J. Plasma Fusion Res. SERIES, Vol.2 (1999) 52V523

Spheromak as a Non-Taylor Relaxed StateDASGUPTA Brahmananda, MYLAVARAPU Janaki S. and DASGUPTA Padmanabha'Saha Institute of Nuclear Physics, I/AF, Bidhanagar, Calcuna 700 064, Indiar
Dept. of Physics, Kalyani tJniversity, West Bengal, India(Received: 8 December 1998 / Accepted: 15 April 1999)

Abstract
The Spheromak Configuration, which is generally believed to be a minimum energy, force-free andzero pressure gradient state, is shown to evolve as a relaxed state with minimum dissipation. The EulerLagrange equation for the minimum dissipation relaxed state can be solved in terms of Chandrasekhar

Kendall eigenfunctions analytically continued in the complex domain. This state is not force-free and has
non zero pressure gradient and further shows the non constancy of the ratio of parallel current to themagnetic field. This is consistent with many experimental measuremets on Spheromak.

Keywords:
relaxation, spheromak, self-organization, minimum dissipation

1. lntroduction

non-flat pressure profile. Also, since competing effects

The Spheromak is a magnetic confinement system,where the magnetic field is self-organized to form a setof closed nested surfaces in a region of space. The
Spheromak configuration corresponds to the classicalare present in any experiment, deviations from a"Hill's Vortex" of fluid dynamics. The toroidal andpoloidal fields of a Spheromak are of equal strengthstate was also noticed [3-6].

uniform, constant )" can be expected. For spheromak,
such pressure profiles have been observed
experimentally and departure from minimum energy

approximately and are generated primarily by internalMontgomery & Phillips [7] utilized the principle ofminimum dissipation rate for the first time in an MHD

plasma currents rather than extemal coils.

problem to predict the relaxed states of a RFP

Important theoretical and experimental results forthe spheromak problem came from the application ofconfiguration. We try to establish here that a spheromak

equilibrium is a non force-free relaxed state, andsupports a significant fraction of perpendicularTaylor's relaxation model [] that conjectured themagnetic fields in a plasma to relax towards a state ofminimum energy subject to the constraint of constantmagnetic helicity. In a closed system, the minimum

component of current as well as a pressure gradient bymodelling the equilibrium as a state of minimum
dissipation rate with constant helicity.

magnetic energy equilibrium satisfies the force freeequation V x B = l.B with l. = constant. Rosenbluth andBusaac [2] obtained the spheromak field configurationfrom the Chandrasekhar Kendal (CK) solution of theforce-free equation and thus attempted to confirm thatspheromak is a Taylor force-free state.
In reality, any plasma confining device must have a2. Euler Lagrange Equation
The ohmic dissipation rate [7] for a magnetofluid isgiven byn = r7

C orre sponding author' s e - mail : das gupta @ p lasma. s aha. e rne t. in520)

i'dt,(1)@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchDasgupta B. el al., Spheromak as a Non-Taylor Relaxed State

where 4 is the plasma resistivity and the integral is overthe entire confinement region. The magnetic helicity

K = | A.Bdr is an invariant of motion in ideal
magnetihydrodynamics. If the turbulence is sufficiently
low, K still serves as a constraint as it decays at a slower
rate compared to R. If the energy dissipation rate given

0.5by eq. (1) is minimized by including helicity as aconstraint on the minimization through the use of
Lagrange's multiplier X. ttre following variationalequation is obtainedaI Oi'+ 1A.8)dr,-0 .5a)

Since this must hold for arbitrary dc we obtain theEuler-Lagrange equation :VxVxYxB =AB,

(3)

0-0.5

-1

where zl =X,/n is a constant.

0.51Fig. 1 The Spheromak

3. Spheromak Solutions of V x V x V x B =wlrere @; are the solutions of (V2 + Ero2\@, = 0 given

ABA solution of eq. (3) is constructed as a linear
combination of the solutions of the force free equation

1D, -V x 86 - LBo which are given in terms ofj^(),air)Pi@os9) ei'c.

(8)

Chandrasekhar-Kendall eigenfunctions. The magnetic
field 86, being a solenoidal field can be decomposed
into its toroidal and poloidal ingredients:

The expression for B given by eq. (6) then satisfies eq.
(3) for A = E. The corresponding flux function 7 isgiven by

Bo =Bor +Bop,>, d,a''ir17airysin2 0.
)(r. 01=+A i=o2

86r=Vx(€A),Boo=VxVx1{Yoy.

