Applications of beltrami functions in plasma physics

NonlmearAnal)sis,

Theov.

Methods

PII: SO362-546X(97)00262-9

APPLICATIONS

OF BELTRAMI

FUNCTIONS

ZENSHO Department

of Quantum

& Applicarions, Vol. 30. No. 6. pp. 3617-3627, ,997 Pmt. 2nd World Congress of,Vmlmear Analysts 0 1997 Elsevier Science~Ltd Printed m Great Britain All rivhts WCPTVC~ 0362-546X197 $17.00 + 0.00

IN PLASMA

PHYSICS

YOSHIDA

Engineering,

Uniwrsity

of Tokyo,

Hongo. Tokyo 113. Japan Key words and phrases: B~ltrami field, plasula physics, htlicity, rohcmology class. rigenfun&m expansion. invariant mrasm~. statistical qnilihri~un, self-organization, dynamo t,heory. sing&x pertnrhation. chaos. reduced model 1.

INTRODUCTION

The nonlinear Inagnetollydrodyrlalllics (MHD) involves a variety of colnplex plwnomena. It, is well nigh impossible to construct physically nontrivial theory from a direct analysis of the basic equations. To elucidate a specific phenomenon, we must apply a reduction of the model with ;q)pealing to scale separations, singular pert.urbations, coarse-graining (averaging), etc. In this paper, we discuss a slow mot,ion (or a steady state) of a low-pressure magnetized plasma. In more specific terms. we consider the following singular limit,. The general MHD equations read, in the standard normalized units, &w

=

-(wV)v+~~‘(VxB)xB-/K’p+rRAv.

dtB

=

v x (v x B) - EL\7 x (V x B).

V~w=O.

(1)

(2)

Unknown variables are t,he magnetic field B: the flow velocit,y z1 and t,he pressure p. The AlfGn number ~‘4, Lundquist number .~il, Reynolds number ERR, and the beta ratio /? are nondimensional positive parameters. The incompressibility condition (V w = 0) may be replaced by an evolution equation for the pressure p in a more sophisticated model. This syst,em of nonlinear parabolic equations (1)~ (2) is a close cousin of the Navier-Stokes system describing neutral fluids (see [l, 21 and papers cited t,herein). The MHD syst~em includes coupling between the magnetic field and the flow velocit,y through the nonlinear induction effect and its reciprocal Lorentz force, which adds a considerable complexit~y to the usual Navier-Stokes system. Surprisingly, however. we observe a more regular and ordered behavior in some MHD systems. Such phenomena are highlighted by a singular pert,urbation of 6: + 0. wit.h fixing the time-scale, in the momentum-balance equation (1). This limit, is amenable t,o slow motion of a strongly magnetized low fl plasma. The determining equat,ion becomes the force-free condit,ion (V x B) x B = 0, which is equivalent to t.he Beltrami condition VxB=XB. Here X is a scalar function. By the solenoidal condition taking the divergence of the both sides of (3) yields

(3) (V . B = 0) and identit,y

V . (V x B) = 0,

B,VX=O. Since (4) means that the function X should be constant along t,he streamline (field line) analysis of the system of equations (3) (4) requires integration of the st,reamline equation

