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Springer Series in Synergetics
Anatoli TurVladimir Yanovsky
Coherent Vortex Structures in Fluids and Plasmas
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Anatoli Tur • Vladimir Yanovsky
Coherent Vortex Structuresin Fluids and Plasmas
123
-
Anatoli TurInstitut de Recherche en Astrophysique
et PlanétologieCNRS, SC. de L’Univ.Toulouse CX 04, France
Vladimir YanovskyInstitute for Single CrystalsNational Academy
of Science of UkraineKharkov, Ukraine
ISSN 0172-7389 ISSN 2198-333X (electronic)Springer Series in
SynergeticsISBN 978-3-319-52732-1 ISBN 978-3-319-52733-8 (eBook)DOI
10.1007/978-3-319-52733-8
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Preface
This book discusses the vortex process of self-organization in
hydrodynamics offluid and plasma. These coherent vortex structures
appear in fluid as a result ofspontaneous self-organization. They
usually have the form of localized vorticesand play an important
role in nature and technology. These localized vorticesare well
observed in shear flows, submerged jets, traces left after the
body, andboundary layers. In these cases, the coherent structures
mainly take the form ofvortex streets. First of all, localized
vortices are responsible for the process ofmacroscopic transport of
momentum, energy, and substance in fluid. They are alsoimportant in
themselves. Some examples are weather-related, such as cyclonesand
anticyclones, typhoons, hurricanes, and tornados, as well as
vortices on otherplanetary atmospheres, such as the Great Red Spot
on Jupiter. It is not surprising,therefore, that numerous books and
articles have dealt with the study of vortexmotions. Paul K.
Newton’s book The N-Vortex Problem (2001) gives the mostcomplete
exposition of the results of the theory of point vortices and
vortex patchesobtained up to the year 2000. But it does not contain
newer results and does notcover many issues related to dynamics,
generation, and interaction of coherentlocalized vortices. Even the
classical problem of the dynamics of two-dimensionalpoint vortices
still holds some unsolved questions, for instance, which
singularitiesare permitted in a 2D Euler equation besides point
vortices? Which other morecomplex localized vortices could contain
the Euler equation? How do point vorticesinteract with potential
waves? Saffman’s Vortex Dynamics (1992) deals mainly withclassical
problems of vortex dynamics, while Chorin’s Vorticity and
Turbulence(1996) focuses on application of vortex structures to
turbulence. Finally, VortexDynamics and Chaotic Phenomena, by V.V.
Melechko and M.Yu. Konstantinov,presents many examples of coherent
vortex structures.
In Chap. 2, we present the general theory of the motion of point
singularitiesin 2D Euler equations. It is shown that in addition to
the usual point vortices, theEuler equation can contain more
complicated moving singularities of dipole type.We consider the
interactions of a usual point vortex and point dipole. We also
studypoint dipole motion in areas with boundaries.
v
-
vi Preface
Chapter 3 covers the interaction of point vortices and sound
waves. Thedynamic of these processes is described by a nonlinear
system that belongs tothe reversible class. It holds an
intermediate position between Hamiltonian anddissipative systems,
because it contains the attractor. Due to these particularities,the
interaction of point vortices and sound waves has numerous
dynamical andchaotic modes. The most interesting of them is a new
mechanism of interactionof resonances with attractor, which is
presented in detail in this chapter.
Chapter 4 contains exact solutions of nontrivial vortex
configurations. First ofall, these are solutions of 2D Euler
equations that have a singularity of point vortextype and a smooth
part of the velocity field that looks like vortex necklaces
aroundthe point vortex. Then we describe in detail stationary exact
solutions with morecomplicated singularities than point vortices.
At the end of Chap. 4, we presentexact solutions in
magnetohydrodynamics (MHD) that give
three-dimensionalconfigurations of velocity and a magnetic field
with nontrivial topology (topologicalsolitons).
One of the key questions is how coherent large-scale vortices
can be generatedby small-scale turbulence. It would seem at first
sight that small-scale turbulencemust destroy such vortices. This
is the subject of Chap. 5. It contains the theoryof generation of
large-scale vortex structures in usual and magnetic hydrody-namics
under the impact of small-scale exterior force (small-scale
turbulence).The development of these instabilities engenders
nonlinear helical periodic waves,vortex solitons, and kinks. We
consider the cases of fluids with stratification andhumidity,
rotating fluids, and electrically conductive fluids. We also
examine thetheory of generation of nonlinear magnetohydrodynamic
vortices in conductivefluids (nonlinear dynamos).
Chapter 6 deals with plasma hydrodynamics. We begin by
introducing pointvortices that are the exact solutions of equations
of two-fluid plasma hydrodynamics.We also present Hamiltonian
equations of motion for them. We present exact three-dimensional
solutions in hydrodynamics of two-component plasma,
describingvortices of nontrivial topology with linkage of stream
lines and magnetic field.We finish the chapter by formulating a
formalism that relates the solutions of 2Dhydrodynamics equations
and 3D magnetostatics and permits the construction of astationary
magnetic configuration for each known 2D hydrodynamics
solution.
This book does not claim to be comprehensive; we mainly examine
questionsthat are close to our own interests. Nevertheless, we hope
that the book will beinteresting to a wide range of readers
interested in the issues of self-organizationand hydrodynamics and
that it will serve as a new contribution to this old field
ofresearch.
We wish to express our deepest gratitude to Hermann Haken for
his interest inour work and his support of this book.
We are very grateful to Christian Caron and Gabriele Hakuba for
their invaluablehelp in editing this book.
We would like to thank Institut de Recherche en Astrophysique et
Planétologie(CNRS, Université Paul Sabatier) and particularly
Philippe Louarn for his support
-
Preface vii
of our project. We also thank Tatiana Tour for her assistance in
the preparation ofthis book.
Toulouse, France Anatoli TurKharkov, Ukraine Vladimir
YanovskyMay 2016
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 11.1 Equations of Motion of Hydrodynamic Media . . .
. . . . . . . . . . . . . . . . . . . . . 21.2 Invariant Properties
of Hydrodynamic Media . . . . . . . . . . . . . . . . . . . . . . .
. . 71.3 Properties of Vortex Motions . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10References . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 13
2 Dynamics of Point Vortex Singularities. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 152.1 Vortex
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2
General Theory of Point Singularities
in Two-Dimensional Nonviscous Hydrodynamics . . . . . . . . . .
