J. Plasma Fusion Res. SERIES, Yol.2 (1999) 516-519

Relaxation of a Magnetized Plasma
to States Other than Force-free
DASGUPTA Brahmananda, DASGUPTA Padmanabhal,

MYLAVARAPU Janaki S.. WATANABE Tomohiko2 and SATO Tetsuya2
Saha Institute of Nuclear Physics, I/AF, Bidhannagar, Calcutta 700 064, India
lDept. of Physics, university of Kalyani, Kalyani 741235, India
2Theory and Computer Simulation Center

National Institute .for Fusion Science, Toki, Gfu 509-52, Japan(Received: 8 December 1998 / Accepted:22 April 1999)

Abstract
A theory of relaxation of a magnetized plasma based on the principle of minimum dissipation andconstancy of global magnetic helicity is presented. The theory explains the occurence of a wide variety ofrelaxed states, which are not necessarily force free, in simulations and laboratory experiments. A solutionof the Euler Lagrange equation describing the states of minimum dissipation is accomplished using theanalytic continuation of the Chandrasekhar-Kendall eigenfunctions in the complex domain. Explicit
forms of the solutions are constructed using appropriate boundary conditions at the boundary. Thesolutions, as expected, support a finite pressure gradient. The distinct feature of this theory is to showthat it is possible to realize MHD equilibria with finite pressure gradient in a single fluid system even
without a long-term coupling between mechanical flow and magnetic field.

Keywords:
relaxation, reversed-field pinches, self-organization, minimum dissipation

1. lntroduction

states are governed by the magneto-hydrodynamic force

The relaxed states of a magnetoplasma, accordingbalance relation J x B = Vp. Several attempts [3,4] haveto Taylor's conjecture [], are obtained by minimizing
the total magnetic energy under the constraint of totalmagnetic helicity defined by K = I e.naV. A variational
technique leads to the Euler-Lagrange's equation V x B

been made in the past to obtain relaxed states which
could support finite pressure gradient, a large number ofthem making use of the coupling of the flow withmagnetic field. The novel feature of our work is to showthat it is possible for a single fluid to relax to an MHD= ,1,8, where i, is a constant. The relaxed state is a forcefree state.

equilibrium with a magnetic field configuration which
can support pressure gradient, even without a long-term
coupling between the flow and the magnetic field.
The concept of minimum dissipation rate was used
for the first time by Montgomery and Phillips [5] in an
MHD problem to understand the steady state profiles of
RFP configuration under the constraint of constant rate

Taylor's theory, although quite successful inexplaining a number of experimental results, including
those of RFP, is viewed as inadequate by many workers.
Relaxed states as envisaged by Taylor have only zeropressure gradient. Extensive numerical works by Satoand his collaborators have established [2] the existenceof self-organized states with finite pressure, i.e. theseCorresponding author's e-mail: dasgupta@ plasma.saha.ernet.in516@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchDasgupta B. et al., Relaxation of a Magnetized Plasma to States Other than Force-freeof supply and dissipation of helicity and the usualphysical boundary conditions for a conducting wall. It
may be pointed out that the principle was also discussed
by Chandrasekhar and Woltzer [6] in a sequel to thecomplete general solution of the force-free equation by
Chandrasekhar and Kendall [7]. It is also our conjecture

conditions and obtain La = 3.Il and ka = 1.23 as theminimum values of La and ka for the m =7 state.
The magnetic field components for the m = 0, k = 0
state are given by

Br=0,

that relaxed states could be characterized as the states oftta,\l|\-ot]

|

,-rl|

lur

Br= 72uolJ, tlrl+ 2Re | 7' a, J, @,drlll ,minimum dissipation rather than states of minimum
energy.

B,= T'aolJo{Lr1+2Re | #at, Jrtilanlll.,1 .
.,"1""

2. Euler-Lagrange Equation
We consider a closed system of an incompressible,
resistive magnetofluid, without any mean flow velocity,
described by the standard MHD equations in presence ofa small but finite resistivity 4. In presence of small scale

''l

@)

4. Toroidal Flux, Field Reversal and Pinch
Parameters

turbulence in the system, we may expect that the rate of

Several quantities that have proven useful inenergy dissipation decays at a faster rate than helicity.

describing the laboratory experiments are the toroidal
flux, @. field reversal parameter F, pinch parameter @and the helicity integral K.

We therefore minimize the ohmic dissipation R =I n jtdV subject to the constraints of helicity I n.naV.

A variational principle leads to the Euler-LagrangeQ, =)ftao 7aequation

VxVxVxB =AB,(l)Bb\where, .rl is a undetermined multiplier.

t+ zne|ft ,r, 0* rl].

<8,>

=+

3. Sofutions of Vx Vx V x B = AB

A solution of eq. (1) can be obtained in terms ofJo(1(i + (u tl a) a2 J o (Loa) + (dzl a) aI o (1.o2 a)I, (1a) + (d, |/ a) aI | @oa) + (azl ao) a2 J r 11a2a)'the Chandrasekhar Kendall eigenfunctions [7] which aresolutions of the equatiol V x 86 = iBo. The solution ofeq. (l) can be written as

B =La,Bn,ls1lal'''ltJ=-Be@)

18,>

_LAQ)

Jt A"a) + (atld)@2J | (hoa) + (a2ta)oJ I A"o2a)Here a,, are constants and, B, are solutions of the forcefree equation for complex l, such that V x Bn= Xo)n Bn.

J, @") + (d tl u) aJ | (lan) + (a2/ ail a2 J | (La2a)'
The toroidal flux parameter is obtained from @. -B,=LanY(D,xYz+Vx1V@, xYz) n =0,1,2. (3)2rlB, rdr. Also, < B. > refers to the volume average of
8.. In cylindrical coordinates the helicity integral iswith @, = J^ (ltnr)expfi(m0 + kz)), It| + k' = Ea2n, a =exp (2nil3). It can be easily demonstrated that theexpression for I given in eq. (2) satisfies eq. (l) for zl =defined by

K =4x2R I tArB, + A,B,)rdr .
where 2rR unl o ui! the lengrh and radius of thecylinder. For the minimum value of La = 3.11,

The constants a2 and a1 as well as the values ofha are fixed by assuming the boundary conditions for a
perfectly conducting boundary wall, given bycorresponding to the cylindrically symmetric state ltt =0,ft=0

B.n=0, jXn=O at r=a.

Kl O: =12.8R/a ,It is to be noted that for the cylindrically symmetric(m = O, ft = 0) state the boundary condition is triviallywhere the unit of KliD! is volt-sec. For values of volt-

satisfied and hence does not determine La. To get thenumerical value of )"for m * 0, we utilize the boundary

dissipation state.

sec 12.8 R/a, m = l, La = 3.11 is the minrmum

517

Dasgupta B. et a/., Relaxation of a Magnetized Plasma to States Other than Force-free

The pressure profile can be obtained from thealso shown plotted in Fig. 2. The current vanishes at theboundary because of the boundary conditions we haverelation j x B = Y p. For the m = O, k = 0 state, the onlynonvanishing component of the pressure gradient exists
in the radial direction, B, being zero. The radial pressure
distribution is obtained fromchosen.

The values of both F and @ at the boundary r = a
are evaluated and F is shown plotted against pinch ratio@ in Fig. 3. It is observed that F reverses at a value of @= 2.4, (Aa = 2.95) whereas for the Taylor state therp0)= 1 (ieB,-i,B)dr.

reversal is achieved at @ = 1.2. However, this field

5. Results

reversed state supports pressure gradient inconstrast toThe toroidal magnetic field profile, B. is plottedagainst rla inFig. I and shows a reversal near the edge

the Taylor state.
The pressure profile is shown in Fig. 4 for the m =of the plasma. The profile of the toroidal current "/. isk = 0 state with La = 3.0 which is the minimum energy

1.00.5

F 0.0
B-0.5

-1.0
Fig. 3 Field reversal parameter F is shown plottedagainst the pinch parameter @. The dotted curve
corresponds to the Taylor state.

Fig. 1 A plot of the 8, against rla lor lhe cylindricallysymmetric state.

0.0 02 0.4 0.6 0.8 1.o

0.0 0.2 0.4 0.6 0.8 1.0

ilailaFig. 2 A plot of the /, against rla lor the cylindricallysymmetric state.

Fig. 4 The radial pressure profile vs /a.

518

Dasgupta B. et al., Relaxation of a Magnetized Plasma to States Other than Force-freedissipation, field reversed state.
To conclude, the principle of minimum dissipation

Plasmas 3,2135 (1996), S. Zhu, R. Horiuchi, T.

is utilized together with the constraints of constantmagnetic helicity to determine the relaxed states of aPlasmas 3,2821 (1996).

Sato and the Complexity Sim. group, Phys.
t3l E. Hameiri and J. Hammer, Phys. Fluids, 25, 1855,

magnetoplasma not driven externally. This relaxed state(1982).

obtained from single fluid MHD supports pressure
gradient. This establishes that a coupling between
magnetic field and flow is not an essential criterion forhaving a non-zero pressure gradient. Further, it isshown that a non force-free state with field reversal

t4l L.C. Steinhauer and A. Ishida, Phys. Rev. Lett. 79,3423 (1997).

t5l D. Montgomery and L. Phillips, Phys. Rev. A. 38,2953 (1988).

t6l S. Chandrasekhar and L. Wolder, Proc. Natl. Acad.
Sci. USA 44,285 (1958).
171 S. Chandrasekhar and P.C. Kendall, Astrophys. J.
126,4s7 (1957).
t8l B. Dasgupta et. al, Phys. Rev. Lett. 81, 3144

properties can exist.

References

lll J.B. Taylor, Phys. Rev. Leu., 33, 139, (1974).

tzl T. Sato and the Complexity Sim. group, Phys.

(l998).

519