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The solenoidal condition (V’B = 0) parallels Liouville’s theorem for the Hamiltonian flow, and hence one can formulate (5) in a canonical form [3]. For a general three-dimensional B, the solution of (5) exhibits chaos. Hence, the general analysis of the syst.em (3)-(4) includes an essent,ial mathematical difficulty. Two special cases, however, can be studied rigorously. One is the case where B has an ignorable coordinate (two-dimensional). Then, (5) becomes integrable, and the system (3)-(4) reduces into a nonlinear elliptic equation [4, 51. The three-dimensional problem involves the nonintegrable streamline problem (5), however, it is decoupled from the Beltrami problem (3)- (4), if we assume a constant X that make (4) trivial. The plan of this paper is as follows. In Sec. 2, we give a concise review of the physical background of the Beltrami condition in plasma physics. The constant-X Beltrami field is considered to be a “ground state” of a turbulent plasma. We define a mathematical problem that characterizes such an equilibrium, and discuss its implication in the “dynamo theory” of astrophysics. Section 3 is devoted to the mathematical analysis of the const.ant-X Beltrami field. In Sec. 4, we develop a statistical mechanics of the MHD equilibrium that is amenable to the constant-X Beltrami condition. Interactions among elements with inhomogeneous X yield chaotic oscillations. In Sec. 5, we introduce a reduced finite-dimension model of such nonlinear dynamics, and present results of numerical analysis. The reduction from PDE into ODE uses a unique technique based on the singular perturbation, which differs from the usual mode truncation. 2. CONSTANT-LAMBDA

BELTRAMI

FIELD

The constant-X condition for the Beltrami field is a strong ansatz based on the following physical reasons. The streamline equation (5) in a, three-dimensional magnetic field is generally non-integrable, and hence, we may aSsume that streamlines (magnetic field-lines) are embedded densely in a volume. Since (4) demands that A is constant along each field line, it is natural to assume a constant X over such a volume. The theory of energy relaxation also derives the constant-J+ condit.ion. Woltjer [7] pointed out the importance of the maguetic helicity h’=

1

J

2 R

A. Bdz.

Here V x A = B, R is the entire volume of the plasma and dx is the volume element. The viscous dissipation does not change the helicity K, while the magnetic energy diminishes toward a “ground state”. The magnetic field self-organized through this energy relaxation is characterized by a minimizer of the magnetic energy W = Jo B”cix/2 subject, to a given helicity. This variational principle reads as 6(W - AK) = 0. where X is the Lagrange multiplier. The formal Euler-Lagrange equation, under appropriate boundary conditions, is identical to (3). Taylor [8] formulated an equivalent variational principle, however, his model is based on a different, hypothesis to just.ify the preferential conservation of the helicity. The energy dissipation proceeds faster than t,he change of the helicity, if the resistive dissipation is dominated by spatially concentrated fluctuation currents (see also Hasegawa [9]). Both effects, the viscous dissipation, resulting in ion heating, and the resistive dissipation, resulting in electron heating, were compared for a specific relaxation process [13]. There are many different, observations suggesting the creation of constant,-X force-free fields in astrophysical, space and laboratory plasmas. Magnetic flux tubes (flux ropes), in which field lines are twisted, are produced through interactions between t.he magnetosphere and interplanetary magnetic fields [lo]. In a laboratory plasma, det,ailed measurement,s of magnetic fields showed that the field produced after self-organization through t,urbulence is closely approximated by a solut.ion of (3) [8]. Galactic jets are also considered t,o have similar configurations of magnetic fields [ll]. The Beltrami field plays an essential role in the so-called “dynamo theory”. To understand the rapid generation of maguetic fields in astrophysical syst,ems, we have t.o invoke a “fast. dynamo action”

