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On ion flow caused by the inductive electric field in a tokamak P. Helander Citation: Phys. Plasmas 8, 4700 (2001); doi: 10.1063/1.1402172 View online: http://dx.doi.org/10.1063/1.1402172 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v8/i10 Published by the American Institute of Physics. Related Articles Transport in the plateau regime in a tokamak pedestal Phys. Plasmas 19, 072512 (2012) Kinetic damping of resistive wall modes in ITER Phys. Plasmas 19, 052502 (2012) Intrinsic rotation with gyrokinetic models Phys. Plasmas 19, 056116 (2012) Experimental measurements of spatial plasma potentials and electron energy distributions in inductively coupled plasma under weakly collisional and nonlocal electron kinetic regimes Phys. Plasmas 19, 033514 (2012) Nonlinear kinetic modeling of stimulated Raman scattering in a multidimensional geometry Phys. Plasmas 19, 056301 (2012) Additional information on Phys. Plasmas Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions PHYSICS OF PLASMAS VOLUME 8, NUMBER 10 OCTOBER 2001 On ion flow caused by the inductive electric field in a tokamak P. Helander EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, United Kingdom 共Received 9 July 2001; accepted 16 July 2001兲 It was recently pointed out that the inductive electric field in a tokamak can give rise to toroidal and poloidal plasma rotation comparable to that sometimes observed in experiments. Here it is shown that the flow velocity of heavy impurity ions, which is normally what is measured, is lower than that of the bulk plasma for rotation produced in this way. If the bulk ions and electrons are in the banana regime, the impurity rotation is at most about two thirds of the bulk plasma rotation and decreases with increasing effective ion charge (Z eff) and distance from the magnetic axis. 关DOI: 10.1063/1.1402172兴 n e ⯝n i . In the experiments reported in Ref. 5 the impurity charge is z⫽19, and as is typical in these and many other experiments, we assume that the bulk ions and electrons are in the banana regime of collisionality while the impurities are collisional. Under the influence of a toroidal electric field each species satisfies the flux-surface averaged 具 ¯ 典 force balance equations2 Understanding the causes of tokamak plasma rotation— both toroidal and poloidal—is important since it sheds light on the transport mechanisms that are operational in the plasma. For instance, transport barriers are frequently observed to be associated with sheared rotation, and it is widely believed that sheared flow is the cause of these barriers. Conversely, plasma transport properties affect the rotation by, e.g., sustaining it against viscous damping. The rotation velocity of each species a in an axisymmetric tokamak is given by an expression of the form 具 B•“•⌸a 典 ⫽ 具 B 共 R a 储 ⫹n a e a E 储 兲 典 , 具 B•“•⌰a 典 ⫽ 具 BH a 储 典 . Va ⫽ a R ˆ ⫹u a B, Following Hirshman and Sigmar, we relate the viscosities ⌸a and ⌰a to the poloidal flux of particles and heat by where a and u a are flux functions, R is the major radius, ˆ the toroidal unit vector, and B the magnetic field.1–3 This is the most general expression for an incompressible flow tangential to flux surfaces. Note that the toroidal rotation has contributions from both a and u a while poloidal rotation is described by u a alone. Standard neoclassical theory relates these quantities to linear combinations of radial density and temperature gradients and 共for a ) the radial electric field. In a recent paper,4 however, it was pointed out that the inductive electric field used to drive Ohmic current also contributes significantly to u a for bulk ions (a⫽i) in a pure hydrogen plasma, and this may be relevant to recent observations in Alcator