ISSN/1120-5598 U I C -^ " 126' 795" 2 ENEIV ENTE PER LE NUOVE TECNOLOGIE L'ENERGIA E L'AMBIENTE Associazione EURATOM-ENEA sulla Fusione Contributions to the 20th EPS CONFERENCE ON CONTROLLED FUSION AND PLASMA PHYSICS (Lisboa, Portugal, 26-30, July 1993) ENEA • Area Energia, Dipartimento Fusione Centro Ricerche Energia Frascati RT/ERG/FUS/93/06 Manuscript received in,final form on April 1993 Printed on July 1993 This report has been prepared, printed and distributed by: Servizio Studi e Documentazione ENEA, Centro Ricerche Energia Frascati, C.P. 65 • 00044 Frascati, Rome, Italy. Published by ENEA, Direzione Centrale Relazioni, Viale Regina Margherita 125, Rome, Italy CONTENTS M. aorn, G. MADDALUNO, A. SESTERO Similarity Solution for "Plasma Shield" in Hard Disruptions D. PRIGIONE. L. PIERONI AND FTU TEAM p. 9 Density Limit in FTU Tokamak During Ohmic Operation F. ALLADIO. P.BURATTI, P.MICOZZI, O.TUDISCO p. 13 Sawtooth-Like Activity with Double Resonant q^2 Surfaces During Current Diffusion in FTU F. BOMBARDA. G. BRACCO, P. BURATTI. B.ESPOSITO.M. LEIGH EB, p. 17 S. PODDA, O. TUDISCO AND FTU TEAM Density Dependence of Energy Confinement in FTU Ohmic Plasma V.ZANZA p. 21 Measurement of the Local Particle Source Induced by Gas-Puffing in the Frascati Tokamak Upgrade M. CIOTTI, C. FERRO, G.FRAN2ONI, G.MADDALUNO AND FTU TEAM p. 25 Heat Fluxes and Energy Balance in the FTU Machine F. ROMANELLI, F.ZONCA p. 29 The Radial Correlation Length of Electrostatic Turbulence G. FOGACCIA. F. ROMANELLI p. 33 The Linear Threshold of the Internal Kink Mode G.VLAD, S.BRIGUGUO, C.KAR, F.ZONCA, F.ROMANELLI p. 37 Linear and Nonlinear Stability of Toroidal Alfuén Eigenmodes Using an Hybrid Code E. BARBATO, F.DE MARCO, E.FJAEGER. M.D.CARTER. DJ.HOFFMAN. p. 41 F.W.BAITY, R.GOLDFINGER, D.B.BATCHELOR Fast Wave at 433 MHz on FTU by a Folded Waveguide Launcher M.LAPICELLA, I.CONDREA, R.DE ANGELIS, G.MA2ZITELLI p. 45 Conditioning of FTU Cryogenic Vacuum Chamber G.GRANUCCLV. PERICOLI RIDOLFINI p. 49 Scaling Laws and Poloidal Asymmetries in the Scrape-Off Layer of FTU I.CONDREA,R.DE ANGELIS, LGABELLIERI p. 53 Impurity Sources and Impurity Concentrations in FTU SIMILARITY SOLUTION FOR "PLASMA SHIELD" IN HARD DISRUPTIONS M. Ciotti, G. Maddaluno, A. Sestero Associazione EURATOM- ENEA sulla Fusione, Centro Ricerche Energia Frascati, C.P. 65 - 00044 Frascati, Rome (Italy) ABSTRACT The behaviour of the material ablated from the wall during thermal quench is investigated in terms of a similarity solution approach. An application to an ITER- type device is carried out. L INTRODUCTION Reported damages due to hard disruptions are never apparently so large as the total amount of energy involved would allow them to be (see, e.g., Refs [1] and [2]). In the literature, Ref. [3] has been the first paper to attempt a theoretical explanation of such feature, offering the suggestion that wall damage is ultimately determined by the effectiveness of radiative energy transport mechanisms acting within the ablation material. Such a suggestion was actually pursued further only a decade later [4]. Amidst recent work [5-7], Ref. [7] is in particular notable, in that it specifically includes magnetic field effects in describing the dynamics of the ionized ablation cloud (i.e., of the so-called "plasma shield"). Reference [8] has a similar scope, except that it further includes finite-^ effects. Moreover, the latter reference shows that atomic physics and radiation physics features can be modelled in a relatively streamlined fashion without much losing in terms of representativeness. The resulting one-dimensional model system of equations is of such nature as to allow for a similarity solution approach. Then, still within the same paper, the similarity solution equations are further simplified, to the point of eventually allowing the derivation of the final result in closed analytical form. In the present work, we take such procedure actually one step back, and directly address the numerical solution of the system of the similarity equations - thus avoiding the error that has been possibly introduced in Ref. [8] while pursuing further approximations. IL MODEL EQUATIONS In Ref. [8] a one-dimensional mathematical model has been derived and justified, which describes resistive magnetohydrodynamical features and radiative energy transport features within the plasma shield. Such model system of equations writes as follows: £) , „, - + —-V) -(1+ Z)NkT+NkW + (l + Z)NkT— + — - JJJ2= 0 . (2) dt ax / 12 J dr ax Ti5« _ F = - 1.1X1056 —— — . (3) - (4) at ax V =- — — [(l + Z)NkT] U n B2 dx (5) The dependent variables in the above equations are