PHYSICS OF PLASMAS
VOLUME 6, NUMBER 10
OCTOBER 1999
Effect of magnetic and electrostatic fluctuations on the runaway electron dynamics in tokamak plasmas J. R. Martı´n-Solı´sa) Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganes, 28911-Madrid, Spain
R. Sa´nchezb) Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8070
B. Esposito Associazione Euratom-ENEA CRE, 00044-Frascati, Italy
共Received 26 January 1999; accepted 9 June 1999兲 A test particle description of the runaway dynamics 关J.R. Martı´n-Solı´s et al., Phys. Plasmas 5, 2370 共1998兲兴 is extended to investigate the behavior of runaway electrons in the presence of fluctuations of electric and magnetic fields. The interaction with the fluctuations is accounted for via a friction force with an effective ‘‘collision’’ frequency determined by the fluctuation induced radial diffusion coefficient. It is shown that both the runaway generation process and the maximum runaway energy can be noticeably affected by magnetic fluctuations. The test particle model is then used to discuss a proposed runaway control scheme via induced magnetic turbulence, with particular emphasis to situations like major disruptions, where a large number of runaway electrons and high runaway energies are expected. It is found that the efficiency of such scheme can in some cases be jeopardized by drift orbit effects as well as by the coexistence of stochastic magnetic regions with good magnetic surfaces. © 1999 American Institute of Physics. 关S1070-664X共99兲03109-2兴
I. INTRODUCTION
derived in Ref. 2, including acceleration in the electric field, collisions with the plasma particles and deceleration due to synchrotron radiation losses. In this paper, these equations are extended to include the effect of the fluctuations via an effective friction force characterized by a ‘‘frequency of collisions with the fluctuations’’ proportional to the fluctuation induced radial diffusion coefficient. The resulting equations are briefly reviewed in Sec. II. The test particle model can be easily applied to a wide variety of interesting problems. In this way, as an example of application, some simple limits of the test equations are used in Sec. III to address the important question of runaway control in tokamaks via magnetic fluctuations. In particular, attention is paid to the problem of runaway control during a disruption in large tokamaks like the Joint European Torus 共JET兲3 or the planned International Thermonuclear Experimental Reactor 共ITER兲.4 The influence on the efficiency of this scheme of both drift orbit effects and the existence of a mixed magnetic topology containing regions of good magnetic surfaces and localized regions of plasma stochasticity is investigated in Sec. IV. Finally, some conclusions are drawn in Sec. V.
It is widely recognized that microturbulent processes play a key role in the anomalous transport observed in tokamak plasmas. Microturbulence leads to intensive electric and magnetic field fluctuations which give rise to fast electron motion across the magnetic field and, therefore, to anomalous radial electron losses. The interaction with the fluctuations is particularly important for runaway electrons, since their Coulomb collision frequency is very small and so their dynamics will be strongly sensitive to the magnitude and nature of the fluctuations. It is, however, of importance to notice that intensive fluctuations of electric and magnetic fields do not only cause anomalous electron losses, depleting the runaway population, but also a substantial change in the runaway electron dynamics. Thus, if the level of the fluctuations is sufficiently high, the electrons may diffuse out before entering into the runaway regime, so that the runaway generation process itself will be altered. Moreover, even when the formation of the runaway population is not affected, fluctuations may determine the runaway lifetime in the tokamak, having an influence on their distribution function and on the maximum energy they can reach. Although for a detailed description of the interaction between the fluctuations and the runaway electrons the kinetic theory should be used,1 it is possible to obtain very useful information by the analysis of a simple test particle model. The basic equations for a test runaway electron were already
II. BASIC EQUATIONS A. Single particle equations
The equations describing the motion of a test particle in momentum space were already obtained in Ref. 2 from the kinetic Fokker–Planck equation for the fast electron distribution, taking into account the effect of synchrotron radiation losses, and following the procedure used in Ref. 5. If the diffusion losses associated with the fluctuations are described
a兲
Electronic mail:
[email protected] Permanent address: Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganes, 28911-Madrid, Spain.
