24T2
Origin and Structure of VORTEX: Frontier of Nonlinear Science
渦の起源と構造：非線形科学の新展開
Zensho Yoshida
吉田善章
The University of Tokyo, Graduate School of Frontier Sciences
5-1-5 Kashiwanoha,Kashiwa 277-8561, Japan東京大学大学院新領域創成科学研究科 〒277-8561 柏市柏の葉5-1-5
A VORTEXE is a dynamic structure that is commonly observed in nature, society, as well as in cyber
space, but, probably, it is the most difficult "thing" to identify, analyze, or even describe. To explore aVORTEX, one has to invoke some non-standard (e.g. nonlinear, non-canonical, non-commutative,
non-Abelian, etc.) frameworks ---a theory of VORTEX will be constructed far beyond the conventional
realm of physics, and plasma physics can be placed at the frontier of this challenge. The aim of this tutorial
lecture is to show how the physics and mathematics of VORTEX is interesting, promising, and essential invarious problems of plasma physics.

1. Introduction
Amongst various "structures" created in the Universe (as well as in society or in cyber space),
VORTEX is the most common, fundamental, but
still an enigmatic structure; the Universe is filledwith vortexes (such as galaxies, accretion disks,stars and planetary systems, etc.) that clump andwind-up with magnetic fields (we will see that thefluid vortical motion is unified with magneticfields). Strikingly absent in this rich narrative ofgrowth and evolution of the cosmic systems, is asatisfactory "universal" identification and understanding of what a VORTEX is.
At the opposite pole to vortexes, there is anothercategory of structures consisting of "waves" withoscillatory, solitary, or stepwise shapes. This rather
well-understood group of structures is characterized
by a phase (or an action). As far as a phase can bedefined as a single-valued regular function, one
may invoke the standard methods of mechanics(known as Fermat's law or the least-action principle). However, VORTEX is a fundamental
breakdown of a simple phase description. An elementary form of vortex may be represented by anangular (multi-valued) phase, and then, the phasesingularity defines a "point vortex." A general
vortex no longer allows any definition of a phase,posing a challenge of geometrical or algebraic characterization.
After the "soliton" has took off from the motherland of plasma physics (and evolved into a sophisticated theory of algebraic analysis), we have had along quiescence in the field of basic, generalscience. VORTEX is certainly a promising paradigm on which we may describe a new discourse ofthe Universe.

2. Vorticity in an Ideal Image
A fundamental difficulty of the standard mechanics in describing a vortex is epitomized in itsvery basic Hamilton-Jacobi equation P =∇ϕ (ϕ isthe action), which forces the momentum P to bevorticity-free (∇× P = 0). This is just fine as far asa single particle pertains. However, to describe a"fluid" (collection of an infinite number of orbits =stream lines), we have to introduce "non- canonicality" that allows P to have a finite vorticity.
A non-canonical Hamiltonian system [1] is formulated by a singular Poisson bracket that has anontrivial kernel consisting of so-called Casimir
elements, which foliates the phase space, i.e., thedynamics is constrained on a level-set of the Casimir element. The Casimir element pertinent to avorticity is the helicity. A rich variety of structures
are obtained on a foliated (constrained) phasespace; even the equilibrium point (minimum of theHamiltonian) has a nontrivial structure (the familiar"Taylor relaxed state" is one of them); we find anintegrable (quantized) nonlinear torsional waves(Alfvén waves) near the equilibrium point [2].
3. Vortex and Entropy
The foregoing ideal framework, however, falls
short to describe a story of "creation" of vorticity,
since the invariance of the helicity poses a fundamental obstacle for a vorticity to emerge from zero.
Here we delineate a basic relation between Kelvin's
circulation law and the thermodynamic first-second
laws, which, in the relativistic regime, provides us
with a universal theory of "creation."
The circulation ∮ δ𝑄, associated with a variation
δ𝑄 of a physical quantity may be zero or finite depending on whether δ𝑄 equals an exact differen-

