Helicities in Geophysics, Astrophysics, and Beyond E-Book
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- Herausgeber: John Wiley & Sons
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- Serie: Geophysical Monograph Series
- Sprache: Englisch
Presents cutting-edge studies of helicities from different research fields Helicities play essential roles in numerous geophysical, astrophysical, and magnetohydrodynamic phenomena, thus are studied from various disciplinary viewpoints. Helicities in Geophysics, Astrophysics, and Beyond draws together experts from different research fields to present an interdisciplinary and integrated approach to helicity studies. This synthesis advances understanding of the fundamental physical processes underlying various helicity-related phenomena. Volume highlights include: * Concise introduction to fundamental properties of helicities * Recent developments and achievements in helicity studies * Perspectives from different fields including geophysics, space physics, solar physics, plasma physics, atmospheric and nonlinear sciences * A cohesive mathematical, physical, observational, experimental, and numerical strategy for helicity studies * A synthesized framework for the application of helicity to real-world problems The American Geophysical Union promotes discovery in Earth and space science for the benefit of humanity. Its publications disseminate scientific knowledge and provide resources for researchers, students, and professionals.
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Table of Contents
Cover
Table of Contents
Geophysical Monograph Series
Title Page
Copyright
List of Contributors
Preface
Part I: Helicity Essentials: Basic and Fundamental Concepts
1 Introduction to Field Line Helicity
1.1 Definitions of Field Line Helicity
1.2 Ideal Evolution
1.3 Non‐Ideal Evolution
References
2 Magnetic Winding: Theory and Applications
2.1 Introduction
2.2 Relative Helicity
2.3 Magnetic Winding
2.4 Applications
2.5 Brief Historical Notes
References
3 Transport in Helical Fluid Turbulence
3.1 Introduction
3.2 Helicity: Definition and Properties
3.3 Theoretical Analysis of Helicity Effects in Inhomogeneous Turbulence
3.4 Turbulence Modeling with Helicity
3.5 Application to Swirling Flow
3.6 Large‐Scale Flow Generation by Inhomogeneous Helicity
3.7 Summary and Conclusion
Acknowledgments
References
Part II: Helicity Manifestations in Nature and Their Observations
4 Observations of Magnetic Helicity Proxies in the Solar Photosphere: Helicity With Solar Cycles
4.1 Introduction
4.2 Magnetic Helicity from Observations
4.3 Electric Current Helicity from Observations
4.4 Magnetic Helicity Injection on the Solar Surface
4.5 Magnetic Helicity as an Index in the High Solar Atmosphere
4.6 Questions about Solar Dynamos with Observation of Magnetic Helicity
4.7 Discussion
Acknowledgments
References
5 Chirality of Solar Filaments and the Supporting Magnetic Field
5.1 Introduction
5.2 Magnetic Chirality of Solar Filaments
5.3 Structural Chirality of Solar Filaments
5.4 Magnetic Configuration of Solar Filaments
5.5 Future Prospects
Acknowledgments
References
6 Solar Flares and Magnetic Helicity
6.1 Introduction
6.2 Basic Concepts and Formulations
6.3 Methods of Helicity Estimation
6.4 Practical Applications
6.5 Numerical Models
6.6 Summary and Discussion
Acknowledgments
References
7 Magnetic Helicity Measurements in the Solar Wind
7.1 Introduction
7.2 The Parker Model as a Guide
7.3 Evaluating Magnetic Helicity
7.4 Helicity Profile in the Solar Wind
7.5 Summary and Outlook
Acknowledgements
References
8 Magnetic Helicity in Rotating Neutron Stars
8.1 Introduction
8.2 Theoretical Background
8.3 Astrophysical Applications
8.4 Conclusion
References
9 Writhing From Biophysics to Solar Physics and Back
9.1 Introduction
9.2 Open Linking and Writhe
9.3 Bounding the Writhe of Proteins
References
10 Kinetic Helicity in the Earth's Atmosphere
10.1 Introduction
10.2 General Information on Helicity
10.3 Helicity in Dynamic Atmospheric Processes
10.4 Helical Turbulence in the Earth's Atmosphere
10.5 Concluding Remarks and Outlook
Acknowledgments
References
Part III: Theoretical and Numerical Helicity Modeling
11 Effects of P‐Noninvariance, Particles, and Dynamos
11.1 Introduction
11.2 Seed Magnetic Fields
11.3 Mirror Asymmetry and Helicity
11.4 Mean‐Field Dynamos in Relativistic Plasmas
11.5 Dynamo Driven by Neutrino Asymmetry in the Early Universe
11.6 Conclusions and Discussion
References
12 Stability of Plasmas Through Magnetic Helicity
12.1 Introduction
12.2 Conservation of Helicity
12.3 Relaxation Equilibrium State
12.4 Field Relaxation Constraints
12.5 Plasma Stability
12.6 Galactic and Intergalactic Media
12.7 Helical Intergalactic Bubbles
12.8 Conclusions
Acknowledgments
References
13 Helicity‐Conserving Relaxation in Unstable and Merging Twisted Magnetic Flux Ropes
13.1 Introduction
13.2 Kink Instability and Relaxation in the Solar Corona
13.3 Merging Flux Ropes
13.4 Discussion and Conclusions
Acknowledgments
References
14 Emergence of Magnetic Structure in Supersonic Isothermal Magnetohydrodynamic Turbulence
14.1 Introduction
14.2 Fourier Diagnostics and Shell‐to‐Shell and Helical Transfers
14.3 Results from Previous Research in Incompressible MHD
14.4 Numerical Method
14.5 Results
14.6 Summary and Conclusion
References
15 Nonlinear Mean‐Field Dynamos With Magnetic Helicity Transport and Solar Activity: Sunspot Number and Tilt
15.1 Introduction
15.2 Mean‐Field Dynamos
15.3 Dynamic Nonlinearity: Transport of Magnetic Helicity
15.4 Formation of Sunspots
15.5 Observations of Proxies of Magnetic Helicity in the Sun
15.6 Nonlinear Model of a Mean‐Field Dynamo and Dynamics of the Sunspot Number
15.7 The Effect of Magnetic Helicity Transport and Prediction of Solar Activity
15.8 The Mean Tilt of Sunspot Bipolar Regions
15.9 Concluding Remarks
Acknowledgments
References
16 The Spatial Segregation of Kinetic Helicity in Geodynamo Simulations
16.1 Introduction
16.2 Importance of Kinetic Helicity in Dynamo Action
16.3 Sources of Helicity
16.4 Recent Results from Numerical Simulations
16.5 A Conjecture for Helicity Segregation in Dynamo Simulations
16.6 Conclusions and Future Scope
Acknowledgments
References
Index
End User License Agreement
List of Tables
Chapter 5
Table 5.1 Correspondence between helicity sign and filament chirality.