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that has a growth rate of the magnetic energy independent of the resistivity (see [6] and papers cited therein). In a highly conductive plasma the evolution of the magnetic field l3 obeys Faraday’s law (2) with CL + 0. A plasma flow 2) with chaotic streamlines (maps with positive Lyapunov exponents), which may have a large length-scale, bring about complex mixing of magnetic flux, and t,he lengthscale of the inhomogeneity cascades toward a small scale, resulting in amplification of the magnetic field. If the length-scale reduces down to the dissipative range, and the resistive damping becomes comparable to the induction effect, then t,he magnetic field energy turns to diminish. In this cla&cal picture of the kinematic dynamo, the magnetic field energy accumulates int,o small scale fluctuations, and the life-time of the amplified magnetic field is limit,ed by the time-scale of the cascade process. To obtain a larger length-scale and a longer life-time of amplified magnetic fields, an appropriate limitation for the scale reduction should occur. The nonlinear effect of the amplified magnet,ic fields, that is the Lorentz back-reaction, plays an essential role in t,his “post-kinematic phase”. Here we aSsume that the plasma achieves a quasi-steady stat,e through the energy relaxation process. Then, the momentum balance equation reduces into (3), and the flow v must be chosen in such a way that B satisfies (3) implicitly. The parameter X characterizes the length-scales of B. Hence, the condition (3) imposes a bound for t,he length-scale of the field, if the magnit.ude of X is restricted by some reason. This bomld avoids scale reduction down to the resistive regime. and extends the life-time of the amplified magnetic field. Through the kinematic dynamo process, the current (X V x B) tends to concentrate in small volumes, which may be disconnected. When the sectional length-scale of such a volume becomes small enough, the Lorentz force dominates (c”, < 1). Let R to be such a “clump” of the magnetic field. Its length-scale is denoted by &. This R may have a complex topology. We want to find a constant-X Beltrami field in Q2. If the parameter X can be chosen such that 1x1 5 X, = O(fJ;‘), then equilibration of the clump into such a Beltrami field results in a lower bound for the length-scale. Here we solve the Beltrami condition (3) for a given helicity and an “external magnetic field”. The external component of B is defined by decomposing B = BE + h, where V x h = 0 and V h = 0. This h, which represents the magnetic field rooted outside 0, is assumed to be a given function. Its complement BE is the unknown variable. We define t,he gauge-invariant. helicit,y by K= We prove the existence of a solution section (Theorem3). The nonvanishing 3. EXISTENCE

Jn

A. Bcdx

(6)

with (XJ 5 X, = O(C;‘) for every h # 0 and K in the next h plays the role of symmetry breaking. THEOREM

AND COMPLETENESS

THEOREM

The constant-X Beltrami condition (3) is regarded as an eigenvalue problem with respect to the curl operator. Interestingly, the topology of the domain plays an essent,ial role in this eigenvalue problem. To study the spectrum the curl derivatives, we need the fundamental theory of vector function spaces. Let R (C R3) be a bounded domain with a smooth boundary dS2 = LJr=,ri (!Ji is a connected surface). We consider cuts of the domain Q. Let Cl.. . , C, (m 1 0) be cuts such that C, n Cj = 0 (i # j), and such that fl \ (UzlCi) becomes a simply connected domain. The number m of such cuts is the first Betti number of 0. When m > 0, we define the flux through each cut by

@c,(u) =

n

uds

(i = 1,2;..,m),

JL

where n is the unit normal vector on Ci with an appropriate orientation. By Gauss’s formula, is independent of the place of the cut, C,, if V . u = 0 in R and n . u = 0 on 80.

@xi (u)

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We denote L2(0) the Lebesgue space of square-integrable (complex) endowed with the standard innerproduct (a, b). We define the following

vector fields in R, which subspaces of L2(fl);

L;(n) L%(n) L%(O)

= = =

{w; V.w=OinR, n.m=Ooni3R, ~c,(‘~))=O(i=l,...,rn)}, {h; V.h=O,Vxh=OinG, n.h=OondO}, {V+; &3 = 0 in !2},

L%(O)

=

{V$;

We have an orthogonal

4 = c, (E C) on ri (i = 1,...,72)}

decomposition L2(Q)

The space of solenoidal

is

[15]

= L;(n)

@ L$(O)

vector fields with vanishing

The subspace L;(0) corresponds to the cohomology and dimL$(R) = m (the first Betti number of 0). L%(Q) = 0. We have the following expression L2,(!2) = {V X 20; w E H’(Q),

63 L&(R) normal

83 L$(O).

component

on 8R is

class, whose member is a harmonic vector field When R is simply connected, then m = 0 and

