les siens ont respecté sa volonté, le maquis
complice a gardé le secret.
Et de tous les racontars entendus dans la châtaigneraie, c’est la
seule histoire dont je veux me souvenir. Mystérieuse et farouche,
avec son allure de défi jeté au delà de la mort, elle me semble mieux
que les coups de fusil et les embuscades, mieux que les meurtres et
les traîtrises, entrer dans le cadre austère du paysage corse.
 LE VILLAGE

Et c’est dans le visage une fierté câline

Faite de grâce hellène et d’ardeur
sarrasine.

J. L.

« Il faut absolument que vous veniez au village un jour de fête.

Vous verrez un peu nos paysans. Je vous les ai montrés sous la
châtaigneraie. Il faut les voir au tir au coq, dans leurs chansons de
cabarets, leurs danses, et puis il ne faut pas manquer la sortie de
l’église. » Et comme j’objectais le fastidieux ennui du voyage dans le
somnolent Decauville, qui fait le service de la Corse, et le trajet déjà
tant de fois effectué de Vizzavona à Ajaccio. « Mais descendez donc
à Bocognano en voiture, la route est superbe. A partir du col vous
découvrez toute la vallée jusqu’au golfe. Je viendrai vous prendre
dans mon break à Bocognano et vous conduirai par la grande route
jusqu’à Ucciani. Vous connaîtrez enfin un peu le paysage. On ne voit
rien sous ce tunnel du Monte d’Oro, toute la contrée vous échappe.
Et puis vous surprendrez les villages au réveil et reverrez peut-être
Bellacoscia sur le seuil de sa porte. » Ainsi parla mon ami, Michel
Tavera, et je me laissai persuader.
 La fête d’Ucciani tombait le surlendemain, la fête annuelle, la
Saint-Antonin, le patron du pays : Il Chianco, le boiteux, comme
l’invectivent dans une dévotion bien italienne les Uccianais déçus
par la sécheresse d’août ou les pluies de septembre, dont n’a pas su
les préserver leur saint.
Le matin de la Saint-Antonin, nous quittions donc Vizzavona.
C’était d’abord la montée au pas dans la clarté verte de la forêt
de sapins, une forêt rajeunie par la nuit, dont les odeurs résineuses
n’étouffent pas encore l’âme végétale des thyms et des menthes,
puis c’était la forêt de hêtres et le friselis de sa verdure plus fraîche
et enfin, comme une vague, le grand souffle d’air pur du col…
Et, dans une aridité de pierres grises, cendreuses, tant elles sont
calcinées c’est, à mesure que l’on descend les lacets de la route, en
amont l’énorme accablement, la soleilleuse et morne solitude des
masses pelées du Monte d’Oro, en aval le dévalement d’eaux vives
et de verdures de ravins ombreux ; le maquis et les torrents nous
escorteront ainsi jusqu’aux châtaigniers de Bocognano, tandis qu’à
notre droite, de l’autre côté de la vallée au ruisseau invisible, tant
elle est profonde, les crêtes déchiquetées de la montagne
continueront à chevaucher sur un ciel de chaleur… et, jusqu’à la
bande de brumes lumineuses, où le doigt tendu de Tavera m’assigne
la place d’Ajaccio, ce sont dans un poudroiement bleuâtre, douze
lieues de vallées, de sommets, de collines et de gorges s’abaissant
insensiblement vers la mer, un horizon d’une vastitude et d’une
tristesse infinies, sous la torpeur de cette matinée déjà chaude… Par
moments, des bouffées de fournaise nous brûlent, tout le paysage
est rempli de fumée… des feux d’herbes ? Non ! Ce sont des forêts
entières qui flambent, allumées par la malveillance des bergers.
Depuis que l’incurie du gouvernement interdit en Corse le
pâturage du domaine de l’État, le paysan, réduit au maquis, met tout
simplement le feu au bois confisqué et pour faire place nette et pour
y trouver au prochain printemps l’herbe nécessaire à ses bêtes.
Nous atteignons Bocognano dans une atmosphère d’incendie.
Maintenant, c’est la plaine.
 L’étouffement s’est fait plus dense et la route court poussiéreuse
entre des vallonnements moutonnés de cystes et de lentisques et
des plantations de maïs ; les montagnes évaporées de chaleur ne
sont plus qu’une vibration lumineuse et quelle solitude !!
