Osaka Central Advanced Mathematical Institute (OCAMI)
Osaka Metropolitan University
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics
OCAMI Reports Vol. 5 (2022)
doi: 10.24544/ocu.20220810-001

Helicity and space-time symmetry— a new perspective ofclassical and quantum systems
Organized by
Yasuhide Fukumoto
Hiroshi Itoyama
Chihiro Matsuoka
October 5–8, 2021
Abstract
The helicity is a key ingredient of the Nambu-bracket for the Euler equation for motion of an ideal ﬂuid. This conference focuses on the helicity,
as the common structure of the non-canonical Hamilton structure and the
Nambu bracket for the ideal ﬂuid motion, and deepens its mathematical andphysical signiﬁcance that transcends the subjects, by enlightening the symmetry behind its preservation. With interdisciplinary fusion, we explore a newtheoretical framework and seek new applications of the helicity.

2020 Mathematics Subject Classiﬁcation.
37K05, 76M60, 37N10, 57M25, 85A30
Key words and Phrases.
helicity, vortex, knots and links, supernovae, Beltrami ﬁeld,
symmetry and consevation laws

© 2022 OCAMI.
OCAMI. Helicity and space-time symmetry — a new perspective of classical and quantum systems.
OCAMI Reports. Vol. 5, Osaka Central Advanced Mathematical Institute, Osaka Metropolitan
University. 2022, 27pp. doi: 10.24544/ocu.20220810-001

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OCAMI Reports Vol. 5 (2022)

Preface“Helicity” is an integral invariant characterizing vortex structures that appear
in a variety of systems, from quantum (micro) to classical (macro) ﬁelds. Sincea vortex, which is a chiral structure with a relatively long life, has odd parity, its
generation cannot be explained by the energy (Hamiltonian) alone, and helicity is the
key. In the joint use / joint research conducted by the proposer as a representative in
FY2020, with the Nambu brackets as the theme, we collected classical and quantum
systems in which vortices and string-like structures play major roles, in order toexplicitly capture the helicity in the non-canonical Hamiltonian structure, and put
in perspective their phenomena and theories that describes the underlying ﬁelds. It
became clear that formulation based on the Nambu brackets is yet completed, andthat there is a gap between its formulation and the framework of the non-canonicalHamiltonian system. In this research, we focus on the helicity, which is locatedin the common part of the non-canonical Hamiltonian structure and the Nambu
mechanics, and deeply dig into the helicity as a basic concept that transcends the
subjects through elucidation of the symmetries behind the phenomena. The purpose
was to explore a new theoretical framework of the helicity and further to explore its
application.
The helicity was originally introduced as the inner product of momentum andspin in the relativistic quantum mechanics. The ﬂuid helicity deﬁned by the integralof the vorticity ﬁeld and the velocity ﬁeld is its macroscopic counterpart, the topological meaning of which was clariﬁed by Moﬀatt (1969). The diﬀerence between
the two can be attributed to the diﬀerence in the hierarchy level, but there remains
a feeling of oversight from the viewpoint of the chiral structure. The dynamics ofa macroscopic system is described by relationships among the vector (tensor) ﬁelds
and their derivatives (the Eulerian description), disregarding the particle view (theLagrangian description). The helicity resides in a common place of the Eulerian andthe Lagrangian descriptions. The aim of this research was to bridge the macro- andmicro-systems, such as classical and quantum ﬂuids, BEC, plasma, crystals, andoptics, by delving into the helicity from the viewpoint of space-time symmetry.
In recent years, there have been a series of interesting discoveries associatedwith the helicity. The Jones and HOMFLYPT polynomials, which are quantum
invariants of knots and links, were derived from ﬂuid helicity (Liu & Ricca 2016), an
experimental realization of knots and links of vortex ﬁlaments in a ﬂuid (Irvine 2012),
possibility of space-time curvature in general relativity for the origin of the cosmic
magnetic ﬁeld and the contribution of the relativistic helicity (Mahajan & Yoshida2010), the generation of chiral nanostructures of materials by laser ablation basedon the spin angular momentum of light (Toyoda et al. 2013), etc. We also discussed
a wide range of applications of the helicity, from mathematics to engineering.