the density N of ions plus neutrals, the temperature T, the radiative energy flux F, the "frozen-in" macroscopic velocity Vp and the "diffusive" macroscopic velocity VD (with the cumulative macroscopic velocity V being defined as Vp-t-Vp). The diamagnetic current density J must be calculated from the derivative of the magnetic field B, which is in turn defined in terms of N and T by the pressure balance relation: 2 B (6) (1 + Z ) NkT + — = 2 where Bo is the vacuum magnetic field. The quantities Z, Z W and q must be expressed as functions of N and T by solving the system of Sana's ionization equilibrium equations (see Ref. [8] for details). In the equations MKS units are used - with however temperatures and single particle energies being measured in eVs. SIMILARITY SOLUTION In Ref. [8] it was shown that by introducing the similarity variables: = x"'tF , (7) V =v"1tV VF 'VF ' the problem simplifies to a system of ordinary differential equations: 1 - V - • (2V- 1) + - | N = 0 (8) Nk -d+Z)T+W T7 + T )[(l+Z)NkTV + F] (9) - 2VF — [(l+Z)NkT] = (10) v 2 -" Bi-^ (11) T15/2 dT F = -2C (12) Z4 Z N2 K where V=VD+V> and C=l.lxl056. Thus, the scope targeted with the present paper was directly addressing the numerical solution of the latter sysyem of equations. To this end, through a further change of variables the system of equations has been cast into canonical form (namely, a form such that anyone equation contains only a single differential term) and then a standard, Runge-Kutta type integration procedure has been applied. Boundary conditions have been prescribed in the form of the requirement of a "physically reasonable" behaviour at either end of the ablation matter cloud (where the simultaneous involvement of both ends has of course called for a suitable iterative approach). In any case, among boundary requirements the controlling role is surely played by power balance at the interface between disrupting plasma and ablation cloud. IV. RESULTS The obtained profiles of the temperature T, radiative energy flux F, ablation plasma "beta" and ablation plasma velocity V are shown respectively in Figs. 1 to 4 - for input data that are representative ot the ITER device, namely: thermal quench 20 —-^ E 15 >^ 10 t 5 Jiat (0 CC 0 2 4 6 8X10-2 0 2 4 6 8X10-2 x(m) x(m) Fig. 1 - Temperature versus spatial Fig. 2 - Radiative flux versus spatial coordinate coordinate I.U £ 0,8 - -§• 400 "O '. "5 0,6 .9 300 | 0,4 .y 200 Q. •/I O E 0,2 S ioo 1 1 1 1 1 1 1 ID 0 2 4 6 8X10-2 0 2 4 t 6 8X10-2 x(m) x(m) Fig. 3 • Plasma shield "beta" versus Fig. 4 • Macroscopic velocity versus spatial coordinate spatial coordinate 8 TABLE I - Comparison of numerical results (first row) with approximate 1 analytical results (second row) (To in eV, zo in mm, Vo in ms' , A in um) To xo Vo Pwl A lev fp* fwl 2.97 73.3 374 0.61 14.9 0.67 0.15 0.18 3.05 52.1 260 1.00 8.32 0.75 0.15 0.10 duration 100 us, average heat load during thermal quench 120 GWm~2, vacuum magnetic field 5 Tesla, wall material graphite. The values plotted are referred to pulse end. The behaviour at earlier times can be inferred from the fact that profiles remain self-similar at all times, while the spatial coordinate changes as t1/2 and the magnitudes of F and V change as t~1/2. Note the marked change of steepness of the profiles near the wall (except for the "beta", that is). This is in part due tò the error that is affecting the model at low temperature. It can be shown, however, that such error does not sensitively affects the conclusions as to the overall plasma shield thickness, and cumulative depth of ablation. Note also that the radiative energy flux has a (shallow) maximum away from the hotter end of the ablation cloud. This can only be explained as the effect of a local energy source - which can indeed be traced, in the model, to the Joule heating produced by diamagnetic currents. In Table I a number of other results are summarized, and compared with the corresponding values obtained instead from the approximate analytical solution derived in Ref. [8]. The data in the Table are referred to pulse end, and the suffix "0" denotes boundary values on the side facing the disrupting plasma - with XQ being thus a measure of the plasma shield thickness. Moreover, Pwj is the value of the plasma shield "beta" near the wall, and A is the produced erosion thickness. Finally, of the total energy that is poured in during the pulse, fev is the fraction that is reflected back into the vessel cavity by blackbody radiation, fwj is the fraction that is transmitted to the solid wall, and fj» is (by difference) the fraction that is deposited inside the plasma shield.