V. zu = 0 in R, n x w = 0 on 80).

This implies that a member of Lc.0) can be expressed as the curl of a vector potential with the boundary condition n x w = 0. We note that a member of L:(0) may not allow such an expression. We also note L*(n) = L:.(R) @ {Vqb; 4 E H’(R)}, which implies that Lz( 0) is the orthogonal complement of the space of potential flout This relation is called the Weyl decomposition. The gauge-invariance of the helicity K defined by (6) follows from this orthogonality. We now study the spectra of the curl operator. A key step of the theory is finding a self-adjoint curl operator [14]. We have the following theorem [14]. Theorem 1 Let R C R3 be a smoothly bounded domain. space L;(O) by su = v x 21. D(S) = {u E L;(R) Then S is a self-adjoint operator. discrete set of real numbers.

The spectrum

We define a curl operator

S in the Hilbert

; v x u E L;(n)}.

of S consists of only point spectra a,(S),

If s2 is multiply connected (m > 0), it is interesting. both mathematically and physically, the domain and range of the curl operator to the space L:(R) [14]. As an intermediate consider another curl operator 7u=vxu,

Lemma

D(7)

= {u E L;(Q)

has a solution.

to extend step, we

: v x u E L:(O)}.

1 FOT every X E C \ oP(S) and for every f E L:(O), (I-X)u=f

which is a

the equation (7)

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(proof) First we show the existence of T’-‘, O-extension off in R3, i.e. j(x)

of Nonlinear

Analysts

i.e., for f E L:(R)

=

fb”’

,“;

3621

we solve 7u = f.

Let i be the

;y

{ By f E L:(R), one observes 0.j = 0 in R 3. We denote by (-A)-’ the vector Newtonian potential. We define wo = V x [(-A)-‘f] in fl. We denote by Px the orthogonal projection in L2(0) onto L:(R), and define uo = P~UJO.Since t:(n) is orthogonal to Ker(curl), we observe v x ul) = v x wo = v x {V x [(-A)-‘j]}. Since V. [(-A)-lj] = 0, we obtain V x {V x [(-A)-‘j]} = -A[(-A)-‘j] = j. We thus have a solution I-‘f = V x ~0. Next we solve (7 - X)U = f, for f E L:(n). We decomposef = g + h with g = Pyf and h E L;(R). Let uo = T-‘h and w = u - ~0. Then (7) reads (T - X)w = g + Au0 E L;(n).

(8)

For A $ZIJ~(S), we may define w = (S - X)-l (g + XUO) E D(S), which solves(8). In summary, we have a solution of (7) u = 7-‘h

+ (S - A)-‘(&f

+ X7-‘h).

(Q.E.D.) Theorem 2 In L:(R)

we define a curl operators by s = v x u,

(i) When dim{&(n) a(S) = OJS). (ii)

D(S)

= {u E L;(R)

= 0, i.e. if fl zs simply

; v x u f L;(R)}.

connected,

then 3 z S, and hence, the spectrum

When dim Li( R ) > 0, i.e. if Q is multiply connected, then 3 is an extension of S. spectmma(S) consists of only spectra a,(s), and ~~(3) = C. Hence, for every X E C, (S - X)u = 0

The

(9)

has a nontrivial solution. (proof) The first part is straightforward. We prove the secondpart. For X E up(S), this has a solution as shown in Theorem 1. We assumeX # aP(S). For a given h E L;(n), the equation (7 - X)w = Ah

has a solution (Lemma 1). Then, the function u = w +‘h [E L:(R)

n H’(0)]

solves (9). (Q.E.D.)

Theorem 2 proves the generalexistenceof the constant-X Beltrami function for every X E C, if R is multiply connected. In the next theorem, we solve the constant-X Beltrami equation (3) for a given helicity K: and harmonic field h E L;(R). Now X is an unknown variable. This problem is related with the magnetic clump discussedin Sec. 2.

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We assume that R is multiply connected. Let {F~} be the complete set of the eigenfunctions the self-adjoint curl operator S (Theorem 1). The corresponding eigenvalues are numbered as .‘.

