a.y.k. chui and h.k. moffatt- instability of magnetic modons and analogous euler flows

8/3/2019 A.Y.K. Chui and H.K. Moffatt- Instability of magnetic modons and analogous Euler flows

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J Plasma P h y s ~ s1996) t o l 56 part 3 p p 677-691

Copyright 0 1996 Cambridge t7nirersiQ Press

677

Instability of magnetic modons and

analogous Euler flows

X . Y . K. C H U I and H. K. M O F F A T T

Department of Applied Mathematics and Theoretical Physics, University of Cambridge,

Silver St reet, Cambridge. CB3 9EW. UK

(Received 23 &lap1996)

ITe construct numerical examples of a 'modon' (counter- rota ting vortices) in an

Euler flow by exploiting the analogy between steady Euler flows and magnetostatic

equilibria in a perfectly conducting fluid. A numerical modon solution can be found

by determining its corresponding magnetostatic equilibrium. which we refer to as

a 'magnetic modon'. Such an equilibrium is obtained numerically by a relaxation

procedure tha t conserves the topology of the relaxing field. Our numerical results

show how the shape of a magnetic modon depends on its 'signature' (magnetic

flux profile). and that these magnetic modons are unexpectedly unstable t o non-

symmetric perturbations. Diffusion can change the topology of the field through

a reconnection process and separate th e two magnetic eddies. We fur ther show

th at the analogous Euler flow (or modon) behaves similar to a perturbed Hill's

vortex.

1. Introduction

By exploiting the analogj7 between steady Euler flows and magnetostatic equilib-

ria in a perfectly conducting fluid. JIoffatt (1985) showed how the steady solutions

of the Euler equation may be obtained by means of magnetic relaxation. In this

approach. an initial magnetic field with arbitrarily prescribed topology is allowed

to relax in an incompressible. perfectly conducting but viscous fluid until a magne-

tostatic equilibrium is reached. For each magnetic field B satisfying the condition

for magnetostatic equilibrium

j A B = V p , (1.1)

where j = V A B is the current density and p is th e fluid pressure, there corresponds

a velocity field U satisfying the steady Euler equation

u A o = V h , (1 .2 )

B - U , j - o , p o - p - h . ( 1 . 3 )

where o = V A U is the vorticity and h = p / p + ku2.via the analogy

Magnetic relaxation can therefore be used as a means to obtain steady Euler flows

with prescribed topology.

This technique is exploited in this paper in order to construct numerical examples

of a 'modon' (a pair of propagating counter-ro tating vortices). I n a frame that

moves with the modon. the flow is a steady solution of the Euler equa tion, having

www.moffatt.tc



678 A. Y .K. Chui an d H . K . Moffatt

a pair of stationary vortices embedded in a uniform, steady background flow. X

modon then corresponds (via the analogy (1.3)) o magnetostat ic equilibrium of a

magnetic dipolar structure. which we shall here describe as a ’magnetic modon’.

Such an equilibrium is obtained numerically by a relaxation procedure (see Sec. 2 )

that conserves the topology of the relaxing field.

The only known analytic solution (sometimes referred to as the Lamb-Batchelor

modon) is obtained by solving

Vz$j= - k 2@ (1.4

in a circle of radius ro, where $j is the streamfunction of the flow, and matching the

boundary condition to a uniform flow past a cylinder. The solution is

where Jo and J1 are Bessel functions. and k is the smallest positive number satisfying

Jl(kr0) = 0. In general, the streamfunction of a non-circular modon satisfies the

Grad-Shafranov equation

0% = F ( $ j ) , ( 1 . 6 )

where F is an arbitrary function that, indirectly, describes the circulation distri-

bution of the rotational fluid. Either F or the shape of the modon. but not both.

can be specified in numerical calculations (see e.g. Boyd and J la 1990: Eydeland

and Turkington 1988).Two cases of non-circular magnetic modons are here studied in detail:

(A ) the magnetic field is weak near the centre of each magnetic eddy:

(B) the magnetic field is strong near the centre of each eddy.