Chapter 10
Table 10.1 Scale and helicity bulk density of different atmospheric vortex ...
Table 10.2 Values of co‐variances obtained experimentally in Koprov et al. ...
Chapter 16
Table 16.1 Parameters in the dynamo simulations. Here,
ν = κ =
...
List of Illustrations
Chapter 1
Figure 1.1 For this link (the Whitehead link), the Gauss linking number, , ...
Figure 1.2 Examples of a pair of magnetic field lines in different domains, ...
Figure 1.3 Examples of closed (left) and open (right) field lines in a domai...
Figure 1.4 Simple example where the boundary has two parallel components, on...
Figure 1.5 Simple example of an initial potential “arcade” being sheared by ...
Figure 1.6 Evolution of field line helicity in a magneto‐frictional model of...
Figure 1.7 Field line helicity in a nonlinear force‐free model of active reg...
Figure 1.8 Evolution of field line helicity during line‐tied resistive relax...
Chapter 2
Figure 2.1 Two closed and linked loops, and , with specified directions....
Figure 2.2 A “tubular” domain between two horizontal planes at and . At...
Figure 2.3 Two discrete and braided flux tubes between and . Both tubes h...
Figure 2.4 A diagram of two crossing flux tubes connected to the same lower ...
Figure 2.5 The emergence and submergence of a twisted flux tube. (a) The tub...
Figure 2.6 Accumulations of the magnetic winding and helicity fluxes (normal...
Figure 2.7 The magnetic winding accumulation for the first 80 hours of the e...
Chapter 3
Figure 3.1 Helicity as the velocity–vorticity correlation. (a) Positive heli...
Figure 3.2 Topological interpretation of helicity. (a) Linked vortex tubes (...
Figure 3.3 Velocity correlations. (a) Longitudinal, (b) transverse, and (c) ...
Figure 3.4 Cyclone swirl accompanied by updraft.
Figure 3.5 Configuration of swirling flow in a straight pipe.
Figure 3.6 Mean axial velocity profile with and without swirl.
Figure 3.7 The radial profile of the mean axial velocity and axial decay of ...
Figure 3.8 The possible physical origin of global flow induction is due to t...
Figure 3.9 Setup of DNSs for validating the helicity effect. A periodic box ...
Figure 3.10 Temporal evolution of turbulent helicity (top) and mean flow (bo...
Figure 3.11 Reynolds stress and helicity gradient at the early stage of evol...
Figure 3.12 Induced mean velocity and turbulent helicity at the developed st...
Figure 3.13 Local Cartesian coordinate .
Figure 3.14 Angular momentum transport in the Sun based on the inhomogeneous...
Figure 3.15 Swirling wind configuration of a tropical cyclone.
Chapter 4
Figure 4.1 Distribution of spiral patterns of active regions in the solar (a...
Figure 4.2 Active region NOAA 6619 observed on 10 May 1991. (a) Photospheric...
Figure 4.3 The active region NOAA 6619, taken at Huairou Solar Observing Sta...
Figure 4.4 Sigmoidal field lines wrapping around a flux rope in active regio...
Figure 4.5 Time variation of the mean slope of linear fit calculated from th...
Figure 4.6 Butterfly diagram of the mean current helicity density . The siz...
Figure 4.7 (a) Distribution of the averaged twist . (b) Current helicity ...
Figure 4.8 (a) (solid line) and (dotted line) for NOAA 11158 at 23:59:54...
Figure 4.9 (a) Distribution of the average scale exponent of the current h...
Figure 4.10 Active region NOAA 10488 on 26–31 October 2003. Left: Grayscale ...
Figure 4.11 Net transfer of helicity into the southern corona and wind (pr...
Figure 4.12 Injective magnetic helicity flux from the northern (solid line) ...
Figure 4.13 (a) Net injective helicity contributed from both hemispheres. Th...
Figure 4.14 The hemispheric handed rule trend of the proportion of soft X‐ra...
Figure 4.15 The mean latitudinal distribution of soft X‐ray loops with left‐...
Figure 4.16 (a) Correlation of of the active region pairs connected by tra...
Figure 4.17 Yearly mean values of and for samples from active regions fr...
Chapter 5
Figure 5.1 Time‐longitudinal graphs of the magnetic chirality of three types...
Figure 5.2 Schematic sketch demonstrating the correspondence between the hel...
Chapter 6
Figure 6.1 Solar flare and CME. (Left) An X2.2‐class flare in NOAA AR 11158,...
Figure 6.2 Schematic diagram that demonstrates relevant examples of
Ftot > 0
...
Figure 6.3 An example of estimating
F
tot
for NOAA AR 8011 with the method ...
Figure 6.4 Long‐term trend of increase in
Δ
H
R
across the photospheric s...
Figure 6.5 Long‐term increasing trend of
H
R
in the coronal volume of NOAA AR...
Figure 6.6 Maps of the relative magnetic helicity flux density for NOAA AR 8...
Figure 6.7 Time profiles of
Δ
H
R
(black line) and
Φ
(blue line) for...
Figure 6.8 (Left) Coronal magnetic field lines extrapolated from the vertica...
Figure 6.9 Flare‐productive probability as a function of
|〈Ftot〉|
...
Figure 6.10 Scatter plots of ARs as functions of free magnetic energy (denot...