b兲
1070-664X/99/6(10)/3925/9/$15.00
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© 1999 American Institute of Physics
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Martı´n-Solı´s, Sa´nchez, and Esposito
Phys. Plasmas, Vol. 6, No. 10, October 1999
by means of an effective diffusion coefficient D r 共whose particular expression will be given in Sec. II B兲, the resulting system of normalized test equations
冉
冊 冉冊
q⬜2 q 兩兩 dq 兩兩 v ⫽D⫺ ␥ 共 ␣ ⫹ ␥ 兲 3 ⫺ F gc ⫹F gy 4 ␥ 4 d c q q
冉
冊 冉冊
q⬜2 4 v q 兩兩 ␥ 2 dq ⫽D ⫺ 2 ⫺ F gc ⫹F gy 4 ␥ d q c q q
3
⫺
3
q 兩兩 q 兩兩 ⫺ , q dr 共1兲
q , dr
共2兲
differs only from that used in Ref. 2 in the last term of the right hand side 共RHS兲 of both equations, which characterizes the effect of the fluctuations by an effective friction force ⫺q/ dr . The variables appearing in Eqs. 共1兲 and 共2兲 have the following meanings: q 兩兩 , q⬜ , and q are the parallel, perpendicular and total electron momenta normalized to m e c, ␥ is the relativistic gamma factor and v is the electron velocity; ⫽ r t, with r ⫽n e e 4 ln ⌳/4 20 m 2e c 3 , and dr ⫽ r d is the normalized diffusion time, being d the characteristic radial diffusion time, d ⫽a 2 / j 20 D r (a is the minor radius and j 0 the first zero of the Bessel function J 0 ); D⫽E 兩兩 /E R is the normalized electric field, with E R ⫽(kT e /m e c 2 )E D , where E D ⫽n e e 3 ln ⌳/4 20 kT e is the Dreicer field 共note that, along this paper, we will refer to the normalized electric field as D, while the symbol D r will be used for the radial diffusion coefficient兲; ␣ ⫽1⫹Z eff , and F gc , F gy are parameters describing the two contributions to the radiation losses coming from the guiding center motion and the electron gyromotion, respectively, whose definitions can be found in Ref. 2. The first term in Eqs. 共1兲 and 共2兲 is the acceleration due to the toroidal electric field, and the second term includes the effect of the collisions with the plasma particles.6 The third term describes the synchrotron radiation losses,2 while the last term represents the effect of the fluctuations, given by a friction force equivalent to a scattering process with a ‘‘col⫺1 , proportional to D r . lision frequency,’’ eff⫽ dr The equation for the perpendicular normalized momentum can be easily obtained combining Eqs. 共1兲 and 共2兲
冋
冉
冊 冉冊 册
q 兩兩2 q⬜2 4 v dq⬜ 1 2 ⫽ ␥ 共 ␣ ⫹ ␥ 兲 2 ⫺ ␥ ⫺ F gc ⫹F gy 4 ␥ q⬜ d q c q q ⫺
q⬜2
dr
.
3
q⬜2
共3兲
D r⫽
To build a radial diffusion coefficient D r that can effectively model the interaction between the runaway electrons and the fluctuations, the following facts are taken into account: 共a兲 Transport in stochastic magnetic fields, associated with the free streaming of the electrons along stochastic magnetic field of lines, characterized by a radial diffusion coefficient7 共4兲
De , v 兩兩
共5兲
˜ /B 0 is the drift electron velocwhere D e ⫽L 兩兩 v E2 , and v E ⫽E ity induced by the fluctuating poloidal electric field ˜E . 共c兲 The interplay between both types of fluctuations which, following Ref. 8, may be written
D r ⫽D m v 兩兩 ⫹2 共 D m D e 兲 1/2cos␦ ⫹
De , v 兩兩
共6兲
where ␦ is the phase of ˜E with respect to ˜b . 共d兲 Drift corrections to the unperturbed orbits of the runaway electrons when their energy increases can change the effective radial diffusion. At large velocities, the runaway electron orbits depart from the flux surfaces and, if such orbit displacement becomes comparable to the radial scale length of the turbulence, the diffusion coefficient may be altered9–11 D r →⌼D r ,
0⬍⌼⬍1,
共7兲
where ⌼ is called the drift correction factor. 共e兲 In a more general situation, both regions of ‘‘good’’ magnetic surfaces and stochastic regions may coexist inside the plasma. This fact can be brought into the model by considering spatially separated regions of magnetic stochasticity imbedded in the plasma volume. The region with good magnetic surfaces, region I, will be characterized by a transport coefficient dominated by electrostatic turbulence 关Eq. 共5兲兴, whereas both magnetic and electrostatic turbulence exist in region II 关Eq. 共6兲兴. Denoting by ␣ s the fraction of plasma volume of good magnetic surfaces, the effective radial diffusion coefficient can be estimated, following Ref. 12:
D r⫽
B. Radial diffusion coefficient
D r ⫽D m v 兩兩 ,
where v 兩兩 is the parallel electron velocity and D m is the mag˜ is the normalnetic line diffusion coefficient, D m ⫽L 兩兩˜b 2 (b ˜ r /B 0 , and ized radial magnetic fluctuation amplitude, ˜b ⬅B L 兩兩 ⬇ q 0 R 0 is the parallel correlation length of the magnetic field fluctuations, with q 0 the safety factor兲. 共b兲 Transport driven by electrostatic fluctuations, which shows an inverse dependence on the electron velocity8
D I D II . ␣ s D II ⫹ 共 1⫺ ␣ s 兲 D I
共8兲
Thus, the effective radial diffusion coefficient used throughout this paper has the general form
D r ⫽⌼
D e D e ⫹2 共 D m D e 兲 1/2cos ␦ v 兩兩 ⫹D m v 兩兩2 , v 兩兩 D e ⫹ ␣ s 关 2 共 D m D e 兲 1/2cos ␦ v 兩兩 ⫹D m v 兩兩2 兴
共9兲
and the associated normalized diffusion time
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Phys. Plasmas, Vol. 6, No. 10, October 1999
dr ⫽
Effect of magnetic and electrostatic fluctuations on the . . .