In Sec. 2. we present details of the magnetic relaxation process. In each case,

the magnetic field rapidly reaches a near-equilibrium state in which the eddies are

elongated, but is then unexpectedly unstable to non-symmetric perturbations. In

Sec. 3. we consider the Euler flows that are analogous t o the near-equilibrium

states. and follow their subsequent time-dependent evolution. In particular. we

examine how the modons response to non-symmetric perturbations. Some physical

explanations of the numerical results are given in Sec. 4 .

2. Magnetic relaxation

2 .1 . Governing equations

In magnetic relaxation, a suitable initial magnetic field is allowed to relax in an

incompressible, perfectly conducting but viscous fluid. Magnetic field lines arefrozen in such a fluid, and the energy

M = /B/’dV‘s (2.1)

is dissipated whenever the fluid is in motion, so an equilibrium state topologically

accessible from the initial field and of minimum energy will be reached.



Xagnetic modons and analogous Euler $ows 679

Any dynamical model th at dissipates energy but conserves magnetic field topol-

ogy may be adopted. We use here a quasi-static ‘porous medium’ dynamical law.

so th at the governing equations are

ku = - V p + j A B , )- V A ( u A B ) , IB

at

V . B = 0 ;

v . u = 0 ,

> ( 2 . 2 )

where k > 0. Rigid-wall boundary conditions U . n = 0. B . n = 0 (or periodic

boundary conditions) are assumed. It follows that

= / V A B . u A B d V = u . B A J d V

( 2 . 3 )

s= J’ U . (-vp - ku) dV = - k 1u12 dV < 0 ,s

so that t he energy decreases whenever the fluid is in motion.

function $ for U by (in Cartesian coordinates)

For two-dimensional fields. we define the flux function x for B and the stream-

B = V A ( x 8 ) ,U = V A ( $ i ) .

so that ( 2 . 2c , d ) are always satisfied. and ( 2 . 2a, ) become

(2 .1)

( 2 . 5 )

where J is the usual Jacobian operator defined by J ( , ) = fzgy- y g z , Sumerical

integration is performed on a uniform rectangular grid in the domain (-a

<x

<a.

-b < y < b) with rigid-wall boundary conditions on y = f b ,

$(x, b ) = const . x ( z . cb) = const ( 2 . 6 )

and periodic boundary conditions at x = f a .

$(-a. ) = w(a.Y) x ( - a . Y) = x ( a , ( 2 . 7 )

2 . 2 . Initial data

The initial magnetic field must satisfy two criteria:

(1) it has the same topology as that of a magnetic modon:

( 2 ) the boundary condition B .n = 0 is satisfied at y = i b .

Let xo be the vector potential of a circular magnetic (Lamb-Batchelor) modon.

normalized via (cf. (15 ) )

(2.8)



680

so t h a t t h e max i mum va l ue of xo ins ide th e ma gne t ic eddies i s I . L et

A. :K. C h u i and H . K. Xoffatf

when r < TO.

f’(0)xo when r 2 TO .(2 .9 )

where f is an y m onotonical lg increas ing funct ion wi th the p roper t i es f (0) = 0 a n df ( 1 ) = 1. N o te t h a t

( i) x1 a n d XO ha r e t he s ame t opo l ogy . becaus e t h e f unc t i on f s imp ly relabels th e

( ii ) th e ’speed of the flow’, (Vx11, s con t inuous acr oss t he bound ar y o f t he r o t a t i ona l

field lines:

region.

However. x1 does not ‘fit’ n th e r ec tangular do main . because i t does not sa t i s fyB . n = 0 o n y = i b .