C-Mod of spontaneous rotation in Ohmic plasmas.5 On the other hand, it is the rotation of impurity ions that is measured in these 共and most other兲 experiments, from Doppler broadening of line radiation, and it is well known that this velocity can be quite different from that of the bulk plasma.6 It is the purpose of this Brief Communication to clarify this point by presenting a calculation which extends that of Ref. 4 to account for the presence of a heavy impurity species in the plasma and, at the same time, simplifies the analysis by using the Hirshman–Sigmar moment formalism.2 From this calculation, it is evident how to make the further generalization to an arbitrary number of impurities of general charges and masses. We thus consider a plasma consisting of electrons (e), hydrogenic bulk ions (i), and heavy impurity ions (z). We treat the impurity charge as a large expansion parameter, z Ⰷ1, and assume that the impurity strength parameter ␣ ⫽Z eff⫺1⫽nzz2/ne is of order unity so that n z z/n e Ⰶ1 and 1070-664X/2001/8(10)/4700/4/$18.00 冔冉 冓 ˆ a1 1 B•“•⌸a ⫽ 2 ˆ a2 具 B 典 B•“•⌰a and 冓 冔 兺冉 1 BR a 储 ⫽ 2 BH B 具 典 a储 ⫽ b ˆ a2 ˆ a3 冊冉 ab l 11 ab ⫺l 12 ab ⫺l 21 ab l 22 冊 u a ⫽Ma •ua 2q a /5p a 冊冉 u b 2q b /5p b 兺b Lab •ub , 冊 共1兲 where q a ⫽qa •“ /B•“ is the contravariant component of the poloidal heat flux and p a ⫽n a T a the pressure of species a. These expressions are obtained by expanding the distribution functions in Sonine polynomials and truncating after two terms; as usual this approximation turns out to be accurate within a few percent, i.e., as accurate as the Coulomb logarithm in the collision operator. In order to isolate the inductive electric field as the single driving term, we have assumed in Eq. 共1兲 that all radial gradients vanish, so that a ⫽0 for each species. 共This assumption is satisfied if the density and temperature gradients are sufficiently weak and, for the gradient of the electrostatic potential, i.e., the radial electric field, if the plasma is viewed from a rigidly toroidally rotating frame of reference.兲 The coefficients ˆ a j ⫽ 3 具共 “ 储B 兲2典 aj 具 B 2典 and 4700 Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions Phys. Plasmas, Vol. 8, No. 10, October 2001 l ab jk ⫽m a n a 冉 j⫺1,k⫺1 M ac ␦ ab 兺 ac c ⫹ On ion flow caused by the inductive electric field . . . j⫺1,k⫺1 N ab ab 冊 Ma •ua ⫽ , which can be looked up in Refs. 2 and 3, summarize all kinetic information needed to evaluate the flows. Neoclassiˆ aj , cal effects are described by the viscosity coefficients which thus depend on collisionality and flux-surface geometry. In the banana regime 冉 冊 冉 冊 冉 冉 冊 3/2 2 2 with ab ⫽3(2 ) 3/2⑀ 20 m 1/2 a T a /n b e a e b ln ⌳, f t the effective 2 fraction of trapped particles, and f c ⫽1⫺ f t . For a tokamak with circular cross section and small inverse aspect ratio, ⑀ Ⰶ1, the trapped particle fraction is f t ⯝1.46⑀ 1/2. The friction coefficients are m in i Lii ⫽⫺ iz 冉 1 3 2 13 & ⫹ 4 ␣ 冉 冊 冉 冊 Lzi ⫽ m in i 1 iz 0 Lie ⫽ m en e 1 ee 0 Lee ⫽⫺ 共 1⫹ ␣ 兲 3 2 0 3 2 0 , 冊 冉 冊 冉 冊 Liz ⫽ Lzz ⫽⫺ m in i 1 iz 0 ⫽ ␣ ⫺1 Lze , m en e ee m in i 1 iz 23 , 冉 1 3 2 3 2 13 & ⫹ 4 1⫹ ␣ 0 0 , 0 ␣ &⫹ 冊 冉 冊 n ae a具 E 储B 典 / 具 B 2典 . 0 For the electrons, this reduces to from which the electron flow velocity can be evaluated by matrix inversion, yielding the usual neoclassical reduction of electric conductivity due to trapping. The ion flows are then obtained by solving the 4⫻4 system of equations ˆ i1 0.533⫹ ␣ mn f ˆ i2 ⫽ i i t ⫺0.625⫺1.5␣ , ii f c ˆ i3 1.386⫹3.25␣ 3 2 Ea ⫽ 兺b Lab •ub ⫹Ea , 共 Me ⫺Lee 兲 •ue ⫽Ee , ˆ e1 0.533⫹ 共 1⫹ ␣ 兲 m en e f t ˆ e2 ⫽ ⫺0.625⫺1.5共 1⫹ ␣ 兲 , ee f c ˆ e3 1.386⫹3.25共 1⫹ ␣ 兲 冊 where 4701 15 2 , 冉 Mi ⫺Lii ⫺Liz ⫺Lzi ⫺Lzz 冊冉 冊 冉 冊 ui Ei ⫺Lie •ue ⫽ , uz ⫺Lze •ue where we have noted that the viscosity is negligible for collisional impurities and that the electric force acting on the impurities is smaller than their friction against electrons, m e j 储 / ␣ ei R ze 储 ⬃ ⬃zⰇ1. n z zeE 储 n z ze 2 E 储 The motion of the impurities is thus simply determined by the balance of the impurity-electron and impurity-hydrogen friction forces. The first element of the right-hand side of Eq. 