dm 共 ␥ 2 ⫺1 兲 1/2 A 2 ␥ 2 ⫹ ␣ s 关共 ␥ 2 ⫺1 兲 cos2 ⫹2A ␥ 共 ␥ 2 ⫺1 兲 1/2cos ␦ cos 兴 cos , ⌼ ␥ A 2 关 A 2 ␥ 2 ⫹2A ␥ 共 ␥ 2 ⫺1 兲 1/2cos ␦ cos ⫹ 共 ␥ 2 ⫺1 兲 cos2 兴
where v 储 ⫽ v cos . For normalization purposes, the quotient A⬅( dm / de ) 1/2 has been introduced, where dm ⬅a 2 r / j 20 D m c and de ⬅a 2 r c/ j 20 D e are, respectively, the characteristic normalized magnetic and electrostatic diffusion times for electrons with v 储 ⫽c. C. Singular points
The essential features of the phase-space structure of the relaxation equations, already described in Ref. 2 for D r ⫽0, are retained when the diffusive friction force associated with the fluctuations is included in the test equations. Hence, two singular points exist in momentum space with a well defined physical meaning: a saddle point, which gives the critical energy for runaway generation, ␥ c , and a stable focus, giving the energy limit for the generated runaway electrons, ␥ l . The precise value of such singular points and energies may be substantially affected by consideration of the diffusive process. An analytical relation between them and the normalized electric field can be obtained considering that the singular points lie at the intersection of the two contours q˙ 兩兩 ⫽0 and q˙⬜ ⫽0 共or q˙ ⫽0). Thus, from q˙ ⫽0, D⫽
␥ s2 cos s 共 ␥ s2 ⫺1 兲
再
共 ␥ s2 ⫺1 兲 3/2 2 共 ␥ s2 ⫺1 兲 5/2 ⫻ 1⫹F gy sin s ⫹F gc
⫹
共 ␥ s2 ⫺1 兲 1/2
cos s dr
␥s
␥s
,
冎 共11兲
3927
共10兲
cause on the first wall structures. To begin with, it is assumed that the radial diffusion process is only determined by the free streaming of the particles along stochastic magnetic field lines 关i.e., the limits ␣ s →0, ⌼→1 and D e →0 of Eq. 共9兲 are considered兴. Examination of the limitations to the efficiency of this control scheme due to drift orbit and magnetic topology effects are postponed until Sec. IV. A. Runaway generation
The good properties of the stochastic magnetic fluctuations as a runaway control mechanism are clearly illustrated in Fig. 1 in which, for given plasma parameters and different levels of magnetic fluctuations, the relation 共11兲 between D and the electron energy ␥ s at the singular points is plotted. For each value of D, branch I in Fig. 1 gives the energy at the saddle point, while branch II gives the energy at the stable focus. The two branches are separated by a minimum which gives the normalized critical electric field, D R , for runaway generation.2 As it is shown in the figure, the value of this minimum increases with the level of fluctuations and, therefore, for a sufficiently large fluctuation level, electrons would escape from the plasma flowing along the stochastic magnetic field lines before relativistic energies could be reached. The above results suggest the possibility of avoiding the generation of runaway electrons by increasing the critical electric field via excitation of magnetic turbulence. Such a method has indeed been proposed, using magnetic coils to disturb the magnetic structure and provide a loss channel for the runaway electrons to reduce the runaway damage to the
with the cosine of the pitch angle at the singular point, cos s , from q˙⬜ ⫽0, determined by the solution of the algebraic equation a dr cos 7 s ⫹b dr cos6 s ⫹c dr cos5 s ⫹d dr cos4 s ⫹e dr cos3 s ⫹ f dr cos2 s ⫹g dr cos s ⫹h dr ⫽0,
共12兲
The coefficients a dr to h dr are given in the Appendix. Note that the two singular point energies, ␥ c and ␥ l , can be obtained as the two roots of ␥ s when solving Eq. 共11兲 for a fixed value of D. Explicit expressions for ␥ l , analogous to those obtained in Ref. 2, can also be derived under proper simplifying conditions ( ␥ 2l Ⰷ1 and ␥ l Ⰷ ␣ /D兲. III. RUNAWAY CONTROL VIA STOCHASTIC MAGNETIC TURBULENCE
The test particle model introduced in Sec. II is now used to address the problem of runaway control in tokamak discharges via magnetic fluctuations. This issue is particularly important in the case of disruption generated runaway electrons, due to the damage that these energetic electrons might
FIG. 1. Normalized electric field vs ␥ s at the singular points: full line, no ˜ ⫽0); dashed line: ˜b ⫽10⫺5 ; dotted line: magnetic fluctuations assumed (b ˜b ⫽2⫻10⫺5 . Plasma parameters: B 0 ⫽3 T, R 0 ⫽3 m, a⫽1 m, n e ⫽0.5 ⫻1019 m⫺3 , Z eff⫽3.
