velocity of the plasma give rise in the average Ohm's law to a dynamo electric field, responsible for driving poloidal currents in the outer region of the plasma: this ...
DRFC-CAD:
EUR-CEA~-FC^15()4
KINETIC MODELING OF FAST ELECTRON DYNAMICS AND SELFCONSISTENT MAGNETIC FIELDS IN A REVERSED-FIELD PINCH
G. GIRUZZI and E. MARTINES !December 1993:.
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KINETIC MODELING OF FAST ELECTRON DYNAMICS AND SELF-CONSISTENT MAGNETIC FIELDS IN A REVERSED-FIELD PINCH G. Giruzzi*, E. Marlines 0 * Association Euratom-CEA sur Ia Fusion, Département de Recherches sur la Fusion Contrôlée. Centre d'Etudes de Cadarache, 13108 Saint Paul-lez-Durance (France) ° Dipartimento di Ingegneria Elettrica, Université di Padova, Via Gradenigo 6/a,
3513l Padova (Italy) rD -, •'
Abstract The dynamics of fast electrons in a reversed field pinch configuration is investigated by numerically solving the appropriate kinetic equation in 3 dimensions (2 dimensions in velocity space and 1 dimension in real space). To this end, a Fokker-Planck code has been developed, including Coulomb collisions, dc electric field, radial diffusion due to magnetic turbulence, ambipolar electric fields, and the self-consistent evaluation of the magnetic fields generated by the plasma itself.
This has allowed the theoretical validation of the
Kinetic Dynamo Model in a realistic geometry. In contrast to fluid-turbulent theories, such model predicts that the radial diffusion of fast electrons associated with stochastic magnetic fields might be able to sustain the reversed-field configuration.
Quantitatively, it is found
that the level of magnetic turbulence necessary to obtain the toroidal field reversal at the plasma edge is compatible with levels typically measured in reversed field pinch devices. In particular, the main parameters of standard discharges in the largest existing facility of this type (RFX1) have been successfully simulated.
PACS numbers:
52.25.Dg, 52.55.Hc
KINETIC MODELING OF FAST ELECTRON DYNAMICS AND SELF-CONSISTENT MAGNETIC FIELDS IN A REVERSED-FIELD PINCH G. Giruzzi', E. Marlines0 * Association Euratom-CEA sur la Fusion, Département de Recherches sur la Fusion Contrôlée, Centre d'Etudes de Cadarache, 13108 Saint Paul-lez-Durance (France) 0
Dipartimento di Ingegneria Elettrica, Université di Padova, Via Gradenigo 6/a,
35131 Padova (Italy)
Abstract The dynamics of fast electrons in a reversed field pinch configuration is investigated by numerically solving the appropriate kinetic equation in 3 dimensions (2 dimensions in velocity space and 1 dimension in real space). To this end, a Fokker-Planck code has been developed, including Coulomb collisions, dc electric field, radial diffusion due to magnetic turbulence, ambipolar electric fields, and the self-consistent evaluation of the magnetic fields generated by the plasma itself.
This has allowed the theoretical validation of the
Kinetic Dynamo Model in a realistic geometry. In contrast to fluid-turbulent theories, such model predicts that the radial diffusion of fast electrons associated with stochastic magnetic fields might be able to sustain the reversed-field configuration.
Quantitatively, it is found
that the level of magnetic turbulence necessary to obtain the toroidal field reversal at the plasma edge is compatible with levels typically measured in reversed field pinch devices. In particular, the main parameters of standard discharges in the largest existing facility of this type (RFX1) have been successfully simulated.
