Parallel transport and profile of boundary plasma with a low recycling wall

1 TH/P4-16 Parallel transport and profile of boundary plasma with a low recycling wall Xian-Zhu Tang 1 and Zehua Guo 1 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. Corresponding Author: xtang@lanl.gov Abstract: With a low reycling wall, the boundary plasma is expected to be of high temperature but low density. The parallel transport and profile of such a low collisionality plasma in open field lines have a number of unusual and interesting behavior. Here we summarize our recent findings in three areas: (1 the parallel heat flux dependence of the parallel variation of plasma temperature, density, and parallel flow; ( the role of kinetic instabilities in enforcing ampipolarity of parallel transport; (3 the parallel variation of ambipolar potential and the collisionality dependence of wall potential. 1 Introduction The low recycling regime [1] is opposite to the usual approach of a radiatively cooled and detached divertor plasma for tokamak fusion reactors. This can be experimentally accessed by a choice of wall material that has intrinsically small recycling coefficient, for example, a flowing liquid lithium surface [1]. The resulting low neutral density near the wall leads to a boundary plasma which is low-density but high-temperature. Similar effect can be obtained with aggressive neutral density control via cryo-pumping. The perceived benefit [1] of such a tokamak reactor operational regime is the reduced plasma temperature gradient across the entire plasma, especially at the edge (hence negating the need of a pedestal, which reduces the primary drive for micro-instabilities that produce anomalously high thermal and particle transport across magnetic field lines. In return, the improved confinement leads to reduced net plasma power load and particle irradiation flux on the divertor and first wall. Equivalently speaking, in the case of a diverted plasma, the power load becomes manageable primarily through the much reduced plasma density, for the plasma temperature is expected to be high. The expected large density gradient, as opposed to a large temperature gradient, is a signature feature in the low recycling regime. The combination of high temperature and low density leads to a low-collisionality boundary plasma. Here we give a summary report of a series of recent theoretical and simulation studies of low-collisionality boundary plasma in the low-recycling regime and elucidate the parallel transport physics and the resulting parallel plasma profile. We are particularly interested

TH/P4-16 in the boundary plasma at the outer divertor plate where significant flux expansion is deployed to spread the heat load. There are three physics issues we will address. The first is on the parallel variation of plasma profile and ambipolar potential to be expected in a low collisionality boundary plasma. The second is on the wall potential and plasma heat flux, and their dependence on plasma collisionality in the long mean-free-path regime. The third is on transport-induced plasma instabilities in a long mean-free-path boundary plasma, and how wave-particle interaction enforces ambipolarity and affects energy flux to the wall. 0.35 0.3 Ion f(v, 8 0.035 0.03 Ion f (ν = 10 7 ω pe Ion f (ν = 10 3 ω pe 6 0.05 0. 0. 0.15 4 0.0 0.015 0.15 0.1 0.1 0.05 0.01 0.005 0.05 0.05 0.1 0.15 0. 0.3 v 0 0 0.005 0.01 0.015 0.0 0.05 0.03 v 0 0.1 v 0. 0.3 0.4 1.5 Electron f(v, 8 6 0.3 0. Electron f.5 1.5 Electron f 1 4 0.15 1 0.1 0.05 0 v 1 1.5 0 0.05 0 0.05 0.1 0.15 0. v 1 0 v 1 FIG. 1: The electron and ion distribution [ln(1 + f e,i ] contour from the VPIC kinetic simulation for two dimensional flux expander with different collisionalities. From left to right, the collisionality increases. The right panel is for a collisional case where the ion and electron distribution functions approach the Maxwellian distribution. Particle orbit and distribution function Most of our findings can be understood from the behavior of the particle distribution function with a plasma sink at the wall. In a long mean-free-path plasma, parallel streaming dominates over the perpendicular drift, except for the trapping effect by both the magnetic field and the ambipolar electric field. The magnetic field in a tokamak can provide trapping for both electrons and ions at the outboard of the scrape-off layer due to B ϕ 1/R, and mirror field acceleration in the divertor region due to both B ϕ 1/R and poloidal flux expansion. Here we focus on the outer divertor region where there is significant flux expansion in order to spread out the heat load. Consequently, the magnetic mirror force accelerates both electrons and ions toward the divertor surface. The ambipolar electric field provides trapping only for the electrons. Taking both into account, one finds

