Impact Of A Uniform Plasma Resistivity In MHD Modelling Of Helical Solutions For The Reversed Field Pinch Dynamo

D. Bonfiglioa, S. Cappelloa and D. F. Escandea,b
aConsorzio RFX, Associazione EURATOM-Enea sulla fusione, Padova, Italy
bCNRS-Université de Provence, Marseille, France
Abstract. Till now the magnetohydrodynamic (MHD) simulation of the reversed field pinch (RFP) has been performed
by assuming axis-symmetric radial time independent dissipation profiles. In helical states this assumption is not correct
since these dissipations should be flux functions, and should exhibit a helical symmetry as well. Therefore more correct
simulations should incorporate self-consistent dissipation profiles. As a first step in this direction, the case of uniform
dissipation profiles was considered by using the 3D nonlinear visco-resistive MHD code SpeCyl. It is found that a
flattening of the resistivity profile results in the reduction of the dynamo action, which brings to marginally-reversed or
even non-reversed equilibrium solutions. The physical origin of this result is discussed in relation to the electrostatic drift
explanation of the RFP dynamo. This sets constraints on the functional choice of dissipations in future self-consistent
simulations.
Keywords: MHD simulations, MHD dynamo and self-organization, Reversed Field Pinch.
PACS: 52.65.Kj, 52.55.Hc.
I. INTRODUCTION
The reversed field pinch (RFP) is a toroidal device for plasma magnetic confinement. Its magnetic configuration is
akin to the tokamak, but the amplitude of the poloidal field is comparable to the toroidal one, which reverses in the
outer region. A self-organized plasma flow provides, through the Lorentz force, the poloidal component of the
electromotive force necessary to sustain the highly sheared configuration: this is the so-called RFP dynamo action.
During the last years, the increasing evidence in different RFP experiments (see Ref. 1 and references therein) of a
tendency to develop regimes characterized by a good degree of helical symmetry has motivated renewed interest in
the modelling of the RFP as a saturated regime of a single kink-type magnetohydrodynamic (MHD) mode. Threedimensional
visco-resistive MHD numerical simulations of the RFP display a transition from turbulent states to
helical laminar equilibria when dissipation is increased2,3. Till now, the RFP has been simulated by assuming timeindependent
dissipation coefficients with axis-symmetric radial profiles. In the case of helical states2,3,4,5, this
assumption is not justified since these dissipations should be constant on helical magnetic flux surfaces. Therefore,
more correct simulations should incorporate self-consistent dissipation profiles. As a first step in this direction, the
effect of uniform dissipation profiles in MHD modelling of helical RFP solutions is considered in this paper.
II. NUMERICAL RESULTS
The RFP is simulated with the SpeCyl 3D non-linear visco-resistive MHD cylindrical code6. SpeCyl provides
numerical solutions of the compressible non-linear MHD model in the constant-pressure constant.