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Martı´n-Solı´s, Sa´nchez, and Esposito
Phys. Plasmas, Vol. 6, No. 10, October 1999
FIG. 2. Critical electric field vs radial magnetic fluctuation level for typical parameters (Z eff⫽3, n e ⫽1020 m⫺3 , T e ⫽5 eV) during disruptions in JET (B 0 ⫽3 T, R 0 ⫽3 m) and ITER (B 0 ⫽5.68 T, R 0 ⫽8.14 m).
FIG. 3. Minimum fluctuation level ˜b min required to block the runaway energy at a given value ␥ l vs the normalized electric field. Full line: ␥ l ⫽3; dashed line: ␥ l ⫽5; dotted line: ␥ l ⫽10. Dimensionless parameters are: ␣ ⫽4, F gy ⫽0.65, and F gy ⫽2.33⫻10⫺8 .
2 ˜b min ⫽
machine following a major disruption, and observations of runaway generation suppression due to enhanced magnetic fluctuations have been reported during disruptive discharge terminations in the Japan Atomic Energy Research Institute Tokamak-60 Upgrade 共JT-60U兲13. The test equations allow to extract useful information in a very straightforward way about this process. In Fig. 2, the critical electric field for runaway generation is plotted as function of the magnetic fluctuation level ˜b for typical plasma conditions during a disruption (Z eff⫽3, n e ⫽1020 m⫺3 , T e ⫽5 eV) in JET (R 0 ⫽3 m, a⫽1 m, B 0 ⫽3 T) and ITER (R 0 ⫽8.14 m, a ⫽2.80 m, B 0 ⫽5.68 T). It is found that a magnetic fluctuation level above ˜b ⬎10⫺3 is required to block runaway generation during a disruption 共critical electric field E R ⭓10 V/m). Notice also that, above a threshold value for the fluctuation amplitude, the diffusion along the magnetic field lines dominates and the critical electric field increases linearly with ˜b .
B. Energy limit
A second way of reducing the runaway damage is by controlling the maximum energy that the generated runaway electrons can gain. This energy is given by ␥ l , the value of ␥ at the stable focus of the test equations. Magnetic fluctuations can be extremely effective in this task, as readily seen from the large sensitivity of branch II in Fig. 1 to the level of the fluctuations. The level of fluctuations required to keep the runaway energy below a certain value ␥ l under given electric field and plasma conditions can be easily estimated within the test particle model. Such a level of fluctuations, ˜b min , from Eq. 共11兲, will be given by
a 2 r␥ l j 20 L 兩兩 c 共 ␥ 2l ⫺1 兲
冋
再
D⫺
␥ 2l cos l 共 ␥ 2l ⫺1 兲
共 ␥ 2l ⫺1 兲 3/2 2 共 ␥ 2l ⫺1 兲 5/2 ⫻ 1⫹F gy sin l ⫹F gc
␥l
␥l
册冎
,
共13兲 where cos l may be written as function of ␥ l and D 关see Appendix in Ref. 2, Eq. 共A5兲兴, cos l ⫽⫺
冋 冉
␣ ␥l ␣ ␥l ⫹ 1⫹ 2 2 2D ␥ l ⫺1 2D ␥ l ⫺1
冊册
2 1/2
.
共14兲
In Fig. 3, ˜b min is plotted as function of D for ␥ l ⫽3,5,10, and plasma parameters as those given in Fig. 1. In the absence of fluctuations, the maximum value D 0 allowed for the electric field in order to keep the runaway energy below ␥ l is given by Eq. 共11兲 with ˜b ⫽0. As the fluctuation level increases, the allowed value for D increases slowly until the fluctuation amplitude reaches a value sufficiently large to provide an effective friction force that can balance the acceleration in the electric field 共see Fig. 3兲. From this point on, the magnetic fluctuations will play a dominant role in limiting the runaway energy and ˜b min , for a given value of D, can be approximated equating the accelerating force in the toroidal electric field to the fluctuation induced friction force: D⬃
␥ 2l ⫺1 ˜ min⬃ ⇒b dm ␥ l
冑
a 2 r
␥ lD
j 20 L 兩兩 c
␥ 2l ⫺1
.
共15兲
As an application of these results, simple estimates can be done of the level of fluctuations required to control the runaway energy during a disruption in JET and the planned ITER project. Thus, for typical conditions during a disruption in JET and ITER, such as those given in Sec. III A, a normalized fluctuation amplitude ˜b ⬃(4⫺10)⫻10⫺4 in JET and ˜b ⬃(6⫺20)⫻10⫺4 in ITER would be enough to limit the runaway energy to 1 MeV for electric fields during the disruption in the range from 10 to 100 V/m.
