Development of a Drift-Kinetic Simulation Code for Estimating Collisional Transport Affected by RMPs and Radial Electric Field

A drift-kinetic δf simulation code is developed for estimating collisional transport in quasi-steady state of toroidal plasma affected by resonant magnetic perturbations and radial electric ﬁeld. In this paper, validity of the code is conﬁrmed through several test calculations. It is found that radial electron ﬂux is reduced by positive radial-electric ﬁeld, although radial diffusion of electron is strongly affected by chaotic ﬁeld-lines under an assumption of zero electric ﬁeld.


Introduction
Understanding of the collisional transport properties in a perturbed magnetic field is important for the control of fusion plasma by employing resonant magnetic perturbations (RMPs) [1]. In recent tokamak experiments, RMPs have been used to mitigate edge localized modes (ELMs) [2]. ELMs were satisfactorily mitigated/suppressed by the RMPs reducing edge plasma pressure gradient. It was found interestingly [2] that the enhanced transport was fairly different from the theoretical estimate based on the field line diffusion derived by Rechester and Rosenbluth [3]. On the other hand, it was confirmed by a test particle simulation [4] that the field line diffusion model was applicable to the evaluations of both the electron and ion diffusion coefficients in a sufficiently ergodized region. Therefore, there is the following open question: why does the difference between the theoretical and experimental results occur? The modifications of field line diffusion have been proposed, but a clear answer to the question has not been obtained [1,5]. An overview of the related theoretical/simulation studies is given in Ref. [6]. Although the drift-kinetic equation is known as a powerful tool for the estimation of collisional transport (neoclassical transport) in toroidal plasma including 3-dimensional field geometry [7,8], it has rarely been used for the studies of the transport in RMP field.
In order to search for the answer, we reconsider the fundamental properties of the transport, based on the driftkinetic equation in 5-dimensional phase space. For this purpose, we have started from examining dependences of the ion thermal diffusivity on several important parameters (the strength of RMPs, collisionality, etc.) by using a δf simulation code, KEATS [9,10], which is programmed to solve the drift-kinetic equation. In our previous studies under an assumption of zero electric field [10], which is the same assumption as in Refs. [3,4], we found the following three main results. 1) The radial thermal diffusivity χ r in the simulation is close to the diffusivity χ RR r derived by Ref. [3] if χ r is evaluated in a temporary state of the guiding-centre distribution function f during an extremely short period Δt Here ω t is the transit frequency and ν eff is the effective collision frequency. The initial condition of f in the simulations is set to f = f M , i.e., δf = 0, at time t = 0, where f M is a Maxwellian. The temporary state relaxes finally to a quasi-steady state of f after being sufficiently exposed to Coulomb scatterings, t ν −1 eff . Note that in the δf simulations in this paper, it is physically meaningful to evaluate χ r in the quasi-steady state rather than the temporary state, which is the same as in the neoclassical theory. 2) The diffusivity χ r evaluated in the quasi-steady state is extremely small compared with χ RR r .
3) The diffusivity χ r has almost the same parameter-dependence as χ RR r . The details are shown in Refs. [9,10]. These results partly explain the difference, but one of the most important effects is ignored; it is an effect of radial electric field E r .
In this paper, we report further improvements of KEATS code to include the effect of E r on the transport, and show the results of test calculations. The scheme of the developing code is explained in the next section. Several tests are executed to check validity of the code in section 3. Finally, summary is given in section 4.
2 Two weight δf method solving the drift-kinetic equation is a magnetic field. In this paper, the number density n, temperature T and electrostatic potential Φ are supposed to be given functions of minor radius r, which are independent of time t.
Here the minor radius r is defined by the label of closed magnetic surfaces of the unperturbed magnetic field B 0 , where B = B 0 + δB and δB is the RMP field. Validity of the supposition with respect to {n, T, Φ} is discussed in section 3. Applying the decomposition f = f M + δf to the drift-kinetic equation, we have the following equation: [11,12].
The Coulomb collision operator is described by C T and C F , where C T is the test-particle collision operator which represents the pitch-angle scattering and C F is the field-particle collision operator which ensures local momentum conservation property [9]. Note that E = constant in this simulation code. Effects of impurities and neutrals on the collisional transport are ignored in this paper.
To solve Eq. (1) by Monte-Carlo techniques, we adopt the two-weight scheme of the δf method [13]: where g is the marker distribution function and both p and w are the weight functions satisfying pg = f M and wg = δf . The Monte-Carlo simulation code, KEATS, is based on the above method, and it seeks a quasi-steady state of δf satisfying Eq. (1) under the assumption of a given electric field. The code is programmed in the so- . In this simulation code, the radial particle and energy fluxes are evaluated by the following equations, respectively: where radial heat flux is given as q r = Q r − (5/2)T Γ r . Here · means the time-average. The time-averaging is carried out in the quasi-steady state of δf after sufficient exposure to Coulomb collisions. Another average · is defined as · = (1/δV) δV · d 3 x, where δV is a small volume and lies between two neighbouring reference surfaces with volumes V(r) and V(r) + δV. Note that a reference surface is labelled by r. www.cpp-journal.org