(4)

In eqs. (7), the constants ai are fixed by assuming theHere ( is a position vector, and Be satisfies the Taylor
equation provided

conducting boundary conditions

B-n=0, JXn=O at r=a,

1Y2 + 121@o = 0, Yo = oo | 1.

(e)where a is the radius of the sphere. For a given vaue ofn and m, the value of Xa can be obtained from theIn spherical polar coordinates (r, 0, Q)

boundary conditions. The eigensolution with lowesttv^(=r, Yo(r, 01=:! i^{7r)P}{cos01ei'o. (5)energy dissipation rate for the n = 0, m = | state has theeigenvalue ).a = 3.99. The Bp lines, given by X =constant are sketched in Fig. I for the lowest energy

Here, j^(A,r) ls a spherlcal Bessel function, Pff (cos9) is
an Associated Legendre function.

The classical spheromak equilibrium solution is givenby n = 0, m = I state and the corresponding fluxfunction Xothat describes the lines of constant poloidal

dissipation state.

4. Results
For values of )La = 3.99, which corresponds to themagnetic field is obtained is ls = -r sin0 dYoldQ.
The solutions of eq. (3) are obtained as the analytic
continuation of the solutions of V x Bo = hBo tocomplex values of ,1,. Expressing B as in eq. (4)B -Vx f|ol*VxVx1(Y;,(6)o =L a,@,,, =+{dio)2io.,a)state with lowest energy dissipation rate, the plots ofjll/
B and j 1lB in the z = 0 plane are shown in Fig. 2. A non

constantjn/B profile is observed which is in contrast tothe usual Taylor state for which i, = constant. Also asignificant amount of perpendicular component ofcurrent is obtained. This is similar to some of theexperimental observations [2-5], where a non constantjnlB profile is obtained. The profiles of toroidal andwith521

Dasgupta B. et al., Spheromak as a Non-Taylor Relaxed Statet.4
L.2

4

tt1"'l

30.8
o.e
0.4

2I-0.520.5

01-0.5-I

00.5

Fig. 2 A plot of ,|i,/B and /r/B against r/a in the z = 0 plane for ,l,a = 3.99.

-0 .5

0.5

-0.2
-0 .4
-0 .6

Fig. 3 The profiles of B, and B, against r/a in the z = 0 plane for the state ia = 3.99. The dotted curves show the field
profiles corresponding to the Taylor state for the same eigenvalue.

65 .75

5.)

2L.75 1' /s
1.5
L.25

Jnls

.25

-0.5

5

I

4.75
4.5
4.25

0 .75

0.5
50.5

1-1

-0.5

00.5

Fig. 4 A plot of y,,/B and ;,/8 against rla in the z = 0 plane for ,l.a = 5.0.

profiles of jxlB and j1/B are shown in Fig. 4. For thisvalue of ),a, tbe fields, however, do not satisfy theboundary conditions. In this case, a double humped
profile for the ratio julB. is sketched in Fig. 4. This statealso supports a perpendicular component of current

poloidal magnetic field components in the z = 0 plane
are shown in Fig. 3. The profiles are similar to thoseobserved in ref. [3].
For ).a = 5 which is higher than the value
colresponding to the lowest energy dissipation state, the522

Dasgupta B. et al., Spheromak as a Non-Taylor Relaxed State

Fig. 5 The profiles of Brand B, against rla inlhez= 0 plane tor La = 5.0.

t3l G.W. Hart et. a/., Phys. Rev. Lett. 51, 1558 (1983).
t4l F.J. Wysocki, Phys. Rev. Lett. 61, 2457 (1988).
t5l S.O. Knox e/. al.,Phys. Rev. Lett. 56,842 (1986).
t6l R. Martin et. al.,Plasma Phys. Contr. Fusion 35,

showing departure from the usual Taylor state. A similar
observation for jxlB was also reported by Hart et. al. [3].
Also, such double humped profile for j11lB was obtainedby solving numerically the Grad-Shafranov equation[3,8] for a spheromak with an oblate conducting

269 (1993\.

boundary. The profiles of toroidal and poloidal magnetic

t7l D. Montgomery and L. Phillips, Phys. Rev. A 38,

field components in the z = 0 plane are shown in Fig. 5.

29s3 (l988).
[8] T. Yeh and G.K. Morikawa, Phys. Fluids 14,781(r97 r\.

References[1] J.B. Taylor, Phys. Rev. Lett.33, ll39 (1974).
[2] M.N. Rosenbluth and M.N. Bussac, Nucl. Fusion
19,489 (1979).

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