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Phys. Plasmas, Vol. 6, No. 10, October 1999
Effect of magnetic and electrostatic fluctuations on the . . .
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IV. LIMITATIONS ON RUNAWAY CONTROL EFFICIENCY
The efficiency of the magnetic fluctuations in blocking the runaway generation and/or the maximum runaway energy can be modified due to different reasons. In this section, the limitations on this efficiency introduced by drift orbit effects and the existence of mixed magnetic topologies, with regions of magnetic stochasticity separated by regions of good magnetic surfaces, are studied within the test model. A. Drift effects
The drift of the runaway orbit away from the magnetic surface, which is an increasing function of the electron energy, can partially average out the effect of the magnetic turbulence, and reduce the effectivity of the control mechanisms described in Sec. III. The importance of this reduction can be quantified removing the constraint ⌼⫽1 from the previous analysis. The drift modification factor, ⌼, was first introduced in Ref. 9 and has been since then calculated for different types of turbulence.11 In the present paper, only the case of randomly phased 共RPA兲 perturbations is considered since there are reasons to doubt that drift effects associated to poloidally localized turbulence would be observable.11 For RPA turbulence, a rigorous calculation shows that when the outward drift d r of the runaway orbit, given by d r共 ␥ 兲 ⫽
冉
冊
m e q 0 ␥ 2 v⬜2 , v ⫹ eB 0 v 兩兩 兩兩 2
共16兲
exceeds the radial width of the turbulence, ⌼ obeys the scaling ⌼⬃ /d r and can be estimated as11
⌼⬃
再
1
if d r ⬍ , ⌼0
2cos
共 ␥ 2 ⫺1 兲 1/2 1⫹cos2
otherwise,
共17兲
where ⌼ 0 ⫽ eB 0 /m e q 0 c. Therefore, for given plasma parameters, the range of values for which there will be an averaging effect on the critical energy for runaway generation, ␥ c , will be given by ⬍ c ⬅d r ( ␥ cb˜ ), where ␥ cb˜ is the critical energy for runaway generation obtained assuming d r ⫽0. Analogously, the range in which the limiting runaway energy ␥ l would be affected will be ⬍ l ⬅d r ( ␥ lb˜ ), being ␥ lb˜ the runaway energy limit for d r ⫽0. Figure 4共a兲 illustrates, as function of the normalized electric field D and for the same plasma parameters as in Fig. 1, the values for both c and l . It is shown that, at the critical energies for runaway generation, the drift orbit displacement d r 共and therefore, c ) is small so that no measurable averaging effects are expected on ␥ c for the values of commonly found in the experiment 共on the order of mm or cm兲, unless the electric field is close to its critical value D R as illustrated in the figure. The expected effect on D R will be equally small. Nevertheless, noticeable effects may be found on the runaway energy limit as in this case the
FIG. 4. 共a兲 Critical values below which drift orbit effects are found on the critical runaway energy ( c ) and on the runaway energy limit ( l ) versus the normalized electric field; 共b兲 Runaway energy limit as function of the radial mode width of the turbulence for D⫽10 and ˜b ⫽10⫺5 . The critical value l is also indicated. Dimensionless plasma parameters: ␣ ⫽4, F gy ⫽0.65, F gc ⫽2.33⫻10⫺8 .
energies are large and so d r , reducing the effect of the magnetic fluctuations on the runaway energy and leading to ␥ l ⬎ ␥ lb˜ . In Fig. 4共b兲, and for the same conditions as in Fig. 3, the effect on the limiting runaway energy ␥ l of the orbit averaging effects is shown as function of the radial width of the turbulence. For low values of , drift effects are important (d r Ⰷ ) and in the limit →0, magnetic fluctuation effects are neutralized and the runaway energy limit in absence of ˜ ⫽0) is recovered; when inmagnetic fluctuations (b creases, the drift effects 共and so the limiting runaway energy兲 diminish, until they disappear for ⬎ l , ␥ l being then fully determined by the magnetic field stochasticity at the given fluctuation level ˜b . As, due to drift effects, ␥ l becomes larger than expected, the efficiency of the magnetic fluctuations to control the runaway energy will be reduced. The corrected minimum level of fluctuations required to keep the runaway energy below a value ␥ l under given electric field and plasma conditions, assuming d r ( ␥ l )⬎ , is obtained analogously to Eq. 共13兲, yielding
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Martı´n-Solı´s, Sa´nchez, and Esposito
Phys. Plasmas, Vol. 6, No. 10, October 1999
FIG. 5. 共a兲 Comparison between the minimum fluctuation level ˜b min required to block the runaway energy at ␥ l ⫽5 with and without including drift effects ( ⫽1 mm), vs the normalized electric field; 共b兲 ˜b min as function of for D⫽10 and ␥ l ⫽5. The critical l value for drift effects is indicated. Plasma parameters are the same as in Fig. 4.
FIG. 6. 共a兲 Normalized critical electric field for runaway generation in a mixed magnetic topology with ˜b ⫽5⫻10⫺5 , v E ⫽103 m/s as a function of the fraction ␣ s of good magnetic surfaces; 共b兲 Runaway energy limit vs ␣ s for D⫽10, ˜b ⫽5⫻10⫺5 , and v E ⫽103 m/s. Dimensionless plasma parameters: ␣ ⫽4, F gy ⫽0.65, F gc ⫽2.33⫻10⫺8 .
B. Mixed Magnetic Topologies drift ˜ ˜b min ⫽b min⌼ 共 ␥ l 兲 ⫺1/2 .
共18兲
drift is increased by a factor ⬃⌼( ␥ l ) ⫺1/2. For Notice that ˜b min drift becomes randomly phased turbulence, ⌼⬃⌼ 0 / ␥ l , and ˜b min essentially insensitive to the precise value of ␥ l . In Fig. 5共a兲, as an example, it is presented a comparison between ˜b min calculated without including drift effects 关Eq. drift obtained assuming orbit averaging effects in 共13兲兴 and ˜b min the RPA approximation 关Eq. 共18兲兴 with ⫽1 mm, as function of D for ␥ l ⫽5. It is shown that, due to the averaging effects, larger ˜b values are needed to keep the runaway endrift 共RPA ergy below a given ␥ l level. The dependence of ˜b min ⫺1/2 ˜ approximation兲 on the radial mode width, b min⬃ , is illustrated in Fig. 5共b兲, in which ˜b min is plotted versus for typical conditions during disruptions in JET and ITER, electron energy ⬃1 MeV, and E 兩兩 ⫽10 V/m. For ⬎d r ( ␥ l ) 关⬃3.5 mm in JET and 1.8 mm in ITER for the conditions given in Fig. 5共b兲兴, drift effects disappear and the stochastic case is recovered, so that ˜b min no longer depends on .
The general picture described in Sec. III can also be importantly modified when mixed magnetic topologies are considered, in which regions of magnetic stochasticity are intertwined with regions of good magnetic surfaces. Runaway electron confinement shows a large sensitivity to the radial extent of the magnetic stochasticity regions because energetic electrons have much smaller transport in the regions of robust magnetic surfaces. Thus, a noticeable dependence of the runaway behavior on the fraction ␣ s of good magnetic surfaces is to be expected. To study this situation using the test particle model, ␣ s ⫽0 will be assumed in the transport coefficient D r . For the sake of simplicity, all drift and correlation effects will be neglected ( ␦ ⫽0, ⌼⫽1兲. Figure 6 illustrates the dependence on ␣ s of the critical electric field for runaway generation 关given by the condition dD/d ␥ s ⫽0, Fig. 6共a兲兴 and the runaway energy limit ␥ l 关Fig. 6 共b兲兴. It is clear that even small regions of good magnetic surfaces may be enough to reduce by a significant amount the effect of the magnetic stochasticity on the runaway dynamics: when ␣ s increases 共i.e., the region of good magnetic surfaces兲 the effective runaway diffusion coefficient diminishes and, correspondingly, the critical electric field for run-
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Phys. Plasmas, Vol. 6, No. 10, October 1999
Effect of magnetic and electrostatic fluctuations on the . . .
a r 2
2 ˜b min ⬃
j 20 L 兩兩 c de
冋 冉 冋 冉
␣ F gy ␥ l ⫹F gc ␥ 4l ⫺ ␥ l D . ␣ F gy 4 D⫺ 1⫹ ␥ l ⫹F gc ␥ l D 共21兲
de D⫺ 1⫹ ␥ l ⫺ ␣ s de
冊册
3931
冊册
Thus, if the denominator in 共21兲 goes to zero, then ˜b min→⬁. This will occur for D⫽D ␣ ⯝
1⫹F gc ␥ 4l ⫹ 共 ␥ l / ␣ s de 兲
再 冋
2
⫻ 1⫹ 1⫹ FIG. 7. Minimum fluctuation level ˜b min required to block the runaway energy at ␥ l ⫽10 versus the normalized electric field in a mixed magnetic topology with v E ⫽103 m/s and ␣ s ⫽0.02. The critical field D ␣ beyond which the energy blocking at ␥ l is not possible is also indicated. Plasma parameters: ␣ ⫽4, F gy ⫽0.65, F gc ⫽2.33⫻10⫺8 .
away generation decreases and the limiting runaway energy increases with respect to the fully stochastic case ( ␣ s ⫽0). It is also expected that, given the large neutralization effect that regions of good magnetic surfaces may have on the radial diffusion driven by the magnetic fluctuations, a remarkable reduction will take place on the efficiency of the magnetic field stochasticity to control the runaway energy. The minimum level of magnetic fluctuations ˜b min needed to keep the runaway energy below a value ␥ l , under given plasma conditions, electric field D and fraction ␣ s of plasma volume of good magnetic surfaces, can be obtained 关using Eq. 共11兲兴 from 2 ˜b min ⫽
a 2 r ␥ 2l
de cos l B min⫺ ␥ l , 共19兲 j 20 L 兩兩 c de 共 ␥ 2l ⫺1 兲 cos2 l ␥ l ⫺ ␣ s de cos l B min
with cosl given by Eq. 共14兲 and B min , B min⫽D cos l ⫺ ⫹F gc
␥ 2l ␥ 2l ⫺1
共 ␥ 2l ⫺1 兲 5/2
␥l
册
冋 .