Figure 6.11 NLFFF extrapolation on NOAA AR 10953. (a, c) Coronal observation...
Figure 6.12 (a, b) Magnetic fields for ground‐truth flux‐emergence simulatio...
Figure 6.13 Examination of helicity ratio (
|
H
j
|/|
H
R
|
) by Pariat et al. ([201...
Chapter 7
Figure 7.1 Normalized magnetic helicity density as a function of the helio...
Figure 7.2 Power spectrum (top panel) and reduced magnetic helicity spectrum...
Figure 7.3 Schematic diagram of reduced magnetic helicity with clear signatu...
Figure 7.4 Reduced magnetic helicity in the solar polar regions evaluated fr...
Figure 7.5 (top) Sunspot number variation. Data courtesy of the World Data C...
Chapter 8
Figure 8.1 The structure of the magnetic field presented in equation (8.59)....
Figure 8.2 The schematic structure of the stellar magnetic field under consi...
Chapter 9
Figure 9.1 A visualization of the Călugăreanu theorem. (a) A ribbon structur...
Figure 9.2 Visual interpretations of the linking of a pair of curves. (a) ...
Figure 9.3 Viewpoints and directions of a trefoil knot, the curve shown in...
Figure 9.4 Open ribbons. (a) An open‐ended ribbon whose endpoints stay fixed...
Figure 9.5 Example closure figures. (a) Relative linking. Fuller, 1978, (b) ...
Figure 9.6 Illustrations of winding. (a) A pair of open‐ended curves embed...
Figure 9.7 Polar writhe calculations of plectoneme geometries. (a) values ...
Figure 9.8 Example DNA applications of the polar writhe . (a) (blue) and ...
Figure 9.9 The ribbon diagram for lysozyme (Protein Data Bank [PDB]: 1DPX). ...
Figure 9.10 A visual example of the contribution of secondary structure to t...
Figure 9.11 In blue, a scatter plot of writhe against the number of secondar...
Figure 9.12 (a) A plot of the local polar writhe for the hypothetical protei...
Figure 9.13 (a) A plot of the local polar writhe for Rhizoctonia solani aggl...
Chapter 10
Figure 10.1 A schematic of helicity balance in intense atmospheric vortices....
Figure 10.2 Frequency spectra of contributions to helicity calculated based ...
Figure 10.3 Triple correlation frequency spectra calculated from two‐hour re...
Figure 10.4 Photo of a tetrahedron composed of four sonic anemometers to mea...
Figure 10.5 Schematic of the rectangular tetrahedron
ABCD
used for helicity ...
Figure 10.6 Turbulent helicity values (in ms) measured during the day in ...
Figure 10.7 Histogram of instantaneous turbulent helicity values measured in...
Figure 10.8 Time dependence of (in Ks
−1
, red line) and (in Ks
−2
...
Chapter 11
Figure 11.1 Evolution of primordial magnetic fields in the universe after th...
Figure 11.2 The evolution of the chiral imbalance (a, b), the magnetic energ...
Figure 11.3 The evolution of the Fourier modes (a, b) for the toroidal field...
Figure 11.4 The evolution of the Fourier modes behind the shock at , (a)
Chapter 12
Figure 12.1 Schematic representation of the initial condition of the numeric...
Figure 12.2 Helical spheromak configuration showing two of the magnetic fiel...
Figure 12.3 Coherence measure of the intergalactic bubbles depending on th...
Chapter 13
Figure 13.1 Magnetic field evolution in a kink‐unstable cylindrical flux rop...
Figure 13.2 Evolution of a reconnecting twisted loop in a gravitationally st...
Figure 13.3 Electron energy spectra in the model of a reconnecting twisted l...
Figure 13.4 Poloidal field lines for the relaxation model of merging flux ro...
Figure 13.5 Contours of axial current in the midplane of an array of 23 para...
Chapter 14
Figure 14.1 Slices of the magnetic helicity density, normalized by its mean ...
Figure 14.2 Mass density properties during the hydrodynamic statistically st...
Figure 14.3 Magnetic helicity spectra compensated by at an instant when ....
Figure 14.4 Alfvénic helicity spectra compensated with , at the same instan...
Figure 14.5 Test of the Alfvénic balance with (a) for low‐Mach‐number runs...
Figure 14.6 Magnetic helicity transfer rates from shell to shell , (com...
Figure 14.7 Filtering the transfer rates displayed in Fig. 14.6 according to...
Figure 14.8 Dominant helical transfer terms (equation (14.14)) from Fig. 14....
Figure 14.9 (a, b, c): Moduli of the geometric helical factor (equations (14...
Figure 14.10 Graphical visualisation of the main results.
Chapter 15
Figure 15.1 Mean observed current helicity for solar active regions (white/b...
Figure 15.2 Comparison of numerical results with observations. Upper panel: ...
Figure 15.3 The long‐term evolution of solar activity. Top panel: butterfly ...
Figure 15.4 Comparison of the results of the one‐month preliminary forecast ...
Figure 15.5 Comparison of the one‐month forecast of the monthly sunspot numb...
Figure 15.6 The mean tilt denoted as (in degrees) as a function of the lat...
Figure 15.7 Butterfly diagrams of (a) the normalized mean tilt (see equati...
Figure 15.8 The mean tilt is determined as (in degrees) against the latitu...
Chapter 16
Figure 16.1 (a) The α
2
‐dynamo. Davidson ([2014]). (b) Parker's lift‐and‐twis...
Figure 16.2 Helicity in two dynamo simulations, S1 (top row) and S2 (middle...
Figure 16.3 Fluctuating relative helicity in four simulations. Simulation p...
Figure 16.4 Helicity sources (left to right): , , , , and for S1 (firs...
Figure 16.5 Gaussian blob under rotation: (a) isosurfaces of
u
z
, (b)
ω
Figure 16.6 (a) Azimuthal root‐mean‐square of the thermal source of vortici...
Figure 16.7 Snapshot in the mid
xz
plane of (a)
−c
, (b)
u
z
, (c)
h
at ...