. . . . . . . . . . 262.3 Motion of Point Vortices . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 332.4 Interaction of a Point Vortex and a Dipole . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Motion of
a Dipole Point Vortex in Areas with Boundaries . . . . . . . . . .
532.6 The Evolution of a Dipole-Type Point Vortex in a Circular
Area. . . . . 63References . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 71
3 Influence of Potential Waves on Point Vortex Motion . . . . .
. . . . . . . . . . . . . . 753.1 The Mechanism of the Influence of
Waves on Vortex Motion .. . . . . . . 753.2 Influence of the Wave
on the Interacting Vortex Pair. . . . . . . . . . . . . . . . . .
783.3 Dynamics of Vortices in Large Wave Envelope . . . . . . . . .
. . . . . . . . . . . . . . 1003.4 Interaction of Resonances with
Attractor in Reversible Systems . . . . . 1073.5 Vortex Motion
Equation in Potential Wave Field Near the Wall . . . . . .
115References . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 128
4 Nontrivial Stationary Vortex Configurations . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 1294.1 Vortex Necklaces . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 1294.2 Vortex Structures
with Complex Point Singularities . . . . . . . . . . . . . . . . .
. 1414.3 Topological Nontrivial Vortices in a Hydrodynamic Medium .
. . . . . . . 161References . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 174
ix
-
x Contents
5 Generation of Large-Scale Vortices . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 1755.1
Nonlinear Vortex Structures in a Stratified Fluid . . . . . . . . .
. . . . . . . . . . . . . 1755.2 Large-Scale Convective
Instability
in an Electroconducting Medium.. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 1945.3 Large-Scale
Instability in a Rotating Fluid
with Small-Scale Force . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2275.4
Nonlinear Vortex Structures in an Obliquely Rotating Fluid . . . .
. . . . . 239References . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 252
6 Vortices in Plasma Hydrodynamics . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 2556.1 Elements
of Plasma Hydrodynamics .. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 2556.2 Point Vortices in Two-Fluid Plasma
Hydrodynamics . . . . . . . . . . . . . . . . . 2626.3 Equations of
Motion of Point Vortices in Plasma . . . . . . . . . . . . . . . .
. . . . . 2716.4 Topological Solitons in Hydrodynamics
of Two-Component Plasmas . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 2836.5 Relation of
2D Hydrodynamics and 3D Magnetostatics . . . . . . . . . . . . . .
290References . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 297
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 301
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 303
-
Chapter 1Introduction
Hydrodynamics is one of the oldest sciences. It has a history of
several centuriesof intense research. Despite this, not only are
numerous problems far from beingsolved, they continue to generate
new directions in theoretical physics.
The main feature of hydrodynamics—classical fields—needs to be
emphasized.It appears that the notion of velocity fields
anticipated the emergence of fields inelectrodynamics and other
sciences, wherein problems related to the nonlinearityand
dissipation of these fields are apparently more complicated than
similarproblems of the contemporary field’s theories. It is
significant that even thetheorem of existence and uniqueness of the
3D Navier–Stokes equation, which isa central hydrodynamic equation,
has not yet been proved and remains an unsolvedMillennium Problem
of the Clay Mathematics Institute.
Vortices and waves are important objects in all hydrodynamic
media. Waves,especially linear ones, are far more studied than
vortices, which are more complexstructures. This is due to the fact
that they are solutions of nonlinear hydrodynamicequations and are
inevitably multidimensional objects, and research methods suchas
linearization and monomerization cannot be applied to them. As
noted by F.Saffmen, an interest in vortices is reborn every 50
years or so [1]. At present,they attract interest for several
reasons. First, vortices are substantially coherentobjects [2, 3],
whose coherence is not destroyed even by a random field such
asturbulence [4]. In addition, vortices are interesting as exact
solutions of quasisolitons[5] or three-dimensional topological
solitons [6], or as the formation of vortex-typepatches [7].
From a physical point of view, the difference between waves and
vortices hasdeep origins. It is known that waves transport energy
and momentum, but not thesubstance of the medium. Unlike waves,
vortices can transport not only energyand momentum, but the
substance of the medium as well. This is an essentialphysical
difference between waves and vortices. For example, it can be
observedin the contrast between the smoke rings produced by a cigar
smoker and devastatinghurricanes. Since vortices and waves are
always present in nature, the crucial issue
© Springer International Publishing AG 2017A. Tur, V. Yanovsky,
Coherent Vortex Structures in Fluids and Plasmas,Springer Series in
Synergetics, DOI 10.1007/978-3-319-52733-8_1
1
-
2 1 Introduction
is to understand the impact of these objects on each other. This
question remainsthe least studied. It should be noted that the
earliest studies of vortex motion wereundertaken a long time ago by
the famous scientists René Descartes, ChristiaanHuygens, Johann and
Daniel Bernoulli, and many others. The main driving forcebehind
this research was an attempt to explain the interaction of bodies
by vortexmotions of ether. These studies made progress in
understanding vortex interaction,but were superseded by Newton’s
theory of gravitation. The detailed description ofthis period of
development of vortex theory is given in [8]. After a decline of
interestin vortices, a new wave of research dates to the middle of
the nineteenth centurythanks to the remarkable works foremost of
Helmholtz, Kelvin, and Kirchhoff. Theirstudies led to fundamentally
new hydrodynamic results.
At the root of the theoretical description of vortex objects was
Helmholtz.His work Über Integrale hydrodynamischen Gleichungen,
welche den Wirbelbe-wegungen entsprechen [9] played a key role in
the development of vortex theory.First of all, he singled out
vortex motion. According to Helmholtz, fluid motionfor which there
is no potential of velocity is called vortex motion.
Helmholtzestablished an analogy between the motion of fluids and
magnetic manifestationsof electric currents. This analogy allowed
him to introduce the rectilinear and ringvortices. In addition, he
proved the main theorems on vortex motions of an idealfluid. The
significance of these studies was highly appreciated later, and
Poincaréconsidered them the most important contribution to
hydrodynamics [8]. The mainfactor allowing Helmholtz to make
substantial progress in the study of vortex motionwas an
understanding of frozen-in vortex lines in medium motions. This
gave himthe possibility to use general laws of classical mechanics
for the vortex objects.Modern notions of hydrodynamic invariants
can be found in [10–12].