1⫹F gy
共 ␥ 2l ⫺1 兲 3/2 2 sin l
␥l
共20兲
4 ␣ F gy ␥ l 共 1⫹F gc ␥ 4l ⫹ 共 ␥ l / ␣ s de 兲兲 2
册冎 1/2
. 共22兲
Nevertheless, in most of the cases, we will be interested in situations in which the runaway dynamics is dominated by the fluctuations and the acceleration in the electric field, being the role played by the synchrotron radiation losses negligible, so that Eq. 共22兲 may be simplified to D ␣⬃
␥l . ␣ s de
共23兲
The above result 共i.e., that if ␣ s ⫽0, for D⬎D ␣ , even the largest levels of ˜b will not be able to keep the electron energy below the value ␥ l ) may be understood from the following considerations: When ˜b min is large, the time spent in the stochastic regions will be close to zero, and the radial diffusion time will be dominated by the regions of good magnetic surfaces, dr ⬃ ␣ s de . Hence, if the electric field force is larger than the diffusive friction force, D ⬎ ␥ l / ␣ s de , the magnetic turbulence will not be efficient enough to keep the electron energy below ␥ l . On the other hand, this could have dramatic consequences on the ability of the magnetic turbulence to control the runaway energy during disruptions as the electric field is large and so very small fractions of good magnetic surfaces ( ␣ s ⬎ ␥ l /D de ) could be sufficient to avoid the blocking of the energy below ␥ l . Thus, for typical parameters during JET and ITER disruptions, for electric fields E 兩兩 ⬃10⫺100 V/m and v E ⫽103 m/s, we find that values of ␣ s as low as ␣ s ⬃(3.5 ⫺35)⫻10⫺4 in JET and ␣ s ⬃(1.4⫺14)⫻10⫺4 in ITER would be enough to ensure that the runaway energy would be larger than 50 MeV even for the largest magnetic fluctuation levels. V. CONCLUSIONS
Figure 7 shows ˜b min as function of D, calculated using Eq. 共19兲, for a case with ␣ s ⫽0.02, ␥ l ⫽10, and a level of electrostatic turbulence v E ⫽103 m/s. The results illustrated in the figure indicate substantial qualitative differences with respect to the cases considered in Secs. III B and IV A: for mixed magnetic topologies ( ␣ s ⫽0) we find a critical electric field D ␣ beyond which is not possible to block the runaway energy at ␥ l 共i.e., ˜b min→⬁). A simple estimate of such electric field D ␣ may be obtained for the conditions ␥ 2l Ⰷ1, ␥ l Ⰷ ␣ /D (cos2 l⯝1). In this case, the relation 共19兲 for ˜b min may be written
In this paper, the single particle equations introduced in Ref. 2 to describe the dynamics of relativistic test runaway electrons have been extended to include the interaction of the runaway electrons with electrostatic and magnetic fluctuations. This interaction has been described by a friction force Fd ⫽⫺p/ d ,where d is a characteristic radial diffusion time associated with the fluctuations. The test equations provide a simple description of the runaway dynamics that can however be applied to a number of interesting problems. As an example, the examination of the general properties of a proposed control mechanism for runaway electron mitigation
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3932
Martı´n-Solı´s, Sa´nchez, and Esposito
Phys. Plasmas, Vol. 6, No. 10, October 1999
via magnetic turbulence has been addressed and the consequences for disruption generated runaway electrons discussed. At the same time, it has been possible to estimate the reduction in the efficiency of this mechanism due to both runaway orbit drifts and the presence of mixed magnetic topologies. Within the purely magnetic turbulence picture two ways of controlling the runaway electron population have been analyzed: 共1兲 Suppression of runaway generation via increasing the critical electric field, and 共2兲 control of the maximum energy that the runaway electrons can gain. The first mechanism has been found to require ˜b ⬎10⫺3 for typical plasma conditions during a disruption in JET and ITER, and is almost insensitive to drift orbit effects. Regarding the second mechanism, values for ˜b ⬃(4⫺10)⫻10⫺4 in JET and ˜b ⬃(6⫺20)⫻10⫺4 in ITER should be enough to limit the runaway energy to 1 MeV for electric fields during disruptions between 10 and 100 V/m. Nevertheless, drift orbit effects will increase the minimum fluctuation level ˜b min required to block the electron energy below a given value ␥ l by a factor ⬃⌼( ␥ l ) ⫺1/2 with respect to the driftless case. Finally, it has been shown that, in mixed magnetic topologies, very small fractions ␣ s of plasma volume with good magnetic surfaces may reduce dramatically the efficiency of magnetic field stochasticity as a runaway control mechanism. Thus, for sufficiently large electric fields, as the electron transport becomes dominated by the regions of good magnetic surfaces, the magnetic fluctuations will not be able to block the electron energy below a given energy ␥ l , even at the largest levels of ˜b . This might be important during disruptions, as the electric field is then high, so that very low fractions ␣ s of regions of good magnetic surfaces could prevent the magnetic fluctuations from efficiently controlling the runaway production and their final energy. After showing the wide applicability and simplicity of the runaway test model, it is now fair to mention some of its shortcomings. As already pointed out in Ref. 12, most of them result from the expression assumed for the runaway diffusion coefficient, D r . Equation 共9兲 does not include a number effects, such as the possible existence of coherent magnetic islands, which provide virtually no insulation against radial transport, or time dependent processes, like growing islands, which could enhance the efficiency of the runaway mitigation scheme by leading to a somewhat larger transport.