Helicity and space-time symmetryiii

Organizers
Yasuhide Fukumoto
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku,
Fukuoka 819-0395, Japan
Email address: yasuhide@imi.kyushu-u.ac.jp
Hiroshi Itoyama
Department of Physics, Graduate School of Science, Osaka City University, 3-3-138,
Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: itoyama@osaka-cu.ac.jp
Chihiro Matsuoka
Department of Physics, Graduate School of Science, Osaka City University, 3-3-138,
Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: cmatsuoka@osaka-cu.ac.jp

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Contents
Y. Matsuo and A. Sugamoto
A description of perfect fluid by Kalb-Ramond field and the duality
with Gross-Pitaevskii . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Z. Yoshida and P. J. Morrison
The kinetic origin of the fluid helicity — a symmetry in the kinetic
phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

H. Keith Moffatt
Invariance of helicity and some topological implications . . . . . . . .

3

Carlo F. Barenghi
Helicity and topology of a tangle of vortices. . . . . . . . . . . . . .

4

N. Yamamoto and Di-Lun Yang
Chiral radiation transport theory of neutrinos. . . . . . . . . . . . .

5

S. Hanai and N. Yamamoto
Gravitational waves caused by chiral transport phenomena in neutronstars and supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Y. Masada, T. Takiwaki and K. Kotake
Convection and dynamo in newly-born neutron stars. . . . . . . . .

7

K. Shimokawa
Untying pathways of vortex knots . . . . . . . . . . . . . . . . . . . .

8

R. Morita
Physics of optical vortex and its application

9. . . . . . . . . . . . . .

X. Liu and R. L. Ricca
Detecting topology of fluid vortex knots in terms of polynomial invariants constructed from helicity . . . . . . . . . . . . . . . . . . . . . . 10
R. L. Ricca and X. Liu and X.-F. Li
Minimal unlinking pathways as geodesics in knot polynomial space . . 11
K. Konno
Review of Chern-Simons modified gravity — its effects on compact
objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
T. Igata
Energy extraction from a rotating black hole . . . . . . . . . . . . . . 13

Helicity and space-time symmetryv

K. Maruyoshi
Surface defects, vortices and integrable systems. . . . . . . . . . . . 14

K. Nishihara, C. Matsuoka, T. Sano, F. Cobos-Campos and V. Zhakhovsky
Richtmyer-Meshkov instability and vortices . . . . . . . . . . . . . . 15
T. SanoLaser-driven interfacial instabilities in magnetized plasmas . . . . . . 16
K. AbeRigidity of Beltrami fields with a non-constant proportionality factor
C. Matsuoka and K. Hiraide
Exact solutions in non-integrable systems

17. . . . . . . . . . . . . . . 18

H. Yoshimura
Dirac geometry and variational structures in nonequilibrium thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
L. Peng
Continuous symmetries of semi-discrete equations and conserved quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Y. Fukumoto and R. Zou
Particle relabeling symmetry and Noether’s first and second theorems
for fluid and MHD equations . . . . . . . . . . . . . . . . . . . . . . 21

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A description of perfect fluid by Kalb-Ramondfield and the duality with Gross-Pitaevskii
Y. Matsuo (U. Tokyo)
A. Sugamoto (Ochanomizu U.)
In this talk, we explain a description of the perfect fluid in terms of a two-form
gauge field (Kalb-Ramond field). Nambu proposed such an idea long ago in 1977,
and one of the authors (AS) developed the idea further.
In the perfect fluid motion, the vortex keeps the circulation and behaves as a“soliton”. In this sense, it is natural to describe the fluid dynamics in terms of thevortices, which were explored long ago by Lund and Regge.
The Kalb-Ramond field couples naturally with the vortex degree of freedom
through the integration over the vortex (string) world-sheet. We study the detailed
relation between the dynamical degree of freedom of the Euler equation with the
Kalb-Ramond field.
We also compare it with the Gross-Pitaevskii equation, giving an alternativedescription of the fluid motion. We explain that the two descriptions are dual,
similar to the electric/magnetic fields in the Maxwell theory. In this sense, we
expect to have the strong/weak duality in the two descriptions.