To study stationary sustainment of helical solutions, Equations 1 are numerically solved in the case of a singlehelicity
spectrum composed by the m =1,n = -10 mode and 79 of its poloidal harmonics. Plasma current and
toroidal magnetic flux are constant, and thus the pinch parameter, ( )/ 1.6 z B a B J Q º = . Resistivity and viscosity
are used with profiles given by ( ) ( ) 0 r 1 rbh
h h =h +a and ( ) ( ) 0 r 1 rbn
n n =n +a . Different values of the dissipative
coefficients 0
h and 0 n are used, corresponding to Lundquist numbers 1
0 S ºh – in the range 1.5´104 £ S £ 3´105 and
magnetic Prandtl number in the range 100 £ P £ 3000 . For each choice of the couple (S,P) , two types of
dissipation profiles are used. Profiles of the first type are characterized by 19 h n a =a = and 10 h n b = b = , which
implies that h and n are practically constant for r £ 0.8 , rising steeply in the edge region (this is the standard shape
of dissipation profiles used in SpeCyl). The second case consists in uniform profiles ( 0) h n a =a = .
The numerical evolution of reversal parameter ( ) z z F º B a B and volume averaged energy of magnetic
fluctuations d EM is shown in Figure 1. A non-reversed axis-symmetric ohmic state is used as the initial
configuration for each simulation, at 1000 A t = t . A small perturbation, corresponding to the unstable m =1,n = -10
mode, is then applied to the system. The initial phase of MHD dynamics is qualitatively the same for standard
profile (Fig. 1a) and uniform profile simulations (Fig. 1b): the mode amplitude grows exponentially, causing F to
change its sign, and non-linearly saturates. In the case of standard profile simulations, a helical RFP equilibrium is
reached just after saturation. The following slow fall of F is caused by the loss of toroidal flux due to numerical
discretization effects ( Q =1.609 equilibria are marked with a cross in Fig. 1a). In the case of uniform dissipation, i.e.
flat profiles simulations, the amplitude of the helical deformation decreases after saturation, leading to a reduction of
the dynamo action. Consequently, the final helical equilibrium turns out to be marginally-reversed or even nonreversed
in the set of simulations with pinch parameter Q =1.6 (simulations at higher pinch parameter,Q =1.9 ,
show deeply-reversed equilibria also with uniform dissipation profiles).
The dependence of the amplitude of the equilibrium helical deformation on the Hartmann number is shown in
Figure 2. In the case of standard profile simulations, the reversal becomes deeper with increasing H and saturates at
F @ -0.3 for large H. Similarly, the magnetic perturbation energy grows and saturates at 5% of the total magnetic
energy. The saturation of F and d EM is found as expected for single-helicity dynamics6,7,8. Saturation for large H
is also shown for uniform profile simulations, but in this case the magnetic fluctuation energy is one order of
magnitude smaller.

IV. SUMMARY AND REMARKS
The impact of a uniform plasma resistivity in MHD modelling of helical RFP solutions has been considered. A
flattening of the resistivity profile is found to cause a reduction of the dynamo action, which brings to a decrease of
the toroidal field reversal. These findings are explained on the basis of the electrostatic nature of the RFP dynamo. In
fact, a finite electrostatic field is required to balance parallel Ohm’s law and the dynamo electric field merely
corresponds to this electrostatic field10,11. We have shown here that with uniform plasma resistivity this balance is
more easily achieved, while, in the case of standard profile simulations, the presence of resistivity gradients along
magnetic field lines requires an additional electrostatic field, which provides a larger dynamo action and hence a
deeper reversal.
Since the essential feature for dynamo reduction is the vanishing of resistivity gradients along magnetic surfaces,
we may argue that future single-helicity simulations with non trivial self-consistent resistivity profiles would also
result in a shallowing of the field reversal. This could explain the difficulty to find deeply-reversed RFP solutions of
the helical Grad-Shafranov equation coupled with Ohm’s law13.
ACKNOWLEDGMENTS
One of the authors (S. C.) wishes to thank J. F. Drake for his valuable suggestion on the effect of resistivity
gradients along magnetic surfaces at the Festival de Theorie 2003 in Aix-en-Provence.
REFERENCES
1. P. Martin et al., Nucl. Fus. 43, 1855 (2003).
2. S. Cappello and D. F. Escande, Phys. Rev. Lett. 85, 3838 (2000).
3. S. Cappello, Plasma Phys. Control. Fusion 46, B313 (2004).
4. J. M. Finn, R. A. Nebel and C. C. Bathke, Phys. Fluids B 4, 1262 (1992).
5. S. Cappello and R. Paccagnella, Phys. Fluids B 4, 611 (1992).
6. S. Cappello and D. Biskamp, Nucl. Fus. 36, 71 (1996).
7. R. A. Nebel, E. J. Caramana and D. D. Schnack, Phys. Fluids B 1, 1671 (1989).
8. D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge: Cambridge University Press, 1993.
9. H. E. Sätherblom, S. Mazur and P. Nordlund, Plasma Phys. Control. Fusion 38, 2205 (1996).
10. D. Bonfiglio, S. Cappello and D. F. Escande, Phys. Rev. Lett. 94, 145001 (2005).
11. S. Cappello, D. Bonfiglio and D. F. Escande, to appear in Phys. Plasmas 13 (2006).
12. R. E. Denton, J. F. Drake and R. G. Kleva, Phys. Fluids 30, 1448 (1987).
13. S. Cappello, F. D’Angelo et al., in Proc. 26th EPS Conf. on Controlled Fusion and Plasma Physics (1999), Vol. 23J, p. 981.

Опубликовано в рубрике Documents