Figure 16.8 Thermal helicity source
〈
H
T
1
〉
y
,
〈
H
T
2
〉
y
,...
Figure 16.9 Isosurfaces of
u
z
colored by (a)
h
, (b) e.m.f, (c)
y
‐averaged e...
Figure 16.10 (top row) and (bottom row) for the simulations S1, S2, a...
Figure 16.11 The evolution of the horizontally averaged values of (a, b)
h
,...
Guide
Cover
Table of Contents
Geophysical Monograph Series
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Preface
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Helicities in Geophysics, Astrophysics, and Beyond
Kirill Kuzanyan
Nobumitsu Yokoi
Manolis K. Georgoulis
Rodion Stepanov
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Library of Congress Cataloging‐in‐Publication Data
Names: Kuzanyan, Kirill, editor. | Yokoi, Nobumitsu, editor. | Georgoulis, Manolis K., editor. | Stepanov, Rodion, editor.
Title: Helicities in geophysics, astrophysics, and beyond / editors Kirill Kuzanyan, Nobumitsu Yokoi, Manolis K. Georgoulis, Rodion Stepanov.
Description: Hoboken, NJ : Wiley, 2024. | Includes bibliographical references and index.
Identifiers: LCCN 2023035109 (print) | LCCN 2023035110 (ebook) | ISBN 9781119841685 (hardback) | ISBN 9781119841692 (adobe pdf) | ISBN 9781119841708 (epub)
Subjects: LCSH: Particles (Nuclear physics)–Helicity. | Geophysics. | Astrophysics.
Classification: LCC QC793.3.H44 H45 2024 (print) | LCC QC793.3.H44 (ebook) | DDC 521–dc23/eng/20231017
LC record available at https://lccn.loc.gov/2023035109
LC ebook record available at https://lccn.loc.gov/2023035110
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Cover Image: © AceClipart_Etsy/Pixabay; Courtesy of Nobumitsu Yokoi
LIST OF CONTRIBUTORS
Arron N. Bale
Department of Mathematical Sciences
Durham University
Durham, UK
Mitchell A. Berger
Department of Mathematics
University of Exeter
Exeter, UK
Philippa K. Browning
Department of Physics and Astronomy
University of Manchester
Manchester, UK
Simon Candelaresi
School of Mathematics and Statistics
University of Glasgow
Glasgow, UK
Jie Chen
National Astronomical Observatories
Chinese Academy of Sciences
Beijing, China
Peng‐Fei Chen
Key Lab of Modern Astronomy & Astrophysics
School of Astronomy and Space Science
Nanjing University
Nanjing, China
Otto Chkhetiani
A. M. Obukhov Institute of Atmospheric Physics
Russian Academy of Sciences
Moscow, Russia
Peter Davidson
Department of Engineering
University of Cambridge
Cambridge, UK
Fabio Del Sordo
Institute of Space Sciences
Barcelona, Spain
and
Catania Astrophysical Observatory
National Institute for Astrophysics
Catania, Italy
Maxim Dvornikov
Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN)
Russian Academy of Sciences
Moscow, Russia
Mykola Gordovskyy
Department of Physics and Astronomy
University of Manchester
Manchester, UK
Alan W. Hood
School of Mathematics and Statistics
University of St Andrews
St Andrews, UK
Nathan Kleeorin
Department of Mechanical Engineering
Ben‐Gurion University of the Negev
Beer‐Sheva, Israel
and
Institute of Continuous Media Mechanics
Russian Academy of Sciences
Perm, Russia
Michael Kurgansky
A. M. Obukhov Institute of Atmospheric Physics
Russian Academy of Sciences
Moscow, Russia
Kirill Kuzanyan
Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN)
Russian Academy of Sciences
Moscow, Russia
and
Institute of Continuous Media Mechanics
Russian Academy of Sciences
Perm, Russia
Jihong Liu
National Astronomical Observatories
Chinese Academy of Sciences
Beijing, China
and
School of Science
Shijiazhuang University
Shijiazhuang, China
David MacTaggart
School of Mathematics and Statistics
University of Glasgow
Glasgow, UK
Wolf‐Christian Müller
Center for Astronomy and Astrophysics
Technical University of Berlin
Berlin, Germany
and
Max‐Planck/Princeton Center for Plasma Physics
Princeton, NJ, USA
Yasuhito Narita
Institute for Theoretical Physics
Technical University of Braunschweig
Braunschweig, Germany
Sung‐Hong Park
Korea Astronomy and Space Science Institute
Daejeon, Republic of Korea
and
Institute for Space‐Earth Environmental Research
Nagoya University
Nagoya, Japan
and
W. W. Hansen Experimental Physics Laboratory
Stanford University
Stanford, CA, USA
Christopher Prior
Department of Mathematical Sciences
Durham University
Durham, UK
Avishek Ranjan
Department of Mechanical Engineering
Indian Institute of Technology Bombay
Mumbai, India
Igor Rogachevskii
Department of Mechanical Engineering
Ben‐Gurion University of the Negev
Beer‐Sheva, Israel
and
Nordic Institute for Theoretical Physics
KTH Royal Institute of Technology and Stockholm University
Stockholm, Sweden
Nikolai Safiullin
Department of Information Security
Ural Federal University
Ekaterinburg, Russia
and
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
Dmitry D. Sokoloff
Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN)
Russian Academy of Sciences
Moscow, Russia
and
Department of Physics
Moscow State University
Moscow, Russia
Jean‐Mathieu Teissier
Center for Astronomy and Astrophysics
Technical University of Berlin
Berlin, Germany
Shin Toriumi
Institute of Space and Astronautical Science
Japan Aerospace Exploration Agency
Sagamihara, Japan
Haiqing Xu
National Astronomical Observatories
Chinese Academy of Sciences
Beijing, China
Shangbin Yang
National Astronomical Observatories
Chinese Academy of Sciences
Beijing, China
and
School of Astronomy and Space Science
University of Chinese Academy of Sciences
Beijing, China
Xiao Yang
National Astronomical Observatories
Chinese Academy of Sciences
Beijing, China
Anthony R. Yeates
Department of Mathematical Sciences
Durham University
Durham, UK
Nobumitsu Yokoi
Institute of Industrial Science
University of Tokyo
Tokyo, Japan
Hongqi Zhang
National Astronomical Observatories
Chinese Academy of Sciences
Beijing, China
PREFACE
Helicities, defined by the volume integral of the inner product of a vector field and its curled counterpart in a generalized sense (including cross helicity and generalized helicity in Hall magnetohydrodynamics [MHD]), are known to play essential roles in several geophysical, astrophysical, and space plasma phenomena, from dynamo actions in geo/planetary magnetism and astrophysical objects to solar and stellar magnetic field evolution, mass condensation in star‐forming regions, accreting jet formation near compact massive objects, amplification of the magnetic field in the Universe, magnetic confinement in fusion plasmas, etc. Aligning with these important roles in various research fields, helicities have been studied using various methods from different viewpoints.