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OCAMI Reports Vol. 5 (2022)

The kinetic origin of the fluid helicity — asymmetry in the kinetic phase space
Z. Yoshida (Natl. Inst. Fusion Sci.)
P. J. Morrison (U. Texas, Austin)
Helicity, a topological degree that measures the winding and linking of vortex
lines, is preserved by ideal (barotropic) fluid dynamics. In the context of the Hamiltonian description, the helicity is a Casimir invariant characterizing a foliation of theassociated Poisson manifold. Casimir invariants are special invariants that dependon the Poisson bracket, not on the particular choice of the Hamiltonian. The total
mass (or particle number) is another Casimir invariant, whose invariance guarantees
the mass (particle) conservation (independent of any specific choice of the Hamiltonian). In a kinetic description (e.g. that of the Vlasov equation), the helicity is nolonger an invariant (although the total mass remains a Casimir of the Vlasov’s Poisson algebra). The implication is that some “kinetic effect” can violate the constancyof the helicity. To elucidate how the helicity constraint emerges or submerges, we
examine the fluid reduction of the Vlasov system; the fluid (macroscopic) system isa “sub-algebra” of the kinetic (microscopic) Vlasov system. In the Vlasov system,
the helicity can be conserved, if a special helicity symmetry condition holds. To put
it another way, breaking helicity symmetry induces a change in the helicity. We
delineate the geometrical meaning of helicity symmetry, and show that, for a specialclass of flows (so-called epi-2 dimensional flows), the helicity symmetry is written as∂γ = 0 for a coordinate γ of the configuration space.

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3

Invariance of helicity and some topologicalimplications
H. Keith Moffatt (Trinity College, Cambridge)
Helicity is a topological invariant of any solenoidal vector field that is transportedby (i.e. ‘frozen in’) a fluid flow. The best known examples are the magnetic field
in a perfectly conducting fluid and the vorticity field in an ideal fluid, the latterproviding an important invariant of the classical Euler equations of fluid flow. In
this lecture, I shall first discuss the historical origin of these invariants, and I willthen comment on two complementary problems in which helicity plays a key role: (i)the spontaneous generation of magnetic field (i.e. dynamo action) by turbulence with
non-zero mean helicity; and (ii) the relaxation of knotted magnetic flux tubes in aperfectly conducting fluid to minimum-energy states constrained by their topology.
Here there is an analogy with the ‘tight-knot’ problem: what is the minimum lengthof rope of unit circular cross-section with which a knot of particular type may betied? There are differences however in that, for the magnetic relaxation problem,
it is the volume of the flux tube that is conserved under deformation (and not thearea of its cross-section), and the internal twist of the magnetic field affects theminimum-energy state. A particular problem here concerns the possibility of two
or more minimum-energy states for a given knot type, and the manner in which atransition from one such state to another may be accomplished. Some aspects ofthis problem, under current consideration, will be discussed, insofar as time permits.

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Helicity and topology of a tangle of vortices
Carlo F. Barenghi (Newcastle U.)
We address the question as to whether turbulence contains vortex knots in acontext which is simpler than ordinary fluids: the context of superfluid helium.
Unlike what happens in ordinary Navier-Stokes fluids (where vorticity is a continuous
field), in superfluid helium quantum mechanics constraints the vorticity to individual(discrete) vortex lines. Quantum turbulence is therefore a disordered tangle of vortex
lines. Since the vortex core thickness is many orders of magnitude smaller than the
average distance between vortex lines in typical experiments, the vortex tangle canbe modelled using the classical Biot-Savart law and the assumption that vortex lines
reconnect [1] when they collide (as seen in the experiments), a mesoscale level ofdescription called the Vortex Filament Model (VFM).
After numerically generating a state of turbulence in a statistical-steady state, we
exploit the discrete nature of the vorticity to instantaneously associate an Alexanderpolynomial to each vortex loop. The degrees of the Alexander polynomials can thus
be used to quantify the complexity of the turbulence, revealing a highly knotted
distribution [3].
In the last part of the lecture I shall discuss the classical definition of helicity [2]in the context of super fluid helium. In particular, I shall show that the classicaldefinition of helicity is consistent with what we know microscopically about thesuper fluid vortex core from N-body quantum mechanics, without the difficulties
which arise in the simpler mean-field approach of Gross-Pitaevskii theory. I shallshow that helicity, compute in the VFM, captures the physical difference [4] between
the two forms of quantum turbulence which have been identified in experiments andnumerical simulations.

References[1] L. Galantucci & al, Proc. Nat. Sci. USA 116: 12204, 2019[2] K.H. Moffatt, J. Fluid Mech. 35: 117, 1969.
[3] R.G. Cooper & al, Sci. Reports 9: 10545, 2019.
[4] L. Galantucci & al, Phys. Rev. A 103: 144503, 2021.