tial 𝑑𝐻 of a state variable H or not. For example, if
δ𝑄 = 𝑇𝑑𝑆 (T: temperature, S: entropy), the circulation is generally finite and measures the work
done in a quasi-static thermodynamic cycle. An
ideal fluid can be viewed as a realization of an infinite number of ideal isolated cycles; in the fluid-mechanical setting, however, the cycle must be
evaluated along a loop L that moves with the fluid.
And, to write a theory in the relativistic regime, wehave to immerse the loop in the 4-d space-time andtransport it by the 4-velocity 𝑑𝑥𝜇 /𝑑𝑠 = 𝑈𝜇 wheres denotes the proper time. The circulation pertinent
to the theory of VORTEX is the loop integral of thecanonical momentum 𝑃𝜇 = 𝑚∗ 𝑐𝑈𝜇 + (𝑞/𝑐)𝐴𝜇 ,where 𝑚∗ 𝑐 2 = ℎ is the molar enthalpy, 𝑞 is thecharge, and 𝐴𝜇 is the EM potential. We obtain𝑑�� 𝑃𝜇 𝑑𝑥𝜇 � = � (𝜕𝜇 𝑃𝜈 − 𝜕 𝜈 𝑃𝜇 )𝑈𝜈 𝑑𝑥𝜇 .
𝑑𝑠 𝐿(𝑠)𝐿(𝑠)On the other hand, the equation of motion reads as(𝜕𝜇 𝑃𝜈 − 𝜕 𝜈 𝑃𝜇 )𝑈𝜈 = 𝜕𝜇 ℎ − 𝑛−1 𝜕𝜇 𝑝. By the firstand second laws, the right-hand side may be writtenas 𝑇𝜕𝜇 𝑆. We thus have𝑑�� 𝑃𝜇 𝑑𝑥𝜇 � = � 𝑇𝜕𝜇 𝑆𝑑𝑥𝜇 .
𝑑𝑠 𝐿(𝑠)𝐿(𝑠)The rate of change of the circulation (left-hand side)parallels the work done ∮ δ𝑊 (with ∮ 𝑑ℎ ≡ 0).
This generalized (relativistic) Kelvin's law can beread as an "antithesis" to the standard Kelvin's lawthat prohibits creation of circulation (vorticity) in abarotropic flow (TdS is an exact differential); writing the Lorentz-covariant circulation law on a reference frame, the vector components reads as

4. Thermal Equilibrium in Foliated Phase Space
Self-organization of a structure is, at its surface,an antithesis of the entropy ansatz. However, disorder can still develop at micro scale while a structure emerges on some macro scale; it seems morecommon in various nonlinear systems that orderand disorder are simultaneous [4], and such coexistence may be possible if the self-organization andthe entropy principle work on different scales. The
self-organization of a magnetospheric plasma vortex is an example.
A systematic formulation of a scale hierarchy isgiven by a foliation of the kinetic phase space interms of actions (adiabatic invariants). Interpretingan action μ (the magnetic moment, for instance) as
the number of quantized "quasi-particles," we canconstruct a grand-canonical distribution function
𝑓𝛼 ∝ 𝑒 −𝛽𝐻−𝛼𝜇 (H: Hamiltonian, β: inverse temperature, α: chemical potential of the quasi-particle)
that maximizes entropy on a macroscopic leaf
where the microscopic action-angle variables areabstracted as quasi-particle numbers. While embedding the macroscopic leaf in the (laboratory-frame)
total phase space, the transforming Jacobian weightforces an inhomogeneous density profile in the laboratory flat space (see Fig. 1).
An inhomogeneous magnetic field (typically adipole) yields a strongly distorted invariant measureof the magnetized particles, creating a self- organized vortex with steep density gradient; this theory
explains "inward diffusion" in magnetospheres, aswell as their laboratory simulations [5].

𝑑�� 𝑷 ∙ 𝑑𝒙� = � γ−1 𝑇∇𝑆 ∙ 𝑑𝒙.
𝑑𝑡 𝐿(𝑡)𝐿(𝑡)Even in a barotropic flow, the kinematic factor γ−1
makes the cycle non-exact and creates a circulation.
The space-time distortion inherent in relativistic
dynamics, thus provides us with a mechanism that,by breaking the topological constraint forbidding
the emergence of magnetic fields (vortexes), allows"general vorticities" ---naturally coupled vortexesof matter motion and magnetic fields--- to becreated in an ideal fluid. The newly postulated relativistic mechanism, arising from the interaction
between the inhomogeneous flow fields and inhomogeneous entropy, may be an attractive universal
solution to the origin problem [3].

Fig.1. The distorted metric of foliated phase space of
charged particles in a dipole magnetic field.
References[1] P. J. Morrison; Rev. Mod. Phys. 70 (1998) 467.
[2] Z. Yoshida; Comm. Nonlinear Sci. Num. Sim. (2011)
doi:10.1016/j.cnsns.2011.04.010.
[3] S.M. Mahajan and Z. Yoshida; Phys. Rev. Lett. 105(2010) 095005.
[4] A. Hasegawa; Adv. Phys. 34 (1985) 1.
[5] Z. Yoshida et al.; Phys. Rev. Lett. 104 (2010)
235004.