1.1 Equations of Motion of Hydrodynamic Media
Before starting to discuss the vortices, let us consider the
important elements of thedescription of hydrodynamic media. A
simple question arises: what is understoodby hydrodynamic media?
First of all, most substances exist in four states ofaggregation:
solids, liquids, gases, and finally, plasmas. It seems to us that
we caneasily distinguish these aggregate states from each other. In
fact, however, this isnot always the case. There are certain
difficulties in the separation of aggregatestates. However, we will
not go into their differences. There is only one propertythat is
important for us that distinguishes solids: the preservation of
their form. Theremaining aggregate states, on being placed in an
arbitrary receptacle, assume itsshape. Thus, the aggregate states
have flow properties. This feature can be takenas a basis for the
concept of hydrodynamic media. The second important featureof
hydrodynamic media is the preservation of their continuity. This
means thatthe study of motion of such media does not take into
account the processes ofspontaneous emergence of areas inside the
medium where the medium is absent
-
1.1 Equations of Motion of Hydrodynamic Media 3
Fig. 1.1 Selected volume ofmedium ˝, confined by thesurface @˝.
Here n is thenormal to this surface
nV
Ω ∂Ω
or another medium appears. Examples of this are the liquid
boiling process andcavitation.
These specific processes in hydrodynamic media are studied by
special methods.Motions of hydrodynamic media are described by the
evolution of certain
fields over time and space. As mentioned previously,
hydrodynamic media givean example of classical field theory. It is
clear that these fields are described byparticular derivative
equations. One of the universal characteristics of
hydrodynamicmedia is a field density of the medium �.t; x/. The
second is the medium’s velocityvector field V.t; x/.
These values are sufficient to obtain the equation that
determines the changein density of the medium [13]. Indeed, we can
imagine some selected volume ofmedium˝0, as shown in Fig. 1.1.
The mean mass of the medium in this selected volume is equal
toR˝�dx. Then,
any change of the mass in this volume is determined by the mass
flow q D �Vacross the boundary of the selected volume. The mass can
flow only in or out of theselected volume. Other mechanisms by
which mass in this volume can change areimpossible, because of the
preservation of the medium’s continuity. We thereforeobtain
Z
˝
@�
@tdx D �
Z
@˝
�Vds:
The minus sign on the right-hand side is associated with the
selection of the externalnormal directed outward from the region
(Fig. 1.1). Now let us use Stokes’s theorem,which, in particular,
allows us to switch from the integral over a closed surface tothe
integral over the volume
Z
@˝
Fds DZ
˝
divFdx;
where @˝ is the closed surface surrounding the volume ˝ . After
a transformation,we obtain
Z
˝
@�
@tdx D �
Z
˝
div�Vdx:
-
4 1 Introduction
Given that the volume was selected arbitrarily, we finally
obtain the followingequation:
@�
@tC div�V D 0; (1.1)
which is called the equation of continuity (see, for example,
[14–16]). This equationis a consequence of the law of mass
conservation and describes the change in themedium’s density as it
flows. Similarly, we can obtain the equation of motion ofan ideal
fluid [17, 18]. According to Newton’s second law, the acceleration
a of amass m is proportional to the force F applied to it. The
proportionality coefficient isassociated with the mass m:
ma D F:
Let us consider a small volume of fluid and determine its change
in velocity. It istrivial to find the velocity derivative:
dV D @V@t
dt C @V@xi
dxi:
From this, it is similarly trivial to obtain
dVdt
D @V@t
C .Vr/V:
Thus, the acceleration of the fluid element consists of the
local accelerationassociated with a change in velocity with time at
a fixed point and the “convective”acceleration associated with a
change in velocity passing from one point to another.An extremely
important object has now appeared, namely the substantial
derivative:
d
dtD @@t
C V � r: (1.2)
This operator gives the total derivative of the field for an
observer moving with thefluid at the point x at time t.
Now we calculate the force acting on the selected volume. Taking
into accountthat the force is due to the presence of pressure in
the fluid, we obtain the resultantforce by calculating the
integral
F D �Z
@˝
Pds:
-
1.1 Equations of Motion of Hydrodynamic Media 5
The integral is calculated over the surface bounding the
selected volume ˝ . Again,we transform it using Stokes’s
theorem:
F D �Z
˝
rPdx:
Going back to Newton’s equation and using the relations above,
we finally obtainEuler’s equation
�
�@V@t
C .Vr/V�
D �rP: (1.3)
Two comments must be made. Equation (1.3) does not take into
account theprocesses associated with internal friction or viscosity
of the medium. It is thereforecalled the equation of an ideal
medium. The pressure in this equation is determinedby the
thermodynamic properties of the medium [19]. The adiabatic
processapproximation is used quite often, in which the equation of
the ideal gas stateP D const�� , where � D CpCv is the adiabatic
index. In an adiabatic process, theexchange of heat inside the
fluid and with the external medium is absent. Whenusing other
equations of state of the medium, we have to introduce into the
mediuma system of hydrodynamic equations to describe the dynamics
of the other fieldsincluded in these state equations. For example,
it is often required to take intoaccount the temperature of the
field or the density of entropy. Naturally, if otherexternal forces
act on the medium, such as gravity �g, they need to be added tothe
right-hand side of Eq. (1.3). For a compressible hydrodynamic
medium with theequation of adiabatic state, the closed system of
equations consists of Eqs. (1.3) and(1.1):
�
�@V@t
C .Vr/V�
D �rP;
@�
@tC div�V D 0;
P D const�� (1.4)
There is another important approximation to describe the
dynamics of a medium thatis an incompressible fluid. In this model,
the density does not change with pressure,and the velocity of sound
correspondingly tends to infinity. This approximation isapplicable
in a hydrodynamic medium if the Mach number is small, M D Vc �
1[16, 20]. Therefore, this model is applicable to a large number of
processes andphenomena. The equation of motion of an incompressible
fluid is given in Eq. (1.3).The pressure entering into it is
determined by the condition of incompressibilitydivV D 0, and not
the equation of state. In order to understand this condition,
we
-
6 1 Introduction
return to Eq. (1.1) and write it in the following form:
@�
@tC .V � r/�C �divV D 0:
Using the total derivative (1.2), we write this equation in the
following form:
d�
dtC �divV D 0:
Preservation of density means d�dt D 0 and consequently leads to
the incompress-ibility condition divV D 0.