Helicity and space-time symmetry

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Chiral radiation transport theory of neutrinos
N. Yamamoto (Keio U.)
Di-Lun Yang (Academia Sinica)
One of the most important properties of neutrinos in the Standard Model of particle physics is the chirality. Although neutrinos are considered to play importantroles in the dynamics of core-collapse supernovae, this property has been completely
neglected in the conventional neutrino transport theory. Recently, we have systematically constructed the radiation transport theory incorporating the effects ofchirality of neutrinos based on the underlying quantum field theory. This is thechiral radiation transport theory [1].
In this talk, we first give a brief introduction to transport phenomena due tothe chirality of elementary particles, called chiral transport phenomena. One ofits consequences is the chiral plasma instability, which generates a magnetic fieldwith magnetic helicity from the chirality imbalance of fermions. We then discuss
applications of the chiral radiation transport theory for neutrinos in corecollapse
supernovae, and in particular, its possible implications for the origins of magnetars
and pulsar kicks [2].

References[1] N. Yamamoto and D. L. Yang, Astrophys. J. 895, 56 (2020) [arXiv:2002.11348(astroph.HE)].
[2] N. Yamamoto and D. L. Yang, [arXiv:2103.13159 (hep-ph)]; work in preparation.

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Gravitational waves caused by chiral transportphenomena in neutron stars and supernovae
S. Hanai (Keio U.)
N. Yamamoto (Keio U.)
Exploring the internal structure of neutron stars is an important issue in astrophysics. Asteroseismology is a method to study the inside of stars by analyzing
their seismic oscillations. These oscillations are classifed according to their physicalorigins, such as p-modes due to pressure, g-modes due to buoyancy, and so on. In
the case of neutron stars, gravitational waves are expected to be emitted inducedby these seismic oscillations.
In recent years, chiral transport phenomena caused by the chirality of relativistic
quarks have been vigorously discussed in the context of quark-gluon plasmas createdin relativistic heavy-ion collisions experiments. Typical examples are the chiralmagnetic effect, which is the current in the direction of the magnetic feld, and thechiral vortical effect, which is the current in the direction of the vorticity. It is
known that these chiral transport phenomena induce collective excitations calledchiral magnetic waves and chiral vortical waves, respectively.
In this talk, we will show that chiral magnetic waves are generated in the electron
matter inside neutron stars, and examine their properties. In particular, we will
discuss the effect of electron mass on the chiral waves. We will estimate the frequencyof new types of gravitational waves produced by these chiral magnetic waves andthe possibility of observations. We will also show the possible existence of the othertype of gravitational waves induced by the propagation of chiral vortical waves in
the core of supernovae.

Helicity and space-time symmetry

7

Convection and dynamo in newly-born neutronstars
Y. Masada (Aichi U. Edu.)
T. Takiwaki (Natl. Astron. Obs. Japan)
K. Kotake (Fukuoka U.)
Neutron stars (NSs) have the most extreme magnetic field in the universe, typically trillion, up to quadrillion times more powerful than Earth’s. Although, weknow, they are formed as an aftermath of massive stellar core-collapse, the origin of
the magnetic field is still an outstanding issue in astrophysics. Mainly, two possible
origins have been proposed: fossil field and dynamo field hypotheses. While theformer regards it as an inheritance from NS’s main sequence progenitor, the later
presumes that it would be generated by some dynamo processes in newly-born NSs,
also known as proto-neutron stars (PNSs). Although significant progress has been
made in the study of the PNS dynamo, the origin of the diversity of NS’s magnetic
fields remains to be solved. “What physics is responsible for the diversity of NS’smagnetic fields ? ” To answer this question, we are now studying the properties of
the convection and resultant dynamo in the PNS with the aid of numerical simulation. In this talk, we will present the results of our latest PNS dynamo simulations
and the physics behind them, and also discuss the Beltrami structure of the flow
and magnetic field, which is a natural consequence of the dynamo process.

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Untying pathways of vortex knots
K. Shimokawa (Saitama U.)
In this talk we will analyze untying pathways of vortex knots by applying knot
theory. The topology of entangled vortices is represented using oriented knots andlinks, and the reconnection of vortices is modeled using a coherent band surgery ofknots and links. For the case where the topology of the vortex is a torus knot or linkT(2,p), we characterize the untying pathway, and determine where the reconnection
occurs.

