J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)465–468
000–000

Exact Periodic Solutions of the Stationary
Hasegawa-Mima Equation
SPINEANU Florin, VLAD Madalina, ITOH Kimitaka1 and ITOH Sanae -I.2
Association Euratom-MEC, 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: 1 April 2004)

Abstract
The small scale vortical structures generated by the ion polarisation drift nonlinearity appear as an intermediatestep in reorganisation of plasma flows (e.g. decays of zonal flows). We find that a lattice of vortices is an exact
solution at stationarity of a model equivalent to the Hasegawa-Mima equation and provide explicit solutions.
Keywords:
ion drift wave, vortices, zonal flow, integrable structure

1. Introduction

known for HM equation. We prove their existence and provide
the exact form.

Due to the strong nonlinear character the ion drift wave
generates turbulent flows and also coherent structures [1-2].
The radially extended eddies of the Ion Temperature Gradientdriven instability (ITG) are a major agent of transport ofenergy in tokamak plasmas. At space scales of the order ofthe ion Larmor radius ( ρs), the polarisation drift nonlinearity
induces condensation of large flows into small vortices, very
similar to the superconducting media. There is also anintermediate space scale, of the order ρs Ln where the flows
are dominated by the scalar nonlinearity [3]. In this range one
finds monopolar vortices which represent a reminiscence of
the anticyclonic monopoles at lower scales and are very robust
without being solitons. At the same scale, exact solutions(with scalar nonlinearity) have precisely the geometry of thezonal flows.
The dynamics on the larger scales is structuraly unstableagainst the increase of presence of the polarisation drift
nonlinearity. Higher level will induce decay of the flow into
small vortices. However these vortices evolve by collisions
and merging and at late time the flow can exhibit a regularpattern, a process similar to the Euler ideal fluid [4]. From
this perspective, this nonlinearity acts to reorganize the flow
and at early stages it traverses states of turbulence and randomvortices. The target of this reorganisation is a state of the fluidwhich associates with the self-duality.
We are interested in the small scale vortical solutions of
the Hasegawa-Mima (HM) equation [1] since the numericalsimulations show that the exact zonal flow solution breaks
down firstly into arrays of vortices. Such solutions are not

2. Structural instability of the zonal flows
pattern and decay into vortices
At intermediate scales the ion drift wave is dominatedby the scalar (or KdV) nonlinearity, which leads to the FlierlPetviashvili (FP) equation [5]ϕα ϕ − βϕ 2T∂
where α = ρ12 (1 – vu* ) and β = 2u2 eBe2 ρ 2s ∂x0s(1)1Ln(v* and u are
ion diamagnetic and respectively the plasma poloidal flow
velocities, B0 is the magnetic field, Ln is the density gradient
length and Te the electron temperature). We have identified
an exact periodic solution [6]ϕ s (x, y ) =α+s2β⎛
3α 2 ⎞⎟ (2)⎜ iay ibx + ω | g 2 =2⎜(s β ) ⎟⎠⎝with the condition a2 + b 2 = sβ/6 where ℘ is the doubly
periodic elliptic Weierstrass function. The first parameter g2of ℘, is fixed by the physical parameters as shown, while sand the second parameter, g 3 are fixed by boundary
conditions. When the FP equation is perturbed with the termof polarisation drift the flow evolves to an ensemble ofvortices initially disposed in a regular array, Fig. 1 ([7] andreferences therein). It is not possible to connect this evolution
to the Kelvin-Helmholtz instability in the absence of many
elements of the Prandtl model (pressure asymmetry, Bernoulilaw) and the most suggestive explanation is connected with©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: spineanu@ifin.nipne.ro1465

Spineanu F. et al., Exact Periodic Solutions of the Stationary Hasegawa-Mima Equation

nonlinear equation (Lax pair). We have found for the AHequation (λ is a constant):⎛⎜
⎜
⎜
⎜
⎜
⎜⎜⎜
⎜
⎜⎜⎝

−−i⎞∂ 1 ⎛ ∂u ∂u ⎞ ⎟⎟− ⎜ +i ⎟ ⎟∂x ⎠ ⎟ ⎛⎜ψ 1 ⎞⎟∂x 4 ⎝ ∂y⎟ ⎜⎟⎟⎟⎜∂u ⎞∂ 1 ⎛ ∂u
λ2⎟ ⎜⎝ψ 2 ⎟⎠i − ⎜ +i ⎟ −exp (u ) − p ⎟⎟∂x ⎠∂x 4 ⎝ ∂y16 p⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Fig. 1

