Belli, E.

Braun, S.

Candy, J.

Helander, P.

MPG.PuRe

Collisional zonal-flow damping in an impure tokamak plasma

Belli, E. A.

Max Planck Society eDoc Server

Collisional zonal-flow damping in an impure tokamak plasmaS. Braun1, P. Helander1, E.A. Belli2, J. Candy21 Max-Planck-Institute for Plasma Physics, Greifswald, Germany2 General Atomics, San Diego, CaliforniaZonal flows are widely known to play an important role in improving tokamak plasma confine-ment by shearing apart turbulent eddies, thus reducing the level of turbulent transport in theplasma. Driven by turbulent Reynolds stress themselves, these flows may be sensitive to othermechanisms which affect either the turbulence itself or the zonal flows. One of these mech-anisms is friction due to Coulomb collisions between circulating and trapped particles whichslow down the poloidal rotation of the plasma. In this work, we study the effect of heavy,highly charged impurities on the zonal flows. Being much more collisional than the lighter bulkions, even small amounts of heavy impurities speed up the damping process by increasing the’effective’ collision frequency of the plasma, although their rotation itself hardly contributes tothe flow [1]. The main focus of this work is on determining how fast this damping process oc-curs, and an analytic formula is derived for the damping timeτp =∞∫0(Er(t)/Er(∞) − 1) dt,which describes the time scale on which the zonal-flow potential approaches its residual value.The analytical results are then compared with the numerical results of the NEO code [2].Following earlier papers by Hinton, Rosenbluth [3] and Xiao, Catto, Molvig [4], the zonal-flow potential is modelled by an initially imposed short-wavelength radial electric field, andthe long-time response (i.e. on time scales longer than a few ion bounce times) of the plasmais calculated. The result can then be generalised to arbitrary source terms by convolution withthe initial-value solution.While the ion distribution function is governed by the drift-kinetic equation, the radial electricfield has to adjust according to the constraint of conservation of angular momentum. Thus, wefirst solve∂fa∂t+ (v‖ + vd) · ∇fa + ẇ∂fa∂w= Ca(fa)for both ion and impurity distribution functions,fi andfz, respectively. Here,vd is the drift ve-locity,w = 12mav2 the kinetic energy and the radial electric field is contained inẇ = −ea(v‖ +vd) · ∇φ, v‖ denoting the velocity along the magnetic fieldB = I(ψ)∇ϕ + ∇ϕ× ∇ψ. Allparticle species (electrons, bulk ions and impurities) are assumed to be in the low-collisionalbanana regime, and we orderZ  Z2nz/ni ∼ Zmi/mz ∼ 1, which allows us to neglectimpurity-ion collisions as their effect is much smaller than the effect of impurity self-collisions.36th EPS Conference on Plasma Phys. Sofia, June 29 - July 3, 2009 ECA Vol.33E, P-4.108  (2009)Self-collisions are modelled byCaa(fa1) = νaaD(L(fa1) +mav‖ua‖fa0/Ta)and impurity-ioncollisions byCiz(fi1) = νizDL(fi1) + miv‖fi0νizDVz‖/Ti, whereνabD denotes the collision fre-quency of speciesa andb, L is the Lorentz scattering operator,Vz‖ the impurity mean flow-velocity andua can be calculated to satisfy momentum conservation in self-collisions.The drift-kinetic equation is then expanded in the smallness of the Larmor radiusδ and theresulting1st order equation is solved. For convenience, we split off the adiabatic part of the1storder distribution function and writefa1 = ga − Iv‖eaφ′fa0/(ΩaTa). There are two differentways of solving the partial differential equation: if one is interested in the complete time evo-lution of the zonal-flow potential, the equation can be solved via an eigenfunction expansionof the bounce averaged self-collision operator. However, the resulting formula for the dampingtime is not very intuitive, and in order to have sufficient accuracy it is advisable to calculatethe eigenfunctions numerically. Another option is to consider only the long-time behaviour ofthe system, as in this limit the time derivative ofga can be shown to be small compared to theother terms and can thus be neglected. Consquently, it is only necessary to solve an ordinarydifferential equation, which can be done easily without expanding in eigenfunctions. A furtheradvantage is that, as the resulting problem strongly resembles a neoclassical Spitzer problem,it