For every B E L:(R),

2 p-2

< ,l-1

< 0 < ,ul 5112

we have an orthogonal-sum B(z)

t) = 2

5 .“.

of

(10)

expansion cj(t)v,(z)

+ h(z,t)t

(11)

where h E L%(Q). The harmonic field h is a given fmiction, which plays an important role of “symmetry breaking” in the following discussion. The first. summation in the right-hand side of (11) is denoted by BE. The energy of B is given by

w = f 1 c; + $lhl12. 3 There

exists g such that h = V x g (Lemma

(12)

1). The vector potential

of B is given by (13)

Denoting

Dj = (~~,g),

the gauge invariant

helicity

(6) becomes (14)

For given K and h, we can solve (3) by the variational

The energy and the helicity

principle

6(W - AK) = 0, and obtain

become

w = c 8(;2:;J2 D; + fll~ll2’

K=c

i

j

‘&t2& 8(pj-X)2

- ‘)D?, 3

We can show that K is a monotone function of X in the range of p-1 < X < ~1 (see definition (lo)), if 03 # 0 (Elj), viz., if we have a “symmetry breaking” h # 0. For every IE E R, the equation K(x) = rc has a unique solution in this range of A. Now we have the following theorem. Theorem 3 Let R (C R3) be a multiply connected bounded domain. Assume that h (E L%(R)) is finite. FOT every K (E R), the Beltrami condition (3) has a unique solution B such that its helicity K = lc, and X such that p-1 < X < ~1. 4.

STATISTICAL

EQUILIBRIUM

The complexity of the nonlinear dissipative dynamics of a plasma invokes a paradigm shift to a “coarse-grained” model. Using the phenomenological variational principle S(W - XX) = 0 (Sec. 2), we develop a statistical mechanical model that reproduces the constant-X Beltrami field at the “zero A finite temperature (in the sense of MHD fluctuation) equilibrium includes temperature limit”.

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fluctuations. The statistical theory predicts the spect,ra of macroscopic physical quantities such as the energy, helicity, etc. A key step is to find an invariant measure of the temporal evolution equation. It corresponds to Liouville’s theorem in the Hamiltonian dynamics. Montgomery et al. [16] used the “ChandrasekharKendall functions”, which are the eigenfunctions of the curl in a cylindrical geometry [17], to expand the solenoidal vector fields B and v, and defined an infinite-dimensional phase space spanned by the expansion coefficients. The formal Lebesgue measure is shown to be invariant, against the nonlinear ideal (ER,~L + co) dynamics. The completeness theorem of the eigenfunctions (Theorem 1) gave a mathematical justification of the expansion and generalized the Hilbert-space approach for an arbitrary geometry. An important development in recent work [18] is the treatment of the harmonic magnetic field, which brings about a symmetry breaking associated with a topological constraint. When we consider a multiply connected domain, the harmonic magnetic fields, which are rooted outside the domain, are represented by the cohomology class. If we impose the ideal conducting boundary conditions, these harmonic fields are invariant. The rest orthogonal complement spans the dynamical phase space. The invariant harmonic component plays the role of an externally applied symmetry breaking. Interestingly, this term yields “power-law spectra” of the energy, helicity and helicity fluctuation. In this section, we give a brief sketch of the statistical mechanics of MHD. Proposition 1 (Invariant B(x, t) obeys

Measure)

Let v(x,t)

&B

Using the eigenfunctions

Then, dC = dE1’ (proof)

in R,

n x (v x B) = 0

on X?.

= C j

d $i

condition

cj(t)Vj(x)

Suppose

that

(17)

(18) field he, we write

(cf..

(11))

+ fJ Ee(t)he(x). e=l

(18), we observe d?e/dt = 0 (V!).

(VX(VXB),~~)=(VXB,VX~~)=X~(VXB,~~)

=

Xj

(19)

= 0

(Vj).