λ2− p
16 p

=0

⎞
∂ 1 ⎛ ∂u ∂u ⎞ ⎟⎟− ⎜ +i ⎟⎟∂x ⎠ ⎟ ⎛⎜ψ 1 ⎞⎟∂y 4 ⎝ ∂y⎟ ⎜⎟ = 0.
⎟ ⎜2⎟λ∂ 1 ⎛ ∂u ∂u ⎞⎟ ⎜⎝ψ 2 ⎟⎠xp (u) − p ⎟⎟ex− ⎜ +i ⎟∂x ⎠ 16 p∂y 4 ⎝ ∂y⎟⎟
⎠

λ2− p
16 p

−

The possible solutions (ψ1, ψ2) must be an intersection of twosets. These equations also determine the spectrum ofeigenvalues p where periodic solutions are possible. When asingle p has a single eigenfunction (i.e. the two eigenfunctions
are confounded) p ≡ pj belongs to the main spectrum, and theWronskian is zero. We assume there are 2N such eigenvalues.
We apply the standard procedure, defining the square
eigenfunctions f, g and h, and introducing the zeroes of g (theauxiliary spectrum). The solution ψ (x, y) is expressed interms of these functions. The equations for the auxiliary
spectrum areA large wavelength inital perturbation evolves to a setof vortices.

the similar phenomenon in the superconducting media. Thisis also supported by the model we will propose below for thestationary HM.

3. The stationary states of the HasegawaMima equation
The Euler equation has the solutions of the sinh-Poisson
equation as stationary states. In the case of the HM eq., there
is a similar model (developed in meteorology) where thedynamics is represented by the motion of point-like vortices
in plane, interacting via a short range potential [2]. Weformalize this model in a continuum version by a “matter”
scalar field and a potential, asking that the density of “matter”
tends at large distance to a condensate of vorticity,
corresponding to the Larmor gyration. On this background,the solutions of the HM equation appear as excitations. The
suitable model for this is the Abelian Higgs (AH) theory(developed for superconducting media) which generates theshort range interaction (Kelvin functions with decay on ρs).
This model clearly exhibits a property which is lesstransparent at the fluid level: the states are governed by anaction functional which at stationarity is extremized at theself-dual point. The self-duality is a topological property
which means that the curvature of a geometrical structure isvanishing. At this point the equation of motion derived fromthe Lagrangian of the collection of point-like vortices
becomes⎡⎛⎞2⎤ ⎡⎤ 1/ 2⎢⎥ ⎢⎛⎞⎟⎜⎢⎥ ⎢ ∏ l =1 ⎝γ k − pl ⎠ ⎥⎥∂γ k
λ 2 ⎜⎝ ∏ l =1 γ l ⎟⎠⎦⎢= −2 ⎢+ γ k ⎥⎥ ⎣2NN
⎢ 16 γ⎥∂x⎛γ − γ ⎞p=k1l∏ l∏ ⎝ k l⎠⎢⎥Nl =1⎢⎣⎡⎢⎢⎢⎢⎢⎢⎢⎣∂γ k
λ= 2i∂y16 γ k2⎛⎜⎜⎝

⎞2⎟l⎟
⎠

l =12Nl =1l⎤ ⎡⎥ ⎢⎥ ⎢⎥ ⎣k ⎥⎥
⎥
⎥⎦

(3)

l ≠k

⎥⎦

∏ γ− γ∏ pN2N⎤ 1/ 2⎞⎥l⎠ ⎥⎦∏ γ −p∏ γ −γ2N⎛
l =1 ⎝ kN⎛
l =1 ⎝l ≠kk

⎞
l⎠

(4)

The solution is based on the fundamental property of γ k (x, y;p) of being defined when p maps the complex plane (of thespectral variable of the Lax operator) to the complex function
given by the square root of the Wronskian. Since the later isa polynomial in p, the square root defines a hyperelliptic
Riemann surface, i.e. a compactified double covering of thecomplex plane with cuts connecting pairs of zeros of theWronskian. These are the points {pk , k = 1, ..., 2N} of themain spectrum, plus the point zero and the point at infinity.
Pairs of zeros pk are joined by cuts and in addition the origin
is connected to infinity. This gives a number of N + 1 cutsand generates a compact Riemann surface of genus g = N.
On this surface there are defined two objects characterising
the differential geometry of the curve: (1) a basis of the one
dimensional cohomology group of the surface; this means two
sets each of N closed paths on the curve (cycles), having
particular intersection properties. The two sets are noted aj ,and respectively bj , j = 1, ..., N. The intersections are aj ° ak= 0, aj ° bk = δjk and bj ° bk = 0. A typical example, for an
elliptic curve g = 1 with the topology of the torus, consists of∆ψ = exp (ψ ) – 1
where ψ is the stream function. We will show that on periodic
domains this equation is integrable and we will give exact
solutions consisting of arrays of vortices.

