J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)89–93
000–000

Stationary Vortical Structures in Electrostatic
Drift Wave Turbulence
SPINEANU Florin, VLAD Madalina, ITOH Kimitaka1 and ITOH Sanae -I.2
Association Euratom-MECT, Bucharest, Romania,
National Institute for Fusion Science, Toki 509-5292, Japan,
2
Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
1(Received: 10 December 2003 / Accepted: 3 August 2004)

Abstract
The conditions of generation of vortical flows in fusion plasma are reviewed with particular interest for the roleof the small scale vortices in the intermittency events of the internal transport barriers (ITB). Exact solutions areprovided for stationary states.
Keywords:
ion drift wave, vortice, zonal flow, Integrable structure

1. Introduction

poloidal asymmetry, etc). We have found that the nonlinear
drift mode equation has, at this scales, an exact, periodic
solution whose pattern is identical to the zonal flows, withgood agreement with the experimental observations andnumerical simulations. This flow, which naturally suppresses
the radial transport, becomes structuraly unstable when thevectorial nonlinearity is significant and decays into an
ensemble of vortices with scale close to rs.
The third aspect is derived from the necessity to assess
the role of these vortical structures, generated under the
Hasegawa-Mima nonlinearity in the phase where the barrier
is destroyed (an intermittent event). It is well known fromnumerical simulations that an ensemble of vortices (which
are not solitons) evolves by collisions and merging, generating larger structures from which the ITG eddies may recover.
We are concerned here with the nature of the solutions
consisting of ensembles of vortices. We prove that the
Hasegawa-Mima equation has stationary periodic solutions
consisting of a lattice of vortices. The basic procedure inthis rather technical analysis consists of the mapping of the
HM dynamics to a system of point-like vortices interacting
via a short range potential. This model is formalized in afield theoretical framework and we search for the states
having a particular property, called self-duality (a minimal
characterization is that they extremize the energy functional).
The outcome of this formalism consists of nonlinear differential equations which must replace at stationarity the HM
equation, precisely as the sinh-Poisson equation replaces theEuler equation for the ideal fluids. We prove that theseequations are exactly integrable.

It is a well known fact, supported by experiments andnumerical simulations, that the drift wave turbulence isaccompanied by generation of cuasi-coherent vorticalstructures. The aim of this work is to examine the role they
can have in the formation and in the intermittent states of
the transport barriers. Three elements are particularly
relevant in this respect.
First, it is recalled that the drift wave has different
nonlinear dynamics on different space scales [1]. At largest
space scale one finds in numerical simulations the dominanceof the eddies driven by the ion temperature gradient (ITG),
which have large radial extension due to the ballooning
character and linear mode coupling, with a particularsensitivity to the shear. The Hasegawa-Mima (HM) equation
governs the electron drift waves at the small space scales, ofthe order of the drift wave dispersion scale, rs = (Te/mi)1/2/W i,
where Te is the electron temperature, mi is the ion mass, W i =eB/mi is the ion cyclotron frequency, e is the ion charge and
B is the magnetic field. At intermediate space scales the scalar(or Korteweg deVries) nonlinearity is dominant. Except for
the large ITG eddies (which are well studied numerically),
the two other scales can be satisfactorily described analytically in closed form and have been the subject of many studies.
Second, the flow structures at mesoscopic scale have astrong influence on the transport properties [2]. This is an
essential part of the theories aiming to explain the highconfinement barriers by the reduction of the radial correlation
length of the turbulent fluctuations in the presence of asheared flow, even if there is not yet an unanimous opinion
on the origin of these flows (ion-orbit losses, Reynolds stress,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: spineanu@ifin.nipne.ro1
89Spineanu F. et al., Stationary Vortical Structures in Electrostatic Drift Wave Turbulence