is possible to compare the analytical results with neoclassical simulations.The solution is then inserted in the equation of angular-momentum conservation, which reads∂∂t〈(mini +mznz)|∇ψ|2B2φ′−∫Iv‖B(migi0 +mzgz0 −Iv‖Bφ′(m2iTifi0 +m2zTzfz0))d3v〉= 0,and can be rewritten in terms of the (Laplace-transformed) neoclassical polarisationP̂ ≡∑a=i,z〈IB∫mav‖ĝa0d3v〉/∑a=i,z〈manaR2〉φ̂′as an equation for the Laplace-transformed potential response,φ̂′(p) =1pφ′(0)〈|∇ψ|2B2〉〈R2〉 (1 − P̂ ).The relation between the neoclassical polarisation and the zonal-flow damping timeτp is foundto be τp = limp→0∞∫0(φ′(t)/φ′∞ − 1)e−ptdt = dP̂/dp(0) and can thus be calculated from the36th EPS 2009; S.Braun et al. : Collisional zonal-flow damping in an impure tokamak plasma 2 of 400.10.20.30.40.50.60.70.80.91τ p/τp(Zeff=1)1 2 3 4 5 6Zeffε = 0.3 (fc = 0.28)TheoryNEO, ConnorNEO, HSFigure 1: Normalised damping time versus effective chargedistribution functionsgi andgz, with the resultτp =I2〈R2〉B20fc{1νiiD + νizD}+fc{νiiDνiiD+νizD}2{νiiD} − fc{(νiiD)2νiiD+νizD}≈I2〈R2〉B20fcν̂ie(4.51Zeff+(0.87 + 2.49fc1 − fc)1Z4eff)where curly brackets denote an average over velocity space,{F (v)} =∫d3vFmv2‖f0/(nT ),fc = 0.75λc∫0dλλ/〈√1 − λB/B0〉is the effective fraction of circulating particles whereλcdenotes the trapped-passing boundary, and the second formula was interpolated in the effectivechargeZeff =∑j(Z2jnj)/∑j(Zjnj).In Figs. 1 and 2, the normalised damping time is plotted versus effective charge and inverseaspect ratioε, respectively, together with the results from the NEO code. In regions where thetheory is valid, the agreement between theory and simulation is quite good. The simulation inFig. 1 was done for a carbon species, thus for largeZeff the impurities cease to be a minorityin the plasma and the assumptions in the theory are violated which explains the discrepancy asZ → Zeff . The discrepancy for small values ofε in Fig. 2 is due to the fact that the plasma is nolonger in the banana regime in this limit as the collision frequency was fixed in the simulation.Clearly visible is also the asymptotic character of the theory asZ → ∞.From both theory and numerical simulations, we find that, when impurity ions are present inthe plasma, the damping time is significantly shorter than the damping time in a pure hydrogenplasma. Although the impurities hardly rotate themselves, they are able to slow down thecirculating bulk ions considerably. The underlying physical mechanism is that impurities are36th EPS 2009; S.Braun et al. : Collisional zonal-flow damping in an impure tokamak plasma 3 of 400.10.20.30.40.50.60.70.80.91τ p/τp(Zeff=1)0 0.1 0.2 0.3 0.4 0.5 0.6εZI = 6 Theory, Zeff = 1.5NEOTheory, Zeff = 2NEOTheory, Zeff = 3NEO00.10.20.30.40.50.60.70.80.91τ p/τp(Zeff=1)0 0.1 0.2 0.3 0.4 0.5 0.6εZI = 100 Theory, Zeff = 1.5NEOTheory, Zeff = 2NEOTheory, Zeff = 3NEOFigure 2: Normalised damping time versusεslowed down on the time scale of impurity collisions, which is much shorter than that of ioncollisions. Thus, when the circulating ions collide, their rotation is not only damped by self-collisions with the trapped population as in the case of a pure plasma, but also by collisionswith the entire, i.e. both trapped and passing, impurity population. Most important is thismechanism in the limit of very large aspect ratio where few trapped particles are present. Inthis case, the rotation in a pure plasma remains nearly undamped, whereas in an impure plasmadamping occurs. For tight aspect ratio, impurities do not matter so much as the number oftrapped particles is high enough to damp the rotation nearly immediately.Insofar as collisional zonal-flow damping plays an important role, this would suggest thatimpurities inhibit zonal flows and could thus have a deleterious effect on plasma confinement.References[1] S. Braun, P. Helander, E.A. Belli and J. Candy, Plasma Phys. Control. Fusion51, 065011(2009).[2] E.A. Belli, J. Candy, Plasma Phys. Control. Fusion50, 095010 (2008).[3] F.L. Hinton, M.N. Rosenbluth, Phys. Rev. Lett.80, 724 (1998).[4] Y. Xiao, P.J. Catto, K. Molvig, Phys. Plasmas14, 032302 (2007).36th EPS 2009; S.Braun et al. : Collisional zonal-flow damping in an impure tokamak plasma 4 of 4

Collisional zonal-flow damping in an impure tokamak plasma

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