3 TH/P4-16 characteristic behaviors in the electron and ion distribution functions in the long meanfree-path regime. In the distribution plots in Reference [], we illustrate the separate roles of a decreasing magnetic field (flux expansion and the ambipolar electric field at the collisionless limit. The example is given for a two-dimensional flux expander, a simplified representation of the magnetic field intercepting the outer divertor plate. With a decreasing magnetic field toward the wall, the mirror force accelerates both electrons and ions. In other words, since magnetic moment is conserved, the perpendicular energy is converted into parallel energy. In the (v, space, this corresponds to a compression of the distribution function in the direction. The bounding curve is roughly given by ǫ µb 0 0 (1 with ǫ the particle energy and µ the magnetic moment. Eq. (1 with equal sign gives a hyperbola (solid lines in the left panel of Fig. 1 in (v, space, below which the distribution is allowed. The ambipolar electric field produces electron trapping, with the trap/passing boundary given by v +v (1 B w /B = e(φ φ w /m e, ( where φ w < 0 is the ambipolar potential at the wall. Inside this eclipse (dashed lines in the left panels of Fig. 1 the electrons are trapped where the electric force dominates over the mirror force. Also shown in Fig. 1 is the gradual transition to Maxwellian distributions for both electrons and ions as the collisionality is increased. 3 Parallel profile variation and parallel heat flux The electron and ion distribution functions constrained by the simple orbit dynamics under the combined force of B and ambipolar electric field as shown in Fig. 1 with the help of Eqs. (1,, suggest unusual behavior for both the plasma profile and parallel heat flux. First, as the plasma approaches the wall, the perpendicular temperature would decrease but the parallel temperature needs not to. Since cold electrons are more likely trapped, the upstream electron temperature can be surprisingly low compared with both the downstream temperature and the source temperature, suggesting a large temperature gradient across the separatrix in the long mean-free-path regime. This finding is opposite to the conventional expectation mentioned in the introduction. Second, the parallel energy flux is carried by the passing particles, which have positive skewness in their distribution independent of the direction of the parallel temperature gradient. Combined with the first finding, we find the peculiar result that parallel heat flux, certainly the component associated with the parallel thermal energy, can climb a temperature hill [3]. Both surprising physics are demonstrated by analytical calculation based on drift kinetic equation and VPIC kinetic simulations solving the kinetic-maxwell equations [4]. We find that the plasma profile variation along an open field line is controlled by two upstream parameters (α 0 and α 0 and the heat flux profile (q n /B and q s /B [5]. The

TH/P4-16 4 α 0 and α 0 are the ratios of the upstream conductive and convective heat flux in the perpendicular and parallel degrees of freedom, respectively, α 0 q s 0/n 0 u 0 T 0 ; α 0 q n 0/n 0 u 0 T 0. (3 The variation of the electron and ion temperature along an open magnetic field with flux expansion toward an absorbing boundary has an explicit dependence on the parallel heat flux profile. This is the case for both the parallel and perpendicular temperature, but with the subtlety that while only q s affects T, the T is affected by both q n and q s. The exact form for T is B [ ( T = T 0 1+α 0 1 q s/b ]. (4 B 0 q s0 /B0 This should be contrasted with the adiabatic limit, in which only the magnetic field variation plays a role, i.e. T = T 0 B/B 0. The variation of T is given by ( qn /B q s /B ( B l T = T l lnu α 0 T 0 l α 0 T 0 q n0 /B 0 q s0 /B0 l. (5 B 0 Without the parallel heat flux terms, q n /B and q s /B, the parallel temperature will be decompressionally cooled with flux expansion for mirror acceleration of the parallel flow u [6]. However, the large direct energy transfer from T to T by q s b or the third term in Eq.(5 produces enough heating such that T increases with flux expansion. The wall-loss-induced ambipolar electric field in the quasi-neutral presheath plasma should be contrasted with the mirror-force-induced ambipolar potential. Both effects fundamentally depends on the much smaller electron mass compared with the ions. Except for a small correction proportional to electron inertia, the ambipolar potential drop along an open magnetic field line can be expressed as e φ = 3 ( Te e e B +(1+α 0 T 0 1 + 1 ( q B 0 α e e n /B 0 Te 0 1, (6 qn0/b e 0 where we have used the notation A A A 0 with A 0 the upstream value of A. The parallel heat flux of perpendicular thermal energy provides an almost one order of magnitude enhancement (α 0 = 3.8 compared with unity to the mirror effect (the second e term on RHS which drives the ambipolar potential drop needed to slow down the parallel electron flow. In the quasi-neutral region, the plasma parallel flow velocity satisfies u = u i = u e due to ambipolarity. The ion momentum equation then gives m i u = 3 T i +(1+α i 0 T i 0 (1 B B 0 α i 0 Ti 0 ( q i n /B 1 e φ. (7 qn0/b i 0 Therefore, the parallel flow acceleration is driven by a combination of the parallel temperature gradient, the mirror force with enhancement by parallel heat flux (α i 0, the local heating/cooling due to divergence of the parallel heat flux q n b, and the ambipolar electric field. There can be a variety of scenarios on driving the parallel flow. Additonal theoretical considerations and numerical demonstrations are given in Ref.[4].