2. The space scales of the drift instability

Petviashvili and Pokhotelov have found the approximate
4 /3
solution f (r) = 42.8ba ÈÎÍsechÊË 43 x 2 + ( y - ut )2 rˆ¯  . The condition imposedto the Flierl-Petviashvili equation is to provide an isolated,finitely extended vortical solution, obeying boundary conditions at infinity (on the 2D) of smooth decay. There is no
known analytical form for an exact solution.
We consider that the insuccess in determining theanalytical vortex solution has a fundamental motivation. Firstof all the equation does not pass the Painlevé test, so it isvery likely that it is not integrable. This, in the definition thatis suggested by the Inverse Scattering Transform method,means that there is no Lax pair of operators for this equation.
This equation has close ressemblance with some approximative form of other nonlinear equations, for example thedifferential equation Df = exp(f) – 1, known as the AbelianHiggs equation (AH), governing the vortices of the superconducting media. This equation has also been derived by Jacobs
and Rebbi. An expansion of the right hand side Df = exp(f)– 1 = f – –21 f2 leads to something close to the FP equation,
especially because the FP equation has actually been derived
under the neglect of the third order powers of f. We have
proved elsewhere [9] that the AH equation is integrable on
periodic domains and we have provided analytical solutionsin terms of Riemann theta functions. It is a known fact that
equations derived as deformations of exactly integrable equations preserve some of their properties: solutions are robustand localised even if they are not exactly solitons. However
it is difficult to give a formal characterisation of this kind ofsoft-nonintegrability.
In conclusion we claim that the monopolar vortex of theFP equation is only a manifestation of the close proximity (infunction space) of an exactly integrable structure, related,most probably, to the Abelian Higgs model. This implies that
we have to look for vortices at smaller scales, corresponding
to the Hasegawa-Mima equation.

We consider the equations of continuity and momentum
conservation for the electrons and ions in two-dimensions [1].
We assume the quasineutrality ni ne and the Boltzmann
distribution of the electrons along the magnetic field line ne|e|f= n0 exp (– —). The equation written for vorticity W = — ¥ nTedis dt– (W + Wi) + (W + Wi) (—^ · n ) = 0. Here —^ is the gradient
operator transversal to the magnetic field.
Using the normalizations t Æ Wit, (x, y) Æ (x/rs, y/rs)
and f Æ ef/Te, the equation for the electrostatic potential f
is obtainedÊ
1 - — 2^ ˆ¯ f - ( -— ^f ¥ nˆ )  vd* + ( -— ^f ¥ nˆ )  vdT f
t Ë(+ ÈÎÍ( -— ^f ¥ nˆ )  — ^  -— 2^f

)
(1)

=0

where nd* = –—^ ln n0 – —^ ln Te and ndT = –—^ ln Te. The twononlinearities that appear in the equation have different
physical effect. However, they are active on distinct spacescales, as shown by a multiple space-time scales analysis[3,4]. By introducing a translation velocity u, the equation isexpressed in a moving frame, h = y – ut. One takes xi = e ix
and ti = e it, f = ef1 + e 2f2 + … for e << 1. One space scale is
specified by the choice f1 = f 1 ( x0, x1, x2, …, h0, h1, h2, …,
t0, t1, t2, … ), with the density and temperature variable on
the i = 1 scale. The characteristic space scale for the potential
is then rs which is the typical extension of the dipolar vortex.
The equation obtained in the first order in e, is the HasegawaMima equation.
A different dynamical equation is obtained on otherspace-time scale f = e 2f2 + e 3f3 + … and the density, thetemperature and the velocity u are assumed to vary on thek (x )
second (slower and larger) scale. Assuming that T (1x ) + nu2 22~ O(e 2), the equation on the time scale of order 5 has theform of the Flierl-Petviashvili (FP) equation.

Df = af - bf 2T

3.2 The exact periodic solution of the FP
equation
In a previous work [10] we have determined an exact
solution of the FP equation. The method consisted in looking
for the trajectories of the one-dimensional solution singularities, in the complex plane of the spatial variables. The exact
solution to the Petviashvilli equation is

(2)

( )

1
where a = —(1 – v–u* ) and b = 2 u 2 eBe 2 r 2 x L1n (v * is ther2s
0 selectron diamagnetic velocity and Ln is the density gradient
length). This equation has been derived by Petviashvili [5] ina study of the Jupiter’s Red Spot. The dominant space variation is here on the scale x1 = rs/e which is much larger thanthe dipolar vortex scale.

f s ( x, y) =

a
+s
2b

Ê
3a 2 ˆ
Á iay + ibx + w | g2 =˜(sb )2 ¯Ë(3)