Test calculations
The toroidal magnetic configuration used in test calculations of the present code is given by the addition of an RMP field δB to a circular tokamak field B 0 , where the total magnetic field is B = B 0 + δB. The unperturbed where R, ϕ and Z are the unit vectors in the R, ϕ and Z directions of the cylindrical coordinate system, respectively. Here q is the safety factor given as q −1 = 0.9 − 0.5875(r/a) 2 , the major radius of the magnetic axis is set to R ax = 3.6 m, the minor radius of the toroidal plasma is a = 1 m and the strength of the magnetic field on the magnetic axis is B ax = 4 T. The RMPs causing resonance with rational surfaces of q = k/ = 3/2, 10/7, 11/7, 13/9, 14/9, 16/11, 17/11, 19/13, 20/13, 22/15, 23/15, 25/17, 26/17 are given by a perturbation field δB = ∇ × (αB 0 ). The function α is given as α = k, a k cos{kθ − ϕ}, which is a similar presupposition to one in Ref. [4]. Here a k = A RMP exp{−(r − r k ) 2 /Δr 2 } and r = r k is the rational surface of q = k/ . We set the parameter A RMP /a = 10 −3 and Δr/a = 5 × 10 −2 so that neighbourhood of resonant surfaces become ergodized. The perturbed region is bounded radially on both sides by the closed magnetic surfaces, and the centre of the perturbed region is around r/a = 0.6, as shown in Fig. 1. This RMP field is applied to the test calculations in subsection 3.2. Note the following points. In order to investigate the properties of the transport affected by only RMP field (i.e., to compare the simulation results with the theory of field line diffusion), the perturbed region is sufficiently away from the plasma edge r/a = 1, because there is a possibility that the particle orbit-loss at the plasma edge influences the transport. The RMPs should be selected to suit well to the comparison, as used in Ref. [4].

Self-consistent electric field in unperturbed magnetic field
In section 3, we attempt to confirm the validity of the code through several test calculations. First, the following drift-kinetic equation of ion is considered in case of δB = 0.
where C is the Coulomb collision operator and Cf = C(f, f ) is the self-collision term. If f = f M and T = constant, then the following is derived from Eq. (6): n exp{eΦ/T } = constant. Therefore, the radial electric field is self-consistently given as E r = (T/e) d ln n/dr, and both particle and energy fluxes are estimated as Γ r = 0 and Q r = 0, respectively. Here the density n is assumed to be a function of r. In this case, δf = 0 should be always satisfied if the initial condition is set to δf = 0 at time t = 0. This test calculation was carried out with the number of markers equal to 1.6 × 10 7 . As shown in Fig. 2, the particle flux with the above electric field is confirmed as Γ r ≈ 0 during the simulation by the present code, and the energy flux is also evaluated as Q r ≈ 0. On the other hand, under the assumption of zero electric field, the self-collision-driven fluxes are seen. See also Refs. [15,16]. 3.2 Dependence of particle flux on radial electric field in an ergodized region When adding RMP field on a circular tokamak field, magnetic flux surfaces around resonant surfaces of q = k/ become often chaotic in the perturbed region, as shown in Fig. 1. In the present code, reference surfaces are defined by closed magnetic surfaces of B 0 , and thus the reference surfaces are labelled by minor radius r. The density n, temperature T and electrostatic potential Φ are supposed to be given functions of minor radius r, and thus the dot product of the RMP field δB and the gradient of {n, T, Φ} is not zero in general, i.e., B · ∇{n, T, Φ} = δB · ∇{n, T, Φ} = 0. Validity of this supposition is confirmed by the facts that after several collision times, 1) quasi-steady state of δf is finally found in the simulations and 2) the condition of |δU/U | := | d 3 v(mv 2 /2)δf /(3nT /2)| 1 is satisfied in the perturbed region in the quasi-steady state, i.e., |δU/U | 10 −3 in the simulations. In this case, the calculations were carried out with the number of markers equal to 6.4 × 10 7 .
Hereafter, we focus on the collisional transport in the perturbed region, then the density and temperature are given as n/ . The radial electric field at r/a = 0.6 is given as E r = Φ 0 /a. Note that radial particle/heat fluxes and radial electric field are zero outside of the perturbed region.
By estimating radial particle fluxes of electron and ion under the assumption of zero electric field, it is found that the electron flux is much larger than the ion flux, i.e., Γ e r Γ i r , because of chaotic field-lines, as shown in Fig. 3. Note that positive radial-electric-field is expected to satisfy the ambipolar condition, Γ e r = Γ i r , from this result. Therefore, we preliminary investigate dependence of electron flux on radial electric field E r and evaluate the value of E r when Γ e r ≈ 0. We find that the electron flux is reduced by radial electric field, as shown in Fig. 4, and that Γ e r ≈ 0 is satisfied in case of E r ≈ +700 V/m at the centre of the perturbed region, i.e., r/a ≈ 0.6. Here we confirm that the radial electron flux in case of the model potential with Φ 0 ≈ 700 V (i.e., E r ≈ +700 www.cpp-journal.org V/m at r/a = 0.6) becomes negligibly small in the quasi-steady state of δf after being sufficiently exposed to Coulomb scatterings, t τ ei (τ ei is the electron-ion collision time), as shown in Fig. 5.

Summary
We have improved the drift-kinetic δf simulation code KEATS for estimating collisional transport affected by RMPs and radial electric field. Several test calculations are attempted in the first trial, and the validity of the code for seeking a quasi-steady state of δf is confirmed. It is found that the radial electron flux strongly affected by RMPs is reduced by positive radial-electric-field. Estimation of radial particle and heat transport of both electron and ion, which is based on the drift-kinetic δf simulation satisfying Γ e r = Γ i r in the perturbed region, and furthermore comparison between the simulation results and the theory of Ref. [3] are not executed in this first report on our code development, and these will be described elsewhere in the near future.