La tête sarrasine d’un paysan corse, en velours marron et le fusil
en bandoulière, est la seule rencontre que nous fassions pendant
quinze kilomètres. Il mène paître ses chèvres armé comme pour une
vendetta.
Un pont. Tavera m’en fait remarquer les proportions hardies.
Le pont d’Ucciani, il a cent ans. Les Uccianais, jaloux de sa
perfection, ne trouvèrent rien de mieux que de noyer l’architecte.
Une fois mort, il ne doterait pas les autres pays de chefs-d’œuvre
pareils : le pont d’Ucciani est unique. Cette férocité n’indigne pas
Tavera.
Et la route enfin remonte ; nous escaladons des lacets dans une
chaleur de brasier ; le maquis pétille de soleil.
Dans une vigne apparaît le cube blanc d’un tombeau, « Le
village », me dit Tavera, le monument funèbre se trouve être celui de
sa famille : sur les pentes d’un coteau de vignobles éclate la
blancheur d’autres mausolées ; c’est bien le village. Le Corse a
l’orgueil de sa sépulture ; l’entrée de tout hameau, de plaine ou de
montagne, se signale par une ceinture de tombes. Nous avions déjà
trouvé cette voie Appienne le long du golfe d’Ajaccio.
Mais des châtaigniers surgissent. Voici la gare… et, par un
sentier de chèvres, qui est aussi celui des voitures, ascensionnent
des groupes de paysans. Ce sont les costumes de velours noir des
fêtes carillonnées et des beaux dimanches, des cavalcades de
mulets, des paysannes dans leur éternelle robe de deuil,
chevauchant à nu comme des hommes. Puis ce sont des chants et
des guitares.
Ajaccio et les environs montent à l’assaut d’Ucciani gaiement.
Et parmi cette foule hâlée, aux yeux aigus et noirs, c’est la
surprise d’adorables visages de blondes. La blonde Corse est
particulièrement désirable. Nulle part je n’ai vu ces ors rouillés et
nuancés de feuille morte dans les chevelures, nulle part ces yeux
d’aigue-marine dans des faces mordorées et mûries par le soleil. La
Corse blonde a la saveur d’un fruit.
Ce sont aussi des théories de femmes, la taille droite ou
balancée sous d’énormes charges, des couples d’ânes et des
cavaliers tenus en croupe par d’autres cavaliers sur des petits
chevaux du pays.
Une débandade de pourceaux, des hottées d’enfants demi-nus,
des jeunes filles groupées autour d’une fontaine, voici la rue.
Rongées d’années et de soleil, les maisons d’Ucciani sont
couleur de tain ; une immense châtaigneraie les domine. Mises en
valeur sur cette verdure fraîche, les maisons d’Ucciani dévalent en
désordre à la rencontre du promeneur, suivies dans la vallée par
l’ombre de la forêt.
Tout un groupe de joueurs nous désigne l’auberge. Des teneurs
de loteries et des bonneteurs sont là, aguichant le montagnard, tous
montés de la ville. Voici le clocher, carré et droit, isolé au milieu du
village, tandis que l’église en contrebas se blottit au pied d’un talus.
Déjà toute cette joie éparse nous dilate et nous rajeunit quand, tel un
hurlement d’orfraie, s’élève et pleure une mélopée lugubre. Une
immense plainte traîne, se lamente et glapit. C’est la troupe des
pleureuses s’énervant autour d’un cadavre dans le ressassement
des voceri. Il y a une morte dans le village.
Une femme y est décédée dans la nuit.
Toute la famille (tous les Corses sont parents entre eux), est là,
derrière les volets clos de la maison, qui hurle et fait ripaille en
veillant le corps. Le deuil a gagné le pays. Il n’y aura ni tir au coq, ni
danses, ni réjouissances dans la rue, il y a une morte dans Ucciani.
La fête se bornera à la grande messe et à la procession.
La grande messe. Impossible d’entrer dans l’église, la foule,
tassée à ne pas y jeter une épingle, reflue, refoulée au dehors. Une
vingtaine de femmes prient sous le portail, à genoux sur les
marches. Au-dessus de leur foulard de soie, à peine distinguons-
nous une marée de têtes encapuchonnées, et, dans le clair obscur
du maître-autel piqué de cierges, les lentes allées et venues d’un
clergé alourdi de chasubles d’or. Toutes les femmes ont
soigneusement caché leurs cheveux, une ferveur adorante les