3. The intermediate-scale structures
where a, b and s are parameters related by the condition a2 +b2 = sb/6 andis the doubly periodic elliptic Weierstrass
function. Here w is half of the period on the real axis of .
Experimental measurements of the characteristics of thezonal flow have been performed on Doublet III-D tokamak[11]. In Ohmic and L-mode plasmas it has been found a> 10V and a flow shear wE ¥ B ~ 2 ¥perturbed potential f̃rms ~5 –1
10 s . The value for the radial wavelength is in the range

3.1 The monopolar vortex
The one dimensional form of the equation (2) has beensolved on an infinite domain, obtaining as solution the KdV
soliton [6]. The two dimensional equation has vorticalmonopolar solutions, well studied numerically [7,8]. By
defining a functional of the solution expressed in onedimensional (radial) geometry and taking the extremum of
the functional under the condition of asymptotic decay,
290Spineanu F. et al., Stationary Vortical Structures in Electrostatic Drift Wave Turbulencekrrs Œ [0.1, 0.6] which means lr Œ (15, 30)rs. Due to thedifferent sensitivity of the solution fs to the parameters, thisset is sufficiently restrictive to determine its form. We haveto take v*/u very close to unity and g3 ~ –1500 which gives:> 17V, lr @ 17.4rs, wE ¥ B ~ 2.2 ¥ 105 s–1. The relativef̃rms ~
amplitude of the perturbation results ~ 4%. An important
experimental result is the radial spectrum of the perturbation.
We have calculated S (kr) from the Fourier transform of thecorrelation of fs (x). The result is very close to Fig. 3 of Ref.
[11], with two symmetric peaks at k0r ~ 4 cm–1. The same
sharp decay for |kr | < k0r is observed, as described in Ref.
[11].

(presently with limitted precision) and analytical considerations suggest that the destruction of the flow pattern does not
immediately lead to an arbitrary random field: a fundamental
process is the generation of small (~ rs) space scale vortices.
We can show that the regular flow structure is replaced by alattice of vortices which is reminescent of the exact solutionof a closely related nonlinear equation. This solution evolves,due to the weak interaction between the vortices, to an ensemble of quasi-independent vortices that collide inelastically andbecome of various amplitudes plus a surrounding drift wave
radiation. From this random field the ITG instability may
regenerate the structure of eddies and reinstate the transport.
From the numerical simulations showing structural
instability of the FP equation we draw the conclusion that amore careful study must be done of the small scale vorticalstructures arising in the dynamics of the Hasegawa-Mima
equation.

3.3 Linear stability of the stationary periodic
solution
The time variation of the stationary periodic solution is
described by the equation from which the FP equation isderived at stationarity. We start from the equation with scalar
nonlinearity, with the presence of a temperature gradient4. The Hasegawa-Mima vortices
The known stable dipolar vortex of the Hasegawa-Mima
equation is the Larichev-Resnik solution consisting of two
monopolar lobes with positive and negative signs. However
this is not an exact solution. We are looking for other possible
vortex solutions. In particular we are interested in thesolutions at stationarity, since the numerical simulations
clearly indicate that large scale vortical structures are createdat late time. There is however an important difficulty in thisapproach. The naive stationary form of the HM equation isextremely general and can be solved by a huge class offunctions. This means that the physics is hardly controlled
and suggests a reformulation of the problem. In the similarcase of the Euler equation, this was possible because the ideal
fluid model is known to be equivalently described as acollection of point-like vortices moving in plane under theaction of a field whose propagator is the inverse of the 2D
Laplacean [12]. This model leads to the sinh-Poisson equation
for the stationary states.
There is a similar situation for the Hasegawa-Mima
equation. In meteorology it has been developed a model
consisting of the motion in plane of point-like vortices with ashort range interaction potential. We start our analysis fromthe basic assumption that the HM equation is equivalent tothis model. In a similar approach as we have developed forthe sinh-Poisson equation, we will formalize this discretemodel as a field theory. In this framework it is possible toidentify an energy functional and in certain situation thefunctional takes a particular form, a sum of squares. Then theextremum is easy to find and results in differential equations
which are simpler than the original equations of motion. Thisis a well known physical property in field theory, known as
self-duality.
In developing this approach we face certain multitude ofpossible choices, which probably will be better clarified infuture. We are constantly led by the condition to attain the
self-dual states of the system.
The model used by Stewart and Morikawa consists of

f
Ê
 Ê 2
1 - — 2^ ˆ¯ f =-— ^ + 4h 2 ˆ¯ f - f
y
t Ë
y Ë

[

]