This book is a synopsis of recent developments and achievements in helicity studies by leading scientists. It grew from a series of gatherings entitled “Online Advanced Study Program on Helicities in Astrophysics and Beyond” (https://helicity2020.izmiran.ru) in Fall 2020 and Spring 2021, which were held online as a substitute for in‐person interactions within the research community during the Covid‐19 pandemic.
These gatherings built on years of scientific studies and previous events. Magnetic helicity has been intensively studied in recent decades from observational, theoretical, and modeling viewpoints in fields such as general astrophysics, solar physics, plasma and fluid dynamics, and pure mathematics. A series of focused events on helicity have also taken place, such as the 1998 AGU Chapman Conference in Boulder, Colorado, USA; the 2009 and 2013 Helicity Thinkshops in China (https://sun10.bao.ac.cn/old/meetings/HT2009/ and https://sun10.bao.ac.cn/old/meetings/HT2013/); the 2017 Helicity Thinkshop in Japan (http://www.iis.u-tokyo.ac.jp/∼nobyokoi/thinkshop/), part of the SEIKEN Symposium (https://science-media.org/conference/23); and the Program on Solar Helicities in Theory and Observations at NORDITA in 2019 (https://indico.fysik.su.se/event/6548/), among others.
The Online Advanced Study Program in 2020 and 2021 was hosted by IZMIRAN, the Russian Academy of Sciences, in Moscow. It featured seminars delivered by international experts in astrophysics, geophysics, and many fields of natural sciences involving observational and theoretical studies of magnetic, kinetic, and other helicities. We deliberately took an interdisciplinary and multidisciplinary approach to attract broader audiences. Some events exceeded 200 online attendees, and we thank all the participants for their energy and for directly and indirectly contributing to the contents of this book.
This book has two primary aims. The first is to provide a perspective on helicities relevant to geophysics, astrophysics, physical and space plasma sciences (including biological and quantum fluids). The second is to propose future directions for helicity studies in these fields using cohesive theoretical, observational, experimental, and numerical strategies for constructing models applicable to real‐world phenomena. Compact, readable introductions in each chapter acquaint the reader with advanced topics and aspects of helicity studies.
We could not cover all topics in the natural sciences that involve helicities, but we have tried to give a reader the broadest possible representation of these fields. Certain important topics are lacking, such as the possible detection of magnetic helicity proxies in observable fast‐rotating stars, a perspective on kinetic and magnetic helicities in the Earth's core, and laboratory fluid and plasma experiments. We hope such topics will be covered in future publications and books.
The book consists of 16 chapters divided into three parts. Part I discusses helicity basics and fundamental concepts. In Chapter 1, Anthony Yeates and Mitchell Berger propose that field‐line helicity provides a finer local topological description of magnetic flux than the usual global magnetic helicity integral, with invariant properties preserved. They present a way to appropriately define field‐line helicity in different volumes. They also discuss the time evolution of field‐line helicity under both boundary motions and magnetic reconnection. In Chapter 2, David MacTaggart introduces the notion of magnetic winding from a theoretical perspective. Magnetic winding is a renormalization of magnetic helicity, directly measuring field‐line topology. This new notion is complemented by an application to observations of solar active regions. In Chapter 3, Nobumitsu Yokoi constructs a turbulent transport model including helicity. The model reveals that the helical contribution may suppress momentum transport with a remarkable feature on the induction of a large‐scale flow caused by an inhomogeneous coupling between helicity and rotation. This flow appears to have numerous applications to astro‐ and geophysical phenomena.
Part II contains several reviews of manifestations of helicities in various natural phenomena and their observations. Chapter 4, by Hongqi Zhang et al., reviews longstanding diagnostic tools available for inferring proxies of both the magnetic and current helicity in the solar atmosphere. They analyze solar atmospheric helicity at short and long timescales, in the latter showing analogs of the butterfly diagram for sunspots using the mean current helicity and twist, that have important implications for the solar dynamo. Chapter 5, by Peng‐Fei Chen, casts doubt on how one can observationally infer the chirality of solar filaments by observing the skewness or bearing of the filament barbs. While this suggestion is gaining traction, it is far from unanimously accepted in the solar physics community but is eloquently presented to spur discussion and debate. Chapter 6, by Shin Toriumi and Sung‐Hong Park, discusses various diagnostic tools for magnetic helicity in local (i.e., active‐region) solar scales, aiming to understand and ultimately predict solar flares. Describing immense magnetic complexity, they make a twofold effort to distinguish populations of flaring active regions from the majority pool of non‐flaring regions in order to understand the separator(s) of these populations and then use this knowledge for prediction purposes. Chapter 7, by Yasuhito Narita, reviews methods of evaluating magnetic helicity in the solar wind in terms of both single‐ and multipoint measurements. Methodologies invariably aim to resolve the transport problems of helicity, magnetic flux, and energy of the Sun into the heliosphere. On short (compared to magnetohydrodynamic turbulence) spatial scales in the ion‐kinetic domain, the observations reveal nonzero helicity as a signature of linear‐mode wave excitation, such as the kinetic Alfven waves and whistler waves. This differs in each solar hemisphere and varies with the solar cycle. In Chapter 8, Maxim Dvornikov reviews the role of magnetic helicity evolution in rotating neutron stars. He utilizes the conservation law for the sum of the chiral imbalance of charged particle densities and the density of magnetic helicity and explores the possibility of X‐ray or gamma bursts observed in magnetars due to this mechanism. He argues that the quantum contribution dominates the classical contribution in the surface terms in standard MHD but only for neutron stars with rigid rotation. He shows that the characteristic time of the helicity change is in accord with the magnetic cycle period of certain pulsars. Chapter 9, by Christopher Prior and Arron Bale, deals with writhing and its prospects for wider interdisciplinary applications and interaction between biophysics, solar physics, and other disciplines. Writhing quantifies a structure's global self‐entanglement (knotting) and plays a fundamental role in DNA compactification (supercoiling). Earlier results on magnetic helicity in solar physics can be used for biophysical applications such as understanding protein structures through their writhing measures. Chapter 10, by Otto Chkhetiani and Michael Kurgansky, explores kinetic helicity in the Earth's atmosphere and its role in atmospheric turbulence. The helicity balance and fluxes are used to analyze atmospheric vortices such as tropical cyclones, tornadoes, dust devils, and Ekman boundary layer dynamics. The helical properties of turbulence within the atmospheric boundary layer have been probed by direct pioneering measurements of turbulent helicity in natural atmospheric conditions.