APPENDIX: CALCULATION OF THE PITCH ANGLE AT THE SINGULAR POINTS
The equation giving the pitch angle value s at the singular points is obtained from q˙⬜ ⫽0 in Eq. 共3兲. The result is an algebraic equation in cos s from which the pitch angle at the singular points may be derived a dr cos7 s ⫹b dr cos6 s ⫹c dr cos5 s ⫹d dr cos4 s ⫹e dr cos3 s ⫹ f dr cos2 s ⫹g dr cos s ⫹h dr ⫽0, with the coefficients a dr to h dr given by a dr ⬅ ␣ s , b dr ⬅2 ␣ s A cos ␦ ␥ s 共 ␥ s2 ⫺1 兲 ⫺1/2, c dr ⬅A 2
␥ s2 ␥ s2 ⫺1
⫺
␣s ␣⫹␥s ␣ s F gc 2 ⫺ 共 ␥ s ⫺1 兲 F gy 共 ␥ s2 ⫺1 兲 3/2 F gy
⫺2 ␣ s , ⌼A 2 2 ␣ s Acos␦ ␥ s 共 ␣ ⫹ ␥ s 兲 共 ␥ s2 ⫺1 兲 ⫺1/2⫺ dm F gy F gy 共 ␥ s2 ⫺1 兲 2
d dr ⬅⫺
2 ␣ s A cos ␦ F gc ␥ s 共 ␥ s2 ⫺1 兲 1/2 F gy
⫺
⫺4 ␣ s A cos ␦
␥s , 2 共 ␥ s ⫺1 兲 1/2
A 2 ␥ s2 共 ␣ ⫹ ␥ s 兲 A 2 F gc 2 2⌼A 3 cos␦ ␥ s ⫺ ␥ ⫺ e dr ⬅⫺ dm F gy ␥ s2 ⫺1 F gy 共 ␥ s2 ⫺1 兲 5/2 F gy s ⫺2A 2
␥ s2 ␥ s2 ⫺1
⫹
␣s ␥s ␣ s F gc 2 ⫹ 共 ␥ s ⫺1 兲 2 F gy 共 ␥ s ⫺1 兲 3/2 F gy
⫹␣s , f dr ⬅⫺
␥ s2 ⌼A 4 ⌼A 2 ⫹ 共 ␥ 2 ⫺1 兲 ⫺1/2 dm F gy 共 ␥ s2 ⫺1 兲 3/2 dm F gy s
␥ s2 2 ␣ s AF gc cos ␦ 2 ␣ s Acos␦ ⫹ ⫹ 2 2 F gy F gy 共 ␥ s ⫺1 兲 ⫻ ␥ s 共 ␥ s2 ⫺1 兲 1/2⫹2 ␣ s A cos ␦
ACKNOWLEDGMENTS
This work was done under financial support from Direc˜ anza Superior 共DGES兲 Project No. cio´n General de Ensen PB96-0112-C02-01. Research supported in part 共R.S.兲 by an appointment to the ORNL Postdoctoral Research Associates Program administered jointly by Oak Ridge National Laboratory and the Oak Ridge Institute for Science and Education.
共A1兲
g dr ⬅
␥ s3 ␥ s2 A2 A 2 F gc 2 2 ⫹ ␥ ⫹A s F gy 共 ␥ s2 ⫺1 兲 5/2 F gy ␥ s2 ⫺1 ⫹
h dr ⬅
␥s , 2 共 ␥ s ⫺1 兲 1/2
2⌼A 3 cos␦ ␥ s , dm F gy ␥ s2 ⫺1
␥ s2 ⌼A 4 . dm F gy 共 ␥ s2 ⫺1 兲 3/2
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Phys. Plasmas, Vol. 6, No. 10, October 1999 1
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