*
where 4h2  1 – v–
u > 0. To study the stability we take f Æ fs
(x, y) + e (x, y, t), where fs (x, y) is the periodic flow solution(3). Taking into account the periodicity of the solution we
obtain the Mathieu equation from which one of the dispersion
relations is(4)where P = –4h2 + –12 fs|min + –12 fs|max and Q = (dx)2 (fs|max –fs|min)2. Evaluating the terms, the linear dispersion relation
becomeswhich at the limit provides an estimation for the typical lengthof fragmentation along the poloidal direction, l^ ~ 2.6rs.

3.4 Relevance for the high confinement
phenomenology
The stability properties change when the polarisation
drift nonlinearity is included, together with the scalar one.
Some perturbations (like, for example, the monopolar vortices
embedded inside a layer) have a very long stability time since
they accomodate with the background flow by reshaping the
distribution of local velocities. However, perturbations thathave a high amplitude relative to the background flow and/or
do not conform to the flow geometry lead to the destabilisation and eventually destruction of the flow pattern. It is
important however to note that the numerical simulations
3
91

Spineanu F. et al., Stationary Vortical Structures in Electrostatic Drift Wave Turbulence

where the nonlinear potential is V(|f |2)  –l4 (|f |2 – v 2)2 (v being
a constant) and Dmf  (m + ieAm)f. The Lagrangean density
gives the equations  mFmn = Jn where the current source is Jn
1  –e2 |f |2 [An – e x n (Nq )] (N is the integer winding numberof the phase of f). At very large distance, the scalar function
f is close to its asymptotic value, |f |2 = v 2, and the integralof this current around the boundary contour is Jn dxn =–e2v 2 Andxn + ev 2  n (Nq) dxn. The gauge potential An allow
to define a magnetic field Bêz = — ¥ A perpendicular on theplane. For a single vortex we can replace the core by a dfunction. Then — ¥ J + e2v 2B = 2pN ev 2d (2) (r) êz. Using theMaxwell equation — ¥ B = J one arrives at the equation

an ensemble of point like vortices moving in a velocity field.
The local value of the velocity is derived from a vectorial
potential. The later is constructed from contributions of all
the vortices, each contribution being expressed in terms ofthe modified Bessel function of the second kind: j = SjwjK0(|r – rj |) . The vortex motion in plane is given by

dr j
dt

(5)

= -—j ¥ nˆThis means that the vortices are shielded since the modified
Bessel function of the second kind K0 decays exponentiallyat large argument.
The interacting N vortices of the Eq. (5) can be describedby a Hamiltonian H = Si > j SjwiwjK0 (|r – rj |). We will constructa Lagrangean density for this system, keeping in mind thatthe major properties must be: (1) short range potential; and(2) topological nature of the elementary vortices.
The fact that the potential j for the HM equation is astatistical potential does not clarify definitively from whatsource it might be derived, except for the fact that we now
know that the Chern-Simons term must be present in the
Lagrangean. We note however that there are at least two ways
to obtain a short range for the interaction field (gauge field):(1) include the Maxwell contribution to the Lagrangean
density; the Maxwell term and the Chern-Simons term bothhave numerical coefficients and a combination of thesecoefficients appears as a mass term in the differential equation
which gives the gauge field. This is a known effect and thefact that the gauge boson aquires mass by simply considering
Maxwell term in addition to Chern-Simons term is independent of the presence of interaction with other (matter) fields.
The latter can simply be represented by a current. (2) By the
Higgs mechanism, a gauge boson aquires mass when thesymmetry is spontaneously broken for a matter field with selfinteraction, after fixing the origin in a vacuum given by adegenerate minimum of the self-interaction potential. Thenthe mass of the boson is dependent on the form of the nonlinear potential.