Part III on theoretical and numerical modeling of helicities opens with Chapter 11 by Victor Semikoz and Dmitry Sokoloff, who review cosmological dynamos and explore the P‐noninvariance of elementary particles. This presents a possibility for nonclassical dynamo generation in the early Universe based on the intrinsic mirror asymmetry of elementary particles. In Chapter 12, Simon Candelaresi and Fabio Del Sordo point out that magnetic helicity constrains the dynamics of plasmas. They discuss how magnetic helicity stabilizes the plasma and prevents its disruption, with reference to observations, numerical experiments, and analytical results. Several illustrative examples in the solar corona are presented, as well as fusion devices, galactic and extragalactic medium, and extragalactic bubbles. Chapter 13, by Philippa Browning et al., addresses the contentious topic of magnetic relaxation in the solar corona and its implications for magnetic helicity. They arrive logically at the concept of magnetic relaxation, likely in terms of the Taylor hypothesis, and show how ideal instabilities and merging magnetic flux ropes in the corona can lead to relaxation. Implications for laboratory plasmas and the elusive coronal heating mechanism(s) are also discussed. Chapter 14, by Jean‐Mathieu Teissier and Wolf‐Christian Müller, deals with the formation and sustainment of magnetic structures in supersonic isothermal magnetohydrodynamic turbulence. They review the first results obtained through direct numerical simulations of isothermal compressible MHD at Mach numbers ranging from subsonic to about 10, finding contributions of the local and nonlocal, direct, and inverse transfer of magnetic helicity in supersonic MHD regimes. Chapter 15, by Nathan Kleeorin et al., discusses nonlinear mean‐field dynamos, with special reference to various mechanisms for sunspot formation and the prediction of solar activity. Based on nonlinear dynamo equations, including the model equation for magnetic helicity, they explain existing observations of magnetic helicity in the Sun and dynamical solar activity. The contributions of magnetic helicity, large‐scale magnetic fields, and differential rotation to the mean tilt angle of sunspot bipolar regions are also discussed. Finally, Chapter 16, by Avishek Ranjan and Peter Davidson, is dedicated to the origin of the spatial segregation of kinetic helicity in dynamo simulations. They discuss various sources of kinetic helicity, including helical waves such as inertial waves. Strong spatial correlations of the segregation pattern of helicity and the source term due to buoyancy exhibited in numerical simulations are interpreted using helical wave propagation.
Together, these chapters present different aspects of helicities in diverse scientific contexts, from basic concepts and fundamental properties to manifestations in several natural phenomena, as well as theoretical and numerical models. The book also presents the possibility of tackling real‐world problems via the many forms and flavors of helicity. We hope the breadth of information and evidence presented in this book will spur discussions and debates that will lead to an enhanced, broader scientific understanding of complexity in various natural systems and networks. We trust that this volume contributes to further developments in this fascinating, challenging, and ever‐evolving subject.
We are grateful to the organizations that have hosted various in‐person and online gatherings relating to helicity studies, such as the National Astronomical Observatories of China, University of Tokyo in Japan, NORDITA in Sweden, and IZMIRAN in Russia. We thank all the participants of these meetings for their contributions and discussions, especially the speakers who gave talks, posters, and online presentations. We would also like to acknowledge the support and hospitality of the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, during the Dynamo Theory Programme (DYT2) held in Fall 2022, where the contents of this book were finalized.
Our thanks go to the AGU Books Editorial Board and the reviewers of our proposal, who encouraged us and helped improve this book. We acknowledge the dedication of each of the chapter contributors for the time they spent preparing their manuscripts. We greatly appreciate the patience and professionalism of those who reviewed chapters to ensure that this book was of the highest standards as well as to increase its readability and appeal to the community. We also acknowledge support from AGU Publications and Wiley, including technical assistance and advice from their staff. Finally, we thank our colleagues, friends, and families for their patience and inspiration while we worked on this project.
Kirill Kuzanyan
Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation Russian Academy of Sciences, Moscow, Russia
Nobumitsu Yokoi
Institute of Industrial Science
University of Tokyo, Tokyo, Japan
Manolis K. Georgoulis
Research Center for Astronomy and Applied Mathematics
Academy of Athens, Athens, Greece
Rodion Stepanov
Institute of Continuous Media Mechanics
Russian Academy of Sciences, Perm, Russia
Part IHelicity Essentials: Basic and Fundamental Concepts
1Introduction to Field Line Helicity
Anthony R. Yeates1 and Mitchell A. Berger2
1 Department of Mathematical Sciences, Durham University, Durham, UK
2 Department of Mathematics, University of Exeter, Exeter, UK
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 interpretation. We also review the time evolution of field line helicity under both boundary motions and magnetic reconnection.
