2D full-wave simulations of different scenarios of ECR plasma heating at the L-2M stellarator

Different scenarios of electron cyclotron resonance (ECR) plasma heating under the conditions typical of L-2M experiments (ηe(0) = 1.75 × 1013 cm−3, Te(0) = 1 keV, λ0 = 0.4 cm) are simulated using a 2D full-wave model with allowance for the nonlocal (differential) thermal correction to the plasma dielectric tensor. It is shown that, under central ECR heating, due to the specific shape of the resonance surface ω0 = 2ωce, a significant fraction of the input microwave power is deflected downward and escapes onto the chamber wall. During off-axis heating at the midradius of the plasma column, about one-half of the microwave power is reflected upward from the resonance surface. Optimal conditions for the deposition of the microwave power in plasma are achieved under ECR heating at the vacuum magnetic axis, when the microwave beam is incident normally onto the resonance surface. In this case, the microwave power is almost completely (≈99.5%) absorbed by the plasma, while the coefficient of microwave reflection into the aperture of the incident microwave beam amounts to ∼0.1%, which agrees with results of 1D full-wave simulations.


Introduction
Electron cyclotron resonance (ECR) heating at the first (0 = ce) or second (0 = 2ce) harmonic of the electron gyrofrequency [1−5] is widely used for plasma heating in toroidal magnetic confinement systems. In particular, an electron temperature of up to ~1 keV at plasma densities of (1−2)  10 13 cm −3 was achieved at the L-2M stellarator under second-harmonic ECR heating by an extraordinarily (X) polarized microwave beam (0 = 0.4 cm, P ≤ 1 MW) launched from the outer side of the torus [6]. As a rule, various versions of the ray tracing method are used to calculate the propagation and absorption of the heating microwave beam [2, 7−9]. Since the microwave power is absorbed in a relatively narrow region (x ~ LBTe/mec 2 << a, where LB = B0/|B0| is the characteristic inhomogeneity length of the magnetic field) near the resonance surface, whereas thermal corrections to the plasma dielectric tensor in the rest of the plasma column are small, a simplified version of the ray tracing method is often used in which thermal corrections are taken into account only through the absorption coefficient, while the ray trajectories are calculated in the cold plasma approximation [2,6]. However, theoretical analysis and computer simulations [10−12] show that such a simplified model can fail under conditions typical of L-2M experiments.
The L-2M device is a classical l = 2 stellarator with a major radius of R = 1 m and an average plasma column radius of a = 11.5 cm [6,13]. Figure 1a shows the L-2M vacuum magnetic configuration in the standard poloidal cross section, in which ECR heating is performed, for the case of central ECR heating. Here, the vacuum magnetic axis is shifted by 2.7 cm to the left (toward the major axis) from the coordinate origin (x = y = 0), which coincides with the minor axis of the vacuum chamber. Due to the specific shape of the resonance surface, the microwave beam launched from the right along the x axis propagates nearly parallel to the resonance surface before it reaches the  [11], in this case, thermal corrections to the plasma dielectric tensor can cause substantial refraction of microwave radiation, due to which an appreciable fraction of the microwave power is deflected downward, not reaching the absorption region. In this study, we consider some other possible scenarios of ECR plasma heating at the L-2M stellarator, differing in the position of the resonance region (the value of the toroidal magnetic field B0): heating on the vacuum magnetic axis and off-axis heating at the midradius of the plasma column. It is shown by means of 2D full-wave numerical simulations that optimal conditions for microwave power deposition in plasma are achieved under ECR heating on the vacuum magnetic axis, when the microwave beam is incident normally onto the resonance surface.

Formulation of the problem
The 2D problem is formulated in the same way as in [11,12]. The radial profiles of the plasma density and electron temperature in terms of the magnetic flux coordinate  = (/max) 1/2 (figure 2) are taken close to those in standard L-2M experiments [6].
It is assumed that the stellarator magnetic field B0 is directed along the z axis (the radial and poloidal components of the magnetic field are neglected). The structure of shifted magnetic surfaces [14] in the L-2M standard cross section for plasma with such density and temperature profiles is shown in figure 1b. The 2D X-polarized microwave beam (0 = 0.4 cm) with a width of 2 cm at a level of e −1 in amplitude is launched from the right (from the outer side of the torus) along the x axis. The complex amplitudes of the wave fields Ex and Ey are calculated by solving the 2D full-wave equation where k0 = 0/c, 0  is the dielectric tensor of cold magnetized plasma [15] and ˆ⊥  is the nonlocal (differential) thermal correction to the dielectric tensor near the resonance 0 = 2ce [10]. The problem is solved in a 24.96  24-cm simulation box divided into 616  600 cells with dimensions x = y = 0.04 cm. Outgoing conditions are imposed on the transmitted and reflected waves at the left and right boundaries, respectively, and smooth absorbing layers are introduced near the lower and upper boundaries to suppress reflection from them. Equation (1) with these boundary conditions was solved by the matrix sweep method [16]. The output parameters are the 2D distributions of the wave fields Ex, Ey, and Bz = i(Ex/y − Ey/x)/k0 in the (x, y) plane; the y-profiles of the transmitted and reflected microwave powers; and the distribution of the absorbed microwave power Q(x, y).

Central heating
Let us remember the results obtained in [11] for the case of central heating (B0(0, 0) = 1.34 T), when the resonance point 0 = 2ce on the x axis coincides with the center of the vacuum chamber. Figures  3a and 3b show the distributions of the wave electric field squared, |E| 2 = |Ex| 2 + |Ey| 2 , and the wave magnetic field Bz, respectively, in the (x, y) plane for the density and temperature profiles shown in figure 2. The profile of the incident microwave power is shown on the right of each panel.
It is seen that, in this case, a fraction of the incident microwave power (about 13%) is deflected downward and escapes onto the wall in the form of a narrow beam. The rest power (about 87%) is almost completely absorbed just behind the resonance surface (the plasma region absorbing 75% of the total absorbed microwave power is marked with purple color). The reason for the downward deflection of microwave radiation is refraction caused by the thermal correction to the plasma dielectric tensor. Analysis of the effective refractive index of hot plasma [11,12] shows that, just below the resonance surface, where 0 > 2ce, the refractive index has a negative vertical gradient, due to which the upper part of the microwave beam, which propagates nearly along the resonance surface, is deflected downward and begins to interfere with its lower part. Since the resonance surface gradually turns down, the deflected radiation is "guided" along this surface and finally escapes downward. The guiding of microwave radiation along the resonance surface is clearly seen in figure  3b, which illustrates the wave structure of the beam. Simulations show that the fraction of the deflected microwave power grows rapidly with increasing plasma density, reaching more than 50% at ne(0) = 3  10 13 cm −3 [11,12].    It is seen that, on the right of the absorption region, the resonance surface is shifted upward compared to the case of central heating, due to which the influence of refraction caused by the thermal correction to the plasma dielectric tensor is substantially reduced and the microwave beam propagates nearly as in cold plasma. Moreover, since the beam is incident on the resonance surface almost normally, no downward deflection occurs. More than 99% of the incident microwave power is absorbed just behind the resonance surface. A small fraction (~0.1%) of the microwave power is reflected back into the aperture of the incident beam, which agrees with results of 1D numerical simulations for similar plasma parameters [17,18].

Off-axis heating at the midradius of the plasma column
At B0(0, 0) = 1.253 T, the resonance on the x axis shifts into the point x = −6.4 cm ( = 0.5). The corresponding distributions of |E| 2 and Bz are shown in figure 5. In this case, in contrast to the case of central heating, the beam is incident obliquely on the convex (rather than on the concave) resonance surface. As a result, only about one-half of the input microwave power passes through the resonance surface and is then absorbed, while the rest power is reflected and escapes onto the wall. The incident and reflected waves produce a clearly pronounced interference pattern in the (x, y) plane.

Hypothetical (yet unfeasible) heating scenarios
In all the above scenarios, the microwave beam was launched from the outer (low-field) side of the torus. Let us now consider some hypothetical heating scenarios in which the microwave beam is launched from the side of the higher magnetic field. There are, in principle, two ways to introduce the microwave bean in the camber from the high-field side between the L-2M stellarator windings: (i) in the equatorial plane from the inner side of the torus and (ii) vertically from the top. Although both these scenarios are yet unfeasible for technical reasons (there is no enough room on the inner side and there are no suitable ports on the top of the L-2M vacuum chamber), it is of interest to compare them with the above results. Figures 6 and 7 show the distributions of |E| 2 and Bz for the cases where the beam is launched from the inner side of the torus and from the top, respectively, for B0(0, 0) = 1.34 T (central heating) and for the same density and temperature profiles as in the previous scenarios (see figure 2). In both cases, the beam before absorption propagates nearly as in cold plasma and is almost completely (>99%) absorbed near the resonance surface. No microwave power is reflected.

Discussion and conclusions
Thus, 2D full-wave simulations of the propagation and absorption of the heating microwave beam in the L-2M stellarator plasma under the conditions typical of experiments on ECR plasma heating at the second harmonic of the electron gyrofrequency show that optimal conditions for the deposition of the microwave power in plasma are achieved under ECR heating at the vacuum magnetic axis. In this case, the microwave power is almost completely (99.5%) absorbed in the region located not far from the center of the plasma column ( ~ 0.2).
In the cases of central heating and off-axis heating, an appreciable fraction of the introduced microwave power is deflected from the resonance surface and escapes onto the wall. Of course, after multiple reflections from the chamber wall, this power will eventually be absorbed by the plasma.