In order to co ns t ruc t a fie ld th a t does sa t i sfy th i s bo und ary condi tion . we modi fyth e f ield x1 as follows. Let ~2 be a uni form ma gne t ic field def ined by

x 2 = Ky , (2 .10)

K = + k f ’ ( O ) J o ( k r o ) . (2.11)

where

S o t e t h a t x1 -+ x2 as r -+cc.We mix th e f ields x1 a n d x 2 v ia

x3 = Xx1+ ( 1 - X ) x 2 , (2 .12 )

where X is a weigh t f unc ti on t h a t t e nd s t o 0 on t h e s i des of t he r ec tang le an d t ends

t o 1 a t th e cent re ( th e or igin) . A sui table choice of X is

X = cos ($) cos ($) (2 .13 )

Then

x 3 ( I c , *b) = &tKb (2 .14 )

so t h a t t h e boundar y cond i ti on x 3 = const (hence B . n = 0) is sat isf ied.

I n th is paper . we cons ider two cases of non-c i rcular mag net i c mo dons :

( A ) the m agnet ic f ield is weak nea r th e cen t re of each magne t ic edd y:

(B) th e magn et ic field i s s t ron g near th e cent re of e a ch e d d y

Fo r Case A. we choose ~ ( x o )t a n h 4xo/ a n h 4. so t h a t

(2 .15 )t an h 4 X o/ t a n h 4 + ( 1 - X)K,y

4XxO/ tanh 4 + (1- X)K,y

when T < TO,

when r 2 T O .X 3 = X A =

de f {where K, = PkJo(kro)/ t a n h 4 . F o r Case B, we choose f(xo) = i x ~ ( l +~ : ) . o t h a t

where Kb = b k J ~ ( k r o ) .he con t our p lo t s of X A a n d XB a r e s hown i n F i gs l ( a ,c).

Figures l ( b , d ) s how t h e signaturefunctionsA (x) of the tw o f ields. where A(xc)s th e

t o t a l a rea of th e r egions in w hich x < xc. These s ignature funct ions are e s t imatedus ing b il inear in terpola t ion on the mesh d a t a . Dur ing t he r e laxat ion . the s ignature



Magnetic modons and analogous Euler $ows 681

3.5

3.02 . 5-2 -1 0 1 2

X X

-1 ."-2 -1 0 1 2 02 -1 0 1 2

Figure 1. Th e ini t ia l mag netic f ie lds: ( a ) he contour plot of th e flux function X A defined inCase A: ( b ) he s igna tu re func t ion of X A : (c) th e contour pl ot of th e f lux function X B definedin Case B: ( d ) he s igna tu re func t ion of X B .

function of the magnetic field should be an invariant (Moffatt 1986a).t Computing

the signature function therefore provides a means of monitoring the accuracy of

the numerical simulation.

2 .3 . Xumerical results

Numerical simulations have been performed on meshes with various sizes: 12 8 x 65.

257 x 65 and 267 x 129. A finite-difference method is used. Derivatives are approxi-

mated by standard formulae up to third order, and the Runge-Kutta (fourth-order)

method with variable time steps is used for time-stepping. Linardatos (1993) re-

por ted some magnetic relaxation calculations for magnetic fields with single elliptic

point. We tested our code on those magnetic fields; the equilibrium states we ob-

tained are in excellent agreement with those of Linardatos.

We first present the results for X B (Case B) calculated on a 129 x 65 mesh equally

spaced in the rectangular region ( -2 < z < 2. -1 < y < 1). The radius of the

rotational region. ' 0 . is set to t . In Fig. 2 ( a ) . he magnetic energy is plotted against

the time. The magnetic energy decreases rapidly from 4.026 (when t = 0) to 3.795

(when t = 0.1): the field evolves towards an equilibrium state (see Fig. 2b). while the

magnetic energy tends to a stationary value. The signature function is invariant

during the relaxation process as expected. The shape of the modon changes from

a circle to an oval with the longer axis parallel to the y axis. The change in the

shape has two effects:

(1) it allows more room for each eddy to become more circular: hence the energycontributed by the eddies decreases:

(2 ) it pushes some field lines towards the edges of the channel; hence the energy

contributed by the background field increases.

t In Moffat t (1986a).th e s ig nat ure of a f ield A ( x ) s def ined as t h e ar ea enclosed by t h eclosed field line w ith label x.Here we exten d th e de f in i tion of the s igna tu re (as in the te x t )so t h a t A ( y ) is always defined, el-en if the field lines are not closed



682 d. .K. Chui and H .K. Xojjaatt

0.02

0-

-0.02

-0.04

(0)

0 0.5 1.0 1.5 2.0 2.5 -0.02

- I.-

Y pg@!gJ0 1 2

1-2 -1

X

0 1 21 __-2 -1

-0.04

-0.06-0.8 -0.4 0 0.4 0.8

X

-0.06JIIIIIIIII/-0.8 -0.4 0 0.4 0.8

Figure 2. The magnetic relaxation of X B (Case B). (a) the energy quickly approaches a

minimum value. ( b ) he initial state to be relaxed: ( c ) the final state of minimum energy: ( d )

the (V'x. x ) catter plot a t the initial state: ( e the V'x, x) scatter plot at the final state.

showing the functional relationship between V x and x.(

The equilibrium state achieves a dynamical balance between these two effects.

In Fig. 2 ( d ) .V'x is plotted against x a t the initial state: each point in the plot

corresponds to a (V'x, x) air evaluated at a mesh point on the rectangular grid.

The points are scattered as expected, since the initial state is not at equilibrium.

Fig. 2 ( e ) shows the scatter plot of V'x against x in the final state. A functional

relationship of the form (1 .6) between V 2 x and x is evident: the final state is

therefore very close to a magnetostatic equilibrium.

Asurprising result is found for Case

A.Figure 3 shows a sequence of contour plots

illustrating how the magnetic modon may disappear in a rapid reconnection process.

Those results are calculated on a 257 x 129 mesh in the rectangular region ( - 2 <x < 2 . -1 < y < 1). The magnetic modon stays at a near-equilibrium (symmetric

about the x axis) for a long period: then there is a symmetry-breaking instability

(presumably triggered by random numeric noise), which tilts the modon. Once the

modon is tilted. the two saddle points collapse to Y points (cf. Linardatos 1993) with

associated current sheets. Owing to unavoidable numerical diffusivity. reconnection

occurs. and quickly annihilates the eddies. Figure 3 ( a ) shows how the energy

changes with time: the plot shows clearly how rapidly the reconnection takes place

when compared with the relatively long lifetime of the near-equilibrium state. We

note an interesting phenomenon observed from the numerical results: the signature

function is invariant (up to the accuracy of the estimate) throughout the relaxation

process, even though the rapid reconnection has occurred (see footnote in Sec. 2 . 2 ) .

To see how close the near-equilibrium state is to the actual symmetric equilibrium

state. we present scatter plots of V'x against x n Fig. 4. The functional relationship

is evident in Fig. 4(b) but is not as good as that in Case B; the loss of accuracy is



Nagnetic modons a n d analogous Eulei $ou*s 683

E 5 3 0 1 Rapid reconnection

(g) Diffusion10 -

AOn I I I

7'" 0 0.02 0.04

1IhJ

1' 0

-1

-2 -1 0 1 2X

-2 -1 0 1 2

0.06 0.08 0.10

1

0

- 1-2 -1 0 1 2

1

0

-1-2 -1 0 1 2

1

0

-1-2 -1 0 1 2

Figure 3.Th e magne t ic re la xa t ion of

X A

(CaseA ) : ( a )

he e ne r gy is plotted against t ime.showing a slow p h a s e of near -equi l ibr ium an d a rap id p hase of magne t ic reconnec tion , ( b - g )

a r e ' sna p - sho t s ' of the magne t ic f ie ld a t r a r io us s tages (each represented by a do t on thee ne r gy p lo t ( a ) ) .

perhaps due to the steep gradient of the vector potential X A near the boundaries

of the eddies.