共2兲 is equal to the sum of the electric force on the bulk ions and their friction against the electron population, and is sometimes written as 具 共 n i eE 储 ⫹R ie 储 兲 B 典 n i e 具 E * B 典 ⫽ , 具 B 2典 具 B 2典 , where zero entries denote quantities of order (m i /m z ) 1/2Ⰶ1 or (m e /m i ) 1/2Ⰶ1. We do not need Lei and Lez as the electron flow velocity is much larger than those of the ion species. As far as the electrons are concerned, the ions are practically stationary and the electron–ion friction force is independent of the ion flow velocity. We thus obtain the following system of equations for each species: 共3兲 where E ⫽E 储 ⫹R ie 储 /n i e is the ‘‘effective electric field’’ felt * by a bulk ion.1 In a pure plasma 共no impurities兲 embedded in a straight magnetic field this vanishes, E ⫽0, and the ions * do not ‘‘feel’’ the electric field; it is exactly cancelled by the friction force from the electrons. The reason for this cancellation is that the electric force acting on the electrons must necessarily equal their friction against ions. As pointed out in Ref. 7, the situation is different in an impure tokamak plasma since the electric force on the electrons is balanced not only by friction against bulk ions, but also by the mirror force 共causing trapping兲 and friction against impurity ions. As a result of these effects, the first row in Eq. 共2兲 equals Eq. 共3兲 with ee ee 兲 e2 ⫺ 共 e3 ⫺l 22 兲 e1 共 e2 ⫺l 12 3.96⫹2.59x⫹ ␣ 共 4.21⫹3.24x 兲 ⫹ ␣ 2 共 1⫹x 兲 ␣ x 具 E *B 典 ⫽ ⫹ , ee ee ee 2 ⯝ 2 1⫹ ␣ 2.59共 0.65⫹x 兲共 1.44⫹x 兲 ⫹ ␣ 共 3.24⫹ ␣ 兲共 1⫹x 兲 1⫹ ␣ 具 E 储 B 典 共 e1 ⫺l 11 兲共 e3 ⫺l 22 兲 ⫺ 共 e2 ⫺l 12 兲 where x⫽ f t / f c . The first term in this expression reflects the fact that the total friction on electrons from both ion species is proportional to Z eff⫽1⫹␣, where the term ␣ represents the contribution from impurities so that, in a straight magnetic field, they take up a fraction ␣ /(1⫹ ␣ ) of the momentum imparted to the electrons by the electric field. The second 共2兲 共4兲 term in Eq. 共4兲 describes the effect of electron trapping. It is clear that for realistic parameters in a tokamak, ␣ ⬃x⬃0.5, the bulk ions ‘‘feel’’ a significant fraction 共more than half兲 of the parallel electric field. We now turn to the ion rotation, which is obtained by solving the system of equations 共2兲. The resulting expres- Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 4702 Phys. Plasmas, Vol. 8, No. 10, October 2001 P. Helander sions are very complicated in general; we give only limiting forms. In the limit of very few impurities, the rotation velocities of bulk and impurity ions, respectively, are û i ⫽ ⯝ û z ⫽ m in i具 B 2典 u e ii 具 E 储 B 典 i 3.98共 1.02⫹x 兲共 1.53⫹x 兲 , 共 0.65⫹x 兲共 1.44⫹x 兲共 2.17⫹x 兲 ␣ →0, m in i具 B 2典 u e ii 具 E 储 B 典 z 0.73共 1.5⫹x 兲共 3.84⫹x 兲 ⯝ , 共 0.65⫹x 兲共 1.44⫹x 兲共 2.17⫹x 兲 ␣ →0. FIG. 1. Ratio of bulk ion and impurity ion poloidal flows as a function of the effective fraction of trapped particles, f t , for different values of the impurity strength parameter ␣ ⫽n z z 2 /n i . When f t or ␣ is large, the bulk ion flow is much larger than that of the impurities. Thus the rotation speed of trace impurities is thus about 2/3 of the plasma rotation near the magnetic axis 共where x ⫽0) and falls off toward the edge. If the impurities are more numerous, their speed is even smaller when compared with that of the bulk ions, as shown in Fig 1. In the Lorentz limit 13/4 , û i ⯝ 1⫹x 