4. The exact solution of the vortex
dynamics
The inverse scattering method on periodic domains
requires the determination of a system of linear differential
equations whose compatibility condition is equivalent to the2466

Spineanu F. et al., Exact Periodic Solutions of the Stationary Hasegawa-Mima Equation

the two possible closed turns around the torus, the short way(a) and the long way (b). (2) A basis in the ring of the onep N – k dp
dimensional differential forms dµk = R ( p ) , k = 1, ..., N.
With these two sets one calculates several quantities
which are invariants of the Riemann surface. Essentially there
are calculated integrals of the elements of the basis ofdifferential forms along the cycles aj and bj . These are called
periods and are organized in two matrices

p N −i dp, i = 1, N , j = 1, N
aj R p( )Aij = ∫ dµi = ∫ajp N −i dp, i = 1, N , j = 1, N
bj R p( )Bij = ∫ dµi = ∫bjIt is useful to work with the inverse of the matrix A, C = A–1.
Using C, the matrix of A periods is reduced at the identity
matrix, and the matrix B becomes τ = CB, the τ-matrix, withpositive imaginary part.
Using this geometrical framework, the solution of the γkequations can be obtained by oparating first a transformationfrom the set {γk} to a set of functions {φk} representing
phases of motion along the cycles of the Riemann surface.
This transformation effectively linearises the motion, which
can be trivially integrated in these new variables.
We have to define the functions of the target set, thephases {φk}. They are integrals of linear combinations of thedifferential one-forms along paths on the Riemann surface,each starting from an initial point γ0 and ending in the point
which corresponds to a function γl . The integrand is acombination of the differential one-forms with coefficientsfrom the matrix C = A–1N

φ k = −∑ ∫l =1γlγ0Fig. 2φk (x, y ) = 2Ck 1 (x − iy ) + φk 0kmm(6)where φk0 are constants of integration, initial phases.
We note from Eq. (6) that the motion on the Jacobi torus
is entirely determined by the main spectrum through thetopological properties of the hyeprelliptic Riemann surface(canonical cycles, differential forms, period matrices).
The next step consists of returning to the variables γk
and from there to the exact solution ψ. This is done as theJacobi inversion problem. The main role is played by the Θfunction. The definition of the Riemann theta function
involves a vector of dimension N (we denote it by φ) and a N
× N matrix τ whose elements have the imaginary part positive.

N∑ C dµThe exact solution of the stationary Hasegawa-Mima
equation, which clearly shows the structure of lattice ofvortices.

(5)

m =1

The mapping that realizes the correspondence from acollection of points {γl , l = 1, N} of the hyperelliptic Riemannsurface to a manifold defined by the collection of points {φk ,k = 1, N} is called Abel map. The manifold generated by thepoints {φk , k = 1, N} has genus g = N (as the initial curve)
and has the topology of a torus. It is called Jacobi torus. Theequations for the phases are∞Θ (φ ,τ ) = ∑m1 =−∞∞⎛⎜NNN⎞⎟mn =−∞⎜⎝k =1i =1 j =1⎟⎠

∑ exp ⎜⎜ 2πi ∑ mkφk + πi ∑ ∑ miτ ij m j ⎟⎟ .

The exact solution is⎡ Θ ⎛⎝φ + 12 I ⎞⎠ ⎤
ψ (x, y ) = 4 ln ⎢⎥ + const⎣ Θ (φ ) ⎦⎞2⎡ 2 ⎛⎜ N⎤⎟N∂φk⎢ λ ⎜⎝ ∏ n =1 γ n ⎟⎠⎥δ (m ) + δ (1 − m )⎥= 2∑ Ckm ⎢2N16 ∏ pn∂xm =1⎢⎥n =1⎦⎣= 2Ck1with φk (x, y) = 2Ck1 (x – iy) + φk0 . A form of the solution isshown in Fig. 2.

5. Conclusion
The intermittent destruction of the regular pattern of theFP solution (zonal flow) exhibits robust vortical structures,strongly suggesting a HM dynamics. As an intersting problem
in itself (and for this particular application), the stationary
states of the HM eq. need to be investigated analytically. We
have shown that these states can be described by a nonlinear
model which has a Lax pair of operators and we haveobtained (using the inverse scattering method) explicitsolutions in terms of the Riemann’s theta function.

⎞2⎤⎡ 2 ⎛⎜ N⎟γN⎜∏n∂φkλ⎥⎢ ⎝ n =1 ⎟⎠δ (m ) − δ (1 − m )⎥= 2i ∑ Ckm ⎢∂y16 ∏ 2 N pnm =1⎥⎢n =1⎦⎣= −2iCk1 .

The equations can be trivially integrated and we obtain the(x, y) dependence of the phases
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Spineanu F. et al., Exact Periodic Solutions of the Stationary Hasegawa-Mima Equation

References

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growth rate scale. The intermittent destruction of the transportbarriers traverses these states where the Reynolds stress isisotropic. This supports the intermittent character of the ITB
dynamics [7].

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