An incompressible ideal fluid is the simplest and most commonly
used model:
�
�@V@t
C .Vr/V�
D �rP;
divV D 0 (1.5)
This model is applicable to fluids and gases in motion with low
Mach number.The equations of motion of ideal hydrodynamic media
will be closer to reality
if we add the dissipative processes, in particular the viscous
contributions to thehydrodynamic equations.
For a description of ideal hydrodynamic media, two approaches
are used: thoseof Euler and Lagrange [17, 18, 21]. We used the
Euler approach above, whereby apoint x is selected and the changes
over time in the field at this point are observed.From the
viewpoint of an Euler medium, motion is known if we know the
functions� D �.t; x/, V D V.t; x/ and other fields that enter the
hydrodynamic model of themedium. The variables t; x are called
Euler variables.
In the Lagrangian approach, the central role is played by the
concept of theLagrangian particle. It is a sufficiently small
volume of the medium that can beconsidered a point. Of course, from
a physical point of view, it must contain manyelementary building
blocks of our medium (i.e., atoms or molecules). In addition,the
microscopic elements of a Lagrangian particle are in a state of
local statisticalequilibrium. Now, numbering these liquid particles
by their positions x0 at t D 0, itis sufficient to trace their
motions in space:
x D x.x0; t/:
For a given value x0, this function of t describes the
trajectory of the Lagrangianparticle, which goes out from x0 at
time t D 0. This information is sufficient todetermine the
characteristics of our medium at any point at the selected moment
oftime. For example, to calculate the velocity at point x at t, we
have to calculate
V.x0; t/ D dx.x0; t/dt
:
-
1.2 Invariant Properties of Hydrodynamic Media 7
The velocity in x at t is determined by this derivative taken
from the Lagrangianparticle that came at this time to the point x.
Thus, the Lagrangian variablesthat describe the evolution of a
continuous medium are .x0; t/, and the remainingvariables are
functions of these values. It should be noted that the choice of
tagsas initial positions of Lagrangian particles is not the only
one. It is only importantto distinguish the Lagrangian particles on
selected tags; their corresponding coordi-nates will be
admissible.
Of course, these approaches are linked to each other. The
transition fromLagrange to Euler variables is made by the
expression of the initial positions ofthe particles through the
coordinates of the point x. For example,
V.x; t/ D V.x0; t/jx0Dx0.x;t/;
where x0 D x0.x; t/ is the initial position of the Lagrangian
particle’s coordinates,which will be at point x at time t. To find
the Lagrange variables at a givenvelocity field V.x; t/ in the
Euler representation, it is necessary to solve a systemof
differential equations of the first order:
dxdt
D V.x; t/;
with the initial condition x D x0 with t D 0. The transition
from Euler to Lagrangevariables is also easy to carry out by
substitution of the relation x from the Lagrangecoordinates:
V.x; t/ D V.x.x0; t/; t/:
Thus, these two approaches are equivalent, but each has a number
of advantageswhen one is considering the various aspects of
hydrodynamics. In the future, wewill use both approaches.
1.2 Invariant Properties of Hydrodynamic Media
As already discussed, the motion of hydrodynamic media is
described by systemsof nonlinear partial differential equations.
Exact solutions can be found only insome particular and very
special cases. Some will be discussed below. Theseequations
generate many very complex problems, and the invariants have
greatsignificance. Of course, it makes sense to speak about
preserved values in theabsence of dissipative processes. In other
words, when discussing the invariantproperties, we restrict
ourselves to the ideal hydrodynamic media. From a geometricpoint of
view, dimensionality of space plays a key role in such field
theories.For hydrodynamics, space dimensionality is also important.
In particular, spacedimensionality determines the number of
invariant types that can appear in the
-
8 1 Introduction
hydrodynamic medium. The most convenient approach to their
description isachieved in terms of invariant differential forms
[10]. However, we shall try toexpress the invariant properties
without the use of invariant form language.
First, we start with geometrically different objects having
different dimensional-ities. The simplest zero-dimensional object
is a point. The next in complexity couldbe a one-dimensional object
such as a line or curve. The basic two-dimensionalobject is a
surface. Finally, the elementary volume is considered a
three-dimensionalbasic object. It turns out that with each of these
type are associated certain invariantsof hydrodynamic media
[10–12]. If, for example, we paint a Lagrangian particlein blue,
then it will move intact to another place only under the influence
of theflow. This means that there may be scalar fields whose
evolution is limited tothe transfer of their values by the fluid
motion. These fields satisfy the universalequation that we
discussed when we were considering the condition of
incompress-ibility:
@I
@tC .V � r/I D 0: (1.6)
Scalar fields satisfying this equation are called Lagrangian
invariants. Examplesof these invariants are the initial positions
x0 D x0.x; t/. Clearly, Eq. (1.6) canbe solved exactly in Lagrange
variables, and the solution has the followingform:
I D I.x0/:
This solution once again reminds us of the meaning of the
Lagrange invariant, asscalar fields transported by media
motions.
The next object such as a line may appear as an integral curve
of the vector field.There are fields in hydrodynamic media in which
integrated lines are transportedonly by medium motion. The integral
curves of these fields are, so to speak, “frozeninto” the medium.
Therefore, such fields have been given the name “frozen-in”fields.
These fields satisfy a universal equation of the following
form:
@J@t
C .V � r/J D .J � r/V;
which, by introducing the vector field commutator ŒA;B� D .Ar/B
� .Br/A, canbe written in a form reminding us of the Poisson
equation of classical mechanics orthe Dirac equation for
observables in quantum mechanics:
@J@t
D ŒJ;V�: (1.7)
The values satisfying this equation are called invariants, or
frozen-in integrals,and they exist in all hydrodynamic media.
Frozen-in invariants are very importantfor understanding many
phenomena in different hydrodynamic media. We will
-
1.2 Invariant Properties of Hydrodynamic Media 9
now discuss them in greater detail. In Lagrangian variables,
this equation can beintegrated exactly. Its solution has the
following form:
Ji D J0.x0/j @xi@x0j
;
where J0.x0/ is the initial vector field; the repeated indices,
as usual, meansummation. This solution also ensures that the
integral lines of this field are frozeninto the medium.