It is puzzling why the magnetic modon is stable in one case but not the other.

We therefore re-examine Case B by extending the integration time: the magnetic

modon appears to remain stable. However, if a small non-symmetric perturbationis added to the initial state then a non-symmetric instability is found. Figure 5

shows the numerical results, and it is evident that the behaviour is similar to that

in Case A . It becomes clear that non-symmetric instability can occur in both cases:

the growth rate of the instability in Case B is. however, much smaller than that in

Case A .

TVe shall discuss these results again in Sec. 4. n the next section, we consider

the time-dependent behaviour of the analogous Euler flows.



( a ) 0 4 I I I I I I

0 2 -

v*x 0- @ - - - . - & ~

-0 2 -

-0 4 I I , I I I I

-0.04I 1 I I I I I

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

Figure 4. T h e ( V ’ X , ~ )cat te r plot in (a) he in i t ia l s t a te and ( b j th e final s tate . T he

functional relationship between V‘x a n d x s still evident in ( b ) .

3. Analogous Euler flows

Even when the magnetostatic equilibria obtained by a relaxation procedure are

stable. the analogous Euler flows may be unstable. The analogy applies only to thestructure of the steady states. but not to their stability properties (Moffatt 1986b).

Only in very special cases can we prove the stability of the analogous Euler flows

(see Davidson 1994). ITe investigate here the behaviour of the analogous Euler

flows using the near-equilibrium states obtained in the previous section.

We adopt the Euler equat ion in the form

where w = -V2$. A simple implementation of a finite-difference method (second-

order accurate) with fixed grid size is used. We use a special formula (Arakawa

1966) to evaluate the Jacobian so th at the enstrophy sw2dV is conserved. This

refinement makes prolonged numerical integration more stable. However. our re-

sults are still only a rough approximation to the t rut h, because our method cannot

resolve the fine scales that develop naturally under Euler evolution.

Figure 6 shows the behaviour of the near-equilibrium state in Case A under Euler

dynamics. The fluid is moving from left to right . The integrat ion continued till

t z 0.6. and snapshots of the evolution up to t = 0.3628 are shown in Fig. 6(b); he

modon can propagate six times its own width during this period. We found tha t thedisturbances are swept in the downstream direction. This kind of response is found

also in a perturbed Hill’s vortex (Moffatt and Moore 1978: Pozrikidis 1986),except

that the spike of rotational fluid found by Pozrikidis (accumulating a t the rear

stagnation point) cannot be observed in our simulation. We monitored the kinetic

energy and the enstrophy of the fluid: over this time of integration. they vary

within 1% of their initial values (Fig. 6a). Careful examination of the data shows.

however. th a t the maximum value of IwI within each vortex has increased by 100%;



Magnetic modo ns and analogous Euler $ows 683

3.8

3.6

3.4

3.2

E 3.0

2.8

2.6

2.4

2.2

-

---

-

-

2.0' ' ' ' ' ' ' ' ' 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

X

-2 -1 0 1 2

-1lrn2 -1 0 1 2

-L -I U 1 L

0 1 2I-2 -1

-2 -1 0 1 2

Figure 5 . The magnetic relaxation of perturbed X B ' ( a ) he energy is plotted against the

time, showing a slow phase of near-equilibrium and a rapid phase of magnetic reconnection

( k g ) re 'snapshots' of the magnetic field a t 17arious stages (each represented by a dot on

the energ:- plot ( a ) )

th e (false) gain of vorticity inside the rotat ional region is due to an accumulation

of truncat ion error in prolonged numerical integrat ion.

We speculate th at the modon would response to disturbance in a similar way ifthis false gain of vort ici ty could have been suppressed. However. developing a more

accurate unsteady Euler equation solver is beyond the scope of this paper.