courbe vers l’autel, des chants profonds traînent en psalmodies.
C’est l’atmosphère d’un sanctuaire espagnol, mais les abords de
l’église sont ceux d’un pardon de Bretagne.
Couchés, assis du côté de l’ombre, tous les hommes sont là, les
garçons surtout, venus pour voir les filles à la sortie de la messe. Ils
devisent entre eux sous le feutre à larges bords, en lourds souliers
ferrés, le bâton à la main, et attendent patiemment le passage des
femmes. Quelques-uns grattent des guitares. Le parapet de pierre,
qui domine le ravin, a été respectueusement laissé aux vieillards.
Toute une bande de vieux, très Bellacoscia d’aspect, y prennent le
frais ; l’un d’eux a quatre-vingt-douze ans, et est père de douze
enfants, il est là avec cinq de ses fils, dont le plus jeune a vingt ans
et l’aîné soixante-six. C’est vous dire la verdeur corse. Fils il est vrai
de différentes femmes. Cet étonnant générateur en a eu quatre. Je
l’examine avec stupeur.
Et la foule s’écoule. On sort de la messe. Passants et citadins
remontent au village, foule recueillie, gardant encore, même dehors,
le silence religieux de l’église. « La population ici est tellement
croyante, si passionnée surtout ! me confie Tavera. Croyez-vous
qu’à la semaine sainte, pendant qu’on lit l’Évangile de la Passion, au
passage des Juifs, lorsque Ponce-Pilate se lave les mains et livre
Jésus à Caïphe, tous les hommes poussent des hurlements,
soufflent dans des cornes de bœuf, démolissent les bancs à coups
de bâton. Et c’est un vacarme de huées et d’injures effroyables à
l’adresse des bourreaux du Christ. Des Espagnols ou plutôt des
Arabes, avec tout le fanatisme sensuel et démonstratif des races
africaines. » Décidément le Corse est bien plus Arabe qu’Italien.
Nous déjeunons maintenant dans une maison corse. Une vieille
maison seigneuriale qui n’est pas sortie de la famille depuis deux
siècles. Les Tavera l’habitent de père en fils, les aïeux y sont morts,
les petits-fils y mourront, les Tavera de l’avenir y sont encore à
naître.
 De vastes pièces, un peu basses de plafond, aux poutrelles
apparentes, aux petites fenêtres s’ouvrant qui sur le village, qui sur
la montagne, et où le service est fait par une lignée d’antiques
serviteurs. Les domestiques y forment une famille à côté de celle
des maîtres. La jolie fille qui nous sert à table est l’arrière-petite fille
de la vieille bonne qui a élevé la mère de Michel Tavera. Elle est
entrée à neuf ans dans la maison, elle ne l’a jamais quittée. Elle y a
vécu, s’y est mariée, y a fait souche de vingt-cinq enfants et petits-
enfants ; elle vit retirée, maintenant, dans la maison des Tavera à
Ajaccio, servante-aïeule, verte et chenue encore sous la neige de
ses quatre-vingt-six ans : soixante-dix-sept ans de service, autre
temps, autres mœurs ! Il faut venir en Corse pour trouver encore de
pareils spectacles. « C’est l’éloge des maîtres et des serviteurs », ne
puis-je m’empêcher de faire remarquer.
Un dernier trait qui fixera les mœurs patriarcales de la race.
Dans l’immense cuisine, que cinq marches séparent de la salle à
manger, il y a aujourd’hui vingt personnes à table, les fils, les filles et
petits-enfants de la servante-aïeule, venus tous d’Ajaccio, célébrer la
fête du pays. Les Tavera les traitent magnifiquement et leur servent
le même repas que nous mangeons à la salle. Il y a là des marins de
la Compagnie Fraissinet, un berger, un forgeron, un maçon même,
tous les corps de métier.
Tous ces braves gens ont quitté Ajaccio à une heure du matin,
empilés sur une charrette à un cheval que leur prête le maître, toute
la nuit ils ont marché au pas sur les routes en chantant : ils sont
arrivés à l’aube au village. Ils repartiront au crépuscule.
Par les fenêtres ouvertes, derrière les persiennes closes, les
voceri des pleureuses, le lamentable chant funèbre de la morte,
pénètrent avec des senteurs de myrte et du soleil.
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