References[1] Proc. Natl. Acad. Sci. USA 110, 20906–20911 (2013)[2] Sci. Rep. 7, 12420 (2017)

Helicity and space-time symmetry

9

Physics of optical vortex and its application
R. Morita (Hokkaido U.)
Optical vortices (OVs), such as Laguerre-Gaussian modes, whose spatial phasedepends on the azimuthal coordinate, have attracted much attention because of
their unique properties. Possessing a phase singularity on their beam center, they
carry the orbital angular momentum proportional to the topological charge, whichis expressed by the value of the line integral of the spatial phase around the singularpoint. Thanks to these properties, they are used in the various applications suchas laser processing, laser trapping/manipulation, quantum information processing,classical communications, super-resolution microscopy, and nonlinear spectroscopy.
In the presentation, giving an overview of physical properties of OVs, we will discuss
the interaction between OVs and materials from the viewpoint of orbital angularmomentum, spin angular momentum, total angular momentum and helicity.

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OCAMI Reports Vol. 5 (2022)

Detecting topology of fluid vortex knots in termsof polynomial invariants constructed from helicity
X. Liu (Beijing U. Tech.)
R. L. Ricca (U. Milano-Bicocca)
Helicity is the most important topological invariant in fluid mechanics, a conserved quantity for the Euler equation and the only integral invariant for volumepreserving diffeomorphic transformations [1]. From vortex filaments viewpoint, helicity provides access to characterize fluid knot topology. In the seminal work of
Moffatt [2], then extended by Moffatt & Ricca [3][4], the relationship between helicity and knot linking numbers was revealed; since then linking number interpretationof helicity has played a key role in the study of the topological hierarchy of entangled fluid vortex knots. With the discovery of highly complex knots in superfluid
turbulence [5] linking numbers and crossing numbers are proven to be no longer
sufficient to detect topological complexity and changes in topology due to reconnections [6]. More powerful tools from knot theory are thus needed; for this reason we
have derived Jones and HOMFLYPT knot polynomials from helicity [7][8]. In thistalk we present a brief introduction to our derivation, emphasizing the role playedby writhe and twist helicity in the polynomial skein relations.
This is joint work with Renzo L. Ricca (U Milano-Bicocca, Italy).

References[1] Enciso A., Peralta-Salas D. & Torres de Lizaur F. 2016 Helicity is the only
integral invariant of volume-preserving transformations. PNAS USA 113, 2035.
[2] Moffatt H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid
Mech. 35, 117.
[3] Moffatt H.K. & Ricca R.L. 1992 Helicity and the Călugăreanu invariant. Proc.
R. Soc. A 439, 411.
[4] Ricca R.L. & Moffatt H.K. 1992 The helicity of a knotted vortex filament. In
Topological Aspects of the Dynamics of Fluids and Plasmas (ed. H.K. Moffatt
et al.), pp. 225–236. Kluwer Acad. Publs.
[5] Cooper R.G., Mesgarnezhad M., Baggaley A.W. & Barenghi C.F. 2019 Knot
spectrum of turbulence, Sci. Rep. 9, 10545.
[6] Kleckner D., Kauffman L.H. & Irvine W.T.M. 2016 How superfluid vortex knots
untie, Nature Phys. 12, 650–655.
[7] Liu X. & Ricca R.L. 2012 The Jones polynomial for fluid knots from helicity.
J. Phys. A: Math. & Theor. 45, 205501.
[8] Liu X. & Ricca R.L. 2015 On the derivation of HOMFLYPT polynomial invariant for fluid knots. J. Fluid Mech. 773, 34–48.

Helicity and space-time symmetry

11

Minimal unlinking pathways
as geodesics in knot polynomial space
R. L. Ricca (U. Milano-Bicocca)
X. Liu (Beijing U. Tech.)
X.-F. Li (Guangxi U. Sci. Tech.)During evolution fluid structures, such as linked and knotted vortex filaments
in classical or quantum systems, decay naturally by a series of reconnection events
through a process of consecutive topological simplifications. The process bears similarities with the DNA unlinking scenario studied in recombinant DNA plasmids[1]. By applying adapted knot polynomials, such as Jones and HOMFLYPT derived
from helicity [2,3], we show that a topological cascade can be described as a geodesic
flow in an appropriate knot polynomial space [4]. Each point of this space coincides
with the knot type given by the polynomial value, that for moderate crossing number(≤ 9) identifies uniquely knot types. For the sake of example we consider adapted
Jones polynomials with metric given by the inner product of Legendre polynomials.
Unlinking pathways can thus be identified with geodesics on this space, and optimal decay paths measured by associated probability. Results are very promising,showing an interesting connection with the logarithmic behavior associated with thegroundstate energy spectrum of tight knots and links [5].
This is joint work with Xin LIU (BJUT, China) and Xinfei LI (Guangxi UST,
China).