PACS numbers:
52.25.Dg, 52.55.Hc
!.Introduction The reversed field pinch (RFP) is an axisymmetric toroidal configuration which constitutes an alternative to the tokamak for the confinement of fusion plasmas2. Hs main feature is that the magnetic field, mostly generated by the plasma itself, has a toroidal component" which" changes sign in the outer region of the plasma; Since the external electric field is applied in the toroidal direction, itJs,not^ possible Jo account for the sustainment of the configuration on times longer than the resistive diffusion time without a mechanism of magnetic field regeneration, generally referred to as dynamo. Due to the presence of this mechanism, the plasma does not obey a local Ohm's law with a resistivity given by the Spitzer-Harm expression 3 : in particular, in the outer region the current density parallel to the magnetic field j, is in direction opposite to that of the local parallel electric field E,,. The nature of the dynamo mechanism in the RFP has not been assessed yet, although it is generally recognised to be related to the high level of magnetic fluctuations present in this configuration. A possible explanation is that coherent fluctuations of magnetic field and velocity of the plasma give rise in the average Ohm's law to a dynamo electric field, responsible for driving poloidal currents in the outer region of the plasma: this model is referred to as MHD dynamo4.
Alternatively, the so-called Kinetic Dynamo Theory (KDT)
starts from the consideration that, at the magnetic fluctuation levels reported in RFP experiments, the magnetic islands generated by resistive modes superpose and the field lines become stochastic, connecting directly the center of the plasma to the edge. The model predicts that hot electrons coming from the center of the plasma flow along the stochastic magnetic lines to the edge region, where they carry a substantial fraction of the local current density, and give rise to a non-local Ohm's law5. An experimental evidence in favor of the KDT is the presence in the edge region of RFP experiments of a population of energetic electrons, which flow quasi-unidirectionally along the magnetic field in a direction opposite to that which would be expected if they were generated by the local E . This population has been detected on 2T-40M by means of an electrostatic energy analyser3, on TPE-1RM15 with soft x-rays target probes and Thomson scattering78, on ETA BETA Il with calorimetric probes9, on MST by means of an electrostatic energy analyser and a soft x-ray target probe10 and on RFX using calorimeiric probes11. The values reported for the temperature of this tail of energetic electrons are of U3 times the central bulk temperature on MST and ZT-40M, whereas on TPE-1RM15 values ranging from 5 times to more than 10 times the central temperature have been found. It is important here to remark that TPE-1RM15 generally operates at a lower density than the other machines. From the theoretical point of view, the possibility of explaining the RFP dynamo by means of the KDT has been shown by means of numerical simulations based on the
simultaneous solution of a kinetic equation for the electron distribution function and one of Maxwell's equations, in order to compute distribution function and magnetic field profiles sell-consistently, either in slab5 or in cylindrical geometry12. These codes used the drastic approximation of an isodense and isothermal background plasma, described electron-ion collisions by a Krook term (i.e., « -v(v)f) neglecting electron-electron collisions, and did not allow radial ambipolar electric fields to build up.
In order to overcome these
limitations, and to investigate with greater accuracy the energetic electrons dynamics in view of a comparison with experimental results, an approach to this problem based on a 3dimensional (20 in velocity and 1D in space) kinetic equation has been developed. In this paper are described the results of simulations performed by means a 3D Fokker-Planck code with magnetic field profiles calculated self-consistenlly with the electron distribution function. This code constitutes a natural evolution of the original approach to the problem5-12, allowing the following improvements: i ) possibility of defining a background plasma with radial profiles of temperature, density and Z effective: ii) better description of electron-ion collisions, including pitch-angle scattering, and addition of electron-electron collisions: iii) inclusion of the effect of a radial electric field {ambipolar field); iv) extension of the class of distribution functions which can be solution of the problem (the distribution function is no more bounded to be odd in the angle between velocity and magnetic field). In this approach, all the relevant known contributions to the fast electron dynamics are included, with no substantial simplification of the geometry of the problem, which is treated in 3 dimensions. This allows, for the first time, a complete assessment of the theoretical validity of the KDT, by ruling out all the uncertainties due to the above discussed approximations. On the other hand, such an approach constitutes a solid basis for detailed simulations of the experimental outputs, including diagnostics specific to fast electrons. The plan of the paper is the following. briefly reviewed.
In Sec. 2, the theoretical and computational model is
In Sec. 3, the results of a few simulations concerning typical 550 KA
discharges realised on the RFX experiment1 are presented. A comprehensive discussion of these results is presented in Sec. 4, whereas the conclusions of this study are summarized in Sec. 5.