Using (17) and (18), we obtain

1

x ~k,‘~j)+g4(vXhe,~j) e=i

‘pj E 0, we find d(dcj/dt)/dcj

in R.

measure.

=

CQ(V k

vector field

‘pj and the harmonic

dE, nj dcj is an invariant

By the boundary

Since (v x cpj)

= V x (v x B)

of the curl operator B(z,t)

be a smooth

(20)

.

Hence the measure

dC is invariant. (Q.E.D.)

The ansatz of the variational principle 6(W - XIC) = 0 suggests that two additive quantities W and K: are the relevant state variables that characterize the statistical equilibrium. The possible ensemble consistent with this variational principle is the Boltzmann distribution P(W, K) c( exp[-/3(W

- X)]

(21)

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where /3 is interpreted as an inverse temperat,ure of the magnet.ic field. The helicity each mode is (cy/p~j + Djc,)/:! and CT/~, respectively. The Boltzmann distribution c3 is CT The ensemble

x -CJ &

2

- XDjcj

averages of W and K over the phase space become

I

.

and the energy of for the amplit,ude

(22)

(23)

These results are compared with (16). The first t,erm of the right-hand side of (23) and that of (24) are the contributions of the fluct,uations. III (23), the energy of the harmonic field, which is constant here, is omitted. This classical st,atistical model suffers from the Rayleigh-Jeans catastrophe, viz.. when we pass the limit, of the infinite smmnation over the all modes, the fluct,uat,ion t,erms diverge. To avoid this divergence. we can appeal to the Bose-Einstein statistics wit,h secorld-quallt,izing the mode amplitude cj and defining bosons MHD fluctuations [18]. 5.

REDUCED

MODEL

OF WEAK

INTERACTIONS

AND

CHAOS

In this section, we consider nonlinear int,cractions among plasma element,s wit.11 inhomogeneous X. Each element satisfies the consta.nt-X Beltrami condition (3). Different elements are separated by a thin layer where the magnetic field lines are weakly chaotic. Hence, t.he connection lengths among different elements are considerably long. If we consider a small deviation from the Beltrami condition (3). and write OxB=XB+a, (25) then (4) receives a small correction

and becomes (B . V)X = -v

a.

(26)

Integrating (26) along a field line, we obtain a finite inhomogeneity in X after a long distance. This allowas us to assume inhomogeneous X in the following discussion. We consider a cylindrical pl~+ma with radial inhomogeneity. We introduce a reduced ODE system. The basic idea of the reduction is the use of the Beltrami condition (V x B = XB) to convert the spatial derivatives in the PDE system (l)-(2) into the multiplying of A. We note that this procedure differs from the Fourier decomposition and truncation, which are usually used to derive reduced models in different problems. For each mode of Fourier decomposition, we can replace derivatives by multiplying of wavenumbers. In the present. method. the conversion applies to the exact function, not. to expansion modes. The X is a dynamical variable to be determined by the evolutions equation. Here we invoke a quasilinear turbulence model of MHD fluctuations (see [19, 20. 221 and papers cited therein). Magnetic field B is decomposed into the fluctuating component b and the ambient component Bo. For the plasma velocity w, we also assume two components; One the a uniform flow V and the other is the fluctuation V driven by the MHD instabilities. Assuming a quasilinear turbulence of resistive instabilities, we may write the ensemble average of the nonlinear term (it x b) in t(erms of the growth rate of the instability, the energy of fluctuations, and some geometric factors. The parallel (with

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0.003 ‘,‘I

0.0025

I! ,, ‘, ‘i, .’ ‘.

I
0.002 0.0015

-

0.001

-

o.ooo5

-

I’

,;: i,! ,,:I, A!, I ‘,: 1, .’ ‘8 ‘i :i .., 81,

O-

.i! ,;t II
/ :;p ,,/

-0.ooo5

-

-0.001

-

:, !

