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...