References[1] Stolz R., Yoshida M., Brasher R., Flanner M., Ishihara K., Sherratt D.J.,
Shimokawa K. & Vazquez M. 2017 Pathways of DNA unlinking: A story of
stepwise simplification. Nature Sci. Rep. 7, 12420.
[2] Liu X. & Ricca R.L. 2015 On the derivation of HOMFLYPT polynomial invariant for fluid knots. J. Fluid Mech. 773, 34–48.
[3] Liu X. & Ricca R.L. 2016 Knots cascade detected by a monotonically decreasing
sequence of values. Nature Sci. Rep. 6, 24118.
[4] Liu X., Ricca R.L. & Li X-F. 2020 Minimal unlinking pathways as geodesics inknot polynomial space. Nature Comm. Phys. 3, 136.
[5] Ricca R.L. & Maggioni F. 2014 On the groundstate energy spectrum of magnetic knots and links. J. Phys. A: Math. & Theor. 47, 205501.

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Review of Chern-Simons modified gravity — its
effects on compact objects
K. Konno (Natl. Inst. Tech. Tomakomai College)
In recent years, many modified gravity theories have been investigated to seek
the ultimate gravitational theory. Among them, Chern-Simons modified gravity isstrongly suggested as an effective theory by two promising quantum gravity theories
and by an extension of the standard model. Furthermore, Chern-Simons modified
gravity is consistent with the recent results of gravitational wave detection.
In the present talk, I will briefly review Chern-Simons modified gravity from thefundamental part of this theory to recent topics. Chern-Simons corrections play
an important role around rotating compact objects. Thus I focus on the effects ofChern-Simons modified gravity on compact objects in this talk.

Helicity and space-time symmetry

13

Energy extraction from a rotating black hole
T. Igata (KEK)No material body or light ray never can be extracted from a black hole. However,Penrose found that energy can be extracted from a rotating black hole [1]. In thistalk, I will introduce an energy extraction mechanism from a rotating black holethat reconciles these seemingly contradictory statements (see, e.g., [2] for a review).
The energy extraction mechanism mediated by a free electromagnetic field, theBlandford-Znajek process [3], has been widely believed to be a viable mechanism for
relativistic jet formation because it achieves a large energy flux. From a spacetime
perspective, the dynamics of magnetic field lines in a force-free magnetic field canbe identified with the dynamics of strings [4, 5]. Using this fact, we will argue thatthe physical picture of the Blandford-Znajek process by a stationary axisymmetricforce-free electromagnetic field is equivalent to the energy extraction process by arigidly rotating Nambu-Goto string [6].

References[1] R. Penrose, Gravitational collapse: the role of general relativity, Rev. Nuovo
Cimento, 1, 252 (1969).
[2] J. P. Lasota, E. Gourgoulhon, M. Abramowicz, A. Tchekhovskoy, and R.
Narayan, Extracting black-hole rotational energy: The generalized Penrose
process, Phys. Rev. D 89, 024041 (2014) [arXiv:1310.7499 [gr-qc]].
[3] R. D. Blandford and R. L. Znajek, Electromagnetic extraction of energy from
Kerr black holes, Mon. Not. Roy. Astron. Soc. 179, 433 (1977).
[4] T. Uchida, Theory of force-free electromagnetic fields. I. General theory, Phys.
Rev. E 56, 2181 (1997).
[5] S. E. Gralla and T. Jacobson, Spacetime approach to force-free magnetospheres,Mon. Not. Roy. Astron. Soc. 445, 2500–2534 (2014) [arXiv:1401.6159 [astroph.HE]].
[6] S. Kinoshita and T. Igata, The essence of the Blandford-Znajek process, Prog.
Theor. Exp. Phys. 2018, 033E02 (2018) [arXiv:1710.09152 [gr-qc]].