1.1. DEFINITIONS OF FIELD LINE HELICITY
We briefly review topological measures of magnetic field structure. Some of these refer to the structure of the total field within a volume (the magnetic helicity), and others to the relationship between individual pairs of field lines (linking and winding). Field line helicity is intermediate between these ideas, as it measures the net linking or winding of one field line with the total field. Helicity integrals depend on both the tangling of field lines with each other and the topology and geometry of the volume in which the field lines reside. A strong rationale for considering topological measures is that field line topology is conserved in any ideal evolution of the field (Moffatt, 1969).
1.1.1. Definitions of Field Line Helicity for Closed Volumes
The simplest situation occurs with two closed field lines residing within a volume where the field lines do not cross the boundary of (or within an infinite space). Historically, Gauss (1809) discovered a double line integral, which measures the linking of two closed curves and . Let positions on the curves be given by and . Then
where . This may be calculated by counting signed crossings (see Fig. 1.1).
For a magnetic field consisting of a finite collection of closed magnetic flux tubes (a very special case), we can define an overall invariant
where represents the magnetic flux of each tube and is the linking number between the tubes. This invariant is called the magnetic helicity. For a more general magnetic field consisting entirely of closed field lines (still a special case), we can take in equation (1.2) so that the sums become integrals. Accounting for the magnetic flux, the tangent vectors in equation (1.1) become magnetic field vectors, and the magnetic helicity may be written
Figure 1.1 For this link (the Whitehead link), the Gauss linking number, , is zero. In general, the linking number equals half the difference between the number of positive crossings and the number of negative crossings, as seen in any plane projection.
Here the integral is over all space or over a volume containing all field lines. In fact, in this equation, we do not need to assume that the field lines close upon themselves – some may ergodically fill a subvolume or twist around a toroidal surface with an irrational winding number. Such fields can be constructed as the limit as of a sequence of fields consisting of thin closed flux tubes (Arnol'd and Khesin, 1998). Thus, regardless of whether the field lines close upon themselves, we can approximate any magnetic field as a collection of such tubes.
Returning to the finite set of flux tubes with fluxes , notice that equation (1.2) can be written as
The limit as of is another topological measure, this time defined for every individual magnetic field line – this is what we call the field line helicity.
Since the field lines do not cross the boundary of the volume, we can now use the Biot‐Savart formula
to find
Historically, invariance of the expression in equation (1.6) was known before the interpretation in terms of Gauss linking number was identified (Moffatt, 1969; Moffatt and Ricca, 1992). For a closed line within the field, the equivalent expression for the field line helicity will simply be
This is just the net magnetic flux encircled by , and since is a closed curve, it is manifestly gauge invariant. We remark that Yahalom (2013) interprets as a magnetohydrodynamic analog of the Aharonov‐Bohm effect from quantum mechanics: if , then , so that a nonzero requires jumps in around closed field lines, with such jumps giving .
Suppose our volume is simply connected (e.g., a sphere, not a torus). Note that because is closed, we can gauge transform for an arbitrary gauge function without affecting . If we have a multiply connected volume, a difficulty arises: the wrong choice of may imply the existence of magnetic flux in the external region that threads through a hole, which makes dependent on the unknown external field. To remedy this, one can restrict the gauge of so that for any closed curve on the boundary encircling a hole the long way around (see also MacTaggart and Valli, 2019).
1.1.2. Definitions of Field Line Helicity for Open Volumes
We now go to volumes with flux crossing the boundary. Suppose a field line forms a loop, with both ends on the same boundary (as, for example, in Fig. 1.2b). An early approach (Antiochos, 1987) involves drawing a geodesic between the two endpoints, thus forming a closed curve for which can be measured. This scheme does not integrate to the full helicity, however. We wish to define field line helicity to be consistent with full helicity but also so that gauge transformations of do not change our results. There are two principal approaches to this problem. One consists of defining topological quantities such as winding numbers and then summing to find the helicities. If the boundary flux does not move, ideal motions of the field lines will not change the windings, so field line helicity will be conserved. If the boundary flux does move, the topological structure can cross the boundary, leading to changes in helicity measures. This will be discussed in section 1.2.
The second approach is to measure helicity relative to a potential field: i.e., a field with zero current in the volume. As the potential field is the minimum energy state for given boundary conditions, its structure depends only on the shape of the boundary and the distribution of flux on the boundary (and for multiply connected volumes, specified new fluxes around each hole). For highly symmetric volumes such as those bounded by planes or spheres, the two approaches give identical results.
Figure 1.2 Examples of a pair of magnetic field lines in different domains, including (a) the volume between two parallel planes; (b) the volume above a plane; (c) a spherical shell (the volume between two concentric spheres); and (d) the volume between two more general surfaces.
For more complicated volumes, the shape of the boundary or boundaries can add extra terms that distinguish the two approaches, as discussed in section 1.1.2. A definition of field line helicity can also be based on relative helicity (Moraitis et al., 2019), to be discussed in section 1.1.3.
Volumes between Parallel Planes
Suppose the volume of interest is the space between parallel planes, say and , as illustrated in Fig. 1.2a. We will first assume that all field lines start at and end at . Then no two field lines link in the Gauss sense; however, they will twist about each other through some angle . The winding number makes this an integer for complete twists:
For the example in Fig. 1.2a, we have .
We can now adopt formulae for the helicity and field line helicity similar to the closed field case, with the winding number replacing the linking number. We denote horizontal positions by . Also, we distinguish positions in the bottom plane by . Let be the field line with foot point at . Also let be the winding number between and . Then the field line helicity of will be
Let us build up one plane at a time – in other words, find expressions for . For now, we make the simplifying assumption that everywhere. There are two contributions to the derivative: first (I), horizontal flux in the plane can wrap around the field line . Second (II), can move horizontally about other field lines. Now is the angular direction at about , with . Also, a field line traveling from to satisfies the equation
Thus for a field line at (writing , etc.),
Next, the field line can move horizontally around other field lines. In this case,
Summing the last two equations, the total change in winding between lines and is
Let be the horizontal plane at height . Suppose we add up contributions from all over the plane: we multiply the previous expression by and integrate. Dividing by to make complete turns into integers,
This expression for the field line helicity can be expressed very simply using a special vector potential – the winding gauge (Prior and Yeates, 2014): for a point on field line at ,
We have
so
Volume above a Plane
In the previous section, we assumed that everywhere. Thus the field lines were braided about each other, and simple winding numbers sufficed to measure pairwise entanglements. If we remove the restriction on , field lines can go up and down. We will also remove the restriction of an upper plane in the following discussion, although that is not strictly necessary. Such a domain is illustrated in Fig. 1.2b.