We now turn to more complex objects: surfaces. These objects may
appear asintegral surfaces or surfaces tangential to the given
field of two-dimensional planes.
This field of planes can be set up for a given vector field.
Indeed, let us imaginethat at every point of our medium is a
certain vector of this field. Let us place at eachpoint the plane
orthogonal to this vector. This can be done in
three-dimensionalspace uniquely. Then the set of planes determines
the field of two-dimensionalplanes.
The tangential surfaces to the field planes are integral
surfaces. In hydrodynamicmedia, there are fields that have a
surprising behavior of their integral surfaces. Theirsurfaces are
transported only by the medium motion. It is clear that this
property canbe considered a specific conservation of integral
surfaces. The equations for thesevector fields have the following
universal form:
@Si@t
C .V � r/Si D Sm @Vm@xi
: (1.8)
Fields satisfying this equation are called S-invariants. It
should be noted that unlikethe integral curves, the fields of
planes are not always integrable; therefore, integralsurfaces do
not always exist. For this existence, the conditions of
FrobeniusS�rotS D0 must be met. Of course, if this condition is
respected at the initial time, it will besatisfied at all
subsequent times. As well as all of the aforementioned
equations,Eq. (1.8) can be integrated in the Lagrange variables.
Its exact solution is
Si D S0.x0/j @x0j@xi
:
Now it is necessary to focus on the final type of invariants
that occur in hydro-dynamic media. We have already met this
invariant: it is related to the medium’scontinuity. It turns out
that there are pseudoscalar fields playing the role of thedensity
of conserved values. An obvious example is the mass density
entering intothe fundamental equation of hydrodynamics. It is clear
that these density invariantssatisfy the continuity equation (1.1).
Like all other invariants, the continuityequation can be solved
exactly in Lagrange variables. Its solution is trivial:
� D �0.x0/D.x0/D.x/
;
-
10 1 Introduction
where D.x0/D.x/ is the Jacobian of the transformation from the
variables x0 to thevariables x [13, 21]. The invariants discussed
above are of great importance for thedescription of the motions of
the different hydrodynamic media. Some examples oftheir use are
given in the following sections.
1.3 Properties of Vortex Motions
Under vortex motion we will understand the movement of fluid at
which the vectoris not equal to zero at least in some part of
fluid. Let V be the velocity field of themedium motion. Then we
will call the vector
! D rotV (1.9)
vorticity. The physical sense of vorticity can be understood by
considering fluidrotation as a whole around an axis �. As is known,
the velocity of this simple flowis determined by the ratioV D 1
2��r. Here � is a constant vector characterizing the
rotation velocity, and r is the vector of rotation of the
orthogonal axis. In calculatingthe rotor (i.e., curl) of this
velocity field, it is easy to derive that rotV D �, andtherefore,
it coincides with the angular velocity of the fluid rotation. Let
us nowconsider the evolution of vorticity in an ideal fluid. It is
simple enough to obtainthe equation describing the variation of
vorticity with time acting with the operatorrot on the equation of
motion of an ideal incompressible fluid. Thus, pressure
iseliminated, and the resulting equation has a rather simple
form:
@!
@tC .Vr/! D .!r/V: (1.10)
This equation is called the frozen-in equation of the vorticity
field !. It should benoted that this equation has a purely
geometric meaning, completely unrelated tothe specific form of the
equations of the medium’s motion. Therefore, the valuesthat satisfy
this equation are present in all hydrodynamic media [10]. The
physicalmeaning of this equation represents the transport of the
field lines by mediummotion fluid flow.
This can be verified by obtaining the equation for an arbitrary
vector field whoselines of force are transported by medium motion
(for example [10]). In order tobetter understand the meaning of a
frozen-in equation, let us consider the curve x Dx.s; t/, which is
transported with time by medium motion (see Fig. 1.2). Here s isthe
parameter that parameterizes the appropriate line in
three-dimensional Euclideanspace, and t is time. Let us introduce
the linear element of this line according to
ı l D @x.s; t/@s
:
-
1.3 Properties of Vortex Motions 11
Fig. 1.2 A parametricallydefined line in space at time twith a
tangent vector ıl. Eachpoint on this line istransported with a
velocityV.x; t/
x(s, t )
δl
V(x,t )
In other words, ı l is a vector tangent to the curve x D x.s; t/
at the point defined bythe parameter s at time t. Then,
differentiating this with respect to time, we get
dı ldt
D ddt
@x.s; t/@s
D @@s
dx.s; t/dt
:
Since by definition, the line x D x.s; t/ is moved by medium
motion, it follows thatdx.s; t/
dtD V.x; t/;
where V.x; t/ is the velocity of medium motion in points of
curve. Using this fact,we transform the previous equation to the
form
dı ldt
DD @@s
V.x; t/ D @xj@s
@
@xjV.x; t/:
Finally, we obtain
@ı l@t
C .V � r/ı l D .ı l � r/V: (1.11)
Comparing the obtained equation (1.11) with the frozen-in
equation (1.10), it iseasy to see their full coincidence. That is
why the geometric meaning of the frozen-in equation is to transfer
the integral curves of the field ! by medium motion,determined by
the velocity field V.
A remarkable feature of this equation is that it is exactly
integrable in Lagrangiancoordinates. Let the motion of Lagrangian
particles be described by the relationsx D x.t; x0/, where x0 is
the initial position of the Lagrangian particles. Then theexact
solution of the frozen-in equation takes the form
!i D !0.x0/j @xi@x0j
:
-
12 1 Introduction
Here !0.x0/ is the initial vorticity at time t D 0. Certainly,
the presence of thissolution does not mean the ability to set the
velocity field. The reason for this is quiteobvious: we do not know
the explicit form of the relation between Lagrangian andEulerian
coordinates. To establish it, we need to solve complex nonlinear
equations,which usually turns out to be more difficult than finding
the solution of Eq. (1.10) inEuler variables. The above-described
properties are of an extremely general natureand apply to all
hydrodynamic equations. Detailed descriptions of the
invariantproperties of hydrodynamic media can be found in [6,
10–12].
Despite its relatively simple form, Eq. (1.10) is nonlinear
(taking into accountthat ! D rotV), and the process of finding its
solutions is not a trivial task. Thisproblem is simplified in some
cases in the presence of special symmetries. Amongthese symmetries,
there many known exact vortex flows.