It is interesting to test how the near-equilibrium st ate responds to non-symmetric

perturbation. Figure 7 shows how quickly th e modon breaks down (near t = 0.06)).

The change of topology is due to t he unavoidable numerical viscosity.

Similarly. Fig. 8 shows how the near-equilibrium state in Case B behaves under

Euler dynamics. and Fig. 9 shows how the near-equilibrium s ta te responds t o non-



1.010

1 00 5

1.000

0 99 5

t = 0 . 1 7 0 8 , E = 5 5 1

, I I , I

- 1

- A

-

5

t = 0.2428, E = 550.1

t = 0.3148, E = 550

0.2 0.3 0.4

t = 0.1054, E = 550.7

t = 0.1948, E = 550.5

t = 0.2668, E = 550.6

t = 0.3388, E = 549.4

t = 0.1462, E = 55 1.8

t = 0.2188, E = 550.4

t = 0.2908, E = 550.4

t = 0.3628, E = 550

Figure 6. The behaviour( a ) he kinetic: energy E

sna psho ts show tha t t h e

of th e near-equi libr ium s ta t e in Case X unde r the Eu le r dynam ics :a n d t h e e n s t r o p h g T va r y w i th in 1 % of the ir ini t ia l values: ( b ) h emod on behaves like a per turbed Hi l l ' s vor tex .

n p in e t r i c disturbances. The results suggest that t he modon initially oscillates

then irrotational fluid enters the modon near the rear stagnation point. a nd finally

the modon moves away from the centred position. An interesting feature is th at

the modon does not break up during the evolution. Again. this behaviour needs to

be verified by a more accurate numerical scheme in future research.



JIagnetic modons and analogous Eule i j h . s 687

1.000

0.995

0.990

0.985

0.980

--

-

r

I , I I I I

0.975; 0.02 0.04 0.06 0.08 0.10 0.12

(b) t = 0, E = 552.6

t = 0.023 87, E = 552.6

t = 0.04787, E = 551.8

t = 0 . 0 7 1 8 7 , E = 5 4 9 . 1

t=0 .007 8 7 1 , E = 5 5 2 . 6

t = 0.031 87 , E = 552.4

t = 0 .0 5 5 8 7 ,E =5 5 1 .3

t = 0.07987,E = 547.5

t = 0.01587, E = 552.6

t = 0.03987, E = 552.2

t = 0.063 87, E = 550.4

t = 0.087 70, E = 545.8

Figure 7 . When a smal l non-s ym metr ic pe r turba t ion is appl ied to the t he near -eyuihbi iunista te in Caqe A t h e modon br e a ks domn qu ick ly unde r the E u le r dyna m ic s ( a ) he kinetice ne r gy E a n d t h e e n s t ro p h y T vary wi th in 2 5% of the i r in i t ia l r a lue i , ( b ) the snapshotsshoir hoir rapidl) the modon responds t o non-symm etr ic pe i turba t io ns



688 A . 1: K. Chui and H . K. X o f f a t t(a) 1.15 I I I , , I I I I

I 1 I I , I I

0 1 2 3 4 5 6.85

( b ) t = 0, E = 2.422

t = 2.253, E = 2.394

t = 4.793, E = 2.535

t = 1.05,E = 2.664

t = 2.739, E = 2.649

t = 3.984, E = 2.388

t = 5.197, E = 2.552

7 8 9

t = 1.74, E = 2.222

0

t = 3 . 1 6 , E = 2 . 4 9 1

t = 4.397, E = 2.544

t = 5.599, E = 2.452

Figure 8. T h e b e h a r i o u r of the near -equi l ibrium s ta te in Case B u n d e r t h e Euler d y n a m i c s(a)he k ine t ic energy E a n d t h e e n s t r o p h y T va r y w i th in 15% of the ir ini t ia l ualues: ( b ) h e

inap shots show t h a t the modon in i tia l ly osc il la te ;, an d become uns table .