5 TH/P4-16 4 Kinetic instabilities and ambipolarity The distribution functions shown in Fig. 1 for the low-collisionality end suggest robust drive for plasma instabilities, which through wave-particle interaction provide collisionless detrapping for the electrons. The self-generated electron instability and the associated electron diffusion across the trap-passing boundary is crucial in maintaining ambipolarity in the parallel transport of open field line plasmas with long mean-free-path electrons [7]. In the absence of perturbations, the ion flux is always greater than the passing electron flux with a general electron and ion source. Analytical calculations with the linearized Vlasov equation of a model trapped electron distribution reveal a robust whistler wave instability below the electron cyclotron frequency for a large range of (Φ Φ w /T. Here Φ w is the wall potential so Φ Φ w is the electrostatic trapping potential. Following the quasilinear theory, the velocity space diffusion of electrons and hence the detrapping rate due to the electron-whistler waves interaction are quantified and shown to be comparable with the simulation result. The actual drive for the observed whistler instability in simulations of both uniform and non-uniform magnetic field is the sharp gradient of trapped electron velocity distribution at the trap-passing boundary. For stability analysis, the initial trapped electron distribution in a 1D sheath bounded plasma is simply modelled by a Maxwellian distribution with cut-offs in the parallel velocity at v = ±v c (the case with uniform B 0, f t (v, = αn 0 πv 3 t e ( +v /v t Θ(1 v /v c, (8 where Θ(1 v /v c is the Heaviside step function that vanishes when v > v c, and α(v c = [erf(v c /v t ] 1 is a normalization factor so f t d dv = n 0. The dispersion relation for the trapped-electron whistler mode is obtained, i.e. D(ω,k = 1 k c ω + αω pe πω ω kv t ˆvc ˆv c ˆv e ˆv ξ dˆv + αω pe ˆv c e ˆv c πω (ˆv c +ξ(ˆv c ξ (9 where ˆv,,c = v,,c /v t, ξ = (ω Ω e /kv t with Ω e being the electron cyclotron frequency, and Θ (x = δ(x (the dirac-delta function has been used. The robust trapped-electron whistler instability is in sharp contrast with the temperature anisotropy whistler mode, which requires a higher temperature anisotropy to be unstable for the same frequency range. To further illustrate the difference between them, the numerical solutions of their growth rate and real frequency for v c = [0.1v t,v t ] are shown in Fig., where the most unstable modes are plotted. The parallel temperature of the bi-maxwellian distribution is determined by T = T [1 αˆv c exp( ˆv c/ π]. Over the entire range of v c = [0.1v t,v t ], it is clearly shown in Fig. that the growth rate of a cutoff-maxwellian is larger than that of a corresponding bi-maxwellian for different cutoff speeds. For the low β e plasma, the conventional whistler mode becomes marginally stable when v c = v t or T /T = 6.4, while the trapped-electron driven mode still has finite growth rate even for v c > v t. Here we have used the parameters in normalized unit (ω pe = 1 where Ω e =, c = 10v t, so that β e = 0.15%.

TH/P4-16 6 0.06 0.05 maximum growth rate vs. v c cutoff double cutoff double 1 0.9 0.8 real frequency vs. v c β e =0.15% 0.04 0.7 γ/ω pe 0.03 0.0 0.01 β =0.15% e β =16% e 0 0.1 0.75 1 1.5 v /v c t ω r /Ω e 0.6 β e =16% 0.4 cutoff double 0.3 cutoff double 0. 0.1 0.75 1 1.5 v c /v t FIG. : The growth rate (top and the real frequency (bottom of the most unstable mode for both cutoff-maxwellian and double-maxwellian distribution with different cutoff parallel velocity v c or equivalent T, for β e = 0.15%(solidline,16%(dashedline. As the whistler waves exponentially grow to large amplitudes, they start to slowly modify the electron equilibrium distribution, which can be described using the standard quasilinear diffusion theory in the literature [8]. The total electron flux from trapped to passing phase space is evaluated, i.e. Γ tp = L 0 n 0 v c v t k c ω c [(1 k cv c ω c g k cv t ω c g ]A kc dl, (10 wheretheresonanceconditionω c Ω e k c v c = 0hasbeenused, g(ˆv isthesaturatedlocal electron distribution function in v with a normalization of g(ˆv dˆv = 1, L (k c L 1 is the total system size, and A kc = (Lq /4πm δê k denotes the perturbation energy density in k space with δêk the kth Fourier component of electric field perturbation. For ω Ω e, the resonance condition of electrons with v 0 requires k c < 0. Therefore, Γ tp is positive when g (ˆv c /g(ˆv c > (ω c / k c +v c /v t. Since the instability drive, namely source injection into the trapped region, always presents, the quasilinear diffusion produces a finite detrapping particle flux. From the simulation (β e = 0.15%, we obtain: v t =, L = 00/, k c, ω c.55, eφ w /T s 0.05 (thus v c 0.3, and values of n 0,g(ˆv c,g (ˆv c,a kc at seven different locations, where the normalization to the Debye length and electron plasma frequency have been applied. The estimated quasilinear diffusion induced flux using Eq.(10 is approximately 19% of Γ i, roughly matching the amount required to satisfy ambipolarity (15%. Even though the trapped electron induced whistler modes tend to isotropize the electron distribution, the residual temperature anisotropy remains substantial at the nonlinearly saturated state. The electron distribution is locally smoothed at the trap-passing boundary but a sufficiently large gradient of the distribution function is still maintained to provide the required ambipolar detrapping flux. As an application, our kinetic-maxwell

7 TH/P4-16 simulation of a magnetic mirror shows that even with finite collisionality, the trapped electron distribution can retain significant anisotropy. From the perspective of collisional transport, one may expect the trapped electrons to be equilibrated into a Maxwellian at steady state, since the confinement time of plasma in magnetic mirror (ion-ion collision time is much longer than the electron-electron collision time. However, the collisional isotropization process is subject to a faster wave-particle detrapping process at low collisionality. An electron can be scattered from the trap-zone into the passing-zone before it suffers significant pitch-angle scattering caused by Coulomb collisions, yet the wave-particle induced detrapping process does not result in energy equipartition between parallel and perpendicular directions. Therefore, the trapped electrons can maintain a significant temperature anisotropy even with finite collsionality, which is confirmed by our kinetic simulations. We also notice that the collisionless detrapping only has little effect on the parallel energy flux to the wall. Further details on how this Φ w (T s 1.3 1. 1.1 1 0.9 0.8 0.7 0.6 upstream T e (T s 0.65 0.6 5 0.45 0.4 0.35 0.3 10 6 10 4 collision frequency (ω pe 10 6 10 4 collision frequency (ω pe FIG. 3: The collisional effects on wall potential. 5 Collisionality dependence of wall potential We also have examined the transition to the collisional limit and found interesting results on the wall potential and average electron temperature with same upstream source. In the collisional limit where the mean-free-path is short compared with gradient length scale, the Coulomb collision produces near-maxwellian distribution for both electrons and ions, see the right panel of Fig. 1. In the intermediate regime of collisionality, the Coulomb collision and wave-particle interaction both contribute to the detrapping of electrons. This shifts the trapping boundary in v outward which corresponds a more negative wall potential. This trend would not last as the drive for instability is also reduced, so the net detrapping rate is not a monotonic function of Coulomb collisionality. The wall potential peaks and then saturates downward into the collisional limit. It is shown in

TH/P4-16 8 Fig 3 the simulation result with uniform magnetic field normal to the wall. The average and upstream electron temperature, when normalized against the source temperature, increases with collisionality and then saturates in the collisional limit. 6 Conclusion The parallel transport and profile of a tokamak boundary plasma with a low recycling wall have a number of interesting behavior which can be understood from the characteristics of the particle distribution function taking into acount the mirror force and the ambipolar electric field. The parallel heat flux is of particular interest in that it explicitly enters the parallel plasma profile variation, and has most unusual behavior with regard to the temperature gradient. The conventional view of a flattened high temperature, which is arrived by extrapolating the collisional result on parallel heat flux, is not consistent with the kinetic analysis. Furthermore, the fact that electromagnetic instability is found to be crucial for satisfying the ambipolar constraint, and the collisionality dependence of the wall potential, suggest that carefully kinetic analysis, including fast time scale electromagnetic instabilties, must be deployed to obtain the correct description. This work was supported by the U.S. Department of Energy Office of Fusion Energy Sciences. References [1] L. Zakharov, et al, Fusion Eng. Design 7 (004 149; L. E. Zakharov, et al, J. Nucl. Mat. 363, (007 453; S. Krasheninnikov, et al, Phys. Plasma 10 (003 1678. [] Z. Guo and X. Z. Tang, Phys. Plasmas 19 (01 06501. [3] Z. Guo and X. Z. Tang, Phys. Rev. Lett. 108 (01 165005. [4] Z. Guo and X. Z. Tang, Phys. Plasmas 19 (01 08310. [5] K. Bowers, et al., Phys. Plasmas 15 (008 055703. [6] X. Z. Tang, Plasma Phys. Control. Fusion 53 (011 0800. [7] Z. Guo and X. Z. Tang, Ambipolar transport via trapped-electron whistler instability along open magnetic field lines, Phys. Rev. Lett. (in press. [8] C. F. Kennel and F. Engelmann, Phys. Fluids 9 (1966 377.