In particular, a significant simplification of the equation
occurs when the vorticityvector field is two-dimensional. For
example, the vorticity vector field mightbe directed along the
z-axis, and the flow in this direction is absent Vz D 0.Such
vortices can be considered vortex tubes, elongated strictly along
the z-axis.Fluid motion in the plane perpendicular to this selected
axis depends only on thecoordinates .x; y/. Then Eq. (1.10) in the
two-dimensional case is considerablysimplified and takes the
form
@!z
@tC .Vr/!z D 0 (1.12)
of a passive impurity transport equation. From the viewpoint of
vorticity, thisequation means that the specified initial vorticity
!z is simply transported by thefluid in the same way as if it were
a passive impurity. This transition to the two-dimensional problem
allows us to simplify the search for exact vortex
solutionsconsiderably. One important change is related to the
possibility of obtaining anonlinear equation for only one scalar
current function ', in contrast to the originalsystem of equations
for the three components of the velocity.
Indeed, using the property of incompressibility of fluid divV D
0 in the two-dimensional case,
@Vx@x
C @Vy@y
D 0; (1.13)
it is possible to write down the components of the velocity
field using a singlefunction '.x; y; t/ as
Vx D @@y'.x; y; t/;
Vy D � @@x'.x; y; t/;
-
References 13
for which no divergence condition of the velocity field will be
automaticallysatisfied. This function '.x; y; t/ is called a stream
function. Thus, it is possibleto unambiguously restore the velocity
field from the known stream function.Trajectories of Lagrangian
particles are found as the solutions of the equation'.x; y/ D
const, at least for stationary flows. Using this representation of
thevelocity field, the frozen-in equation (1.10) can be written
down in terms of thestream function. Let us recall that in terms of
the stream function ', the vorticity !is determined in a known
manner as
! D ��'; (1.14)
and after substituting into Eq. (1.10), we obtain
@�'
@tC f�'; 'g D 0: (1.15)
Here � is the two-dimensional Laplace operator, fA;Bg D "ik
@A@xi @B@xk is thePoisson bracket, and the indices i; k D 1; 2 are
number coordinates in two-dimensional Euclidean space. The tensor
"ik is the unit antisymmetric tensor. Thisequation is often called
the Helmholtz equation. This form of the equations of
anincompressible fluid is extremely useful, and it is often used to
find exact solutionsin two-dimensional hydrodynamics. A decisive
role is played here by the transitionfrom several unknown functions
to a single function '. As noted above, assumptionsabout the
increased symmetry of solutions lead to certain simplifications and
are themain source of exact solutions. For these cases, certain
methods of solution havebeen developed for the equations in the
Euler variables [22] and the Lagrangiancoordinates [23] as well. It
should be noted that the evolution of two-dimensionalHamiltonian
systems satisfies this equation exactly. This is a useful analogy,
and wewill return to it below.
References
1. Saffman, P.G.: Phys. Chem. Hydrodynamics. 6(5/6), 711–726
(1985)2. Williams, J.E.F., Kempton, A.J.: Structure and mechanisms
of turbulence II. In: Fiedler,
H. (ed.) Proceedings of the Symposium on Turbulence. Springer,
Berlin (1977)3. Levich, E.: Coherence in turbulence: new
perspective. In: Concepts of Physics, vol.VI, 3
(2009). On-line4. Cantwell, B.J.: Ann. Rev. Fluid Mech. 13,
457–515 (1981)5. Petviashvili, V.I., Pokotelov, O.A.: Solitary
Waves in Plasma and the Atmosphere. Gordon and
Breach Sciences Publishers, Amsterdam (1992)6. Sagdeev, R.Z.,
Moiseev, S.S., Tur, A.V., Yanovsky, V.V.: Problem of the strong
turbulence and
topological solitons. In: Nonlinear Phenomena in Plasma and
Hydrodynamics, pp. 137–182.Mir Publishers, Moscow (1986)
7. Newton, P.K.: The N-Vortex Problem. Springer, New
York/Berlin/Heidelberg (2000)8. Poincare, H.: Theorie des
Tourbillions. Carre, Paris (1893)
-
14 1 Introduction
9. Helmholtz, H.: J. Reine Angew. Math. 55, 25–55 (1858)10. Tur,
A.V., Yanovsky, V.V.: J. Fluid. Mech. 248, 67–106 (1993)11.
Moiseev, S.S., Sagdeev, R.Z., Tur, A.V., Yanovsky, V.V.: Sov. Phys.
J. 56(1), 117–123 (1982)12. Volkov, D.V., Tur, A.V., Yanovsky,
V.V.: Phys. Lett. A 203, 357–361 (1995)13. Chorin, A.J., Marsden,
J.E.: A Mathematical Introduction to Fluid Mechanics. Springer,
New
York (1993)14. Acheson, D.J.: Elementary Fluid Dynamics. Oxford
University Press, Oxford (1990)15. Patterson, A.R.: A First Course
in Fluid Dynamics. Cambridge University Press, New York
(1983)16. Faber, T.E.: Fluid Dynamics for Physicists. Cambridge
University Press, New York (1995)17. Tritton, D.J.: Physical Fluid
Dynamics. Van Nostrand Reinhold Company, New York (1989)18. Kundu,
P.K.: Fluid Mechanics. Academic, San Diego (1990)19. Shapiro, A.H.:
The Dynamics and Thermodynamics of Compressible Fluid Flow, vols. 1
and
2. Wiley, New York (1953)20. Lighthill, M.J.S.: An Informal
Introduction to Theoretical Fluid Mechanics. Oxford University
Press, Oxford (1986)21. Meyer, R.E.: Introduction to
Mathematical Fluid Dynamics. Dover, New York (1982)22. Ovsyannikov,
L.V.: Group Analysis of Differential Equations. Nauka, Moscow
(1978) (in
Russian)23. Abrashkin, A.A., Yakubovich, E.I.: Vortex Dynamics
in Lagrangian Description. Fizmatlit,
Moskow (2006) (in Russian)
-
Chapter 2Dynamics of Point Vortex Singularities
This chapter focuses on localized vortices in an incompressible
fluid. We consider indetail a class of point vortices that can
serve as an example of single hydrodynamicquasiparticles. The
interaction and movement of even a small finite number of
thesevortices generate complex hydrodynamic flows. We develop the
general theory ofthe motion of complex point vortex singularities.
We also examine the interactionof dipole vortices with ordinary
point vortices. In addition, we study the motion ofpoint dipoles in
a restricted domain. The first section presents the main
propertiesof a large number of known localized vortices.
2.1 Vortex Structures
In this section, we describe the main types of vortex
structures, their models, andtheir properties. It is known that
partial solutions with high symmetry are a goodsource of vortex
structures. As noted above, nonlinear hydrodynamic equations donot
belong to the class of integrable ones. For a sufficiently high
symmetry, onecan apply group-theoretic methods to find particular
solutions for hydrodynamicequations (see, for example, [1]).
Two-dimensional flows were studied first. Forthese cases, using the
methods of the theory of functions of complex variables looksrather
attractive. Poincaré in [2] used the method of conformal mapping to
studyvortex motion (see, for example, [3]). An interesting version
of this approach forfinding exact solutions of an incompressible
fluid, but in Lagrangian variables, wasproposed in [4, 5]. Using
this approach, some classes of exact solutions of the Eulerequation
have been found. A detailed review of this topic is presented in
the book[6]. The most interesting class of solutions that this
method permits is that of dual-frequency solutions describing
Ptolemaic flows. These solutions are determined by
two analytic functions G.z/ and F.Nz/ that satisfy the
conditionˇˇˇ dG.z/dz
ˇˇˇ2�ˇˇˇ dF.Nz/dNz
ˇˇˇ2 ¤ 0.
In fact, this value determines the magnitude of the vorticity of
two-dimensional
© Springer International Publishing AG 2017A. Tur, V. Yanovsky,
Coherent Vortex Structures in Fluids and Plasmas,Springer Series in
Synergetics, DOI 10.1007/978-3-319-52733-8_2
15
-
16 2 Dynamics of Point Vortex Singularities
flows. Particular cases of these solutions, such as Gerstner
waves, were knownearlier [7]. Nevertheless, a significantly wider
range of solutions has been obtainedin Euler variables. A few
extremely important exact solutions had been derivedby the
beginning of the twentieth century. In the list of these solutions
shouldbe included elliptic Kirchhoff vortices [8], Hill [9] and
Chaplygin–Lamb [10, 11]vortices, and the Maxwell toroidal vortex
(see, for example, [11]). A search fortwo-dimensional solutions is
based on the solution of Eq. (1.15). It is clear that thestationary
solutions are the simplest ones. In order to find these solutions,
a simplerequation should be solved. There are roughly three classes
of stationary solutions:
�' D Const; (2.1)�' D Const � '; (2.2)�' D F.'/: (2.3)
It is easy to see that the solutions of these equations are
exact stationary solutionsof the Poisson bracket (1.15). The class
of solutions that satisfy the first equationcorresponds to the
flows with constant vorticity in some domain. These domains canbe
limited by rigid walls or partially bounded, such as, for example,
the solutionsin [12, 13]. The same class of solutions includes the
elliptic Kirchhoff vortexsurrounded by potential flow [8], the
Rankin circular vortex [14], the Chaplyginelliptical vortex
surrounded by shear flow [15], and the vortex in the velocity
fieldthat depends linearly on the coordinates [16, 17].
Some famous solutions such as the Lamb [11] and Chaplygin
vortices [10] satisfythe linear equation (2.2). The basic set of
known solutions of the nonlinear equation(2.3) is limited to a
small choice of functions on the right-hand side. So the solutionis
found when the right-hand side is chosen as F.'/ D e' . For
example, thesesolutions describe the path of circular vortices in a
shear flow [18]. Another exampleis the Kelvin–Stewart solution (cf.
[19]). The stream function of this solution has theform
' D � ln�
c � cosh y Cp
c2 � 1 � cos x�:
This is a one-parameter family of (see Fig. 2.1) solutions
periodic over x with theconstant c � 1. The solutions with a
different choice of functions are not wellknown. So F.'/ D ˛' ln' C
ˇ' gives the exact solutions found in [20], and asolution with a
power function was considered in [21].
Other classes of localized vortex flows are associated with
axisymmetric vortexflows. They can be considered “two-dimensional.”
This class of vortices was firstconsidered by Helmholtz [22]. For
these vortices, his equation takes the simple form
d!
dtD 0:
-
2.1 Vortex Structures 17
X0
y
0
Fig. 2.1 Level lines of Kelvin Stewart vortex solutions. It is
obvious why this solution is oftencalled cat’s eyes
0
z
ρ
Fig. 2.2 Streamlines of the Hill vortex inside a sphere and
streamlines of potential flow surround-ing the sphere
Here ! is the angular component of the vorticity in cylindrical
coordinates. ThenHill found the existence of a localized vortex in
an unbounded medium [9]. Hisparticular exact solution in
cylindrical coordinates has the form
vr D 2 ka2
r.z � Z/;
vz D �2 ka2
r.z � Z/2 � 2 ka2.2r2 � a2/� PZ;
! D 10 ka2
r:
This solution is localized in the area of the radius a, whose
center is moving withvelocity PZ. Here Z is an arbitrary function
of time. Outside, the vortex is surroundedby a potential flow.
Figure 2.2 shows a view of the streamlines of this vortex. TheHill
vortex is, in a sense, the definitive type of a whole family of
toroidal vortices
-
18 2 Dynamics of Point Vortex Singularities
Fig. 2.3 The streamline of fluid flow of two vortices from the
toroidal family are shownschematically
(see Fig. 2.3). An approximate solution in the form of a vortex
ring was consideredin works of Hicks [23, 24]. Other solutions for
vortex rings were obtained in [25–28]. Therefore, there was a
problem of reconciling these results.
Meanwhile, the correct solution was proposed by Dayson [29].
This remarkablework describes completely all the characteristics of
the steady motion of thevortex ring of the final section. The
ultimate verification of these results wasperformed in [30]. A
detailed numerical analysis of the motion of a vortex ring
[30]confirmed the reliability of Dayson’s asymptotic development.
Most of the vorticesabove discussed are the exact solutions of
ideal hydrodynamics. To conclude thediscussion of vortices in a
finite area, we present the Sullivan vortex [31], which isan exact
solution of the Navier–Stokes equations. The velocity field of this
vortexhas the form
Vr D ˛r2
C 6�r
�
1 � e ˛r24��
;
V� D �2r
�H�˛r2
4�
�
H.1/ ;
Vz D ˛z�
1 � e ˛r24��
;
where � is circulation at infinity, ˛ is a constant, and � is
kinematic viscosity.This solution is a two-celled stationary
vortex. More recently, in [32], the Sullivansolution was
generalized to the nonstationary case. Families of vortices of this
typeare considered in [33–36]. The elliptical vortices of Moore and
Saffman [17] shouldalso be noted. It is interesting that these
classical solutions are still relevant (seerecent works [37, 38]).
Generalizations of these solutions are models of variouscoherent
structures, vortex patches and vortex crystals (see, for example,
[39–43]and references therein), which are easily observed in
numerical and laboratoryexperiments (see [44–51]). Other classes of
smooth exact solutions are based on the
-
2.1 Vortex Structures 19
equation of sinh–Poisson type for stream functions [52–54]. We
should also note aninteresting class of Kida–Neu vortices [55, 56].
There are also many examples ofsolutions with a singular vorticity
distribution (see [16]), as well as solutions thatcontain both the
smooth part of the vorticity field and point singularities [57,
58],which are usually arranged in symmetric configurations, as will
be discussed inChap. 4.
Another important class of two-dimensional solutions consists of
point vortices.These solutions relate to the singular generalized
solutions of the Helmholtzequation. Such vortices appeared in [22].
The importance of these vortices inhydrodynamics cannot be
overestimated. A huge number of problems have beensolved with the
use of these vortices. The main element of the idealization in
thetransition to these vortices is a consideration of infinitely
thin vortex filaments.
The vorticity of the vortices is concentrated on these lines.
When these straightfilaments coincide with the direction of the
z-axis, it is sufficient to consider themotion of the intersection
points of these vortices with the plane, for example .x; y/.The
image of such a rectilinear vortex is a point, which is why they
are called pointvortices. The vorticity of this vortex in the plane
is characterized by the followingformula:
!z D !0ı.x � a/; (2.4)
where the vector x belongs to the plane .x; y/, and the vector a
determines theposition of the vortex. However, using point vortices
allows us to consider a set ofhydrodynamic flows as being induced
by a system of interacting point vortices [22].In a sense, the set
of hydrodynamic problems is reduced to problems of
classicalmechanics of particle motion with a special law of
interaction. This approach wasalso suggested by Helmholtz, and this
formulation of the problem has been appliedwidely for the study of
complex vortex flows. However, there is no evidence thatall real
vortex flows can be modeled solely by a system of point vortices.
The mainassumption for the modeling of complex vortex flows is a
theorem of frozen-invortex lines in a medium. This means that every
vortex line remains constantlycomposed of the same fluid particles
and moves in the fluid with them. Thus, bysetting the location of
the vortex and the values of their vorticity, the fluid flow can
berestored without studying the interaction of complex vortex
structures. The velocityfield of a point vortex and particular
cases of their interactions were consideredby Helmholtz [22]. He
examined the problem of interaction of two vortices, theinteraction
of vortex rings, and the evolution of a point vortex in an ideal
fluidbounded by a plane, and he also studied the impact of
boundaries on vortex motion.In his work, Helmholtz laid the
foundations of vortex theory and also defined theprincipal
directions of its development.
However, the modern form of equations of motion of point
vortices was given bythe outstanding scientist Gustav Kirchhoff. In
his lectures on mathematical physics[8] he derived the equations of
motion for an arbitrary number of point vortices andobtained them
in Hamiltonian form. In addition, he obtained all the
conservationlaws for this system. Obviously, he examined in detail
the case of the motion of two
-
20 2 Dynamics of Point Vortex Singularities
vortices, whose study was initiated by Helmholtz. Using the
point vortex approach,Kirchhoff considered an elliptical domain
filled with point vortices. This solutionis known as the Kirchhoff,
or elliptical, vortex [8]. The model of point vortices hasbeen
generalized to some more complex media as well. The equations of
motionfor point vortices in a stratified incompressible ideal fluid
and the Hamiltonian formof the equations of motion were obtained in
[59]. The equations of motion of pointvortices in a fluid with a
free surface were obtained in [60]. Attempts to transfer thismodel
to plasma hydrodynamics succeeded only recently [61].
Another important type of vortex is called the vortex sheet. In
the case of a pointvortex, vorticity is localized only on the line,
but in the case of a vortex sheet,vorticity is concentrated in an
infinitely thin layer along a two-dimensional surface.Analytically,
this means that the vorticity of this object has the form
! D �ı.xn/:
The vector � D � � determines the vorticity’s magnitude and
direction, the vector� is tangent to the vortex line, and xn is the
coordinate transversal to the sheet. Thevelocity field of the
vortex sheet is given by
V D � 14
Z
S
.r � r 0/ � �jr � r 0 j3 ds.r
0
/:
Here ds.r0
/ is the element of surface area S, and � denotes vector
multiplication.It should be noted that the vortex sheet problem
arises very frequently. Simpleexamples of it are the flows around
the wings of aircraft, and around different planesand various
objects. A flat sheet is unstable and therefore rolls into spiral
vortexstructures [16, 22, 62]. This instability is the basis for
explaining various effectsobserved in nature.
Some progress has been made in the search for new vortex
structures andhas led to the introduction of a new type of singular
point vortex in the three-dimensional case. These vortices were
named vortons [63]. Critical discussionconcerning vortons and
comparative analysis of different types of them can befound in
[64]. The introduction of vortons as three-dimensional point
vortices isconnected with great difficulties, because the velocity
field of the object must havea number of internal parameters more
than those of a point vortex. Attempts toavoid such difficulties
led to the introduction of vortex objects whose velocity
fieldcoincides with the remote field of three-dimensional localized
vortices [65, 66].However, such an idea, though physically
understandable, has certain shortcomingsof a purely mathematical
nature. Strictly speaking, they cannot be consideredexact solutions
of the hydrodynamic equations. So, these studies point clearlyto
the need to search for new types of vortices. The necessity of
these objectsto explain various hydrodynamic phenomena remains
topical. It should be notedthat the exact solutions of
two-dimensional hydrodynamic equations are importanteven for
magnetohydrodynamic issues, which seem quite distant. A theorem
onthe existence of these relations was proven in [67]. It appears
that the presence