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Surface defects, vortices and integrable systems
K. Maruyoshi (Seikei U.)
In this talk, we discuss surface defects in a certain class of four-dimensional supersymmetric gauge theories, and a related vortex solution, in relation with the
integrable lattice model. A surface defect has a support on a two-dimensional surface in the space-time. A typical surface defect is defined by the coupling with
two-dimensional field theory on the surface or by the singular condition on thegauge field around the origin of the perpendicular direction. It is known that sucha surface defect in four-dimensional supersymmetric gauge theory is realized as thelow energy limit of the vortex solution of certain U(1) gauge theory associated withthe original four-dimensional theory. By using this construction, we compute thepartition functions of the supersymmetric gauge theory with the surface defect. It
turns out that certain partition functions are identified with the transfer matrices
of some integrable lattice models. We argue the reasons of this correspondence,and its application. This talk is based on the collaboration with Junya Yagi, andon arXiv:1606.01041, and the collaboration with Toshihiro Ota and Masahito Yamazaki, work in progress.

Helicity and space-time symmetry

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Richtmyer-Meshkov instability and vortices
K. Nishihara (Osaka U.),
C. Matsuoka (Osaka City U.)
T. Sano (Osaka U.)
F. Cobos-Campos (U. Castilla-La Mancha)
Vasily Zhakhovsky (Rosatom)
When an incident shock wave hits a corrugated interface, ripples are induced on
the transmitted and reflected shocks. Due to the refraction of fluids crossing therippled shock fronts, the velocity shear left at the interface behind the shocks. The
Richtmyer-Meshkov instability occurs due to the shear flow at the corrugated interface. In linear regime, i.e., the distances among the interface and shock fronts are
not large, the rippled shock fronts and the corrugated interface interact each otherthorough sound wave and entropy wave. However it is known that the shock ripples
and sound wave decay exponentially and the perturbation becomes incompressible.
There exists an asymptotic linear growth rate. But fluids are not irrigational andvortices remain in the bulk between the shocks [1,2].
We investigated the nonlinear dynamics of RMI by developing a non-uniform vortex sheet model [3]. In RMI, bulk vortices are generated behind the rippled shocks.
We here consider more generally nonlinear interactions between a non-uniform vortex sheet with point vortices. It will be shown that the nonlinear interaction between
a non-uniform vortex sheet and point vortices as unstable and the mixing will beenhanced [4].

References[1] R. D. Richtmyer, Commun. Pure Appl. Math, 13, 297 (1960); E. E. Meshkov,
Fluid Dyn. 4, 101 (1972); Y. Zhou, Phys. Rep. 720–722, 1 (2017).
[2] K. Nishihara et al Philos. Trans. R. Soc. A 368, 1769 (2010); G. Wouchuk andK. Nishihara, Phys. Plasmas, 3, 3761 (1996); ibid 4, 1028 (1997).
[3] C. Matsuoka, K. Nishihara and Y. Fukuda, Phys. Rev. E 67, 036301 (2003),
C. Matsuoka and K. Nishihara, Phys. Rev. E, 73, 053304 (2006).
[4] C. Matsuoka and K. Nishihara, Phys. Plasmas, 27. 052305 (2020).

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OCAMI Reports Vol. 5 (2022)

Laser-driven interfacial instabilities in magnetizedplasmasT. Sano (Osaka U.)
Extreme plasma states observed in astrophysical phenomena can be created in
the laboratory using large laser facilities such as the GEKKO laser at Osaka University. For example, the verification of the metallic hydrogen state under ultra-highpressure comparable to that of Jupiter’s interior and the formation of collisionlessshocks are being actively pursued. We are conducting laser experiments and MHD
simulations on interfacial instabilities in magnetized plasmas associated with supernova explosions. Such phenomena are very important processes in laser fusion
plasmas to realize ideal implosion. Therefore, there is an urgent need to understandthe mechanisms that suppress the growth of interfacial instabilities. In this talk, I
introduce our laser astrophysics experiments and theoretical understandings on theamplification of magnetic fields by the Richtmyer-Meshkov instability [1-3].