兲␣ 共 1 , û z ⯝ 共 1⫹x 兲 ␣ 2 û z ⯝ 3.78⫹3.83␣ ⫹ ␣ 2 , 共 1⫹ ␣ 兲共 2.41⫹ ␣ 兲共 0.75⫹1.95␣ ⫹ ␣ 2 兲 f t →0. ␣ →⬁, ␣ →⬁, and close to the magnetic axis, The impurity rotation is always in the same direction as the bulk ion rotation but is considerably smaller if ␣ 共or, to a lesser extent, f t ) gets large. In contrast, the poloidal rotation caused by radial gradients usually calculated in neoclassical theory does not depend strongly on the impurity content and stays finite in the limit ␣ →⬁. These flows are6 û i ⯝ 3.25共 0.71⫹1.24␣ ⫹ ␣ 2 兲 , 共 1⫹ ␣ 兲共 0.75⫹1.95␣ ⫹ ␣ 2 兲 u i⫽ 1.17⫹3.64␣ ⫹1.99␣ 2 d ln T i IT i , 2 e 具 B 典 1⫹0.46x⫹ 共 2.58⫹1.65x 兲 ␣ ⫹1.33共 1⫹x 兲 ␣ 2 d u z⫽ IT i d ln p i 1.17⫹0.69x⫹ 共 3.64⫹2.47x 兲 ␣ ⫹1.99共 1⫹x 兲 ␣ 2 d ln T i ⫺ ⫹ , e 具 B 2典 d 1⫹0.46x⫹ 共 2.58⫹1.65x 兲 ␣ ⫹1.33共 1⫹x 兲 ␣ 2 d f t →0, 冋 where I⫽RB and is the poloidal flux function, so that B⫽I( )“ ⫹“ ⫻“ . Hence it was noticed in Ref. 6 that in the edge pedestal, where the pressure gradient is very steep, the poloidal rotation of impurities is larger than and in the opposite direction to the bulk ion rotation. In conclusion, our results may be summarized as follows. Considering a tokamak plasma consisting of collisionless electrons and ions and collisional impurities, we have calculated the ‘‘effective electric field’’ 共4兲 acting on the ions and its effect on ion flow. Although this effective electric field vanishes for a pure plasma in a straight magnetic field, this is not at all true in a realistic tokamak discharge. In fact, for typical tokamak parameters E /E 储 ⲏ0.5. As pointed out * in Ref. 4, this electric field drives a poloidal and toroidal ion flow, but as we have seen here the corresponding flow of heavy impurities 共which is what is measured兲 is significantly smaller than that of the bulk plasma. This behavior is the 册 opposite to that of rotation caused in the more orthodox neoclassical way by radial gradients, which tends to be larger for impurities than for bulk ions, making the detection of rotation driven by the Ohmic field correspondingly more difficult. The fact that E /E 储 is so large implies that it is not * unreasonable to expect that ion runaway may sometimes occur in tokamaks, as suggested by Furth and Rutherford.7 During normal tokamak operation the toroidal electric field is too low for this, but just like runaway electrons tend to be produced in tokamak disruptions, one could expect that ion runaway could occur during reconnecting instabilities where the toroidal electric field becomes very large. Work is in progress to assess whether this can explain recent observations of bursts of high-energy ions following internal reconnection events in the Mega-Ampere Spherical Tokamak 共MAST兲.8 Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions Phys. Plasmas, Vol. 8, No. 10, October 2001 ACKNOWLEDGMENTS Helpful discussions with P. J. Catto and R. J. Hastie are gratefully acknowledged. This work was funded jointly by the U.K. Department of Trade and Industry and Euratom. 1 2 F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 共1976兲. S. P. Hirshman and D. J. Sigmar, Nucl. Fusion 21, 1079 共1981兲. On ion flow caused by the inductive electric field . . . 3 4703 P. Helander and D. J. Sigmar, Collisional Transport in Magnetized Plasmas 共Cambridge University Press, Cambridge, in press兲. 4 P. J. Catto, R. J. Hastie, I. H. Hutchinson, and P. Helander, Phys. Plasmas 8, 3334 共2001兲. 5 I. H. Hutchinson, J. E. Rice, R. S. Granetz, and J. A. Snipes, Phys. Rev. Lett. 84, 3330 共2000兲. 6 Y. B. Kim, P. H. Diamond, and R. J. Groebner, Phys. Fluids B 3, 2050 共1991兲. 7 H. P. Furth and P. H. Rutherford, Phys. Rev. Lett. 28, 545 共1972兲. 8 A. Sykes, J.-W. Ahn, R. Akers et al., Phys. Plasmas 8, 2101 共2001兲. Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions