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coils.

Device Axisymmetry?

ToroidalField

Topology

FRC yes no spheroidal

Spheromak

yes yes spheroidal

RFP yes yes toroidal

Tokamak yes yes toroidal

Stellerator

no yes toroidal

Table 1: A summary of various confinement schemes.

The primary difference between the FRC andthe spheromak is that the former has zerotoroidal field everywhere, while the latter

creates its own toroidal field by self-inducingcurrents within the plasma in a attempt toachieve a least-energy configuration, inaccordance with the laws of MHD. Inaddition, spheromaks have no toroidalmagnetic field at the bounding surface, whichtranslates to zero toroidal field at the wall inlaboratory applications. Figure 1 shows somecommon field configurations, with their associated flux surfaces. The configuration

labeled ST applies to both stellarator andtokamak geometries.

Of the common laboratory confinementschemes, only the FRC is simpler than thespheromak. Unfortunately, the FRC isinherently MHD unstable due to its complete

lack of toroidal field. This gives thespheromak the distinction of being thesimplest laboratory configuration that iscapable of achieving MHD equilibrium.

Spheromak plasmas are highly dependent onthe topology of the J ×

B hydromagneticforce. In toroidal magnetic fields, charged

particles move along flux surfaces,generating a plasma current J . These

particles then experience a force, directedradially in the J × B direction, whichcauses expansion. Confinement schemestherefore try to minimize this J × B force inan attempt to establish or maintainequilibrium. In 1950 Lundquist (cite this)

investigated J × B equilibria in solar andmagnetospheric plasmas and showed that for small hydrodynamic pressure J × B≈ 0 . If we make this assumption and take the curl of Ampere's law, we get what is known as thethe “force-free” MHD equation:

∇× B= B (1)

Lundquist proceeded to solve this equation,

arriving at the equilibrium solution (cite):

B= B J 1 J 0 z (2)

Where J 1 , J 0 are Bessel functions of thefirst kind. This is known as either theLundquist solution or the Bessel functionmodel (BFM), and requires a helical toroidalfield.

Spheromaks:Historically, spheromak research grew out of some unanticipated results from the ZETAreverse field pinch (RFP) operated at Culhamin the UK during the 1950's. An RFP is nearlyidentical to the tokamak in its generalconstruction, but the externally generatedtoroidal field is much weaker, while toroidal

Figure 1: Common Field Configurations

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currents are much larger. Researchers notedthat when this toroidal field dipped belowsome minimum value, the initially unstable

plasma would settle into a stable state. Inaddition, if the ratio of toroidal current totoroidal field was high enough, the toroidalfield would reverse its polarity at the wall. In1974, Taylor proposed an explanation thatrelied on one primary assumption; theconservation of magnetic helicity.

We can define magnetic helicity as follows. Ina force-free system subject to the laws of ideal MHD, there are two invariants [2].

The total energy:

W = ∫ B2

8

p− 1

dV (3)

and the infinite set of integrals given by:

K l = ∫ A⋅ BdV ; l = 0,1,2,3. .. (4)

Where the integrals are over the volumes of each flux tube. In equations (3) and (4) p is

the pressure, A is the magnetic vecot potential, and γ is the ratio of specific heats.Therefore the total magnetic flux of thesystem is conserved. In equations (3) and (4)

p is the pressure, A is the magnetic vector potential, and γ is the ratio of specific heats.

According to variational theory, the preferredstate of the MHD system can determined byminimizing W with respect to the infinite set

of integrals K l . Taylor observed that,according to equation (4), that the final,least-energy configuration of the systemdepended in a very detailed and complicatedway on the initial state. This contradictedobservations, which tended to support thatthe final state of the system was independent of its initial conditions. Taylor proposed that,in a plasma with large but finite conductivity,

bounded by a surface with infiniteconductivity (or in the case of laboratory

plasmas, an extremely large conductivity),all of the integrals represented by (4) wouldno longer retain their invariancy with theexception of the flux tube tangent to the

bounding surface, K o. There is then thesingle invariant

K o= ∫ A⋅ BdV

where the integration is now taken over onlythe volume of the entire plasma volume. Thequantity K o is the magnetic helicity of thesystem. Minimizing W (3) with respect tothis single invariant leads directly back to (1)

∇× B= B (1)

with the associated Lundquist solution in (2).In this way Taylor showed that Lundquist'sBessel function model was appropriate todescribe the behavior of the ZETA RFP.

The ZETA RFP is an example of a doublyconnected toroidal geometry, and as such is

different from spheromak geometry, which issimply connected and spherical. In 1979,Rosenbluth and Bussac extended Taylor'sRFP formulation to a spherical geometry, withthe result that the minimum energy statesnow have no external current, and thereforeeliminate the need for external coils. Thetopology then reduces to that of a simplyconnected sphere, and the authors coined the

phrase “spheromak” to describe these

systems. They were not, however, the first toderive spherical solutions to force-free MHDequations. That distinction goes to theinimitable Chandrasekhar, who hadinvestigated these solutions more than adecade earlier.

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Spheromak plasmas thus have the followingdistinctive features.

1) Spheromaks are simply connected andspherical.

2) Spheromaks have zero toroidal magneticfield at the the wall.

3) Spheromaks have no external field coils;

internal toroidal fields are the result of plasmaself-organization that results from the plasmaseeking a minimum energy state. This state isknown as the Taylor state.

4) The flux surfaces within spheromaks areentirely the consequence of instabilitieswithin the plasma.

5) Due to their self-creating nature,spheromaks do not have to be as meticulouslyengineered as other confinement schemes.Indeed, there are several methods that areused to create spheromaks; all result in thesame characteristic plasma configuration.

How to make a spheromak-

References:

[1] Paul Bellan, "Spheromaks", Imperial College Press, 2000

[2] S. Ortolani, Dalton D. Schnack, “Magnetohydrodynamics of plasma relaxation” ,World Scientific, 1993

[3] http://www.frascati.enea.it/ProtoSphera/ProtoSphera%202001/1.%20General%20framework.htm

[4] http://ve4xm.caltech.edu/Bellan_plasma_page/howto.htm

Figure 2: Examples of simply and doublyconnected topology

Simpl y Connected Doubly connected