References[1] T. Sano, Alfven number for the Richtmyer-Meshkov instability in magnetizedplasmas, Astrophys. J. in press (2021)[2] T. Sano et al., Laser astrophysics experiment on the amplification of magnetic
fields by shock-induced interfacial instabilities, Phys. Rev. E 104, 035206 (2021)[3] T. Sano et al., Suppression of the Richtmyer-Meshkov instability due to adensity transition layer at the interface, Phys. Rev. E 102, 013203 (2020)

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Rigidity of Beltrami fields with a non-constant
proportionality factor
K. Abe (Osaka City U.)
Beltrami fields curl u = f u, div u = 0 appear as steady states of ideal incompressible flows or plasma equilibria. I will discuss existence and non-existence issues
on them for non-constant factor f . In the first half of the talk, I will explain existence of axisymmetric Beltrami fields forming vortex rings and their construction
via a variational principle. In the second half, I will discuss a rigidity problem on
symmetry of u for symmetric f and a relation with Grad’s conjecture. This talk isbased on preprints arXiv:2008.09345, arXiv:2108.03870.

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OCAMI Reports Vol. 5 (2022)

Exact solutions in non-integrable systems
C. Matsuoka (Osaka City U.)
K. Hiraide (Ehime U.)
We present exact solutions in non-integrable systems, taking the Henon map as
an example. The solutions are constructed using the Laplace transform. Unlike integrable systems such as soliton systems, the obtained solutions cannot be describedby any existing functions. We visualize a boundary of KAM tori and the Henon
attractor by use of the obtained functions, in which chaotic orbits are captured
accurately.

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Dirac geometry and variational structures innonequilibrium thermodynamics
H. Yoshimura (Waseda U.)
A Dirac structure is a geometric object of unifying symplectic and Poisson structures. In mechanics, for the case of degenerate Lagrangian systems with nonholonomic constraints, which are linear in velocities, given as a distribution on a configuration manifold Q, it is known that an induced (almost) Dirac structure over the
Pontryagin bundle P = T Q ⊕ T ∗ Q can be defined from the given distribution.
On the other hand, a general class of nonholonomic systems with nonlinear
constraints may be formulated by a general Lagrange-d’Alembert principle by introducing variational constraints CV ⊂ T Q ⊕ T ∗ Q as well as nonlinear kinematicconstraints CK ⊂ T Q, in which case one cannot define any appropriate geometricstructure such as Poisson, symplectic as well as Dirac structures for such a generalclass of nonholonomic systems with nonlinear constraints.
In this talk, we study a class of nonholonomic systems with nonlinear onstraintsof thermodynamic type, in which CK can be mathematically deduced from CV . For
the cases of nonholonomic systems with nonlinear constraints of thermodynamic
type, we show that an induced Dirac structure DP ⊂ T P ⊕ T ∗ P can be defined fromCV and therefore a Dirac dynamical system can be introduced by using DP . We also
show that there exists an associated variational structure with the Dirac dynamicalsystem. In thermodynamics, such a class of nonlinear nonholonomic constraints
typically appears in the relation of entropy production regarding the second law ofthermodynamics. Finally, we show that We illustrate our theory with some examplesof nonequilibrium thermodynamic systems for the case of adiabatically closed andopen cases.
This is a joint work with François Gay-Balmaz in École Normale Supérieure in
Paris.

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OCAMI Reports Vol. 5 (2022)

Continuous symmetries of semi-discrete equations
and conserved quantities
L. Peng (Keio U.)
During the1880s, Sophus Lie started his study on what role continuous transformations play in the solvability of ordinary differential equations. Nowadays, weknow it well that solvability or integrability of differential equations is greatly due
to the existence of sufficient symmetries, i.e., continuous transformations moving
one solution to another solution, or sufficient conserved quantities. For variational
equations, they are connected to each other according to the celebrated Noether’s
theorem in a one-to-one manner.
On the other hand, many mechanical and physical systems can be modeled assemi-discrete equations, such the Toda lattice. However, symmetries of semi-discrete
equations have not been well established because prolonging a transformation tovariables involving both derivatives and shifts has been the essential challenge, thathas bothered scholars for decades. In this talk, we will show how this problem is
eventually solved that allows us to extend Noether’s two theorems to semi-discrete
equations.

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Particle relabeling symmetry and Noether’s firstand second theorems for fluid and MHD equations
Y. Fukumoto (Kyushu U.)
R. Zou (U. Hawai’i)
For the ideal fluid dynamics and magnetohydrodynamics (MHD), Noether’s theorem states that the topological invariant associated with the particle relabelingsymmetry is the cross helicity, the volume integral of the scalar product of the velocity field and a frozen-in field. A proof to it is given in terms of variation of the
Lagrangian label as a function of the Eulerian position. According to Noether’s
second theorem, the existence of the topological invariant implies redundancy of thegoverning equations. The generalized Bianchi identity reflecting it is derived for
MHD.