‘,. !i!/J

-0.001 *8olm

22000 a

24000

”

26000

30000

28000

32000

34000

36000

38000

R

Figure

1: Feigenbaum

diagram

for R in the range of 2.0 x lo4 - 3.8 x 10”.

respect to the mean magnet,ic field Bo) component of (V x 6) makes an essential contribution, is denoted by -E/1*). The quasilinear turbulence theory [20] yields E;,*’

=

-V

which

(q(2)Vj,,,0),

where q(*) is the hyper resisGvity given by

77P)

(27)

-= PO

Here the subscript I; indic&es a Fourier component of t.he fluctuation, yk is the growth rate and l;(, is the parallel wavenumber with lcll(rk) = 0. The ensemble average of Faraday’s law (2) becomes &b = -eLV We assume Bo = Vz. The Beltrami

(28)

x (V x b) + V x (V x b - E;%). condition

Vx(Vxb) vx (Vxb)

= M

reads V x b = X(b + Bo). X2b+VXx(b+Bo). -(V x b) x v = -Xb

We obtain

x v.

We consider a low pressure plasma, so that the parallel component of 6 is neglected. In the cylindrical coordinates such that. VP x Vr = Vz (p and r are the angle and radial coordinates, respectively), we write p = b, and q = h,. Inhomogeneity of the plasma is assumed in t,he direction of Vr. We may write the q~ and T components of (28) as &p

=

-e&p

- vxq,

&q

=

-cLX2q

+ VXp

(29) - &.Ef)

(30)

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4 N

2: Dependence

Analysts

5

of the Ljapunov

6

dimension

The last term in the right hand side of equation (30) represents We can write 8E;,2’ _ 2 a?p a2x - aT - -zy$T-par”’

on N.

the effect of the MHD

fluctuations.

Two unknown variables p and 9 represent, the amplit,ude of the magnetic perturbation b. The third unknown variable X characterizes the “curl” of b. The equation obeyed by A, which is also derived from (28), becomes [22] a*x = c(p)2x2$

+ E/J2,

where & is the external driving electric field. represents the diffusion of the helicity induced by the MHD We note that the factor a2x/aT2 fluctuations. We discretize the radial coordinate into points where some different instabilities are resonant. Replacing the spatial derivatives of the helicity diffusion by difference quotients, we obtain the reduced model equations =

-E&Pn

- v&l9,,

(31)

971 = L =

-d&n

+VX,P,+Q,(P,~-~,P,+~,X,-I.X,,X,+I),

(32) (33)

pn

x~(L,(P,,x,-l,x,,x,+~)+Eh),

where n (= 1,2,. . . N) indicates Qnh-1,~

the nth radial

position, (Pn+d2

n+lrXn-l?hL,&l+l)

Ln(p”~,Xn-l,X,,,~n+l) A is a radial distance between different helicities.

=

2c

=

CO?n)

-

(Pn-d2

L+1

+ h-1

2Xn+1+

- 2L

A2

A

b-1 A2

’

- 2&l

the radial grid points, and N is the number

’ of interacting

islands with

Second

The model equations

(31)-(33)

x C

- 1, N 1,

= =

lo-’ 10-l

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chaotic orbits of the solution.

lpi, IQ1 = lo-* A N lo-‘,

- 1o-2, V = lo-”

The typical

EL = 10-6 - 10-5, N 10-3, Eh = lo-*

-

parameters

are

lo-‘.

Changing the Lundquist number R = ei’ (magnet,ic Reynolds number), we observe bifurcation and inverse cascade in the chaotic behavior of the soluteion. Figure 1 shows the Feigenbaum diagram, where we plot peak points of the time series with changing R. Here we fix other parameters as N = 3, C = 1.0, A = 0.1, V = 7.72 x 10e5 and Eh = 1.0 x 10-s. We observe two branches bifurcate into chaos. For a larger R, the chaos quenches and periodic attractor appears. The total number N of modes defines the freedom of the model equations, which is 3 x N. Figure 2 shows the dependence of Ljapunov dimension Df on N.

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