4. Discussion

In Boyd and RIa (1990). an elliptic modon is determined by first specifying its

shape. and then evaluating the required circulation distribution. It is found tha t as

such a modon becomes more elliptic. the circulation becomes stronger a t its bound-

a r y This phenomenon can now be explained physically via the magnetic analogy.

Note tha t the magnetic field defined by t he vector potential X A s concentrated near



Xa gne tic modons and analogous Euler f lows 689

(a) 1.2

1 .o

0.8

0 1 2 3 4

t = 0, E = 2.424 t = 0.9359,E = 2.334

t = 1.711, E=2 .50 1 t = 2.091, E = 2.499

t = 2.593, E = 2.816 t = 2.785, E = 2.785

t = 3.184, E= 2,564 t = 3.645,E = 2.728

5 6 7

t = 1.327,E = 2.47

t = 2.383,E = 2.779

t = 2.978,E = 2.671

t = 4.071,E = 2.492

Figure 9. (a ) The kinetic energj E and the enstrophy T vary within 25% of their initial

values ( b )When a small non-symmetric perturbation is applied to the the near-equilibriumstate in Case B the instability of the modon is evident The modon does not break up but

moves away from the centred position

the boundary of the eddies. The magnetic pressure near the centre of the eddies is

therefore lom-, and t he eddies are liable to be compressed. The Lorentz force near

the rigid walls pushes the fluid ‘inwards’. an d th us t he eddies are compressed in the

y direction (Fig. 3 ) .



690 A . I.:K . C h u i an d H .K. X o f f a t t

The other case is also interesting. In Case B, the magnetic field is concentrated

near the centre of the eddies. and the background field is relatively weak. The lines

of magnetic force inside the eddies tend to be more circular. and this dominating

effect causes the eddies t o elongate in the y direction (Fig. 2 ) .

The unexpected non-symmetric instabil ity can be interpreted as follows. Fi rs t,

the two magnetic eddies repel each other because they behave like two parallel con-

ducting elements carrying opposite current density (in the z direction). Secondly.

the background magnetic field has the effect of pushing the upper eddy downwards

and the lower eddy upwards. \Then the magnetic modon is perturbed by a non-

symmetric displacement. the torque generated by these magnetic forces rapidly

tilts the inodon.

It is worth noting that a similar tilting instability is also found in axisymmetric

magnetic containment devices like SPHEROJIAK. Rosenbluth and Bussac (19i9)

considered a force-free magnetic field B confined in a slightly perturbed spherical

region, satisfying j = V A B = kB. where k is a constant. and has the same topology

as a Hill’s vortex. When the perturbation makes the sphere prolate (cf. Case A ) . an

internal tilting instability (i.e. within the sphere) is found: when the perturbation

makes the sphere oblate (c f . Case B). an external tilting mode is found unless the

spherical region is bounded by a conducting shell of sufficiently sinall size. These

results are rer y similar t o ours. although th e magnetostatic equilibria th at we have

considered in this paper are two-dimensional and are not force-free.

To summarize. magnetic modons are in general non-circular. the direction of

elongation of th e eddies being different in th e two cases considered. Kumerical

experiments have revealed t hat these magnetic modons are unexpectedly unstable:

non-symmetric perturbation drives the structure to rotate. and reconnection takes

place near the saddle points of the field owing to (unavoidable) numerical diffusion.

Eventually. the two magnetic eddies separate. and they either hit the boundary or

disappear altogether.

The near-equilibrium states referred to above are close to steady (analogous)

Euler flows (or modons), whose stability has also been investigated numerically.

Our results show that the modon may respond to disturbances in a manner similar

to th at of a perturbed Hill’s vortex.

Acknozcledgements

This work is supported by the EPSRC under Grant GR1J21.139 and UK/Hong

Kong Joint Research Grant J R C 94/24.

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