— ¥ — ¥ B + e 2 v 2 B = 2pNev 2d ( 2 ) (r) eˆ zThis equation has the solution (Nielsen-Olesen)

B = ev 2 N K0 ( mV r ) eˆ zwhere the mass of the gauge boson is identified mV  ev =e|f |• . Generalizing for a number of parallel vortices, withvorticities of signs na = ±1 the equation becomes— ¥ — ¥ B + e 2 v 2 B = 2pev 2az

and the solution is

B = ev 2

Â n K (m r - r ) eˆ
a

0

a

V

z

a

From this derivation we learn that the Higgs mechanism
leads to a finite mass for the photon, mV which must be mV 
e|f |• = 1/rs and e = cs.
This model is able to support the self-dual states. For
the explicit form one has to write the squared differential
operator at stationarity as shown by Bogomolnyi

1where the spatial part of the current is, as above Jj = 2i– [f*(Djf) – f (Djf)*]. The static energy functional e = d2x [ –12 B2+ |Df |2 + l–4 (|f |2 – v2)2] is for l = 2e2, e = d2x [E 12 + E 22]
ev 2B, where E1  B e (|f |2 – v2), E2  |D±f |, and D±  D1 ±iD2. The self-duality means here the vanishing of the squared
terms plus the condition between the constants of the theory(implicitely between the masses)

The Lagrangean density must have the standard structure: gauge field Lagrangean, matter field (sometimes calledin the following the Higgs field) and interaction (between thematter field f and the gauge field Am. The scalar nonlinearity
is not arbitrary although it leaves certain freedom: we haveto obtain the short range potential and the self-duality. The
second of the method mentioned above for the photon to getmassive and the interaction become short range, can berealised with only the Maxwell gauge field Lagrangean
density plus the Lagrangean density of the self-interacting
complex scalar (matter) field f
2

2

a

a

4.1 From the discrete system of vortices tocontinuum

14

Â n d ( ) (r - r ) eˆ2

D±f = 0 , B = ± e ÊÁË f - v 2 ˆ˜¯At this point the energy is bounded from below by the magnetic flux e >> v 22pN. We write the complex scalar field f =÷r exp (ic) and then the vanishing of the covariant derivatives D± yields eAj = –jc –21 eijjlnr. This must be substituted
in the equation of B

— 2 ln r = 2e 2 ÊË r - v 2 ˆ¯

( )L AH = - Fmn F mn + Dmf - V f 2

492

Spineanu F. et al., Stationary Vortical Structures in Electrostatic Drift Wave Turbulence

5. Conclusion

With the substitution r = exp (y) (and rescalings) the
self-duality equation becomes

Dy = exp(y ) - 1

Robust vortical structures appear in a transition fromzonal flows to the random field, from which the ITG eddies
may grow again. The possibility of such solution is nontrivial
since they are different of the only known dipolar vortex(Larchev-Resnik) of the HM equation. We have proved thatsuch solutions exist at stationarity and we have provided theparticular forms the HM equation takes in this case. The
equations are exactly integrable and exhibit lattices ofvortices.

(6)

The physical vorticity is obtained as w = |f |2 = r and, in
physical units, r = —2y + Wi. The Eq. (6) can be solved
exactly on periodic domains and gives a lattice of vortices inplane, which is what the numerical simulations show.
The most general model of a field mediating the interaction of vortices (represented by a density) and which is able
to generate fields with short range and to present self-dual
stationary states is the nonabelian Maxwell Charn-Simons
Higgs theory. The Lagrangean density isÊ - 1 F F mn + k e mnr F A
mn
mn r
4
L = ÁÁ 4†
Á - tr Dm f Dm f - V f , f †
Ë

[(

)(

)] (

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)

It has been found that the only possiblity this model has toreach self-dual states is to choose a scalar field nonlinear self1
interaction given by a sixth order potential V(f, f†) = 4k 2 trY †Y) where Y = [[f, f†], f] – v2f and the trace is definedover the finite dimensional representation of the Lie algebra
to whom A and f belong.
The self-duality equations lead to  + –ln|f |2 = k22 |f |2(2 |f |2 – v2). After several transformations, with the substitution ln|f |2 = y the equation becomes

Dy = exp(3y )sinh(y )This equation is also exactly integrable and its solutions
consists of lattices of vortices In Ref. [13] a graphicalrepresentation of the solution of Eq. (6) is provided.

593