If field lines can form loops that go up and down, we can measure topological structure in several ways. One method is to cut the field lines at any local minima or maxima where and then sum the winding numbers as before (Berger and Prior, 2006). The winding gauge will still work in this situation, as the formulae for calculating do not require .
A second method, when considering the winding between two loops, consists of examining the angles of the quadrilateral formed by the four foot points of the loops (Berger, 1986; Demoulin et al., 2006). Consider the upper half space , shown in Fig. 1.2b. The foot points of loops 1 and 2 are labeled , , , and , where at and . If the loops cross as seen from above, we assume that loop 1 is the upper loop. Consider the quadrilateral . Let and be the angles at vertices and , respectively. Then (from considering the net change of helicity found when bringing the two loops in from infinity) Berger (1984) showed that
Poloidal‐Toroidal Representation
The winding gauge employed in the previous sections is equivalent to the vector potential found when employing a poloidal‐toroidal representation for the field (Moffatt, 1978; Berger and Hornig, 2018). Because the magnetic field is a three‐vector subject to one condition (), we can often express it in terms of two scalar functions. Here we let and be the poloidal and toroidal functions. Let the horizontal surface Laplacian be . In any plane , is determined by the vertical flux,
while is determined by the vertical current,
We assume , , , and all vanish at infinity. Then the surface Laplacians have unique solutions,
We can now create a vector potential
Note that the horizontal components of only involve the poloidal function . Furthermore, the horizontal divergence of vanishes: . One can then employ equations (1.21) and (1.22) to show that this vector potential is identical to the winding gauge, , provided that the volume considered is infinite in the and directions. We can again employ this vector potential to calculate field line helicities. The total helicity can also be regarded as the net linking of toroidal and poloidal fields (Berger and Hornig, 2018),
Volumes Bounded by a Sphere
Spherical boundaries – such as in Fig. 1.2c – can, for the most part, be treated the same way as planar boundaries, with the radial unit vector replacing the vertical vector . In essence, winding angles become azimuthal angles. Suppose we consider three points , , and on a sphere. Rotate the spherical coordinates so that is at the North pole and is at the azimuthal coordinate . Then the angle equals the azimuthal coordinate of .
The poloidal and toroidal flux functions are (Kimura and Okamoto, 1987)
where
is the spherical distance between and . With these functions,
Some care must be taken in spherical geometries. One must ensure that no magnetic monopoles are hiding inside the sphere – i.e., the net flux through the sphere must vanish (similarly for the net current).
Also, in the special case of a spherical shell geometry where flux enters at the inner sphere and exits at the outer sphere (as in Fig. 1.2c), the separation of helicity into self helicity and mutual helicity can be ambiguous (Campbell and Berger, 2014). Self helicity measures the twist and writhe of individual tubes, while mutual helicity measures linking and intertwining between tubes. This will not affect the field line helicity, however.
More General Volumes
For volumes bounded by planes or spheres, there are natural definitions of angle and winding angle depending only on the Euclidean metric. The winding gauge, or the poloidal‐toroidal vector potential, captures this natural definition. We could also consider a volume with boundaries consisting of a bottom plane and side boundaries, where flux only crosses the bottom plane. Such a volume could represent a closed active region. We could choose winding angles without regard to the side boundaries, using the same winding gauge. An alternative will be described later when using relative measures of helicity.
Next, consider a volume bounded by one or two simply connected surfaces that lack the symmetry of planes or spheres. An example is shown in Fig. 1.2d. Here, definitions of angle are less obvious. However, we can still employ a generalization of the poloidal‐toroidal representation (Berger and Hornig, 2018) to define the winding gauge.
Let our volume be sliced into nested surfaces. Employ coordinates , where labels one of the nested surfaces. The toroidal field (which is responsible for the current perpendicular to surfaces of constant ) can be written in terms of a toroidal function as before:
However, the poloidal field is more complicated. Let the normal component of the curl be given by the operator ,
This operator has an inverse . To make the inverse unique, we require that the inverse normal curl gives a divergence‐free vector field parallel to the surface. Thus
Let be the normal magnetic field. Ordinarily, we would write the poloidal field as
However, without spherical or planar symmetry, this magnetic field may have nonzero normal current – but the toroidal field has already taken care of . Thus (Berger and Hornig, 2018) we must add a shape term
where
Putting it all together, the generalized winding gauge becomes
1.1.3. Relative Field Line Helicity
In the previous sections, helicity and field line helicity have been defined in terms of geometrical quantities such as winding and linking. The helicity of open fields in a volume can also be defined in terms of how much the magnetic structure within contributes to the helicity of all space (Berger and Field, 1984). Let the field inside be called . Also consider a potential (zero current) field inside with the same boundary conditions . A potential field minimizes the magnetic energy in the volume subject to the constraint of the given boundary flux distribution. Hence potential fields can be said to have the minimum structure (more accurately, their structure only depends on the boundary conditions, so they add zero additional structure to the field).
Here we compare the total helicity of all space with the helicity that would be obtained if, inside , were replaced by . All integrals are gauge invariant, so the difference between the two helicities will also be gauge invariant. Let space external to be . Then the relative helicity is
It is important to note that the calculation of does not require knowledge of the external field (Berger and Field, 1984; Finn and Antonsen, 1985).
For symmetric volumes (i.e., boundaries consisting of a plane or parallel planes or one or two concentric spheres), the topological and relative definitions give the same answers (Berger and Field, 1984; Berger and Hornig, 2018