2.Description of the code The Fokker-Planck code used in this work has been originally developed for the study of suprathermal electrons properties in tokamaks13. It solves the equation
eE,, 3f = C(f) + D(f) m 3V,/
df 3t
(D
where f(t,r,v,,,Vj_) is the electron distribution function, averaged on both the cyclotron motion and the guiding center orbit, C(f) is the collision term and D(f) is a diffusive term describing the effect of magnetic field stochasticity and ambipolar electric field.
The
collision term used is the high velocity limit of the Fokker-Planck operator (see, e.g., Réf.
14)
_ 4ne 4 A c n v 3v m~
C(f) =
(2)
where vth=VT/m, 6=arccos(v,//v) and A C is the Coulomb logarithm. In this expression 2 is the effective ion charge number and T and n refer to temperature and density of both ions and electrons, assumed to be equal. It is known that this collision term is not able, in absence of diffusion, to reproduce the classical Spitzer-Harm conductivity3. This is taken into account in the calculation of the current density from the distribution function by adding a corrective term, which represents a fraction of current density carried by low energy due to tail-to-bulk momentum transfer' 4 : these electrons, being highly
electrons
collisional, do not play a significant role in the diffusion process. The radial diffusion term has the form15 1S
D(f) =
LrDyLf
(3)
with f
L=
3
eE, 3
--- 3r ITiV11 3v/;
--
(4)
where DM(r) is the magnetic field lines diffusion coefficient and E A (r) is the ambipolar electric field, arising because of the different diffusion coefficient of electrons and ions. In the following, DM(r) is assumed to be uniform in space, due to the lack of knowledge about its radial profile, whereas the ambipolar field is calculated by imposing the radial particle flux to vanish:
J|v // |D M Lfd 3 v = 0
(5)
In order to simplify the calculations, equation (5) has been solved substituting to the actual electron distribution function the background Maxwellian, thus reducing to the simple relationship
_eE^ = J_dn J^dT T ~ n dr 2T dr
(6)
In fact, the problem of the radial electric field in toroidal machines is excessively complex. As known, predicting just even the sign of the radial electric field in the strong gradient region at the plasma edge may be difficult.
Subtle phenomena, such as the L-H mode
transition in tokamak and stellarators, are believed to be strongly dependent on radial electric fields, although the theoretical understanding of the mechanisms generating such fields is far from being satisfactory17.
For these reasons, even an exact self-consistent
computation of the ambipolar field by means of Eq. (5) might be a poor approximation of the actual radial field and Eq. (6) is preferred in view of its simplicity. While in tokamaks, where most of the magnetic field is externally imposed, equation (1) is generally solved with a fixed E profile (usually flat), here field profiles need to be calculated self-consistently with the distribution function. This is achieved by the following two-stage procedure: a) equation (1) is evolved in time with a fixed E. profile, calculated assuming that the magnetic field obeys the Bessel Function Model (BFM)18, which is the simplest model to describe the RFP configuration. The evolution starts from a Maxwellian and proceeds until stationary suprathermal tails are created, which usually happens in about 40 tc, where tc is the bulk collision time evaluated at the plasma center. The time step used for the evolution is 0.01 T C , ; b) the evolution is then carried on, but every 100 time steps the parallel current density is calculated from the f via its definition
J//=-eJv//fd'v The current density is in turn used to calculate the magnetic field profiles by means of Ampere's law V x B = nj
( s)
neglecting the effect of perpendicular currents. The E. profile is then evaluated and replaced into equation (1), and the time evolution continues for 100 further time steps. This process is iterated until a stationary state is achieved, which usually happens after 20 TC. The distribution functions obtained by this procedure have the property of being self-consistent with the magnetic field profiles, which are, in turn, an output of the code.
The choice of the BFM as starting magnetic field configuration speeds up the convergence of the code to a stationary condition, but does not influence the final result. In fact, simulations similar to those described in this paper have been performed starting from a uniform toroidal field configuration, and no significant change has been found in the results obtained.
Equation (1) is solved on a 64x64x12 grid in the (v,e,r) space, where 9 is the
angle between velocity and magnetic field. The velocity.range covered is 0
